Lie Groups and Lie Algebras I: Foundations of Lie Theory Lie Transformation Groups

Encyclopaedia of Mathematical Sciences

Volume 20

Editor-in-Chief: R. V. Gamkrelidze

A. L. Onishchik (Ed.)

Lie Groups and Lie Algebras 1

Foundations ofLie Theory Lie Transformation Groups

Springer-V erlag Berlin Heidelberg GmbH

Consulting Editors of the Series: . A. A. Agrachev, A. A. Gonchar, E. F. Mishchenko, N. M. Ostianu, V. P. Sakharova, A. B. Zhishchenko

Title of the Russian edition: Itogi nauki i tekhniki, Sovremennye problemy matematiki,

Fundamental'nye napravleniya, VoI. 20, Gruppy Li i Algebry Li 1

Publisher VINITI, Moscow 1988

Mathematics Subject Classification (1991): 17Bxx, 22-XX, 22Exx, 53C30, 53C35, 57Sxx, 57Txx

ISBN 978-3-540-61222-3

Library of Congress Cataloging-in-Publication Data Gruppy Li i algebry Li 1. English. Lie groups and Lie algebras II A. L. Onishchik, ed. p. cm. - (Encyclopaedia of mathematical sciences; v. 20) Translation of original Russian, issued as v. 20 ofthe serial: Itogi nauki i tekhniki.

Sovremennye problemy matematiki. Fundamental 'nye napravieniiil. Includes bibliographical references and index. Contents: Foundations ofLie theory I A. L. Onishchik, E. B. Vin

berg - Lie groups oftransformations/V. V. Gorbatsevich, E.B. Vinberg. ISBN 978-3-540-61222-3 ISBN 978-3-642-57999-8 (eBook) DOI 10.1007/978-3-642-57999-8

1. Lie groups. 2. Lie algebras. 1. Onishchik, A.L. II. Onishchik, A.L. Foundations ofLie theory.I993. III. Gorbatsevich, V. V. Lie groups of transformations. 1993. IV. Title. V. Series.

QA387.G7813 1993 512'.55-dc20

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List of Editors, Authors and Translators

Editor-in-Chiif

R. V. Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul. Vavilova 42,117966 Moscow, Institute for Scientific Information (VINITI), ul. Usievicha 20 a, 125219 Moscow, Russia

Consulting Editor

A L. Onishchik, Yaroslavl University, Sovetskaya ul. 14, 150000 Yaroslavl, Russia

Authors

v. V. Gorbatsevich, Moscow Institute of Aviation Technology, 27 Petrovka Str., 103767 Moscow, Russia

AL.Onishchik, Yaroslavl University, Sovetskaya ul. 14,150000 Yaroslavl, Russia E. B. Vinberg, Chair of Algebra, Moscow University, 119899 Moscow, Russia

Translator

A Kozlowski, Toyama International University, Toyama, Japan

Contents

I. Foundations of Lie Theory A. L. Onishchik, E. B. Vinberg

1

II. Lie Transformation Groups V. V. Gorbatsevich, A. L. Onishchik

95

Author Index 231

Subject Index 232

I. Foundations of Lie Theory

A. L. Onishchik, E. B. Vinberg

Translated from the Russian by A. Kozlowski

Contents

Introduction

Chapter 1. Basic Notions

§1. Lie Groups, Subgroups and Homomorphisms 1.1 Definition of a Lie Group 1.2 Lie Subgroups 1.3 Homomorphisms of Lie Groups 1.4 Linear Representations of Lie Groups 1.5 Local Lie Groups

§2. Actions of Lie Groups 2.1 Definition of an Action 2.2 Orbits and Stabilizers 2.3 Images and Kernels of Homomorphisms 2.4 Orbits of Compact Lie Groups

§3. Coset Manifolds and Quotients of Lie Groups 3.1 Coset Manifolds 3.2 Lie Quotient Groups 3.3 The Transitive Action Theorem and the Epimorphism Theorem 3.4 The Pre-image of a Lie Group Under a Homomorphism 3.5 Semidirect Products of Lie Groups

§4. Connectedness and Simply-connectedness of Lie Groups 4.1 Connected Components of a Lie Group 4.2 Investigation of Connectedness of the Classical Lie Groups 4.3 Covering Homomorphisms 4.4 The Universal Covering Lie Group

4

6

6 6 7 9 9

11 12 12 12 14 14 15 15 17 18 18 19 21 21 22 24 26

2 A. L. Onishchik, E. B. Vinberg

4.5 Investigation of Simply-connectedness of the Classical Lie Groups .......... . 27

Chapter 2. The Relation Between Lie Groups and Lie Algebras 29

§1. The Lie Functor ............ 29 1.1 The Tangent Algebra of a Lie Group 29 1.2 Vector Fields on a Lie Group . . . . 31 1.3 The Differential of a Homomorphism of Lie Groups 32 1.4 The Differential of an Action of a Lie Group 34 1.5 The Tangent Algebra of a Stabilizer . . . . 35 1.6 The Adjoint Representation . . . . . . . . 35

§2. Integration of Homomorphisms of Lie Algebras 37 2.1 The Differential Equation of a Path in a Lie Group 37 2.2 The Uniqueness Theorem . . . . . . . . . . . . . 38 2.3 Virtual Lie Subgroups .............. 38 2.4 The Correspondence Between Lie Subgroups of a Lie Group

and Subalgebras of Its Tangent Algebra 39 2.5 Deformations of Paths in Lie Groups 40 2.6 The Existence Theorem 41 2.7 Abelian Lie Groups . . . . 43

§3. The Exponential Map 44 3.1 One-Parameter Subgroups 44 3.2 Definition and Basic Properties of the Exponential Map 44 3.3 The Differential of the Exponential Map 46 3.4 The Exponential Map in the Full Linear Group 47 3.5 Cartan's Theorem .............. 47 3.6 The Subgroup of Fixed Points of an Automorphism

of a Lie Group . . . . . . . 48 §4. Automorphisms and Derivations 48

4.1 The Group of Automorphisms 48 4.2 The Algebra of Derivations . 50 4.3 The Tangent Algebra of a Semi-Direct Product of Lie Groups 51

§5. The Commutator Subgroup and the Radical 52 5.1 The Commutator Subgroup . . . . . . 52 5.2 The Maltsev Closure ......... 53 5.3 The Structure of Virtual Lie Subgroups 54 5.4 Mutual Commutator Subgroups 55 5.5 Solvable Lie Groups 56 5.6 The Radical 57 5.7 Nilpotent Lie Groups 58

Chapter 3. The Universal Enveloping Algebra 59

§1. The Simplest Properties of Universal Enveloping Algebras 59 1.1 Definition and Construction ............ 60

1. Foundations of Lie Theory 3

1.2 The Poincare-Birkhoff-Witt Theorem 61 1.3 Symmetrization 63 1.4 The Center of the Universal Enveloping Algebra 64 1.5 The Skew-Field of Fractions of the Universal Enveloping

Algebra 64 §2. Bialgebras Associated with Lie Algebras and Lie Groups 66

2.1 Bialgebras 66 2.2 Right Invariant Differential Operators on a Lie Group 67 2~3 Bialgebras Associated with a Lie Group 68

§3. The Campbell-Hausdorff Formula 70 3.1 Free Lie Algebras 70 3.2 The Campbell-Hausdorff Series 71 3.3 Convergence of the Campbell-Hausdorff Series 73

Chapter 4. Generalizations of Lie Groups 74

§1. Lie Groups over Complete Valued Fields 74 1.1 Valued Fields 74 1.2 Basic Definitions and Examples 75 1.3 Actions of Lie Groups 75 1.4 Standard Lie Groups over a Non-archimedean Field 76 1.5 Tangent Algebras of Lie Groups 76

§2. Formal Groups 78 2.1 Definition and Simplest Properties 78 2.2 The Tangent Algebra of a Formal Group 79 2.3 The Bialgebra Associated with a Formal Group 80

§3. Infinite-Dimensional Lie Groups 80 3.1 Banach Lie Groups 81 3.2 The Correspondence Between Banach Lie Groups and

Banach Lie Algebras 82 3.3 Actions of Banach Lie Groups on Finite-Dimensional Manifolds 83 3.4 Lie-Frechet Groups 84 3.5 ILB- and ILH-Lie Groups 85

§4. Lie Groups and Topological Groups - 86 4.1 Continuous Homomorphisms of Lie Groups 87 4.2 Hilbert's 5-th Problem 87

§5. Analytic Loops 88 5.1 Basic Definitions and Examples 88 5.2 The Tangent Algebra of an Analytic Loop 89 5.3 The Tangent Algebra of a Diassociative Loop 90 5.4 The Tangent Algebra of a Bol Loop 91

References 92

Introduction

The theory of Lie groups, to which this volume is devoted, is one of the classical well established chapters of mathematics. There is a whole series of monographs devoted to it (see, for example, Pontryagin 1984, Postnikov 1982, Bourbaki 1947, Chevalley 1946, Helgason 1962, Sagle and Walde 1973, Serre 1965, Warner 1983). This theory made its first appearance at the end of the last century in the works of S. Lie, whose aim was to apply algebraic methods to the theory of differential equations and to geometry. During the past one hundred years the concepts and methods of the theory of Lie groups entered into many areas of mathematics and theoretical physics and became inseparable from them.

The first three chapters of the present work contain a systematic exposition of the foundations of the theory of Lie groups. We have tried to give here brief proofs of most of the more important theorems. Certain more complex theorems, not used in the text, are stated without proof. Chapter 4 is of a special character: it contains a survey of certain contemporary generalizations of Lie groups.

The authors deliberately have not touched upon structural questions of the theory of Lie groups and algebras, in particular, the theory of semi-simple Lie groups. To these questions will be devoted a separate study in one of the future volumes of this series.

In this entire work Lie groups, as a rule, will be denoted with capital Latin letters, and their tangent algebras with the corresponding small Gothic letters, In addition the following notation will be used:

GO - connected component of the identity of a Lie group (or a topological group) Gj

G' = (G, G) - the commutator subgroup of a group Gj G(p) = (G(p-l) , GP-l))j

Rad G - the radical of a Lie group Gj rad g - the radical of a Lie algebra gj )<I - the semidirect product of groups (normal subgroup on the left)j -e - the semidirect sum of Lie algebras (ideal on the left)j 1I' - the group of complex numbers of modulus Ij exp - the exponential mappingj Ad - the adjoint representation of a Lie groupj ad - the adjoint representation of a Lie algebraj Aut A - the group of automorphisms of a group or algebra Aj Int G - the group of inner automorphisms of a group G; Der A - the Lie algebra of derivations of an algebra Aj Int g - the group of inner automorphisms of a Lie algebra gj GL (V) - the group of all automorphisms (invertible linear transforma

tions) of a vector space V j


Ln(K) - the associative algebra of all square matrices of order n over a field K;

GLn(K) - the group of all non singular matrices of order n over K; SLn(K) - the group of all matrices of order n with determinant 1; PGLn(K) = GL(K)j{.XE} - the projective linear group; GL~ (lR) - the group of all real matrices of order n with positive determi-

nant; On(K) - the group of all orthogonal matrices of order n over K; SOn(K) = On(K) n SLn(K); SPn(K) - the group of all symplectic matrices of order n over K (n even): Ok,l - the group of all pseudo-orthogonal real matrices of signature (k, l); SOk,1 = Ok,l n SLn(lR); O~ I - the group of pseudo-orthogonal matrices of signature (k, l) whose

~inor of order k at the top left corner is positive; Un - the group of unitary complex matrices of order n; Uk,l - the group of pseudo-unitary complex matrices of signature (k, l); SUn = Un n SLn(C); SUk,1 = Uk,l n SLk+I(C).

Finally we would like to mention a piece of non-standard terminology: we use the term "the tangent algebra of a Lie group" instead of the usual "the Lie algebra of a Lie group". We do so with a view to emphasise the construction of this Lie algebra as the tangent space to the Lie group. This seems to be appropriate here since, in particular, the tangent algebra of an analytic loop is not, in general, a Lie algebra. We reserve the term "Lie algebra" for its algebraic context.


Chapter 1 Basic Notions

We will assume familiarity with the basic concepts of manifold theory. However in order to avoid misunderstandings some of them will be defined in the text. The basic field, by which we mean either the field lR of real numbers or the field C of complex numbers, will be denoted by K. Unless stated otherwise, differentiability of functions will be understood in the following sense: in every case there exist as many derivatives as are needed. Differentiability of manifolds and maps is understood in the same sense. The Jacobian matrix of a system of differentiable functions h, ... ,f m of variables Xl. ... ,Xn will

be denoted by ~~~:"."", ',~:~. For m = n its determinant will be denoted by D(/l,oo. ,In) D(Xl,oo. ,xn)

The tangent space of a manifold X at a point x will be denoted by Tx(X) and the differential of a map f : X -t Y at a point x by dxf. In many cases, when it is clear which point is being considered, the subscript will be omitted in denoting a tangent space or a differential.

All differentiable manifolds will be assumed to possess a countable base of open sets.

§l. Lie Groups, Subgroups and Homomorphisms

1.1. Definition of a Lie Group. A Lie group over the field K is a group G equipped with the structure of a differentiable manifold over K in such a way that the map

I-L: G x G -t G, (x, y) I---> xy

is differentiable. In other words, the coordinates of the product of two elements have to be differentiable functions of the coordinates of the factors.

With the aid of the implicit function theorem it is easy to show that in any Lie group the inverse

£ : G -t G, x I---> X-I

is also a differentiable map. Lie groups over C are called complex Lie groups and Lie groups over lR - real Lie groups. Any complex Lie group can be viewed as a real Lie group of twice the dimension.

One can also consider analytic groups by requiring that the manifold G and the map I-L be analytic over the field K. Clearly, every complex Lie group is analytic, but even in the real case it turns out that in any Lie group there exists an atlas with analytic transition functions, in which the map I-L is expressed in terms of analytic functions (see 3.3 of Chap. 3).


Examples. 1. The additive group of the field K (we will denote it also by K).

2. The multiplicative group K X of the field K. 3. 'The circle' 'll' = {z E ex : Izl = I} is a real Lie group. 4. The group GLn(K) of invertible matrices of order n over the field K,

with the differentiable structure of an open subset of the vector space Ln(K) of all matrices, i.e. (global) coordinates are given by the matrix entries.

5. The group GL(V) of invertible linear transformations of an n-dimensional vector space over the field K can be regarded as a Lie group in view of the isomorphism GL(V) ~ GLn(K), which assigns to each linear transformation its matrix with respect to some fixed basis.

6. The group GA(S) of (invertible) affine transformations of an n-dimensional affine space S over the field K possesses also a canonical differentiable structure, which turns it into a Lie group. Namely, with respect to the affine coordinate system of the space S affine transformations can be written in the form X 1---+ AX + B, where X is a column vector of coordinates of a point, A an invertible square matrix and B a column vector. The entries of the matrix A and the column vector B can be taken as (global) coordinates in the group GA(S).

7. Any finite or countable group equipped with the discrete topology and the structure of a O-dimensional differentiable manifold.

The direct product of Lie groups is the direct product of the corresponding abstract groups endowed with the differentiable structure of the direct product of differentiable manifolds.

The Lie group Kn (the direct product of n copies of the additive group of the field K) is called the n-dimensional vector Lie group. The Lie group ~ (the direct product of n copies of the group 'll') is called the n-dimensional torus.

1.2. Lie Subgroups. A subgroup H of a Lie group G is said to be a Lie subgroup if it is a submanifold of the underlying manifold of G.

Let us recall that by am-dimensional submanifold of an n-dimensional manifold X we mean a subset Y c X such that for each of its points y one of the following equivalent conditions is satisfied:

(1) in a local coordinate system in some neighbourhood U of the point y the subset Y n U can be described parametrically in the form

(i=l, ... ,n)

where CP1,' .. , CPn are differentiable functions defined in some domain of the space Km and the rank of the matrix ~~ t : ...... :t::? at all points of this domain is equal to m.

(2) in a local coordinate system in some neighbourhood U of the point y the set Y n U can be given by equations of the form

(i=l, ... ,n-m),


where /1, ... ,fn-m are differentiable functions and the rank of the matrix 8(/1, ... ,In-Tn) at all points of the neighbourhood U is n - m.

8("1, ... '''n) (3) in a suitable local coordinate system in some neighbourhood U of the

point y the subset Y n U is given by equations

Xm+l = ... = Xn = o.

(Sometimes the terms 'submanifold' and correspondingly 'Lie subgroup' are understood in a wider sense. In this book this wider meaning is referred to by the term 'virtual Lie subgroup'; see 2.3 of Chap. 2. Lie subgroups in our sense are also known as 'closed Lie subgroups'.)

Every m-dimensional submanifold of a differentiable manifold carries the structure of a m-dimensional differentiable manifold, as local coordinates on which we can take, for example, the parameters tl, ... ,tm from condition (1). Every Lie subgroup, endowed with this differentiable structure is itself a Lie group.

From the topological and the differential geometric viewpoints every subgroup H of a Lie group G looks at any point h E H the same as at the identity, since it is transformed into itself by a translation (left or right) by h, which is a diffeomorphism of the manifold G. Therefore in order to verify that a subgroup H is a Lie subgroup it suffices to establish that it is a submanifold in some neighbourhood of the identity.

Examples. 1. Any subspace of a vector space is a Lie subgroup of the corresponding Lie group.

2. The group l' (see Example 3 of 1.1) is a Lie subgroup of the group ex, viewed as a real Lie group.

3. Any discrete subgroup is a Lie subgroup. 4. The group of non-singular diagonal matrices is a Lie subgroup of the

Lie group GLn(K). 5. The group of non-singular triangular matrices is a Lie subgroup of the

Lie group GLn(K). 6. The group SLn(K) of unimodular matrices is a co dimension 1 Lie sub

group of the Lie group GLn(K). 7. The group On(K) of orthogonal matrices is a Lie subgroup of dimension

n(n2-1) of the Lie group GLn(K). 8. The group SPn(K) (n even) of symplectic matrices is a Lie subgroup of

dimension n(n2+1) of the Lie group GLn(K). 9. The group Un of unitary matrices is a real Lie subgroup of dimension

n2 of the Lie group GLn(C). A Lie subgroup of the Lie group GLn(V) (and in particular of GLn(K) =

GL(Kn» is called a linear Lie group. As any submanifold, a Lie subgroup is an open subset of its closure. How

ever, any open subgroup of a topological group is at the same time closed, since it is the complement of the union of its own cosets, which, like the


subgroup itself, are open subsets. Hence any Lie subgroup is closed. For real Lie groups the converse is also valid, see Theorem 3.6 of Chap. 2.

1.3. Homomorphisms of Lie Groups. Let G and H be Lie groups. A map I : G ---+ H is a homomorphism if it is simultaneously a homomorphism of abstract groups and a differentiable map. A homomorphism I : G ---+ H is called an isomorphism if there exists an inverse 1-1 : H ---+ G, i.e. if I is simultaneously an isomorphism of abstract groups and a diffeomorphism of manifolds (however, in connection with this, see the corollary to Theorem 3.4).

Examples. 1. The exponential map x f-t eX is a homomorphism from the additive Lie group K to the Lie group K x

2. The map A f-t det A is a homomorphism from the Lie group GLn(K) to the Lie group K X

3. For any element 9 of a Lie group G the inner automorphism a(g) : x f-t

gxg- 1 is a Lie group automorphism. 4. The map x f-t eix is a homomorphism from the Lie group lR to the Lie

group T. 5. The map assigning to each affine transformation of an affine space its

differential (linear part) is a homomorphism from the Lie group GA(S) (see Example 6 of 1.1) to the Lie group GL(V), where V is the vector space associated with S.

6. Any homomorphism from a finite or a countable group to a Lie group is a homomorphism in the sense of the theory of Lie groups.

Obviously the composition of homomorphisms of Lie groups is also a homomorphism of Lie groups.

1.4. Linear Representations of Lie Groups. A homomorphism from a Lie group G to the Lie group GL(V) is called its linear representation in the space V.

For example, if to each matrix A E GLn(K) we assign the transformations Ad(A) and Sq(A) of the space Ln(K), defined by the formulas

Ad(A)X = AXA-l, Sq(A)X = AXAT , (1)

then we obtain linear representations Ad and Sq of the Lie group GLn(K) in the space Ln (K).

Sometimes one considers complex linear representations of real Lie groups or real linear representations of complex Lie groups. In the former case, it is understood that the group of linear transformations of a complex vector space is being considered as a real Lie group, in the latter - that the given complex Lie group is being considered as a real one.

Let Rand S be linear representations of some group G in spaces V and U respectively. Recall that, by the sum of representations Rand S, is meant the linear representation R + S of the group G in the space V E9 U, defined by the formula

10 A. 1. Onishchik, E. B. Vinberg

(R + S)(g)(v + u) = R(g)v + S(g)u (2)

by the product of the representations Rand S the linear representation RS of the group G in the space V ® U, defined on decomposable elements by the formula

(RS) (g)(u ® v) = R(g)v ® S(g)u (3)

The sum and product of an arbitrary number of representations are defined analogously.

By the dual representation of a representation R we mean the representation R* of the group G in the space V* - the dual of V, given by the formula

(R*(g)f)(v) = f(R(g)-lV) (4)

It is easy to see that, if Rand S are linear representations of a Lie group G, then the representations R + S, RS and R* are also linear representations of it as a Lie group (i.e. they are differentiable).

For any integers k, l 2 0 the identity linear representation I d of the group GL(V) in the space V generates its linear representation Tk,l = Idk(Id*)1 in the space V ® ... ® V ® V* ® ... ® V* of tensors of type (k, l) on V. We ~ '-...-'"

k I will give convenient interpretations of representations Tk,l in the two most commonly met cases: k = 0 and k = 1. Tensors of type (0, l) can be viewed as l-linear forms on V. For any such form f we have

(5)

Tensors of type (1, I) can be viewed as I-linear maps V x ... x V ~ V. For any such map F we have

(6)

The representations Ad and Sq of the group GLn(K) considered above, are just its representations in the spaces of tensors (on Kn) of type (1,1) and (2,0) respectively, expressed in the matrix form.

If R is a linear representation of some group G in a space V and U c V is an invariant subspace, there is a natural way to define the subrepresentation Ru : G ~ GL(U) and the quotient representation Rv/u : G ~ GL(VjU). Clearly, every subrepresentation and every quotient representation of a linear representation of a Lie group G are linear representations of it as a Lie group.

A special role in group theory is played by one-dimensional representations, which are precisely the homomorphisms from the given group to the multiplicative group of the base field. They are referred to as characters 1 of the group G. Characters form a group with respect to the operation of multiplication of representations; the inverse of an element in this group is its dual representation. We will denote the group of characters of a group G by

1 Here the word character is being used in its narrower sense. In its wider sense character refers to the trace of any (not necessarily one-dimensional) linear representation.


X(G). Traditionally additive notation is used to denote its group operation, thus by definition

(Xl + X2)(g) = Xl (g)x2(g) (Xl,X2 E X(G)).

In the context of the theory of Lie groups characters are assumed to be differentiable.

1.5. Local Lie Groups. In certain situations it turns out to be useful to have a local version of the concept of a Lie group. By a local Lie group we mean a differentiable manifold U together with a base point e, its neighbourhood V and a differentiable map (multiplication)

JL : V x V ----t U, (x, y) r-t xy

satisfying the conditions ex = xe = x and (xy)z = x(yz) for X,y,Z,XY,yz E V. These conditions imply the existence of a neighbourhood of the identity We V and a differentiable map (inversion)

t: W ----t W, X r-t X-I

such that xx-l = x-Ix = e for wE W. Every Lie group G can be viewed as a local Lie group by taking V = U = G.

Replacing U and V by neighbourhoods of the identity Ul and VI C V n Ul , satisfying the condition VI VI C Ul , one obtains also a local Lie group, called a restriction of the original one. By transitivity restriction generates a certain equivalence relation of local Lie groups. Strictly speaking, by a local Lie group one understands an equivalence class defined in this way. Two local Lie groups are said to be isomorphic, if for some of their restrictions (Ub eb Vb JLl) and (U2, e2, V2, JL2) there is a diffeomorphism f : Ul ----t U2 satisfying the conditions f(ed = e2, f(Vl ) = V2 and f(xy) = f(x)f(y) for x, y E VI. One can easily see that isomorphism of local Lie groups is an equivalence relation.

The concepts of Lie subgroup, homomorphism of Lie groups, etc. have natural local analogues and many theorems from the theory of Lie groups can be formulated for local Lie groups (some of them even turn out to be simpler). However the theory oflocal Lie groups does not have an independent status for the reason that every local Lie group a posteriori turns out to be a restriction of some Lie group. (This is a corollary of the theorem on theexistence of a Lie group with a given tangent algebra: see Theorem 2.11 of Chap. 2).

Within the theory of Lie groups the significance of the concept of a local Lie group lies basically in that it enables us to use local terminology. For example, two Lie groups are said to be locally isomorphic if they are isomorphic as local Lie groups. This definition is a precise interpretation of the intuitive notion that two given Lie groups "look the same in a neighbourhood of the identity" .


§2. Actions of Lie Groups

2.1. Definition of an Action. A homomorphism a from a Lie group G to the group Diff X of diffeomorphisms of a differentiable manifold X is called its action on X if the map G x X ----) X, (g, x) f---t a(g)x is differentiable.

Examples. 1. For any Lie group G one can define the following three actions l, r, a on itself:

l(g)x = gx, r(g)x = xg-l, a(g)x = gxg- 1

2. The natural action of the group GLn(K) on the projective space p(Kn) is a Lie group action.

3. Every linear representation R : G ----) GL(V) of a Lie group G can be viewed as its action on the space V. This kind of action is called linear.

4. Analogously, every homomorphism f : G ----) GA(S) can be viewed as an action of the Lie group G on the affine space S. Such an action is called affine.

Clearly, the composition of an action f : G ----) Diff X and a homomorphism f : H ----) G is an action of the Lie group H on the manifold X.

In cases where there is no danger of confusion we will write simply gx in place of a(g)x.

Actions of Lie groups will be considered in detail in the second part of this volume. We will use without any additional explanations certain common terms which are defined there.

2.2. Orbits and Stabilizers. Suppose we are given an action a of a Lie group G on a manifold X and let x be a point of this manifold. Consider the map ax : G ----) X, 9 f---t a(g )x. Its image is precisely the orbit a( G)x of the point x, and the pre-image of the point x is its stabilizer

G x = {g E G: a(g)x = x}

The pre-images of the other points of the orbit are the cosets of Gx .

From the definition of a Lie group action it follows that the map ax is differentiable, and from the commutativity of the diagram

ax G -----; X

I(g) 1 1 a(g)

ax G -----; X

for any 9 E G, that it has constant rank. It is known (see, for example, Dieudonne 1960), that a differentiable map

f : X ----) Y of constant rank k is linearizable in a neighbourhood of any point of the manifold X. From this it follows that:

1) the pre-image of any point y = f(x) is a submanifold of co dimension k in X, with TxU-1(y)) = Kerdxf;


2) for each point x E X there is some neighbourhood U such that its image is a submanifold of dimension k in Y, with Tf(",)U(U)) = d",f(T",(X));

3) if f(X) is a submanifold in Y, then dimf(X) = k

Proof. The last part is proved in the following way: if we had dim f(X) > k, then in view of (2) the manifold f(X) would be covered by a countable number of submanifolds of lower dimension, which is impossible. 0

Applying this to the map a", constructed above we obtain the following theorem:

Theorem 2.1. Let a be an action of a Lie group G on a differentiable manifold X. For any point x E X the map a", has a constant rank and if this constant rank is k, then:

1) the stabilizer G", is a Lie subgroup of codimension k in G and Te ( G "') = Kerdea",;

2) for some neighbourhood U of the identity in the group G the set a(U)x is a submanifold of dimension k in X, and T",(a(U)x) = dea",(Te(G));

3) if the orbit a( G)x is a submanifold in X, then dim a( G)x = k.

We remark that the orbit is not always a submanifold. (A counter-example will be given below).

Assertion 1) of the theorem can be used to prove that a given subgroup H of a Lie group G is a Lie subgroup. For this purpose it suffices to realize H as the stabilizer of some point for a certain action of the Lie group G. Moreover, if the orbit of the point turns out to be a manifold of known dimension, then assertion 3) makes it possible to compute the dimension of the subgroup H.

Applying these considerations to the representations Tk,l of the group GL(V) in tensor spaces (see 1.4) we find, in particular, that the group of non-singular linear transformations, preserving some given tensor, is a linear Lie group.

Examples. 1. By considering the representation of the group GL(V) in the space B+(V) of symmetric bilinear forms (symmetric tensors of type (0,2)) we see that the group O(V, f) of non-singular linear transformations preserving a given symmetric bilinear form f is a linear Lie group. If the form f is non-degenerate, then its orbit is open in B+(V) and, therefore,

dimO(V,f) = dimGL(V) - dimB+(V) = n(n2-1)

where n = dim V. 2. Analogously, by considering the representation of the group GL(V) in

the space B_(V) of alternating bilinear forms, we see that the group Sp(V, f) of non-singular linear transformations preserving a given alternating bilinear form f is a linear Lie group. If the form f is non-degenerate, then

dim Sp(V, f) = dim GL(V) - dim B_ (V) = n(n 2+ 1)


3. By considering the representation of the group GL(V) in the space of bilinear operations on V (tensors of type (1,2)) we see that the group of automorphisms of any algebra is a linear Lie group.

2.3. Images and Kernels of Homomorphisms. Let f : G - H be a homomorphism of Lie groups. Define an action a of G on the manifold H by the formula

a(g)h = f(g)h,

where the right hand side is the product of elements in H. In other words, a is the composite of the homomorphism f and the action l of H on itself by left translations.

Let e be the identity of the group H. Then a e = f, a(G)e = f(G) and the stabilizer of the point e under the action a is just the kernel Ker f of the homomorphism f. Applying Theorem 2.1 to the action a and the point e E H, we obtain the following theorem

Theorem 2.2. Let f : G - H be a homomorphism of Lie groups. Then f is a map of constant rank and if this rank is equal to k, then

1) Ker f is a Lie subgroup of codimension kin G, and Te(Ker f) = Kerdef· 2) For some neighbourhood U of the identity in the group G the set f(U)

is a submanifold of dimension k in Hand Te(f(U» = def(Te(G)). 3) if f(G) is a Lie subgroup of H, then dimf(G) = k.

Example. Consider the homomorphism det : GLn(K) - KX. Its kernel is the group SLn(K) of unimodular matrices. Since det GLn(K) = K X we have rkdet = 1 and hence SLn(K) is a Lie subgroup of co dimension 1 in GLn(K).

Clearly, if f(G) is a submanifold, then f(G) is a Lie subgroup in H. The following example shows that f(G) is not always a submanifold. Let f : IR.-1m be a homomorphism given by the formula

f(x) = (eia1 "', .•. ,eian "') (ab'" ,an E IR.)

It is known (see, for example, Bourbaki 1947), that if the numbers ab'" ,an are linearly independent over Q, then the set f(lR.) is dense in 1m (this is the so called dense winding of the torus), and therefore, for n > 1 is not a submanifold. In order that the set f(lR.) be a submanifold it is necessary and sufficient for the numbers al, ... ,an to be commensurable.

2.4. Orbits of Compact Lie Groups. The preceding example makes the following assertion particularly interesting.

Theorem 2.3. Every orbit of an action of a compact Lie group is a submanifold.

Proof. Let a be an action of a compact Lie group G on a manifold X and let x E X. We will prove that the orbit a(G)x is a submanifold in X. For this purpose it is enough to verify that it is a submanifold in a neighbourhood of

I. Foundations of Lie Theory 15

the point x. Let U be a neighbourhood of the identity in the group G such that a(U)x is a submanifold of X. The orbit a(G)x is a union of nonintersecting sets a(U)x and a(C)x, where C = G\UG",. Since the set UG", is open in G its complement C is closed and thus compact; but then the set a(C)x = a",(C) is also compact and therefore closed in X. Thus the intersection of the orbit a( G)x with the open set X \ a( C)x containing the point x, is a submanifold.

D

Corollary. A homomorphic image of a compact Lie group is a Lie subgroup.

The most important examples of compact Lie groups (besides the finite ones) are the n-dimensional torus 'II""', the orthogonal group On (= On (IR)) and the unitary group Un. In order to prove the compactness of the group On we note that it is the subset of the space Ln (IR) of all real matrices determined by the algebraic equations Ek aikajk = bij, and hence is closed in Ln(IR). The same equations imply the inequalities laij I :::; 1 which show that the group On is bounded in Ln{lR). Analogously one proves the compactness of the group Un·

§3. Coset Manifolds and Quotients of Lie Groups

3.1. Coset Manifolds. On the set of cosets of a Lie subroup of a Lie group one can naturally introduce the structure of a differentiable manifold. In giving an axiomatic description of this structure we will make use of several definitions.

Let X and Y be differentiable manifolds and let p be a differentiable map from X onto Y. For any function f defined on an open submanifold U C Y we define a function p* f on p -1 U by the formula

(p* f)(x) = f(p(x))

The map p is called a quotient map if 1) A subset U C Y is open if and only if p-1(U) is open in X. 2) A function f defined on an open subset U c Y is differentiable if and

only if the function p* f is differentiable. The map p is called a trivial fibre bundle with fibre Z (where Z is also a

differentiable manifold) if there is a diffeomorphism v: Y x Z ~ X satisfying the condition p(v(y, z)) = y.

The map p is called a locally trivial fibre bundle with fibre Z if the manifold Y can be covered by subsets, such that the map p is a trivial fibre bundle with fibre Z over each of them.

Every locally trivial fibre bundle is a quotient map. (It suffices to check this for trivial fibre bundles.) As an example of a quotient map, which is not a fibre bundle, one can take the map z --+ Z2 of the complex plane onto itself.

A quotient map possesses the following universality property: given a commutative triangle


in which p is a quotient map and q a differentiable map, the map ¢ is differentiable. If q is also a quotient map and ¢ is bijective, then ¢ is a diffeomorphism. The latter fact can be interpreted in the following way: if we are given a map p from a differentiable manifold X to a set Y, then Y posesses at most one differentiable structure for which p is a quotient map.

Theorem 3.1. Let H be a Lie subgroup of a Lie group G. The set G / H of left cosets of H in G possesses a unique differentiable structure for which the canonical map

p: G -+ G/H, 9 t--t gH

is a quotient map. In addition 1) the map p is a locally trivial fibre bundle 2) the canonical action of the group G on G / H (by left translations) is

differentiable.

Proof. We introduce a topology on the set G / H so that a subset U c G / H is open if and only if p-1(U) is open in G. It is easy to see that this makes the map p continuous and open, and the space G / H Hausdorff.

Let now 8 c G be a submanifold transversal to H at the point e. Consider the map

v: 8 x H -+ G, (s, h) t--t sh

Since v(s,e) = sand v(e,h) = h

d(e,e) v(ds, dh) = ds + dh

so that d(e,e)v is an isomorphism of the tangent spaces. Hence, there exist neighbourhoods 8 1 and V of the point e in 8 and H respectively, such that v maps 8 1 x V diffeomorphic ally onto an open subset of the group G. Since v(s,hh') = v(s,h)h', the map v is a local diffeomorphism everywhere on 81 xH. Let 8 2 be a neighbourhood of the point e in 8 1 such that 8 2 -1 82nH c V. Then v is injective on 8 2 x H. Thus by initially choosing a suitable cross section 8 we can assume that v maps 8 x H diffeomorphically onto an open subset of the group G.

Under the map p the cross section 8 maps bijectively onto some neighbourhood U of the point p( e) in the space G / H. We transfer by means of this map the differentiable structure of 8 onto U. The map p becomes then a trivial fibre bundle over U.

Furthermore, for any 9 E G we transfer the differentiabie structure from U to gU via the left translation leg). Since the map pis equivariant with respect to left translations, this definition of a differentiable structure on gU turns the map p into a trivial fibre bundle over gU and hence into a quotient map. For gl, g2 E G the differentiable stuctures, defined on gl U and g2 U coincide on the intersection gl Un g2 U, since over it the map p is a quotient map with


respect to any of these structures. Thus we defined a differentiable structure on G I H with respect to which the map p is a locally trivial fibre bundle (and consequently a quotient map).

The natural action of the group G on G I H is defined by the map

A: G x GIH -t GIH, (g',gH) f---+ g'gH

which fits into a commutative diagram

GxG G

idxp 1 G x GIH

where JL is the multiplication in the group G. The map id x p is a locally trivial fibre bundle and therefore a quotient map. From the universality of quotient maps it follows that A is a differentiable map. 0

From assertion 1) of the theorem it follows that the map

is surjective and has Te(H) as its kernel. Therefore the space Tp(e) (G I H) can be canonically identified with Te (G) ITe (H).

3.2. Lie Quotient Groups

Theorem 3.2. Let N be a normal Lie subgroup of a Lie group G. Then the quotient group GIN with the differentiable structure defined in Theorem 3.1 is a Lie group.

Proof. The differentiability of the multiplication JL N on GIN is proved analogously to the proof of Part 2) of Theorem 3.1, with the help of the commutative diagram

GxG

GIN x GIN -----> GIN o I-'N

Under the canonical bijection between the subgroups of a group G containing N and subgroups of the group GIN, Lie subgroups in G correspond to Lie subgroups in GIN and conversely.

If the group G acts on a differentiable manifold X so that the action of N is trivial, then the induced action on X of the quotient group GIN is differentiable.


3.3. The Transitive Action Theorem and the Epimorphism Theorem

Theorem 3.3. Let a be a transitive action of a Lie group G on a differentiable manifold X. Then for any x E X the map

{3,,: GIG" ---t X, gG" I-t a(g)x

is a diffeomorphism, which commutes with the action of G. (It is assumed that the group G acts on GIG" by left translations. )

Proof. We have a commutative triangle

G

X

GIG"

where a,,(g) = gx. Since p is a quotient map the map (3" is differentiable. According to Theorem 2.1

rka" = dimX = dim GIG"

so that dea,,(Te(G)) = T,,(X) and dp(e){3" is an isomorphism of tangent spaces. Thus (3" is a diffeomorphism. 0

Theorem 3.4. Let f : G ---t H be an epimorphism of Lie groups and let N = Ker f. Then the map

¢: GIN ---t H, gN I-t f(g)

is an isomorphism of Lie groups.

Proof. It suffices to apply Theorem 3.3 to the action a of the group G on H constructed in 2.3. 0

Corollary. A bijective homomorphism of Lie groups is an isomorphism.

3.4. The Pre-image of a Lie Group Under a Homomorphism

Theorem 3.5. Let f : G ---t H be a homomorphism of Lie groups and let HI be a Lie subgroup in H. Then GI = f-I(Hd is a Lie subgroup in G and

Te(Gd = (de!)-l(Te(HI)).

Proof. Consider the composite a = {3 0 f of the homomorphism f and the canonical action (3 of the Lie group H on HI HI. The subgroup GI is then the stabilizer of the point pee) E HIHI (where p is the canonical projection of H onto HIHI) under the action a. By Theorem 2.1 it is a Lie subgroup and

Te(G I ) = Kerdeap(e) = Ker (deP 0 de!).

Since KerdeP = Te(HI ), we have


D

Corollary 1. Let HI, H2 be Lie subgroups of a Lie group G. Then HI n H2 is also a Lie subgroup and

Te(HI n H 2) = Te(Hd n Te(H2)'

Proof. This is proved by applying the theorem to the inclusion HI C G and the subgroup H2 C G. D

Note that the intersection of submanifolds is not, in general, a submanifold. Corollary 1 can be trivially extended to any finite number of subgroups.

It also holds for an infinite family of subgroups (see Theorem 4.2).

Corollary 2. Let R : G ----; GL(V) be a linear representation of a Lie group G and let U C V be any subspace. Then

G(U) = {g E G : R(g)U C U}

is a Lie subgroup of G and

Te(G(U)) = {~ E Te(G) : (dR)(OU C U}.

Proof. The proof consists of applying the theorem to the homomorphism R and the subgroup

GL(V; U) = {A E GL(V) : AU C U}

of the group GL(V), which is an open subset of the space

L(V; U) = {X E L(V) : XU C U}. D

Corollary 3. Under the assumptions of Corollary 2 let W C V be any subspace contained in U. Then

G(U, W) = {g E G : (R(g) - E)U c W}

is a Lie subgroup of G and

Te(G(U, W)) = {~ E Te(G) : (dR)(~)U C W}.

Proof. The proof consists of applying the theorem to the homomorphism R and the subgroup

GL(V; U, W) = {A E GL(V) : (A - E)U c W}

of the group GL(V), which is an open subset of the space

L(V; U, W) = {X E L(V) : XU C W}. D

3.5. Semidirect Products of Lie Groups. In many cases it is most convenient to describe the structure of a Lie group by means of the concept of semi-direct product.

Let us recall that by the semi-direct product of abstract groups G1 and G2 we mean the direct product of the sets G1 and G2 equipped with a group operation by means of the formula

(7)

where b is some homomorphism from the group G2 to the group Aut G1 of automorphisms of the group G1 . We will denote the semi-direct product by G1 )<J G2 or, more correctly, by G1 )<J G2 •

b

Elements of the form (gl, e) (respectively (e, g2)) form a subgroup of G1 )<J G2 isomorphic to G1 (respectively G2 ). The subgroup G1 is normal and

(8)

The subgroup G2 is normal if and only if the homomorphism b is trivial; in this case G1 )<J G2 coincides with the direct product G1 x G2 •

It is said that a group G decomposes as a semi-direct product of subgroups G1 and G2 (written: G = G1 )<J G2 or G = G2 ~ Gt), if

1) the subgroup G1 is normal; 2) G 1G2 = G; 3)G1 nG2 ={e}. In this case there is indeed an isomorphism

G1 )<J G2 ~ G, (gl, g2) f---+ glg2, b

where b: G2 ---7 Aut G1 is the homomorphism defined by the formula (8).

(9)

The semi-direct product of Lie groups is defined as the semi-direct product of the underlying abstract groups with the differentiable structure of the direct product of differentiable manifolds. Moreover, the homomorphism b is required to define a differentiable action of the group G2 on G1 . (In particular, the automorphism b(g2) of the group G1 should be differentiable for any g2 E G2.) This ensures the differentiability of the operation (7).

We say that a Lie group G decomposes as a semi-direct product of Lie subgroups G1 and G2 if it decomposes as their semi-direct product as an abstract group. In this case the action b of the group G2 on G1 , defined by the formula (8), is differentiable and the abstract isomorphism (9), by the corollary of Theorem 3.4, is an isomorphism of Lie groups.

Examples. 1. Let R : G ---7 GL(V) be a linear representation of a Lie group G. Then one can form a semi-direct product V)<J G, where V is being

R considered as a vector Lie group.

2. Let Id be the identity linear representation of the group GL(V) in the space V. Then there is an isomorphism

V )<J GL(V) ~ GA(V), Id

which takes each vector v E V to the parallel translation tv : x f---+ x + v of the space V.


3. Every Lie subgroup G c GA(V) containing all parallel translations decomposes as a semi-direct product of the group of parallel translations and a linear Lie group H = dG C GL(V). In particular, the group of isometries of the Euclidean space En decomposes as a semi-direct product of the group of parallel translations and the orthogonal group On.

4. The Lie group of non-singular triangular matrices decomposes as a semidirect product of the normal Lie subgroup of un i-triangular matrices (triangular matrices with units on the diagonal) and the Lie subgroup of non-singular diagonal matrices.

§4. Connectedness and Simply-connectedness of Lie Groups

The properties of connectedness and simply-connectedness play an important role in the very foundations of the theory of Lie groups (see Theorems 2.2 and 2.10 of Chap. 2). Because of this we devote to them a separate section.

The definition of the fundamental group and proofs of the topological theorems used in this chapter (the existence of the universal covering space and the exactness of the homotopy sequence of a fibre bundle) can be found, for example, in (Spanier 1966).

4.1. Connected Components of a Lie Group. A topological space is said to be connected if it cannot be decomposed into two non-empty open subsets, and pathwise connected if any two of its points can be joined by a (continuous) path. For a differentiable manifold these concepts coincide. Moreover, any two points of a connected differentiable manifold can be joined by a differentiable path. Connected components of a differentiable manifold are closed and open. From our assumption of the existence of a countable base it follows that a differentiable manifold has at most a countable number of connected components.

Theorem 4.1. A connected component GO of a Lie group G, which contains the identity element, is a normal Lie subgroup. The remaining connected components are the cosets of GO. Every open subgroup of the group G contains GO.

Proof. Since both right and left translations are automorphisms of the group manifold, they can only permute its components. Hence it follows that the decomposition into connected components coincides with the decomposition into cosets of a normal subgroup, which obviously is the connected component containing the identity. The last assertion of the theorem follows from the fact that every open subgroup is closed (see 1.2). 0

Let us note that a Lie subgroup H eGis open if and only if dim H = dimG or, equivalently, Te(H) = Te(G).


Corollary. A connected Lie group is generated (as an abstract group) by any neighbourhood of the identity.

Proof. Indeed, the subgroup generated by any neighbourhood of the iden-tity is open and, therefore, coincides with the entire group. 0

The quotient group GIGo is obviously discrete. It is known as the group of components of the group G.

In 3.5 we showed that the intersection of a finite number of Lie subgroups is a Lie subgroup. Now we can generalize this statement.

Theorem 4.2. The intersection H = nH" of an arbitrary family {H,,} of " Lie subgroups is a Lie subgroup, and Te(H) = nTe(H,,).

" Proof. The subspace n Te(H,,) coincides with the intersection of a finite

" number of the subspaces Te(H,,), say, Te(H"J ... Te(H"k) and, according to Corollary 1 of Theorem 3.5, is the tangent space of the Lie subgroup if = H"I n· . ·nH"k' For any 1/, the Lie subgroup ifnH" has the same tangent space as if and hence is contained between if ° and if. Therefore the subgroup H must also be contained between if ° and if. Hence, it is a Lie subgroup with Te(H) = nTe(H,,) 0

" 4.2. Investigation of Connectedness of the Classical Lie Groups. In study

ing connectedness of concrete Lie groups the following theorem is usually used.

Theorem 4.3. Let a be a transitive action of a Lie group G on a connected differentiable manifold X. Then

1) The group GO also acts transitively on X 2) GIGO '::C Gxl(Gx n GO) for any point x E X 3) If the stabilizer Gx of a point x E X is connected, then the group G is

connected.

Proof. According to Theorem 2.1 rkax = dimX for any point x E X. Applying the same theorem to the restriction of the action a to the subgroup GO we find that the orbit a(GO) x contains a neighbourhood of the point x. Hence, all orbits of the group GO are open in X. Since X is connected, there is in fact only one orbit. From this one can easily deduce the remaining assertions of the theorem. 0

Let us denote by GL~ (JR.) the group of real matrices with positive determinant, by SOn the group of orthogonal matrices with determinant 1 and by SUn the group of unitary matrices with determinant 1.

Proposition 4.4. The groups SLn(K), GLn(CC), GL~ (JR.), SPn(K), SOn(K), Un and SUn are connected.

Proof. We prove by induction on n that the group SLn(K) is connected. For n = 1 there is nothing to prove, since the group SL1 (K) is trivial. For n ~ 2 consider the natural action SLn(K) : Kn. It is transitive on the open subset K n \ {O}, which is connected. The stabilizer of a point of this subset is isomorphic to a semi-direct product K n - 1 )<J SLn- 1 (K) and, by the inductive hypothesis, is connected. By Theorem 4.3 this implies that the group SLn(K) is also connected. In an analogous way one can prove that the groups GLn(q, GL~(IR) and SPn(K) are connected.

Let us prove, also by induction on n, that the group SOn(K) is connected. For n = 1 there is nothing to prove. For n ~ 2 consider the natural action SOn(K) : sn-l(K), where sn-l(K) is the unit sphere in Kn. This action is transitive, and the stabilizer of a point of the sphere is isomorphic to the group SOn-l (K), which by the inductive hypothesis is connected. The sphere sn-l(K) is also connected: this is clear for the real sphere sn-l(IR) = sn-l and the complex sphere sn-l (q is diffeomorphic to the tangent bundle of sn-l. Hence the group SOn(K) is connected.

In an analogous way one can prove that the groups Un and SUn are con-nected. 0

Corollary. The groups GLn(IR) and On(K) consist of two connected components, distinguished by the sign of the determinant.

Let us consider more complicated cases. Let k, l > 0, k + l = n. A real matrix of order n is called a a pseudo-orthogonal matrix of signature (k, l), if the corresponding linear transformation preserves the quadratic form

q(x) = X1 2 + ... + Xk 2 - Xk+1 2 - ••• - xn2 •

The group of pseudo-orthogonal matrices of signature (k, l) will be denoted by Ok,l' This is a Lie group of dimension n(~-l) (see the example in 2.2). Clearly Ok,l ~ Ol,k'

The determinant of a pseudo-orthogonal matrix is equal to ±l. Just as in the case of orthogonal matrices, the subgroup SOk,1 of pseudo-orthogonal matrices of determinant 1 is an open subgroup of index 2 of the group Ok,l' However we will see now that it is not connected.

Let {el' ... en} be the standard basis of the space IRn. Let

IR~ = (el, ... ek), IR":. = (ek+1, ... en).

Since the form q is positive definite on IR+ and negative definite on IR":., so for any A E Ok ,I we have AIR+ n IR":. = O. This means that the top left corner minor ~k (A) of order k of the matrix A is non zero. It is not difficult to show that, for matrices A E SOk,1 (even diagonal ones), the sign of ~k(A) can be arbitrary. Hence, the group SOk,1 has at least two connected components, distinguished by the sign of ~k'

Analogously, a complex matrix of order n is called pseudo-unitary matrix of signature (k, l) if the corresponding linear transformation preserves the quadratic hermitian form


q(x) = IXll2 + ... + IXkl2 -IXk+112 - ... -lxn I2 .

The group of pseudo-unitary matrices of signature (k, l) is denoted by U k,l.

It is a real Lie group of dimension n2 . Its subgroup of unimodular matrices is denoted by SU k,l.

Proposition 4.5. The group SOk,1 consists of two connected components distinguished by the sign of the determinant and the top left corner minor of order k, and its component group is isomorphic to Z2 x Z2. The groups U k,l

and SU k,l are connected.

Proof. We will prove, by induction on n, that the group SOk,1 has at most two connected components. For n = 2 (and k = l = 1), with respect to the base of]R2 in which the form q has the form YlY2, transformations in SOl,l

are expressed by matrices (~ >. ~l ) , >. E ]R*. Therefore, the group SOl,l

consists of two connected components (distinguished by the sign of >.). For n :::: 3 either k :::: 2 or l :::: 2. Supposing that k :::: 2, consider the action

of the group SOk,1 on the hyperboloid

Sk-l,l = {x E ]Rn : q(x) = I}.

This action is transitive, and the stabilizer of a point of the hyperboloid is isomorphic to the group SOk-l,l, which, by the inductive hypothesis, has at most two connected components. Since the hyperboloid Sk-l,l is connected (it is diffeomorphic to Sk-l x ]Rl ), by Theorem 4.3 the group SOk,1 also has at most two connected components.

An analogous argument shows that the groups Uk,l and SUk,1 are con-nected. D

Corollary. The set

O~,l = {A E Ok,l : ~k(A) > O}

is a subgroup of index 2 of the group Ok,l'

4.3. Covering Homomorphisms. Locally isomorphic Lie groups may, nevertheless, be non-isomorphic globally. As an example one can take the onedimensional abelian Lie groups ]R and T. The homomorphism x f---+ eix relating these groups is a local isomorphism, since its kernel 2nZ is discrete. Such homomorphisms are called covering homomorphisms.

More exactly, a homomorphism f from a Lie group G onto a Lie group H is called a covering homomorphism if it satisfies any of the following equivalent conditions:

1) f maps diffeomorphic ally some neighbourhood of the identity of the group G onto a neighbourhood of the identity of H;

2) the kernel of f is discrete; 3) f is a covering map in the topological sense (i.e. it is a locally trivial

fibre bundle with a discrete fibre);


4) def is an isomorphism of the tangent spaces. The equivalence of these conditions follows from Theorems 3.4, 3.1 and

2.2.

Examples. 1. The homomorphism

f : e ----> ex, x f-+ e27rix ,

is a covering homomorphism with kernel Z, with f(lR) restriction of f to lR defines a covering homomorphism

fo : lR ----> 1['.

2. There is a covering homomorphism

f: SL2 (C) ----> S03(C)

1I', so that the

with kernel {E, -E}, whose restriction to SU2 defines a covering homomorphism

Proof. In order to construct this homomorphism, consider the linear representation Ad of the group SL2 (C) in the 3-dimensional space Lg (C) of traceless matrices, given by the formula

Ad (A)X = AXA-I .

The operators Ad (A) preserve, in Lg(C), the non-degenerate quadratic form q(X) = det X and, since the group SL2 (C) is connected, they all have determinant 1. This defines the homomorphism f : SL2 (C) ----> S03(C). One verifies directly that Ker f = {E, -E}. Since dimSL2(C) = dimS03(C) = 3, f is a covering homomorphism.

For A E SU2 the operator Ad (A) preserves the real form of the space Lg(C) consisting of skew-hermitian matrices (with trace zero), on which the quadratic form q is real and positive definite. With respect to an orthonormal basis of this real form, we have f(SU2 ) = S03. 0

3. There exists a covering homomorphism

f: SL2 (C) x SL2 (C) ----> S04(C)

with kernel {( E, E), ( - E, - En, whose restriction to SU 2 X SU 2 determines a covering homomorphism

Proof. One constructs this homomorphism analogously to the one above, beginning with the linear representation R of the group SL2 (C) x SL2 (C) on the 4-dimensional space L 2 (C) of all matrices, defined by the formula

R(A,B)X = AXB-I .

For A, B E SU2 , the operator R(A, B) preserves the real form of the space

L 2 (C), consisting of matrices of the form (u_ ~). 0 -v u


Proposition 4.6. Every discrete normal subgroup N of a connected Lie group G is contained in its centre.

Proof. For any n E N consider the map

G -t N, 9 t--+ gng- l .

Its image is connected and hence it consists of just one point n, which means that n belongs to the centre of the group G. 0

Thus, for a given connected Lie group G, the description of the covering homomorphisms G -t H reduces to the description of discrete central subgroups of G.

4.4. The Universal Covering Lie Group. A topological space is said to be simply connected if it is connected and every closed path in it can be contracted to a point. It is known that, for any connected differentiable manifold X, there exists a simply connected differentiable manifold X and a (differentiable) covering p : X -t X. Such a covering is called universal. It has the following functorial property:

(F) Let f : X -t Y be a differentiable map between connected differentiable manifolds, p : X -t X and q : Y -t Y their universal coverings. Then, for any points Xo E X and Yo E Y satisfying the condition f(P(xo)) = p(fjo), there exists a unique differentiable map j : X -t Y, such that the diagram

pl lq f

X ----+ Y

is commutative and j(xo) = Yo. (In this situation one says that j covers f.) Let p : X -t X be a universal covering. Automorphisms of the manifold

X covering the identity automorphism of X form a group r(p), called the group of the covering p. By property (F) for any points Xl, X2 E X satisfying the condition p(Xl) = P(X2) there exists a unique element of the group r(p) which maps Xl to X2.

The group r(p) is isomorphic to the fundamental group 7l'l(X) of the manifold X and the isomorphism is constructed in the following way. Let xo be a base point of the manifold X and let Xo = p(xo). Then to every element "I E r(p) there corresponds a class of closed paths on X beginning at xo, which are images of paths on X connecting Xo with 'Y(xo),

If p : G -t G is a covering homomorphism between Lie groups and the group G is simply connected, then the group r(p) coincides with the kernel N of the homomorphism p, acting on G by translations, and hence 7l'1 (G) ~ N.

Theorem 4.7. Every connected Lie group G is isomorphic to a quotient G / N where G is a simply connected Lie group and N a discrete subgroup.


The pair ( G, N) is determined by these conditions up to an isomorphism, i. e. if (G, N 1 ) and (G2 , N 2 ) are two such pairs, then there exists an isomorphism G1 ~ G2, taking N1 to N2.

Proof. Let p : G -T G be the universal covering of the group manifold G and let e E G be any pre-image of the identity e of G. The map p x p : G x G -T G x G is the universal covering of the manifold G x G. We define the multiplication iJ, : G x G -T G to be the map covering the multiplication in G and taking the point (e, e) to e, and the inversion 'i : G -T G as the map covering the inversion L in G and taking the point e to itself.

Since each of the maps

covers the map

G x G x G -T G, (x, i), z) ~ iJ,(iJ,(x, i)), z), (x,i),z) ~ iJ,(x,iJ,(fj,z)),

G x G x G -T G, (x,y,z) ~ xyz,

and takes the point (e, e, e) to e, the multiplication iJ, is associative. Analogously, we can verify all the other group axioms. From the definition of iJ, it follows that p is a homomorphism. Its kernel N is a discrete central subgroup of G (Proposition 4.6) and G ~ G/N.

Let now G1 and G2 be simply connected Lie groups, N1 and N2 their central subgroups and f : GdN1 -T G2/N2 an isomorphism of Lie groups. Then the diffeomorphism j : G1 -T G2 covering f and taking the unit of G1 to that of G2 is a group isomorphism and takes N1 to N2. D

A group G satisfying the conditions of the theorem is called the universal covering Lie group of the Lie group G. For example, the Lie groups C and IR are the universal coverings of C* and 1l' respectively, see Example 1 of 4.3. (As we will see in 4.5, the covering groups of the other examples in 4.3 are also universal).

Corollary. The fundamental group of a connected Lie group G is abelian.

In addition we can show that 71"1 (G) has a finite number of generators.

4.5. Investigation of Simply-connectedness of the Classical Lie Groups. When studying simply-connectedness of concrete Lie groups one usually makes use of a fragment of the homotopy sequence of the locally trivial fibre bundle p: G -T G/H (see Theorem 3.1).

Theorem 4.8. Let G be a connected Lie group and H a Lie subgroup. Then there is the following exact sequence of groups and homomorphisms:

The homomorphism 71"1 (G/H) -T H/Ho in sequence (10) is defined in the following manner. Let f3 be a closed path in G / H with initial point p( e). Then


there is a path a in G with initial point e such that p(a) = (3. A connected component C of the group H containning the end point of the path a depends only on the homotopy class of the path (3. To this class we assign the element C-l E HjHo.

Corollary. If7rl(GjH) = 7r2(GjH) = 0, then 7rl(G) ~ 7rl(H).

Note that the condition "7r2(X) = 0" for any pathwise connected topological space X means that any continuous map of a two-dimensional sphere into X is contractible to a point.

We will next give a table of fundamental groups of some classical Lie groups.

Table. Fundamental groups of some classical Lie groups

G

SLn(lC), SUn, SPn(iC) GLn(lC), Un, SPn(lR) SOn(lC), SOn, G L;t (JR), SLn (JR)

7q(G)

o z

Z:! for n 2': 3, Z for n = 2

Proof. For the groups GLnUC), SLn(C) and SPn(C) consider, as in the proof of Proposition 4.4, their action on Cn \ {o}. Noting that 7rl (Cn \ {o}) = 7r2(Cn \ {o}) = 0 for n 2: 2 we find, with the help of the corollary of Theorem 4.8, that

7rl(GLn(C)) = 7rl (GLI (C)) = 7rl(C X ) ~ Z,

7rl(SLn(C)) = 7rl(SL1(C)) = 7rl({e}) = 0,

7rl(SPn(C)) = 7rl({e}) = O.

For the group SOn(K) for n 2: 3 we consider its action on the sphere sn-l(K). Noting that 7rl(sn-l(K)) = 7r2(sn-l(K)) = 0 for n 2: 4 we see that

7rl(SOn(K)) = 7rl(S03(K)).

An analogous argument shows that

7rl(Un) = 7rl(Ut} = 7rlClI') ~ Z,

7rl(SUn) = 7rl(SUt} = 7rl({e}) = O.

Moreover, since the groups SL2(C) and SU2 are simply connected it follows from Example 2 of 4.3 that

7rl(S03(K)) ~ Z2.

Let us now consider the action of the group GL~ (lR) on the manifold Pn of positive definite symmetric matrices, defined by the linear representation Sq (see 1.4). This action is transitive, with the group SOn as the subgroup of the unit matrix. Since the manifold Pn is an (open) convex cone in the vector space of all symmetric matrices, it is diffeomorphic to lRn(n+l)/2. Hence

1. Foundations of Lie Theory

Furthermore, as GL~(lR) = SLn(lR) x {.AE : A > O} so

7rl(SLn(lR)) = 7r(GL~(lR)).

29

Finally, consider the action of the group SPn(lR) on IRn \ {O}. As 7rl(lRn \ {O}) = 7r2(lRn \ {O}) = 0 for n ~ 4 so

o

Chapter 2 The Relation Between Lie Groups and Lie Algebras

The basic method of the theory of Lie groups, which makes it possible to obtain deep results with striking simplicity, consists in reducing questions concerning Lie groups to certain problems of linear algebra. This is done by assigning to every Lie group G its "tangent algebra" g, which to a large extent determines the group G, and to every homomorphism f : G -t H of Lie groups a homomorphism df : 9 -t ~ of their tangent algebras, which to a large extent determines the homomorphism f. In the language of category theory we have a functor from the category of Lie groups into the category of Lie algebras, whose properties are very close to those of an equivalence of categories. In honour of the founder of the theory of Lie groups we will call this functor (following M. M. Postnikov (1982)) the Lie functor.

§ 1. The Lie Functor

1.1. The Tangent Algebra of a Lie Group. The most direct method of defining the tangent algebra of a Lie group G consists in the following.

Choose a coordinate system in a neighbourhood of the identity e of the group G so that the point e is the origin. The column vector of coordinates of a point x will be denoted by x. Consider the Taylor series of the coordinates of the product xy. Since ey = y and xe = x we have

(1)

where Q is a bilinear vector-valued form.


Switching the order of x and y we obtain

(2)

We see that non-commutativity of the multiplication in the group G is reflected only in terms of degree 2: 2. The group commutator (x, y) = xyx-1y-l serves as a measure of non-commutativity. Terms of degree two in the Taylor expansion of the commutator (x, y) can easily be found from the relation (x, y)yx = xy. Comparing (1) and (2) we obtain

(3)

where

'}'(x,1)) = a(x,1)) - a(y, x). (4)

Let us now define on the tangent space Te(G) a bilinear "commutator" (~, 11) 1--+ [~, 11] by the formula

(5)

where "( denotes the column of coordinates of a tangent vector ( in the coordinate system of the space Te (G) associated with the chosen local coordinate system on G.

The above definition can be given a coordinate free form. Let g(t) and h( s) be differentiable paths on G such that

g(O) = h(O) = e, g'(O) = ~, h'(O) = 11. (6)

Then {j2

[~,11] = 8tas (g(t), h(s))lt=s=o. (7)

(The right hand side of this equality has an invariant meaning since differentiation with respect to t gives, for any s, an element of the tangent space Te (G) and the subsequent differentiation with respect to s is a differentiation of a path in Te(G).)

The space Te (G) with the operation [ , ] defined in this way, is the tangent algebra of the group G and will be denoted by g. In general, the tangent algebra of a Lie group denoted by any upper case Latin letter is denoted by the corresponding lower case Gothic letter.

It is clear from the definition that the tangent algebra is anticommutative, i.e. it satisfies the identity

(8)

The tangent algebra of an abelian Lie group is an algebra with trivial multiplication.

Let A be a (finite-dimensional) associative algebra with an identity e and G = A x its multiplicative group of invertible elements. The group G has a natural differentiable structure as an open subset of the vector space A.


Moreover, it is a Lie group and its tangent space at any point can be naturally identified with A. The identity

(e + a)(e + b) = e + a + b + ab

shows that a(a, b) = abo Therefore the commutator in the algebra 9 has the form

[a, b] = ab - ba.

In particular, if A = L(V) is the algebra of linear operators on the space V, then A x = GL(V). Thus the tangent algebra of the Lie group GL(V) is the space L(V) with the commutator

[X,Y] = XY - YX. (9)

It is denoted by gl(V). In the matrix case we find that the tangent algebra of the Lie group

GLn(K) is the space Ln(K) of matrices with the commutator given by (9). It is denoted by gln(K).

Obviously, the tangent algebra of a Lie subgroup of a Lie group G is a subalgebra of the algebra g. In particular, the commutator in the tangent algebra of any linear Lie group is given by the formula (9).

1.2. Vector Fields on a Lie Group. It is possible to give a definition of the tangent algebra of a Lie group in which the commutator arises from the commutator (Lie bracket) of vector fields.

With the help of left or right translations one can construct natural isomorphisms between tangent spaces of a Lie group G at different points. Let l(g) denote left translation by 9 and r'(g) right translation by g. Then for any ~ E Th(G) let

ge = (dl(g))(e) E Tgh(G), eg = (dr'(g))(e) E Thg(G).

From the associativity of group multiplication we derive the following identities:

(gh)~ =g(h~), (g~)h =g(eh), (eg)h= ~(gh)

for any g,h E G, e E T(G) If, in particular, G = AX is the group of invertible elements of an associative algebra A, then the "products" ge and ~g coincide with the products in the sense of the algebra A.

In a local coordinate system in a neighbourhood of the identity formula (1) after differentiation with respect to the first or the second factor at the point e gives

~y = ~ + a(~, 1]) + 0(11112) (e E Te(G)).

X1J = Tj + a(x, m + 0(lxI 2 ) (1J E Te(G)).

(10)

(11)

For every ~ E Te(G) consider the right invariant vector field ~* on G given by the formula

(12)


Clearly, the map e f-7 e* is an isomorphism from the vector space Te (G) to the vector space T* (G) of all right invariant vector fields on the group G, with e = e*(e).

As the commutator of vector fields is invariant under arbitrary diffeomorphisms, the commutator of right invariant vector fields is itself right invariant. Thus T* (G) is an algebra with respect to the operation of taking the commutator of vector fields 1. This algebra can be viewed, by definition, as the tangent algebra of the group G. We shall next prove that the map e f-7 e* is an isomorphism of algebras.

Proof. According to (10) in a local coordinate system in a neighbourhood of the identity we have:

(13)

Therefore

o It is well known that the commutator of vector fields satisfies the Jacobi

identity. Therefore the commutator in the tangent algebra 9 of a Lie group G also satisfies the Jacobi identity:

[[e, 1]], (l + [[1], (], el + [[(, el, 1]l = 0 (14)

Every algebra with an operation [ , 1 satisfying the anti-commutativity identity (8) and the Jacobi identity (14) is called a Lie algebra. Thus the tangent algebra of any Lie group is a Lie algebra (usually called the Lie algebra of the Lie group).

1.3. The Differential of a Homomorphism of Lie Groups. Let f : G --> H be a homomorphism of Lie groups. From any definition of the tangent algebra it easily follows that the map def : 9 --> ~ is a homomorphism of Lie algebras. In cases where this cannot lead to misunderstanding we will denote it simply by df. Let N be the kernel of the homomorphism f and let neg be its tangent algebra. According to Theorem 2.2 of Chap. 1 n is the kernel of the homomorphism df and therefore an ideal of the algebra g.

Any normal Lie subgroup N of a Lie group G is the kernel of the canonical homomorphism p : G --> GIN. Hence, its tangent algebra is an ideal of the algebra g. By considering the homomorphism dp we see that the tangent algebra of the quotient Lie group GIN is canonically isomorphic with the quotient algebra gin.

1 Definitions of the commutator (Lie bracket) of vector fields in different texts may differ as to sign. Here we take the following definition: [€,1J]i = L(1JjOj€i -€jOj1Ji), i.e. [€,1J] =

1J~ - fr;o


A homomorphism from any Lie algebra 9 to the Lie algebra g[(V) is called its linear representation in the space V. The differential of a linear representation of a Lie group is a linear representation of its tangent algebra in the same space.

Examples. 1. Consider the homomorphism

det : GLn(K) ----t K X •

Using the explicit expression for the determinant we find that

(dE det)(X) = tr X.

Hence, the tangent algebra s[n(K) of the group SLn(K) consists of all matrices with trace zero.

2. The differentials of linear representations Ad and Sq of the group GLn(K), defined in 1.4 of Chap. 1, have the form

(dAd(X))Y = XY - YX, (dSq(X))Y = XY + YXT.

Proof. To prove, let us say, the first of these formulas, consider the path E + tX in the group GLn(K). We have

(E + tX)-l = E - tX + O(t2 ),

so that

(Ad(E + tX))Y = (E + tX)Y(E - tX + O(t2)) = Y + t(XY - Y X) + O(t2).

o Let Rand S be linear representations of a Lie group G in spaces V and

U respectively. Then

d(R + S)(~)(v + u) = dR(~)v + dS(~)u, (15)

d(RS)(~)(v 0 u) = (dR(~)v) 0 u + v 0 (dS(~)u), (16)

(dR*(~)f)(v) = -f(dR(~)v). (17)

Proof. Let us prove, for example, formula (16). Let g(t) be a differentiable path in the group G satisfying the conditions g(O) = e, g'(O) = ~. Then

d d(RS)(~)(v 0 u) = dt (RS)(g(t))lt=o(v 0 u) =

d d = dt (RS)(g(t))(v 0 u)lt=o = dt (R(g(t))v 0 S(g(t))u)lt=o

= (dR(Ov) 0 u + v 0 (dS(~)u). 0

With the help of these formulas one can compute the differential of the product of an arbitrary number of linear representations and their duals and, in particular, the differential Tk,l of the natural linear representation Tk,l of the group GL(V) in the space of tensors of type (k, l) (see 1.4 of Chap. 1)


Let us give interpretations of the representations TO,I and T1,1, obtained by differentiating formulas (5) and (6) of Chap. 1:

(To,1 (X)f(v1,"" VI) = - L f(vI, ... , XVi"'" vd (18)

1.4. The Differential of an Action of a Lie Group. Even though the group of diffeomorphisms is not a Lie group, it does indisputably possess the tangent algebra - the algebra of vector fields. Correspondingly, the differential of an action of a Lie group G on a manifold X ought to be a homomorphism of the algebra g into the algebra of vector fields on X. The precise definition is given below.

Let a be an action of a Lie group G on a differentiable manifold X. To every element, E g we assign a vector field da(') = e on X defined by the formula

(20)

where g(t) is any differentiable path in G satisfying the conditions g(O) = e, g'(O) = ,. The field e is called the velocity field of the action a corresponding to the element, E g. The map da is a homomorphism of the algebra g to the algebra of vector fields on X.

Proof. Let g(t) and h(s) be differentiable paths in G, satisfying conditions (6). Then by (7)

- (j2 [',77Hx) = {)t{)s (g(t), h(s»xlt=s=o.

Differentiating with respect to t we obtain a vector

e(x) - dh(s)(e(h(s)-1x» E T",(X).

Differentiating with respect to s we obtain

[[;j](x) = (iJe - ei])(x) = [e, iJ](x). o

Examples. 1. If a = R is a linear action of a group in a vector space V, then

e(v) = dR(,)v,

where dR in the right hand side is viewed as the differential of a linear representation.

2. The group SL2 (K) acts naturally on the projective line Kp1 = KU{ oo}:

( ac b)x=ax+b. d cx+d

(21)

The algebra s[2(K) has a basis


E+ = (~ ~), H = (~ ~1)' E_ = (~ ~). Differentiating (21) with respect to a, b, c, d we obtain

E+(x) = 1, H(x) = 2x, E_(x) = _x2.

35

1.5. The Tangent Algebra of the Stabilizer. The following theorem is a reformulation of one of the assertions of Theorem 2.1 of Chap. 1.

Theorem 1.1. Let Q be an action of a Lie group G on a differentiable manifold X, and let gx be the tangent algebra of the stabilizer Gx of a point x E X. Then

gx = {~ E 9 : dQ(~)(x) = O}.

This theorem is a very effective tool for determining tangent algebras of Lie subgroups. In particular, with its help we can determine the tangent algebra of a linear Lie group defined by the requirement of preservation of some tensor.

Examples. 1. The group G of non-singular linear transformations of a space V, preserving a given bilinear form f, is the stabilizer of the form f under the linear representation TO,2 of GL(V). Applying formula (18), we find that the tangent algebra of G consists of all linear transformations of V antisymmetric with respect to f.

2. In the case when V is a complex vector space, the analogous assertion holds also for any sesqui-linear form f. In particular, the tangent algebra of the group Un of unitary matrices consists of all skew-Hermitian matrices.

3. The group Aut Qt of automorphisms of a finite-dimensional algebra Qt is the stabilizer of the structure tensor of the algebra Qt under the linear representation T1 ,2 of the group GL(Qt). Applying formula (19), we see that the tangent algebra of the group Aut Qt consists of all linear transformations D of the space Qt satisfying the condition

D(ab) = (Da) b + a (Db).

Such transformations are known as derivations of the algebra Qt. They, consequently, must form an algebra with respect to the commutator (which, of course, can also be verified directly). This algebra is denoted by DerQt.

1.6. The Adjoint Representation. Every Lie group G has a natural linear representation in its tangent algebra g. Namely, for any element 9 E G we consider the inner automorphism

a(g) : x 1-+ gxg- 1

of G. Its differential at the point e we will denote by Ad(g). It is an automorphism of the algebra g. Since a(glg2) = a(gI)a(g2)' the map

Ad: G 1-+ GL(g), 9 1-+ Ad (g),

is a linear representation of G. It is referred to as the adjoint representation.


In the notation of 1.2 we can write

(22)

In particular if G = A x is the group of invertible elements of an associative algebra A, then Ad(g) is simply the conjugation by the element 9 in the algebra A.

The differential of a linear representation of a Lie group G is a linear representation of its tangent algebra 9 in the space g. It is known as the adjoint representation of the algebra 9 and is denoted by ad. From formulas (10) and (11) it follows that

(23)

As Ad(G) C Autg, so ad(g) c Derg (see Example 3 of 1.5). On the other hand, for every algebra 9 with an operation [ , lone can

define by means of formula (23) a linear map ad : 9 --) 9[(9). It is easy to see that, in the presence of anticommutativity, the Jacobi identity is equivalent to any of the following properties

1) the map ad is an algebra homomorphism; 2) ad(g) c Der 9 Thus we obtain yet two more proofs (and two interpretations) of the Jacobi

identity in the tangent algebra of a Lie group. The following standard facts about centralizers and normalizers are con

nected with the adjoint representation.

Proposition 1.2. For any element 9 E G its centralizer Z(g) is a Lie subgroup with Lie algebra

3(9) = {e E 9 : Ad(g)e = 0 (24)

(known as the centralizer of the element 9 in the algebra g).

Proof. The subgroup Z(g) is just the stabilizer of the point 9 under the action a of G on itself by inner automorphisms. It is, therefore, a Lie group. By Theorem 1.1 its tangent algebra consists of e E 9 such that da(e)(g) = 0; but it is easy to see that

o Proposition 1.3. For any element e E g, its centralizer Z(e) in G given by

the formula Z(e) = {g E G : Ad(g)e = 0,

is a normal Lie subgroup with tangent algebra

(known as the centralizer of the element e in the algebra g).

(25)

(26)

Proposition 1.4. For any subspace ~ c g, its normalizer N(~) in G defined by

N(~) = {g E G : Ad(g)~ C ~}, (27)

is a Lie subgroup with tangent algebra

n(~) = {~ E 9 : [~, ~l c ~} (28)

(known as the normalizer of the subspace ~ in the algebra g).

Proof. The assertion is proved by applying Corollary 2 of Theorem 3.5 of Chap.! to the adjoint representation of G. 0

§2. Integration of Homomorphisms of Lie Algebras

In this and the following sections we shall be considering paths in differentiable manifolds. By a path we shall be understand a differentiable map of a connected subset of the real line which is not just a single point, into the given manifold (real or complex). In the majority of cases, this subset (the domain of definition of the path) will not be explicit ely given.

2.1. The Differential Equation oCa Path in a Lie Group. For any path g(t) in a Lie group G we have, according to 1.2,

g'(t) = ~(t)g(t); (29)

where ~(t) E g. The path ~(t) in the algebra 9 is called the velocity of the path g(t).

The identity (29) can be viewed as an equation determining the path g(t) in terms of its velocity ~(t). In a local coordinate system it turns into a system of ordinary differential equations of the first order with respect to the coordinates of g(t). Therefore the path g(t) is uniquely determined by its velocity ~(t) and the initial condition g(to) = go. On the other hand, the path g(t)h for any h E G also satisfies equation (29). Hence, all solutions of this equation can be obtained from one another by right translations.

Let us consider now the question of existence of solutions to equation (29).

Proposition 2.1. Suppose we are given a differentiable map t f--t ~(t) from a connected subset T C lR to the algebra g. Then there exists a solution of equation (29) defined for all t E T.

Proof. It suffices to prove the assertion for the case when T is a segment. Further, it is enough to prove that there exists an c > 0 such that, for any to E T, there exists a solution of equation (29), defined for It - tol < c. Moreover, in view of the invariance of the set of solutions with respect to right translations one can suppose that g(to) = e.

In a coordinate system in a neighbourhood of the identity of G equation (29) takes the form

g'(t) = F(~(t),g(t)), (30)

where F is a differentiable vector valued function, depending only on the chosen local coordinate system. Assuming that the identity is the origin of the coordinate system, we shall denote by R a positive number for which the chosen coordinate neighbourhood contains the ball IXI :::; R. Let also

C = max 1~(t)l, M = max IF(X, Y)I. tET IXI::;c,IYI::;R

Then by the well known theorem on the existence of solutions of systems of differential equations (see for example (Dieudonne 1960)) equation (30) has a solution defined for It - tol < ~, t E T. Since ~ does not depend on to it can be chosen as €. 0

2.2. The Uniqueness Theorem

Theorem 2.2. A homomorphism I, from a connected Lie group G to a Lie group H, is uniquely determined by its differential.

Proof. Any element 9 E G can be connected with the identity by a path g(t),O :::; t :::; 1 Let g(t) satisfy equation (29) with initial condition g(O) = e. Applying to this equation the homomorphism I, we find that the path h(t) = I(g(t)) in the group H satisfies the equation

h'(t) = dl(e(t))h(t)

with initial condition h(O) = e. This determines the element I(g) = h(l). 0

2.3. Virtual Lie Subgroups. As we have seen in 2.3 of Chap. 1, the image of a Lie group under a homomorphism is not always a Lie subgroup. The more general subgroups obtained in this way can serve in some cases as surrogates of Lie subgroups.

Let us call a subgroup H of a Lie group G, which is equipped with the structure of a Lie group in such a way that the identity inclusion i : H ~ G is a homomorphism of Lie groups, a virtual Lie subgroup. In this situation we shall consider the algebra ~ as embedded in the algebra 9 by means of the homomorphism di.

Clearly, any Lie subgroup (with the induced Lie group structure) is a virtual Lie subgroup. If 1 : H ~ G is any homomorphism of Lie groups, the group I(H), equipped with the Lie group structure of the quotient HIKer I, is a virtual Lie subgroup of the group G with tangent algebra dl(~).

The topology of a virtual Lie subgroup may be different from the topology induced from the ambient Lie group. This is clearly seen from the example of the dense winding of the torus 1m, which carries the structure (and, in particular, the topology) of the Lie group lR but intersects any nonempty open subset of the torus in a subset which is unbounded in lR.

However, from Theorem 2.2 of Chap. 1 it follows that any virtual Lie subgroup H contains a neighbourhood V of the identity, which is a submanifold


of the ambient Lie group (and, in particular, possesses the induced topology), and moreover Te(V) = ~. Global topological structure of virtual Lie subgroups is clarified by the following

Proposition 2.3. Let H be a virtual Lie subgroup of a Lie group G. There exists a neighbourhood V of the identity in H and a submanifold S C G, containing the identity, such that v : S x V --+ G, (s, h) f-+ sh is a diffeomorphism of the direct product S x V onto some neighbourhood U of the identity in G. In addition H n U = TV, where T = H n S is at most a countable set. If the neighbourhood V is connected, then it is the connected component of the identity of the intersection H n U in the induced topology.

Proof. The neighbourhood V and the submanifold S can be constucted as in the proof of Theorem 3.1 of Chap. I. The count ability of T follows from the fact that H can contain at most a countable family of mutually nonintersecting open subsets. To prove the last assertion one makes use of the fact that every countable subset of]Rn is totally disconnected. 0

The following theorem makes it possible to give a topological characterization of virtual Lie subgroups of real Lie groups.

Theorem 2.4 (Yamabe 1950). Every path connected subgroup of a real Lie group is a virtual Lie subgroup.

Corollary. Virtual Lie subgroups of real Lie groups coincide with subgroups having (in the induced topology) at most a countable set of path connected components.

2.4. The Correspondence Between Lie Subgroups of a Lie Group and Subalgebras of Its Tangent Algebra

Theorem 2.5. Let G1 and G2 be virtual Lie subgroups of a Lie group G. If G1 C G2, then G1 is a virtual Lie subgroup of the Lie group G2 and 91 C g2. Conversely, if 91 C 92 and the group G1 is connected, then G1 C G2 •

Proof. To prove the first assertion of the theorem we have to show that the identity inclusion of G1 in G2 is differentiable. With the help of Proposition 2.3, applied to G2 , we see that a sufficiently small connected neighbourhood of the identity in G1 is contained in a neighbourhood of the identity in G2 ,

which is a submanifold in G. From this follows the required differentiability. To prove the second assertion consider a path g(t) in G1 with velocity '(t)

and initial condition g(t) = e. Since '(t) C 91 C 92 one can find in G2 a path with the same velocity and the same initial condition (Proposition 2.1). Being a path in G it must coincide with g(t). Thus g(t) E G2 • 0

Corollary 1. If virtual Lie subgroups coincide as subsets, they carry the same Lie group structure.


Corollary 2. A connected virtual Lie subgroup is uniquely determined by its tangent algebra (as a subalgebra of the tangent algebra of the ambient Lie group).

Not every subalgebra of the tangent algebra is the tangent algebra of some Lie subgroup. However, consideration of virtual Lie subgroups makes the picture of the correspondence between Lie subgroups and Lie subalgebras more complete.

Theorem 2.6. Every subalgebra of the tangent algebra of a Lie group is the tangent algebra of some (uniquely defined) connected virtual Lie subgroup.

A proof of this theorem will be given in 5.3.

Theorem 2.7. The normalizer N(H) of a connected virtual Lie subgroup H of a Lie group G is a Lie subgroup, the tangent algebra of which coincides with the normalizer n(~) of the subalgebra ~ in the algebra g.

Proof. As gHg- 1 (g E G) is a connected virtual Lie subgroup with tangent algebra Ad(g)~, we have N(H) = N(~) and the assertion of the theorem follows from Theorem 1.4. 0

Corollary. A connected virtual Lie subgroup H of a connected Lie group G is normal if and only if the subalgebra ~ is an ideal of the algebra g.

Theorem 2.8. The centralizer Z(H) of a connected virtual Lie subgroup H of a Lie group G is a Lie subgroup, whose tangent algebra coincides with the centralizer 3(~) of the subalgebra ~ of the algebra g.

Proof. Replacing G with N(H) we can suppose that H is a normal subgroup. In virtue of Theorem 2.2, the inner automorphism a(g) of G is the identity on H if and only if its differential Ad(g) is the identity on ~. Thus, Z(H) is the kernel of the linear representation Ad~ of G. Hence, it must be a Lie subgroup and its tangent algebra coincides with the kernel of the linear representation ad~ of the algebra g, with 3(~). 0

Corollary. The centre Z (G) of a connected Lie group G is a (normal) Lie subgroup, whose tangent algebra coincides with the center 3(9) of g.

(The center of a Lie algebra is the collection of elements whose commutators with all alements of the algebra are zero.)

2.5. Deformations of Paths in Lie Groups. By a deformation of a path in a differentiable manifold X we shall mean a differentiable map

T x S ---+ X, (t, s) f---> x(t, s),

where T, S c lR are connected subsets, not consisting of a single point. We shall view s as the deformation parameter and the map t f---> x( t, s), for a fixed s, as the deformed path.


Proposition 2.9. Let (t, s) f--+ g(t, s) be a deformation of a path in a Lie group G. Let the elements e(t,s),,,,(t,s) E g be defined by the equations

Then

{ 8g~~,8) = e(t,s)g(t,s),

(31) 89~~8) = ",(t, s)g(t, s).

(32)

Proof. Differentiating in a coordinate system the first equation of (31) with respect to s and the second with respect to t and comparing the results we obtain:

ae(t,s)* a",(t,s)* as +",(t,s)*e(t,s)* = at +e(t,s)*",(t,s)*, (33)

where e (t, s) * and ",{ t, s) * are right invariant vector fields corresponding to e{t,s) and ",{t,s) (see 1.2). Since for e,,,, E g

",*e* - e*",* = [e*, ",*] = fe, ",]*,

(33) is equivalent to (32). o The elements e (t, s) and ",{ t, s) have the following interpretation: for a fixed

s, e{t, s) is the velocity of the path which is being deformed, and ",(t, s) is the velocity of the deformation. Equation (32) can be viewed as a differential equation with respect to t for the velocity of the deformation, which makes it possible to determine it from the velocity of the path and a given initial condition ",{to, s) = ",o{s).

2.6. The Existence Theorem

Theorem 2.10. Let G and H be Lie groups with G simply connected. Then for every homomorphism ¢ : g ~ b there is a homomorphism f : G ~ H, such that df = ¢.

Proof. In order to define the image of an element 9 E G we connect it with the identity by a path g(t), 0 ~ t ~ 1, and find the velocity e{t) of this path. Further, we consider a path h{t), 0 ~ t ~ 1 in H with velocity ¢(e(t)) and initial condition h{O) = e. The element 1{g) will be taken, by definition, as h{l).

Since the path g{t) is not unique we must show that the above definition does not depend on it. This is the most difficult part of the proof.

Let go{t) and gl{t) be two paths in G which connect e with g. We shall denote by ho{t) and hl{t) the corresponding paths in H. We have to show that ho{l) = h1 (1).

As the group G is simply connected there exists a deformation of go{t) into gl{t) i.e. a differentiable map (t, s) f--+ g{t, s) of the square Q = [0,1] x [0,1] into the group G, possessing the following properties:


1) g(t,O) = go(t), g(t, 1) = gl(t); 2) g(O, s) = e, g(l, s) = g.

Let ~(t,s),'T}(t,s) E 9 be the elements defined by the equations (31). According to Proposition 2.9 they are connected by the relation (32). In

addition, from the property 2) it follows that

'T}(O,s) = 'T}(l,s) = 0.

Next we define a differentiable map (t, s) f--+ h( t, s) of the square Q into H as the solution of the differential equation with respect to t

Oh~; s) = ¢(~(t, s))h(t, s)

with initial condition h(O, s) = e. This is a deformation of the path ho(t) into the path hl (t). Let (( t, s) E I) be the velocity of this deformation, i.e.

oh(t, s) = r( )h( ) os ." t, s t, s

According to Proposition 2.9 we have

O(~; s) _ o¢(~:' s)) = [¢(~(t, s)), ((t, s)].

View the last equality as a differential equation with respect to t for (( t, s). Applying the homomorphism ¢ to (32), we see that this equation is satisfied by ¢('T}(t, s)). Since

((0,8) = ¢(ry(O, s)) = 0,

we have ((t, s) = ¢('T}(t, s)). In particular, ((1, s) = ¢('T}(1, s)) = 0. This means that h(l, s) = const and, therefore, ho(l) = hd1).

Thus we have defined a map f : G ~ H. Let us prove that f is a homomorphism.

Let gl (t) and g2 (t), ° ~ t ~ 1 be paths in G, connecting e with gl and g2 respectively, with 6 (t) and 6 (t) as their velocities. The path connecting e with glg2 can be defined by the following equalities

g(t) _ {g2(2t), ° ~ t ~ ~, - gl(2t-1)g2, ~~t~1.

(with a suitable choice of the paths gdt) and g2(t) the map t f--+ g(t) will be differentiable.). Its velocity ~(t) is defined by the equalities

t _ { 26 (2t), ° ~ t ~ ~, ~ ( ) - 26 (2t - 1), ~ ~ t ~ 1.

Consequently, if hl (t),h2(t) and h(t) are paths in H corresponding to the paths gl(t), g2(t) and g(t), then

h(t) _ {h2(2t), ° ~ t ~ ~, - hl(2t-1)h2' ~~t~1.


In particular

f(glg2) = h(l) = h1(I)h2(1) = f(gl)f(g2).

With the help of a change of parameter, we obtain that any path g(t) E G with velocity ~(t) and initial condition g(O) = e is taken, by the map f, to a path h(t) E H with velocity ¢(~(t)) with initial condition h(O) = e. Consequently, the map f is differentiable and def = ¢. 0

Corollary. A simply connected Lie group is determined up to an isomorphism by its tangent algebra.

There is also the following theorem, various proofs of which will be given in one of the future volumes of this series (see also Postnikov 1982).

Theorem 2.11. Every finite-dimensional real (complex) Lie algebra is the tangent algebra of some real (complex) Lie group.

2.7. Abelian Lie Groups. The vector Lie group Kn is the unique simply connected Lie group whose tangent algebra is abelian 2. Hence, every connected Lie group is isomorphic to a Lie group of the form K n If, where f is a discrete subgroup of the group K n (see Theorem 4.7 of Chap. 1). If f1 and f 2 are two discrete subgroups of the group Kn, then the Lie groups K n If 1

and and Kn If 2 are isomorphic if and only if there exists an automorphism of the Lie group Kn (i. e. some nonsingular linear transformation of the vector space Kn), which takes f1 to f 2.

Every discrete subgroup of the group IRn can be taken, by a suitable automorphism, to one of the subgroups

fk = {(Xl, ... ,Xk,O, ... ) E IRn : Xl,··· ,Xk E Z},

where k = 0, ... ,n (Bourbaki 1947). Thus we obtain the following classification of connected abelian real Lie groups.

Theorem 2.12. Every connected abelian real Lie group is isomorphic to a Lie group of the form uk x IRI.

The classification of abelian complex Lie groups is considerably more complicated. For example, every connected one-dimensional complex Lie group is isomorphic to one of the Lie groups

C, C/Z ~ C* and A(u) = Cj(Z + Zu),

where u E C, Im( u) > 0, with the Lie groups A( u) and A( v) isomorphic (as complex Lie groups) if and only if

v=:::~, (~ ~)ESL2(Z). Thus connected compact one-dimensional complex Lie groups are parametrized by points of the quotient space of the complex upper half-plane by the

2 An abelian Lie algebra is an algebra with trivial multiplication.


action of Klein's modular group, which, as is well known, can be given in a natural way the structure of the complex plane C

§3. The Exponential Map

3.1. One-Parameter Subgroups. A path g(t) in a Lie group G, defined for all t E JR, is called a one-parameter subgroup if

g(t + s) = g(t)g(s)

(and then automatically g(O) = e, g( -t) = g(t)-l). In other words, a oneparameter subgroup is a homomorphism from the Lie group JR into G. Sometimes, however, the term one-parameter subgroup is used to denote the image of such a homomorphism. A one-parameter subgroup in this sense is a virtual Lie subgroup (but need not be a genuine Lie subgroup).

Proposition 3.1. A path g(t) in a Lie group G is a one-parameter subgroup if and only if its velocity ~(t) is constant and g(O) = e.

Proof. Let g(t) be a path with velocity ~(t) and initial condition g(O) = e. For any s E JR the path gs(t) = g(t + s) has velocity ~s(t) = ~(t + s) and satisfies the initial condition gs(O) = g(s). From this it follows that if ~(t) = const, then gs(t) = g(t)g(s). Conversely, if gs(t) = g(t)g(s) for all s E JR, then ~s(t) = ~(t) for all s E JR, i.e. ~(t) = const. 0

For any ~ E 9 we shall denote by g€(t) the one-parameter subgroup with velocity ~(t) == ~. We shall refer to the vector ~ as its direction vector.

If G = A x is the group of invertible elements of an associative algebra A, then

ga(t) = expta,

where the exponential is understood as the sum of the series:

00 n

expa= L:;. n.

n=O

(34)

(In the case when A is the matrix algebra this fact amounts to the contents of the theory of systems of linear differential equations with constant coefficients. )

One-parameter subgroups of the vector Lie group Kn are one-dimensional subspaces of the vector space Kn. More precisely, gv(t) = tv (with the usual identification of To (Kn) with Kn).

3.2. Definition and Basic Properties of the Exponential Map. For any Lie group G we set by definition

exp~ = ge(1) (~E g).


The map exp : 9 ----t G defined in this way is known as the exponential map. In the case when G is the group of invertible elements of an associative algebra, it coincides with with the map defined by means of the series (34). In the case when G is the vector Lie group, the exponential map is the identity.

By means of a linear change of the parameter t we see that

ge(t) = expt~ (35)

The theorem about the differentiable dependence of the solutions of a system of differential equations on the parameters shows that the map exp is differentiable and from (35) it follows that its differential at zero is the identity map. From this, in turn, follows

Proposition 3.2. The exponential map exp : 9 :----t G maps a certain neighbourhood of zero in the tangent algebra 9 diffeomorphically onto a neighbourhood of the identity of G.

By an analogous method we can prove a more general assertion.

Proposition 3.3. Let 9 = al E9 ... E9 ak be a decomposition of a Lie algebra 9 as a direct sum of subspaces. Then the map

6 + ... + ~k f-t exp ~i •.• exp ~k (~i E ai),

maps some neighbourhood of zero in the algebra 9 diffeomorphically onto a neighbourhood of the identity in G.

These properties of the exponential map make it possible to choose certain special coordinate systems in a neighbourhood of the identity of G. Namely, let {ell ... ,en} be a basis of the algebra g. Then each of the maps

(tb ... , tn) f-t exp(tlel + ... + tnen),

(tl, ... , tn) f-t exphel ... exptnen,

defines a diffeomorphism of some neighbourhood of zero in the space Kn onto a neighbourhood of the identity in G. Defined in this way coordinates in a neighbourhood of the identity in G are called canonical coordinates of the first and second kind respectively.

In general, the exponential map does not posssess any good properties globally. As we shall see in the following paragraphs it need not be surjective, injective, open etc.

A homomorphism f of Lie groups takes the one-parameter subgroup with direction vector ~ to the one-parameter subgroup with direction vector df(~). Consequently,

f(exp~) = expdf(~). (36)

(This means that in canonical coordinates of the first kind, every Lie group homomorphism is expressed as a linear map.) In particular

Ad(exp~) = expad(~) (37)

46 A.1. Onishchik, E. B. Vinberg

Example. Consider the homomorphismdet: GLn(K) ----+ KX. Since ddet = tr (Example 1 of 1.3) so

det exp X = etrX (X E Ln(K)).

The property of multiplicativity, which characterizes the usual exponentiation, is satisfied in the case of the exponential map in a Lie group only in a restricted sense.

Proposition 3.4. If [~, TJ 1 = 0, then

exp(~+TJ) =exp~exPTJ·

Proof. If [~, TJ 1 = 0, then there exists a homomorphism f : K2 ----+ G for which df(a, b) = a~ + bTJ (Theorem 2.10). Hence, it suffices to prove the assertion for a vector group; but in this case it is obvious. 0

In particular, if a Lie group G is abelian, the map exp is a homomorphism of the vector group 9 into G.

3.3. The Differential of the Exponential Map. In order to compute the differential of the map exp : 9 ----+ G at a point ~ E g, we consider the deformation of the path in G, defined by the formula

g(t, s) = expt(~ + STJ) (38)

(so that for any S the deformed path is a one-parameter group). We have (d{exp) (TJ) = TJ(l)exp~, where TJ(t) = TJ(t,O) is the velocity of deformation (38) for s = O. As the velocity of the deformed path is e(t, s) e + STJ, according to Proposition 2.9, we have,

TJ'(t) = [~, TJ(t) 1 + TJ

with initial condition TJ(O) = O. The solution of the equation (39) can be written in the form

() _ exptad(~) -1( ) TJ t - t ad (~) tTJ ,

(39)

where exp;: ~ 1 for a linear operator A is understood as the sum of the series

expA - 1 = f An. A n=o(n+1)!

In particular, for t = 1 we obtain

exp ad (~) - 1 (d{ exp )( TJ) = ad (~) ( TJ) exp ~ . (40)

(This formula is a special case of the formula of Helgason (1964), for the differential of the exponential map in an arbitrary linear connection space.)

From formula (40) it follows that the kernel of the linear map d{exp in the case K = IC is the sum of the eigenspaces of the operator ad (0 corresponding

to eigenvalues of the form 2rrik, where k E Z, k i=- 0, and in the case K = lR the real part of this sum. We shall denote the dimension of this sum by v(e).

Theorem 3.5 (Nono 1960). The map exp : 9 -+ C is a local diffeomorphism at a point e E 9 if and only if the operator ad(e) has no eigenvalues of the form 2rrik, k i=- o. If this condition is not satisfied, then not only the map exp is not a local diffeomorphism but it also is not open at the point e. The set exp-l(expe) is a closed submanifold 3 of the algebra g. Its connected component containing e coincides with the connected component of the set Ad( Z (exp e)) e and has dimension v( e).

3.4. The Exponential Map in the Full Linear Group. It is easy to see that the exponential of a Jordan block with eigenvalue A is similar to a Jordan block with eigenvalue eA. Hence, the exponential map in GLn(C) is surjective. The exponential map in GLn(lR) is not surjective, its image consisting of matrices which have an even number of Jordan blocks of every order corresponding to each negative eigenvalue. This image is neither open nor dense in GLn(lR).

The exponential map in SLn(C) is not surjective. Its image is dense, but it does not contain, for instance, a Jordan block with eigenvalue different from 1 (but which is an n-th root of 1). However, in PSLn{C) = PGLn(CC) the exponential map is surjective, just as in GLn(C). The image of the exponential map in SLn{lR) is described just as in GLn(lR).

In each of the groups GLn(C), GLn(lR), SLn(lR) the exponential map defines a diffeomorphism of the open subset of the tangent algebra, consisting of matrices all of whose eigenvalues A satisfy the condition IImAI < rr, onto the open subset of the group, consisting of the matrices without negative eigenvalues (Morinaga 1950).

3.5. Cartan's Theorem. One of the applications of the exponential map is the proof of the following theorem, which gives a topological characterization of Lie subgroups of real Lie groups.

Theorem 3.6 (E. Cartan's Theorem). Every closed subgroup of a real Lie group is a Lie subgroup.

Proof. Let H be a closed subgroup of a real Lie group C. Let us denote by T the set of elements e E 9 for which there exist sequences en E 9 and Cn E lR such that en -+ 0, cnen -+ e and exp en E H. It is easy to see that the numbers Cn can be taken to be integers. In this case we obtain:

expe = lim (expen)Cn E H.

Moreover, for any a E lR we have acnen -+ ae so that ae E T. Let e, 'Tl E T. Consider the path

h(t) = expte exp t'Tl E H.

3 This submanifold can have connected components of different dimension.


For sufficiently small t we have h(t) = exp((t), where ((t) is a path in g, with

((0) = 0, ('(0) = h'(O) = ~ + 'fl. Consequently,

~ + 'fI = lim n (( .!.) E T. n

Thus T is a subspace in 9 and expT c H. Let S c 9 be a complementary subspace. Consider the map

¢ : 9 ---? G,

According to Proposition 3.3, it gives a diffeomorphism of some neighbourhood U of zero in the algebra 9 onto a neighbourhood of the identity in the group G. We shall prove that, for a sufficiently small neighbourhood U,

H n ¢(U) = ¢(T n U) (= exp(T n U)). (41)

Let us suppose that equality (41) does not hold for any choice of U. Then there must exist a sequence 'fin E S \ {O}, such that 'fin ---? 0 and exp 'fin E H. Passing to a subsequence we can ensure that Cn'fln ---? 'fI E S \ {O} for some Cn E JR, but in this case 'fI E T which is impossible. 0

The complex analogue of Cartan's Theorem is false, since any real Lie subgroup of a complex Lie group is closed, but is not necessarily a complex Lie subgroup.

3.6. The Subgroup of Fixed Points of an Automorphism of a Lie Group. In the special case when H is the subgroup of fixed points of some automorphism, the exponential map makes it possible not only to show that H is a Lie subgroup, but also to find its tangent algebra, and moreover this applies in equal measure to real and complex Lie groups.

Theorem 3.7. Let a be an automorphism of a Lie group G. Then

Gtr = {g E G : a(g) = g}

is a Lie subgroup with tangent algebra

Proof. The assertion of the theorem follows directly from the fact that the automorphism a commutes with the exponential map (formula (36)). 0

§4. Automorphisms and Derivations

4.1. The Group of Automorphisms. Let G be a Lie group, Aut G-the group of its automoprphisms (as a Lie group), Autg-the group of automorphisms of its tangent algebra.


If G is connected, then the map d : Aut G --; Aut g, which assigns to each automorphism of G its differential, is an injection (Theorem 2.2), and if G is simply connected, then it is an isomorphism (Theorem 2.10). The group Autg is a linear Lie group (Example 3 of 2.2 of Chap. 1). Using these facts one can, in the case of a simply connected group G, transfer the Lie group structure of Aut 9 to Aut G. This makes the action of the group Aut G on G differentiable. In the general case we have

Proposition 4.1. For any connected Lie group G the group d Aut G is a Lie subgroup of the group Aut g.

Proof. By Theorem 4.7 of Chap. 1 we have G = GIN, where G is the simply connected Lie group with the same tangent algebra and N is a discrete central subgroup of it. The group Aut G can be naturally identified with a subgroup of Aut G consisting of the automorphisms which preserve N. It contains a subgroup H consisting of the automorphisms which fix N. By Theorem 4.2 of Chap. 1 H is a Lie subgroup of Aut G (as the intersection of the stabilizers of the points of N), and from the discreteness of N it follows that in some neighbourhood of the identity of Aut G the subgroups Aut G and H coincide.

D

Thus, for any connected Lie group G the group Aut G possesses a natural structure of a Lie group.

The inner automorphisms of the group G form a normal subgroup in Aut G, which is isomorphic to G I Z, where Z is the center of G, and which is denoted by Int G.

If G is connected, then dInt G = Ad G depends only on the algebra 9 (see 4.2) and is a normal subgroup of Aut g. It is called the group of inner automorphisms of the algebra 9 and is denoted by Int g. Being the image of the group G under the adjoint representation the group Int 9 is a virtual Lie subgroup (but not necessarily a genuine Lie subgroup) of Aut g. Correspondingly, the group Int G is a virtual Lie subgroup of Aut G.

The quotient group Aut GlInt G (which can be given the structure of a Lie group when Int G is a Lie subgroup of Aut G), is referred to, somewhat loosely, as the group of outer automorphisms of the Lie group G. Analogously, the quotient group Aut glInt 9 is called the group of outer automorphisms of the algebra g. In the case of a simply connected group G there is a natural isomorphism Aut GlInt G '::: Aut glInt g.

Examples. 1. Let G be a connected abelian Lie group. Then Aut 9 = GL(g), Intg = {E}, and the group dAutG consists of the automorphisms of the algebra 9 preserving the kernel of the exponential map (which can be any discrete subgroup of the vector group g).

2. Let G be the Lie group of matrices of the form


o

Xn Z

o YI

1 Yn 1

Its tangent algebra 9 has a basis {Xl,'" X n , Yb ... , Yn , Z} for which [Xi, YiJ = Z and the remaining commutators of the basis vectors are O. (This Lie algebra is known as the Heisenberg algebra.) The subspace 3 = (Z) is the centre of g. Any automorphism has to preserve it, i.e. multiply Z by a number c #- 0, and induce in g/3 a linear transformation which multiplies by c- l the skew-symmetric bilinear form f, defined by the following rule: f(X i , Yi) = 1 and is 0 on the remaining pairs of basis vectors. Conversely, any linear transformation of 9 with these properties is an automorphism. As for the inner automorphisms, they have the form

The group Int 9 is a Lie subgroup of Aut 9 and is isomorphic to the vector group K2n. The quotient group Aut glInt 9 is an extension of the group SP2n (K) by the group K*.

3. Let G be the group of affine transformations of the line. This Lie group

is isomorphic to the group of matrices of the form (~ ~) (where a =1= 0),

whose tangent algebra consists of matrices X = (~ ~) and Y = (~ ~) , satisfying the relation [X, YJ = Y. One verifies directly that Aut 9 = Ad( G) ~ G. In the complex case the group G is connected and Intg = Ad(G) = Autg. In the real case G consists of two connected components (distinguished by the sign of a) and Int 9 = Ad(GO) is a subgroup of index 2 in Aut g. In both cases AutG = IntG.

4.2. The Algebra of Derivations. The tangent algebra of the group Aut 9 is the algebra Der 9 of derivations of the algebra 9 (Example 3 of 1.5). The tangent algebra of the group Int 9 is the image of the algebra 9 under the homomorphism ad = dAd. This, in particular, shows (see Corollary 2 of Theorem 2.5) that the group Intg does not depend on which G is chosen from among the connected Lie groups having 9 as their tangent algebra.

Derivations of the form ad(~), ~ E 9 are called inner derivations of the algebra g. They form an ideal in the algebra Der g. More precisely,

[D, ad(~)J = ad(D~) (42)

for any D E Derg, ~ E g.

4.3. The Tangent Algebra of a Semi-Direct Product of Lie Groups. Corresponding to semi-direct products of Lie groups (see 3.5 of Chap. 1) we have semi-direct sums4 of Lie algebras.

By a semi-direct sum of Lie algebras 91 and 92 we mean the direct sum of the vector spaces 91 and 92, with the Lie algebra structure given by the formula

where (3 is some homomorphism of the Lie algebra 92 into the Lie algebra Der 91. We shall denote it by 91 -9 92 or more precisely by 91 -9 92·

f3 Elements of the form (6,0) (respectively (0,6)) form a sub algebra of

91 -992 isomorphic to 91 (respectively 92), which we shall identify with 91 (respectively 92). The sub algebra 91 is an ideal and

(43)

The sub algebra 92 is an ideal if and only if (3 = 0, in this case the semi-direct sum coincides with the direct sum 91 EB 92·

Example. Let V be some vector space which we consider as an abelian Lie algebra. Then Der V = 9[(V). For any linear representation p : 9 -t 9[(V) of a Lie algebra 9 one can form the semi-direct sum V -9 9, in which V is an

p

abelian ideal. We say that a Lie algebra 9 decomposes as a semi-direct sum of sub algebras

91 and 92 if 1) the subalgebra 91 is an ideal 2) the algebra 9, as a vector space, is a direct sum of the subspaces 91 and

92· In this case we have the isomorphism

where (3 : 92 -t Der 91 is the homomorphism given by formula (43). In this situation we shall write 9 = 91 -992 or 9 = 92 Et 91

Proposition 4.2. The tangent algebra of a semi-direct product G1 )<J G2 of b

Lie groups G1 and G2 is the semi-direct sum 91 -992 of their tangent algebras f3

91 and 92. Moreover, (3 = dB, where B : G2 -t Aut 91 is a homomorphism of Lie groups, given by B(g2) = d(b(92)).

Examples. 1. Let R : G -t GL(V) be a linear representation of a Lie group G. The tangent algebra of the semi-direct product V)<J G is the semi-direct

R sum V -9 9, where p = dR.

p

4 Though one could equally well refer to them as semi-direct products.

2. The Lie group GA(V) of affine transformations of a vector space V is isomorphic to the semi-direct product V ~ GL(V) (Example 2 of 3.6 Chap. 1).

Id Correspondingly, its tangent algebra is isomorphic to the semi-direct sum V -e g[(V), where id is the identity linear representation of g[(V) in V.

id

Proposition 4.3. Let G1 and G2 be simply-connected Lie groups. For any homomorphism f3 : g2 --t Der g1 there exists a homomorphism b : G2 --t

Aut G1 such that the action of G2 on G1 defined by it is differentiable and the tangent algebra of the semi-direct product G1 ~ G2 is g1 -e g2.

b (3

Proof. The required homomorphism b is obtained from f3 by "integration", the inverse procedure to the one described in the statement of Proposition 4.2. D

§5. The Commutator Subgroup and the Radical

5.1. The Commutator Subgroup. Recall that the commutator subgroup of a group G is a subgroup (G, G) = G' generated by all the commutators (g, h) = ghg- 1h-1 (g, hE G). This subgroup is normal and is the least normal subgroup with an abelian quotient group.

By the commutator subalgebra of a Lie algebra 9 we mean the subspace [g, g] generated by all the commutators [e,1]] (e, 1] E g). This is the least ideal with an abelian quotient.

Theorem 5.1. The commutator subgroup G' of a connected Lie group G is a connected virtual Lie subgroup with tangent algebra g'. If the group G is simply-connected, G' is a genuine Lie subgroup.

Proof. Let us first suppose that G is a simply-connected Lie group. Consider the quotient algebra g/ g'. It is abelian and hence is the tangent algebra of some vector Lie group V. By Theorem 2.10 the canonical isomorphism cp : 9 --t gig' is the differential of some homomorphism f : G --t V. Let H denote the kernel of this homomorphism. It is a normal Lie subgroup whose tangent algebra coincides with the kernel of the homomorphism cp, i.e. with g'. As the quotient group G / H ~ V is abelian, H J G'. From the exact sequence (10) Chap. 1 it follows that H is connected. We shall now show that some of its neighbourhoods of the identity is contained in G'; from this it will follow that H = G' (see the Corollary of Theorem 4.1 of Chap. 1).

Lemma 5.2. For any e,1] E 9 there exists a path g(t) of class C1 in G, defined in some neighbourhood of zero and possessing the following properties:

1) g(O) = e,g'(O) = [e,1]]; 2) for any t the element g(t) is a commutator in G.

Proof. Let x(t) and y(t) (0 ::; t < 10) be paths in G such that x(O) = y(O) = e,x'(O) = e,y'(O) = 1]. Then one can take

{ (x(v't),y(v't)) for t ~ 0

g(t) = -1 (x( A), y( A)) for t ::; 0 o

Let us now choose a basis {(1, ... , (n} of the space g' over JR, consisting of commutators. Let gi(t), It I < Ci be a path satisfying the conditions of the lemma for [e,17] = (i. Let us denote by U the neighbourhood of 0 in JRn cut out by the inequalities Itil < Ci and consider the map

From the properties of the path gi(t) it follows that dof is an isomorphism of tangent spaces over R Therefore f(U) and hence G' contains a neighbourhood of the identity in H.

For an arbitrary connected Lie group G the assertion of the theorem follows from the existence of a universal covering p : a -t G and from the already proved fact that a' is a Lie subgroup of a. 0

Corollary. If G is a connected Lie group and g' = g, then G' = G.

Examples. 1. Let Eij (l ::; i,j ::; n) be matrix units. The relations [Eu -Ejj, Eij] = 2Eij, [Eij , Eji ] = Eii - Ejj show that s[n(K)' = s[n(K). Since tr [X, Y] = 0 for any matrices X, Y we have g[n(K)' = s[n(K). Hence, the commutator subgroup of GLn(K) is equal to SLn(K) and the commutator subgroup of SLn(K) is the whole group.

2. Let H be the group of unitriangular real matrices (i. e. triangular matrices with l's on the diagonal) of degree three. The commutator subalgebra of its tangent algebra is one-dimensional: it is generated by the matrix unit E13 .

Consider the Lie group G = (H x 'If)/N, where N is the cyclic group generated by the element (exp E 13 , c) E H x T. If c is an element of infinite order in the group 'If, then the commutator subgroup of G is the "dense winding" of the two-dimensional torus ({exptE13 } x 'If)/N. This example shows that the commutator subgroup of a non-simply connected Lie group need not be a genuine Lie subgroup.

5.2. The Maltsev Closure. The tangent algebra of a Lie group may contain subalgebras not corresponding to any Lie subgroups. For example, the onedimensional subalgebra of the tangent algebra of the group ~, spanned by the element (ial, ... ,ian), where a1, ... ,an E JR, is the tangent algebra of some Lie subgroup if and only if the numbers a1, ... , an are commensurable. Nevertheless, as we shall see presently, there is always a Lie subgroup whose tangent algebra is only "slightly larger" than the given subalgebra.

Let I) be any subalgebra of the tangent algebra of a Lie group G. According to Theorem 4.2 of Chap. 1 there exists the least one among the Lie subgroups of G whose tangent algebras contain I). We shall refer to its tangent algebra as the Maltsev closure of the subalgebra I) and denote it by I)M.


Theorem 5.3 (A. 1. Maltsev 1945, 1946). Let IJ be a subalgebra of the tangent algebra of a Lie group and let IJM be its Maltsev closure. Then (IJM)' = IJ'.

Proof. Apply Corollary 3 of Theorem 3.5 of Chap. 1 to the adjoint representation of G, taking as U and W the subspaces IJ and IJ' respectively. We obtain that

HI = {g E G: (Ad(g) -E)IJ c IJ'} is a Lie subgroup with tangent algebra

IJI = {~ E 9 : (ad(~))1J C IJ'}.

Since IJ C IJI we have IJM C IJI i.e. [IJM, IJJ c IJ'. Taking now as U the subspace IJM we analogously obtain [IJM, IJMJ C IJ'. 0

5.3. The Structure of Virtual Lie Subgroups. Below we shall prove, by making use of the Maltsev closure, the theorem on the existence of a virtual Lie subgroup with a given tangent algebra (Theorem 2.6) and simultaneously we shall obtain a description of all virtual Lie subgroups.

Proof. Let IJ be a subalgebra of the tangent algebra of a Lie group G. We shall show that there exists a connected virtual Lie subgroup H C G with IJ as its tangent algebra. Let f = IJM. By Theorem 5.3 f' = IJ'. Let F C G be a connected Lie subgroup having f as its tangent algebra, and let P be its universal covering group. Since P / P' is a vector group, it contains a connected Lie subgroup (a vector subspace) with tangent algebra IJ/IJ' C flf'. Hence, the group P itself contains a connected Lie subgroup iI with tangent algebra IJ. The image of this group in F is the required virtual Lie subgroup ofG. 0

In the process of proof we have in fact obtained the following description of virtual Lie subgroups.

Theorem 5.4. For any connected virtual Lie subgroup H of a Lie group G there exists a connected Lie subgroup F of G and a connected Lie subgroup iI of the universal covering Lie group P, containing its commutator subgroup P', such that H = p( iI), where p : P -t F is the covering homomorphism. (It is easy to see that iI is the universal covering Lie group of H.)

Making use of this description we can easily prove the following analogue of Theorem 4.2 of Chap. 1 for virtual Lie subgroups.

Theorem 5.5. The intersection H = n H /I of an arbitrary family {H /I} of /I

virtual Lie subgroups is a virtual Lie subgroup with tangent algebra IJ = n IJ/I. /I

The proof reduces to the case of two subgroups in exactly the same way as the proof of Theorem 4.2 of Chap. I. Moreover, if one of the two subgroups happens to be a genuine Lie subgroup, then the theorem follows from Theorem 3.5 of Chap. 1 (cf. the proof of Corollary 1 of that theorem).


To complete the picture we also give the following theorem.

Theorem 5.6. The subgroup H genemted by an arbitmry family {Hv} of connected virtual Lie subgroups is a connected virtual Lie subgroup, and its tangent algebm ~ coincides with the subalgebm genemted by the family of subalgebms {hv }.

Proof. Let 6 be the sub algebra generated by the family of subalgebras {~v} and if the connected virtual Lie subgroup having 6 as its tangent algebra. Clearly H C if. Further, by generalizing Lemma 5.2 to the case of commutators of arbitrary length we can show, just as in the proof of Theorem 5.1, that H contains some neighbourhood of the identity in if. From this it follows that H = if. D

By virtue of the theorem ofE. Cartan (Theorem 3.6) a virtual Lie subgroup (of a real as well as complex Lie group) is a genuine Lie subgroup if and only if it is closed. We note the following result (Maltsev 1945, 1946): A virtual Lie subgroup is closed if and only if it contains the closure of anyone-parameter subgroup contained in it.

5.4. Mutual Commutator Subgroups. By the mutual commutator subgroup of normal subgroups G1 and G2 we mean the subgroup (G1, G2), generated by all the commutators (g1, g2), where gl E G1, g2 E G2. It is also a normal subgroup of G.

By the mutual commutator subalgebm of ideals 91 and 92 of a Lie algebra 9 we mean the subspace [91,92] generated by all the commutators [6,6], where 6 E 91 and 6 E 92. It is also an ideal of the algebra 9.

Theorem 5.7. The mutual commutator subgroup (G1 , G2) of connected normal Lie subgroups G1 and G2 of a Lie group G is a connected (normal) virtual Lie subgroup with tangent algebm [91, 92].

Proof. Let H be the connected virtual Lie subgroup with tangent algebra ~ = [91,92]. Let us show that (G1 ,G2 ) c H.

Since ad (91)92 C ~, we have (by Corollary 3 of Theorem 3.5 of Chap. 1)

for all gl E G1 ·

Lemma 5.S. Let a be an automorphism of a connected Lie group G which preserves a normal virtual Lie subgroup H. If da is the identity on 9/~, then a is the identity on G / H .

Proof. Let g(t) E G be a path with velocity ~(t) and initial condition g(O) =

e, and let da(~(t)) = ~(t) + 'T/(t), 'T/(t) E ~. Consider the path h(t) E H with velocity Ad(g(t))-l'T/(t) and initial condition h(O) = e. One verifies directly that the velocity of the path g(t)h(t) is da(~(t)). Hence, a(g(t)) = g(t)h(t).

D

Applying this lemma to the automorphism Ad(gd of the group G2 , we see that g1g2g1 1g2 1 E H for all g1 E Gl,g2 E G2 •

From Lemma 5.2, just as in the proof of Theorem 5.1, one deduces that (G 1, G2 ) contains some neighbourhood of the identity in H. Hence, (G 1 ,G2 )=H. 0

An argument analogous to the proof of Theorem 5.3, shows that

[gf1,g~] = [g1,g2] (44)

for any ideals g1, g2 of the tangent algebra of a Lie group G.

5.5. Solvable Lie Groups. Recall that the higher commutator subgroups G(k) (k = 0,1,2, ... ) of a group G are defined by the inductive rule

G(O) = G, G(k+1) = (G(k))'.

A group G is called solvable if there exists an m such that G(m) = {e}. Every subgroup and every quotient group of a solvable group is solvable. Conversely, if a normal subgroup NeG and the quotient group GIN are solvable, then the group G is solvable.

Analogously, higher commutator subalgebras g(k) (k = 0,1,2, ... ) of a Lie algebra 9 are defined by the inductive rule

g(O) = g, g(k+1) = (g(k)),.

A Lie algebra 9 is called solvable ifthere exists an m such that g(m) = O. Every sub algebra and every quotient algebra of a solvable Lie algebra is solvable. Conversely, if an ideal neg and the quotient algebra gin are solvable, then the algebra 9 is solvable.

By induction we can show that the higher commutator subgroup G(k) of a connected Lie group G is itself a connected virtual Lie subgroup with tangent algebra g(k). From this follows

Theorem 5.9. A connected Lie group G is solvable if and only if its tangent algebra 9 is solvable. More precisely, G(m) = {e} if and only if g(m) = O.

Example. An important and, in a certain sense, basic example of a solvable Lie group is the group Tn(K) of nonsingular triangular matrices of order n over the field K. Its tangent algebra is the algebra tn(K) of all triangular matrices. It is not hard to see that the k-th commutator subalgebra of this algebra consists of matrices (Xij), satisfying the condition Xij = 0 for j - i < 2k~1. Corresponding to this, the k-th commutator subgroup of the group Tn (K) consists of matrices (aij) satisfying the condition aij = bij for j - i < 2k~1.

Every non-trivial solvable Lie algebra 9 can be decomposed as a semidirect sum of an ideal n of co dimension 1 and a one-dimensional subalgebra o. Namely, one can take as n any subspace of co dimension 1 containing g' and as 0 any complementary subspace. Applying Proposition 4.3, we obtain, by induction on dim 9 the following theorem


Theorem 5.10. For any solvable Lie algebra 9 there exists a Lie group having 9 for its tangent algebra.

Simultaneously we have established that every non-trivial simply connected solvable Lie group decomposes as a semi-direct product of a normal Lie subgroup of co dimension 1 and a one-dimensional Lie subgroup. In the real case this statement can be generalized to arbitrary connected solvable Lie groups.

5.6. The Radical. Since the sum of two solvable ideals of a Lie algebra is a solvable ideal, every Lie algebra 9 contains the greatest solvable ideal. It is known as the radical of the algebra g. We shall denote it by rad g.

The analogous construction can be carried out also in the case of Lie groups.

Theorem 5.11. Every Lie group G contains the greatest connected solvable normal Lie subgroup. Its tangent algebra coincides with rad g.

Proof. Consider the Maltsev closure t of the radical of the algebra g. By Theorem 5.3 t' = (rad g)'. Hence, t = rad g, i.e. G contains a connected normal Lie subgroup R whose tangent algebra coincides with rad g. By Theorem 5.9 it is solvable. On the other hand, every connected solvable normal Lie subgroup of G is contained in R, since its tangent algebra, being a solvable ideal of 9 is contained in rad g. D

The subgroup satisfying the conditions of this theorem is known as the radical of the Lie group G. We shall denote it by RadG.

A Lie group G (respectively Lie algebra g) is called semisimple if Rad G = {e} (respectively radg = 0). From Theorem 5.11 it follows that a Lie group is semi-simple if and only if its tangent algebra is semi-simple. For any Lie group G (respectively Lie algebra g) the quotient group G /Rad G (respectively the quotient algebra g/rad g) is semi-simple.

In order to prove semi-simplicity of Lie groups or algebras one often makes use of

Proposition 5.12. A Lie algebra is semi-simple if and only if it does not contain non-zero abelian ideals.

Proof. If a Lie algebra is not semi-simple, then the last non-zero term in the series of the higher commutator subalgebras of its radical is its abelian ideal. D

Examples. 1. The Lie group SLn (K) is semi-simple. Moreover, its tangent algebra s(n(K) is simple, i.e. does not contain any non-trivial ideals. Indeed, let 0 be an ideal of s(n(K) and let X = (Xij) E o,X i- o. For i i- j we have:

[Eij , [X, Eij]] = 2EijXEij = 2XjiEij Eo.

By acting on X with inner automorphisms one can insure that Xij i- O. Hence, Eij Eo. As [Eij,Eji ] = Eii - E jj , so 0= s(n(K).


2. As g[n(K) = s[n(K) EB PEl, so radg[n(K) = PEl. Correspondingly, the radical of the Lie group GLn(K) is the connected component ofthe group of non-singular scalar matrices.

3. Consider the Lie group G consisting of all matrices of the form

( Ao Be) A E SLn(K), BE SLm(K).

Its subgroup R, consisting of the matrices for which the blocks A and B are the identity matrices, is a connected abelian normal Lie subgroup. As G/R ~ SLn(K) x SLn(K), so R = RadG.

5.7. Nilpotent Lie Groups. Recall that by the decreasing (or lower) central series

G = Go ::) G1 ::) G2 ::) ...

of a group G is defined by the rule G k+ 1 = (G, G k)' A group G is called nilpotent if there exists an m such that Gm = {e}. Every subgroup and every quotient group of a nilpotent group is nilpotent. Clearly, G(k) c Gk; hence every nilpotent group is solvable.

Analogously, the decreasing (or lower) central series

G = go ::) gl ::) g2 ::) ...

of a Lie algebra 9 is defined by the rule gk+1 = (g, gk). A Lie algebra 9 is called nilpotent if there exists an m such that gm = O. Every subalgebra and every quotient algebra of a nilpotent Lie algebra is nilpotent. Clearly, g(k) C gk; hence every nilpotent Lie algebra is solvable.

Let G be a connected Lie group. From Theorem 5.7 it follows that Gk is a connected virtual Lie subgroup with tangent algebra gk. From this follows

Theorem 5.13. A connected Lie group G is nilpotent if and only if its tangent algebra 9 is nilpotent. Moreover, Gm = {e} if and only if gm = O.

Example 1. The basic example of a nilpotent Lie group is the group UTn(K) of unitriangular matrices of order n over the field K. Its tangent algebra is the algebra ut.,(K) of triangular matrices with zeros on the diagonal. It is easy to see that the k-th term of the decreasing central series of the algebra utn(K) consists of matrices Xij satisfying the condition Xij = 0 for j - i :::; k. Correspondingly, the k-th term of the decreasing central series of the group UT n (K) consists of matrices (aij) satisfying the condition aij = 8ij for j - i :::; k.

The property of nilpotency just as the property of solvability leads to the construction of a certain "radical".

Using the fact that the terms of the decreasing central series of any ideal of a Lie algebra are also ideals of that algebra one can easily show that the sum of nilpotent ideals is a nilpotent ideal. Whence it follows that any Lie alge bra 9 contains the largest nilpotent ideal (obviously contained in rad g).


For Lie groups there is the following theorem, which can be proved analogously to Theorem 5.

Theorem 5.14. In every Lie group G there exists the greatest connected nilpotent normal Lie subgroup. Its tangent algebra coincides with the greatest nilpotent ideal of the algebra g.

Example 2. Consider the Lie group G consisting of all matrices of the form

(~ ~), A E GLn(K), B E GLm(K).

Its radical consists of the matrices for which the blocks A and B are scalar matrices (in the case K = IR. - with positive coefficients), and the greatest connected nilpotent normal Lie subgroup consists of the matrices for which the blocks A and B are both identity matrices.

Chapter 3 The Universal Enveloping Algebra

As is well known (see §19 of Encycl. Math. Sc. 11) every associative algebra A can be turned into a Lie algebra L(A) by replacing its multiplication (a, b) f-+ ab by the commutator [a, b] = ab - ba. Clearly, every homomorphism of associative algebras is automatically a homomorphism of the corresponding Lie algebras, i.e. we have a functor L from the category of associative algebras to the category of Lie algebras. In this chapter we shall consider a functor U, which acts in the opposite direction. In this case a Lie algebra 9 is embedded into the corresponding associative algebra U(g) as a subalgebra (with respect to the commutator) and generates U(g) as an associative algebra. The algebra U(g) is called the universal enveloping algebra of the Lie algebra g. It was first considered in the year 1899 by Poincare, who introduced it as a certain algebra of differential operators on the corresponding Lie group (see 2.2 below). The universal enveloping algebra makes it possible to look from a different viewpoint at the Lie functor considered in Chap. 2. In particular, in this way one proves the equivalence of the categories of local analytic Lie groups and finite-dimensional Lie algebras. An important role is played by the Campbell-Hausdorff formula (see § 3).

§1. The Simplest Properties of Universal Enveloping Algebras

In this section we consider the definition and certain algebraic properties of universal enveloping algebras. The details can be found in (Post-


nikov 1982), (Bourbaki 1971), (Diximier 1974), (Humphreys 1972), (Jacobson 1962), (8erre 1965).

1.1. Definition and Construction. Let 9 be a Lie algebra over an arbitrary field K. An associative algebra U(g) with identity over K, equipped with a map 0: : 9 ----t U(g) is called a universal enveloping algebra if the following conditions are satisfied:

1) 0: : 9 ----t L(U(g)) is a homomorphism of algebras; 2) if A is an associative algebra with identity over K and h : 9 ----t L(A)

is an algebra homomorphism, then there exists an algebra homomorphism k : U(g) ----t A such that k(l) = 1 and h = ko:.

Example 1. Let the Lie algebra 9 be abelian. Then the symmetric algebra 8(g) over the vector space 9 (i.e. the algebra of commutative polynomials in the basis elements of the vector space g) together with the natural embedding 0: : g81 (g) ----t 8(g) is a universal enveloping algebra of g. Indeed, any linear map h of the vector space 9 into an associative algebra A with identity over K such that h(x)h(y) = h(y)h(x) for all x, y Egis extends uniquely to an identity preserving algebra homomorphism 8(g) ----t A.

The following theorem shows the existence and uniqueness of the universal enveloping algebra. We note that the homomorphism 0: of property 1) is indeed injective as will be established in 1.2.

Theorem 1.1. For any Lie algebra 9 over K there exists a universal enveloping algebra (U(g),o:). If (U1 (g),o:d is another universal enveloping algebra of g, then there exists a natural isomorphism k : U1 (g) ----t U (g) such that k0:1 = 0:.

Proof. The uniqueness of the universal enveloping algebra follows directly from its definition. In the proof of existence one can make use of the following construction, generalizing the construction of the symmetric algebra, to which it reduces in the abelian case (see Example 1). In the tensor algebra T (g) over the vector space 9 (see page 68 of Encycl. Math. Sc. 11) consider the ideal P, generated by the elements of the form x 0 y - y 0 x - [x, y] (x, y E g). Let U (g) = T (g)/ P and let 0: : 9 ----t U (g) be defined by the formula o:(x) = x+ P. It turns out that (U (g), 0:) is a universal enveloping algebra for g. 0

We note the following properties of the universal enveloping algebra, which follow easily from the construction.

1. Let 9 be a Lie algebra and V some vector space over the field K. Any structure of a g-module on V (i.e. any linear representation of the Lie algebra 9 in V) defines a U (g)-module structure on V such that

o:(x)V = xv (x E g,v E V).

Conversely, any U (g)-module structure on V can be obtained in this way from some (uniquely determined) g-module structure.

Thus universal enveloping algebras playa role in the theory of linear representations of Lie groups which is analogous to that played by group algebras in the theory of group representations.

2. Any Lie algebra homomorphism ¢ : 9 ---) gl determines a unique homomorphism U (¢) : U (g) ---) U (gd such that U (¢)(1) = 1 and the diagram

 9 -------+ 9 1

U (g) u (<1»

-------+

whose vertical arrows are the homomorphisms coming from the definition of a universal enveloping algebra, is commutative. The correspondence 9 1--*

U (g), ¢ 1--* U (¢) is a covariant functor from the category of Lie algebras to the category of associative algebras with unity over K.

3. The algebra U (g) is generated by the vector space a(g). 4. There exists a natural anti-isomorphism u 1--* u~ of the algebra U (g)

such that x~ = -x (x E g). Note that all the definitions and results in this subsection carryover to the

case when K is an arbitrary commutative and associative ring with identity.

1.2. The Poincare-Birkhoff-Witt Theorem. In the study of the structure of the algebra U (g) there arises the following question: how to construct a basis of the algebra U (g) starting with a given basis of the algebra g? The answer is provided by the Poincare-Birkhoff-Witt theorem.

We may assume that U (g) = T (g)/ P, where P is the ideal defined in the proof of Theorem 1. Let 7r : T (g) ---) U (g) be the natural homomorphism (which coincides with a on T 1 (g) = g.) Suppose that in 9 there is given a basis (et)tET, whose index set T is totally ordered. Let Yt = aCed E U (g). For any ordered choice of indices I = (i1, ... , ir) E Tr such that i1 ::; ... ::; ir we set YI = Yi 1 ••• Yir (we assume that Y0 = 1). The more usual way of writing the elements YI is in the form YiI ... Yi r = y'!'l ... y;nt ... , where mt 2 0 are integers, only finitely many of which are not equal to 0, and L: mt = r. If 9

tET is an abelian Lie algebra, then U (g) = S (g) is the algebra of polynomials in Yt (see Example 1), so that the monomials YI form a basis in U(g). It turns out that this property is preserved in the general case.

Theorem 1.2 (Poincare-Birkhoff-Witt). The elements YI, where I runs over all non decreasing ordered choices of indices from T, form a basis of the algebra U (g). The map a : 9 1--* U (g) is injective.

Proof. Let us first clarify how one shows the linear independence of the terms YI, from which also follows the injectivity of a. Consider the algebra S of polynomials over K in the variables Zt (t E T). Its basis consists of monomials ZI = Zi1 ... Zir, where I = (i1, ... ,ir) E Tr,i1 ::; ... ::; ir, with

Z0 = 1. By induction we can show that S possesses a structure of a g-module, satisfying the following conditions:

et(ziw· . ZiJ = ZtZi, ... ZiT if t ::::; i1 ::::; ... ::::; ir; Yt1 = Zt (t E T) for all t E T. Making use of property 1 of §1.1 we obtain a U (g)-module structure on S such that

Yt(Zi" ... zd = ZtZi, ... ZiT (t ::::; i1 ::::; ... ::::; ir); Yt1 = Zt (t E T) If I: ClYl = 0, where Cl E K, then multiplying both sides of this equality

I by 1 E S we obtain I: ClZl = 0 from which it follows that all Cl = O. In order

I to show that U (g) is spanned by the elements Yl it is convenient to define an increasing filtration of U (g) by subspaces

(1)

Clearly,

U (g)(O) C U (g) (1) C ... C U (g)(p) c ... ,

U (g)(p) U (g)(q) c U (g)(p+q),

so that U (g) is a filtered algebra. Clearly, 7r(x ® Y - Y ® x) E U (g) (1) for any x, Y E g. From this it follows that if Q is the ideal in T(g) generated by the elements of the form x ® Y - Y ® x(x, Y E g), then 7r(Q n TP(g)) C U (g)(P-1) for all p 2: O. 8ince the symmetric algebra 8 (g) = T (g)/Q is the algebra of polynomials in et E 81(g) = 9 it easily follows by induction that U (g)(p) is generated by the elements of the form Yl = Yi, ... i r with r ::::; p. D

The statement of Theorem 1.2 can be formulated also in an invariant form, without using a basis. For this purpose we consider the filtration of the algebra U (g) (see (1)) and construct the associated graded algebra

gr U (g) = EB gr(p) U (g), where gr(p) U (g) = U (g)(p)/U (g)(p-1). P?:O

The multiplication in it is given in the following way: if a E U (g)(p), b E U(g)(q) and a,b are the corresponding elements of grpU(g) and grqU (g),

then ab = ab + U (g)(p+q-1) E grp+qU (g). As can be seen from the proof of Theorem 1.2 given above, 7r determines a

homomorphism of graded algebras ir : 8 (g) -+ gr U (g) given by the formula

ir(u + Q) = 7r(u) + U (g)(p-1) (u E TP(g)).

Corollary 1. The homomorphism ir : 8 (g) -+ gr U (g) is an isomorphism of graded algebras.

Corollary 2. The algebra U (g) has no zero divisors.

Corollary 3. If dimKg < 00, then the algebra U (g) is Noetherian from the left and from the right.


From now on we shall identify 9 with a subspace of the algebra U (g) by means of the injection Q. In particular, in the notation of Theorem 1.2 we shall write Yt = et, YI = el = eiI ... ei r •

Let g1, g2 be two Lie algebras, 9 = gl EB g2 their direct sum. The natural inclusions gi -+ 9 can be extended to inclusions U (gi) -+ U (g). Consider the map /1: U (gl) 0 U (g2) -+ U (g) given by the formula /1(U1 0 U2) = U1U2.

Corollary 4. The map /1 is an isomorphism of algebras.

Let us briefly consider the case when 9 is a Lie algebra over a commutative ring K with unit. If 9 is a free K-module, the proof of Theorem 1.2 carries over in its entirety. In (Lazard 1952) it is shown that Corollary 1 continues to hold if K is a principal ideal domain. It is false in general (Shirshov 1953).

1.3. Symmetrization. We continue to use the notation of 1.2. If K is a field of characteristic 0, then the symmetric algebra S (g) = T (g)/Q is isomorphic to the algebra of symmetric tensors ST (g), which is embedded in T (g) as the complementary subspace to Q. This makes possible to give a construction of a basis in U (g) which is different from the one described in Theorem 1.2.

Let STm(g) = ST (g) n Tm(g). Then 7f maps STm(g) isomorphic ally onto some subspace of U (g)m. We define a vector space isomorphism

Wm = 7flJ'm : sm(g) -+ U (g)m,

where

1 IJ'm(6··· ~m) = I L ~s(l)'" ~s(m) (6,···, ~m E g). m.

sESrn

Then for any i 1 , ... , im E T we obtain

Let M = (mt)tET be a family of integers mt ~ 0, of which only a finite number are not equal to O. We shall write IMI = L mt. Let

tEl

(2)

From what has been said follows

Theorem 1.3. Let char K = O. The maps Wm defined above determine an isomorphism of graded vector spaces

W : S (g) -+ U(g).

The elements e(M) for all possible choices of M form a basis of the algebra U (g).

Note that the map W is an isomorphism of algebras only when 9 is abelian.


1.4. The Center of the Universal Enveloping Algebra. In the notation of 1.3, to each element ~ E g there corresponds a derivation ad ~ ofthe algebra U(g), given by the formula

(ad~)u = [~,uJ = ~u - u~ (u E U(g)).

Moreover, ad is a linear representation of the Lie algebra g in U (g). The vector space

U(g)9={uEU(g)l(ad~)u=0 V~Eg}.

of invariants of this representation coincides with the center Z(U(g)) of the algebra U (g). One verifies that ad~ (~E g) preserves the subspaces U (g)(m) and U (g)m. From this it follows that ad induces a linear representation ad of the Lie algebra g in gr U (g) and moreover (gr U (g))9 = gr Z(U(g)), where the right hand side is the graded algebra associated with the filtration of the center Z(U(g)) by subspaces Z(U(g)(m) = Z(U(g) n U (g)(m). On the other hand the operator ad in the algebra g uniquely extends to a derivation of the algebra S (g). This gives rise to a linear representation of g in S (g), which defines the subalgebra S (g)9 of invariant elements of the algebra S (g). It turns out that the maps it of 1.2 and w of 1.3 are isomorphisms of g-modules. From this follows

Theorem 1.4 (1. M. Gelfand, see Kirillov 1972). If char K = 0, then the map

w : S (g)9 --+ Z(U(g))

is an isomorphism of vector spaces, and

it : S (g)9 --+ gr Z(U(g))

is an isomorphism of algebras.

The map w only in rare cases turns out to be an isomorphism of algebras (for example, if g is nilpotent, see Dixmier 1974, Proposition 4.8.12). Nevertheless, there holds the following non-trivial

Theorem 1.5 (Duflo, see Diximier 1974). For any finite-dimensional Lie algebra g over a field of characteristic 0 the algebras Z(U(g)) and S (g)9 are isomorphic.

1.5. The Skew-Field of Fractions of the Universal Enveloping Algebra. Let g be a finite-dimensional Lie algebra over K. By the theorem of GoldieOre (see Jacobson 1962, Chap. V, §3), Corollaries 2 and 3 of Theorem 1.2 imply the existence of a skew-field P(g), containing U(g) as a subring, in which every element can be represented in the form u-1v (and also in the form vu-1), where u, v E U(g), u =f. O. The skew-field of fractions P(g) is called the enveloping skew-field (or the Lie skew-field). Its centre C(g) = Z(P(g)) satisfies the condition C(g) n U(g) = Z(U(g)). There is a conjecture of Gelfand-Kirillov (see Gelfand and Kirillov 1966, Kirillov 1972) concerning


the structure of the skew-field P(g) in the case when 9 is the tangent algebra of a certain complex linear algebraic group.

To formulate this conjecture we introduce the following concept. Let A be an associative algebra with identity over K. For any finite system a = {aI, ... ,as} C A and any N > 0 we shall denote by d( a, N) the dimension of the subspace of A spanned by the products of ~ N elements ai. By the Gelfand-Kirillov dimension of the algebra A we mean

. -. logd(a,N) DlmKA=sup hm 1 N '

a N--+oo og

where a runs over all finite subsets of A. For any skew field P containing the field K set

. .. log d( ab, N) DlmK P = sup mf hm 1 N '

a bEP\{O} N--+oo og

where a runs over all finite subsets {al,"" as} C P and ab stands for {al b, ... , asb}. From Theorem 1.2 one deduces

Theorem 1.6 (Gelfand and Kirillov 1966). For any finite-dimensional Lie algebra 9 over K we have DimKP(g) = DimKU(g) = dimg.

Let us~so note that, if G is a complex linear algebraic group, then the number DimKC(g) coincides with the minimal co dimension of an orbit of the representation Ad* of the group G on the vector space g*, which is the dual of the adjoint representation.

Let Rn,k(K) be the Weyl algebra over the field K associated with the skew-symmetric bilinear form b of rank 2n on the vector space V over K of dimension 2n + k, i.e. the quotient algebra of the tensor algebra T(V) by the ideal generated by elements of the form x ® y - y ® x - b( x, y) . 1, and let Dn,dK) be its field of fractions. It turns out that DimKDn,k(K) = 2n + k.

The Gelfand-Kirillov conjecture consists in the following: If 9 is the tangent algebra of a complex linear algebraic group then,

where 2n + k = dimg, k = DimC(g).

Theorem 1.7. The Gelfand-Kirillov conjecture holds in the following cases: a) 9 = g(n(C) or s(n(C) (Gelfand and Kirillov 1966); b) 9 is a solvable Lie algebra (Joseph 1974, McConnell 1974).

Theorem 1.8 (Gelfand and Kirillov 1969). For any complex semi-simple Lie algebra 9 some finite extension of P(g) is isomorphic to Dn,k(C), where 2n + k = dimg, k = Dim C(g).

66 A.1. Onishchik, E. B. Vinberg

§2. Bialgebras Associated with Lie Algebras and Lie Groups

In connection with this section see also (Bourbaki 1971), (Serre 1965).

2.1. Bialgebras. Let A be a vector space over K. By a co-multiplication in A we mean any linear map A ----) A 129 A. As usual by a multiplication in A we mean a linear map A 129 A ----) A, that is a binary operation in A which turns this vector space into an algebra over K. A triple (A, JL, 8), where JL is a multiplication and 8 a co-multiplication, is called a bialgebra if 8 : A ----) A 129 A is a homomorphism of the algebra A with multiplication JL into the tensor product of this algebra with itself.

Sometimes (usually when certain additional conditions are satisfied) bialgebras are called Hopf algebras. This is due to the fact that the analogous concept in the category of graded algebras was first considered by Hopf in the case of the cohomology algebra of a Lie group. There are natural definitions of homomorphisms and isomorphisms of bialgebras.

Example 1. Let 9 be a Lie algebra over K. Consider the diagonal homomorphism ~ : x f--+ (x, x) of the algebra 9 into 9 EB g. By virtue of Corollary 4 of Theorem 1.2 we can identify the algebra U(gEBg) with U(g) 129 U(g). By Corollary 2 of 1.1 ~ extends to a homomorphism ~ : U(g) ----) U(g) 129 U(g), which is called the diagonal map. The coproduct ~ turns U(g) into a bialgebra.

In the finite-dimensional case there is a remarkable duality between bialgebras: to every finite-dimensional bialgebra (A, JL, 8) there corresponds a dual bialgebra (A*,8T ,JLT ), where 8T , JLT are the dual homomorphisms. In the general case, the existence of a natural inclusion A * 129 A * c (A 129 A) * implies that if 8 is a comultiplication in A then, 8T is a multiplication in A *, but the map JLT : A* ----) (A 129 A)* dual to the multiplication JL in A not always defines a comultiplication in A *. As an example one can take the object dual to the bialgebra U(g) from Example 1, which will be considered in 2.3 (see Corollary 1 of Theorem 2.3).

Let (A, JL, 8) be a bialgebra with an identity 1 (with respect to the multiplication JL). An element a E A is said to be primitive if 8a = a 129 1 + 1 129 a. It is easy to show that the set II(A) of all primitive elements of a bialgebra A is a sub algebra of A with respect to the commutator [x, y] = xy - yx.

Theorem 2.1. Let 9 be a Lie algebra over K. Consider U(g) as a bialgebra with comultiplication ~ (see Example 1). Then 9 c II(U(g)), and if char K = 0, then 9 = II(U(g)).

Proof. Obviously we have an inclusion 9 c II(U(g)). The existence of the opposite inclusion can easily verified in the case when 9 is an abelian Lie algebra, i.e. when U(g) = S(g) (see Example 1 of 1.1). In the general case we consider the filtration (1) of the algebra U(g) by subspaces. It turns out that ~ induces a homomorphism A : gr U(g) ----) gr U(g) 129 grU(g), which, under the isomorphism *-1 : gr U(g) ----) S(g) see Corollary 1 of Theorem 1.2)


transforms into the above described diagonal map for the algebra 8(g). From this one easily deduces that, if x E U(g)(p) is a primitive element, then p = 1, i. e. x E K EB g. As the non-zero elements of K are not primitive we have x E g. 0

2.2. Right Invariant Differential Operators on a Lie Group. Let G be a local analytic Lie group (real or complex) and 9 its tangent algebra. We shall be considering differential operators with analytic coefficients acting on local analytic functions on this group. These operators form in a natural wayan associative algebra with identity over the field K = lR or C, depending on whether we are considering a real or a complex Lie group. Let us denote by V( G) its subalgebra consisting of right invariant operators, i.e. operators, which commute with all transformations R(g)* induced by right translations R(g) (g E G).

To every element u of the tangent algebra 9 = Te (G) of a group G there corresponds a right invariant vector field u* on G such that u* (e) = u. In its turn, to the field u* there corresponds an operator Lu. (the Lie derivative) which belongs to V(G) (see 1.2 of Chap. 1 of Part II). Moreover, the correspondence u 1--+ Lu. is an injective homomorphism of the tangent algebra 9 into V(G) (see 1.2 of Chap. 2). By the definition of the universal enveloping algebra this homomorphism uniquely extends to an algebra homomorphism p : U(g) ---+ V(G). Our aim, in particular, consists in proving that p is an isomorphism of algebras.

Let us consider the algebra Oe of germs of analytic functions at a point e on a Lie group G. Let m = {f E Oe I f(e) = O} be the maximal ideal of this local algebra. Denote by (Oe)~ the subspace of 0; consisting of all linear forms which are zero on one of the ideals mT • Define a linear map a : V( G) ---+ (Oe); by the formula

a(8)(¢) = (8¢)(e) (8 E V(G), ¢ E Oe).

Putting T = ap we obtain a commutative diagram

U(g)~(Oe)~

p'\./a V(G)

Theorem 2.2. The map p is an isomorphism of algebras, and a and T are isomorphisms of vector spaces.

Proof. Let Xl .•• ,Xn be a local coordinate system in a neighbourhood of the point e of G (such that e = (0, ... ,0)). For any ordered choice M = (ml,'" ,mn) of non-negative integers mi we define a linear form 8M E (Oe)~ by the formula

M 1 81M1¢ l1 (¢) = M! 8X'{'1 ... 8x~n (e),


where M! = ml!'" mnL One can easily show that the elements llM for all possible choices of M form a basis of the vector space (Oe)~ over K = IR or Co Clearly, II M (cjJ) is the coefficient of xM = X;"l ... x~n in the Taylor series of the function cjJ E Oe at the point e = O. The elements llM depend on the choice of coordinates. In what follows, we shall suppose that Xl, ... ,Xn is the canonical coordinate system of the first type. This means that the coordinates Xl, ... ,Xn of an element X E G can be found from the relation

where el, ... ,en is a fixed basis of the vector space 9 (see 3.2 of Chap. 2). If cjJ is an analytic function in a neighbourhood of a point g E G, then for sufficiently small e E 9 we have

00 1 cjJ((expe)g) = L I" (C'cjJ)(g)

m=om. (3)

(see formula (9) of 2.1 Chap. 1 Part II). In particular, with g = e it follows from (3) that for sufficiently small Xi

,",1M ~ M!x e(M),

IMI=m

where the summation is done over all ordered choices of M = (ml, ... , m n ),

and e(M) is defined by formula (2). Therefore the Taylor series ofthe function cjJ E Oe has the form

( l)IMI cjJ(x) = L -MI r(e(M))xM,

M .

whence r(e(M)) = (-l)IMIM!llM for any M. By Theorem 1.3 the elements e(M) form a basis of the algebra U(g). Hence r is an isomorphism.

To complete the proof of the theorem it suffices to show that Ker (j = O. But this follows easily from the permutability of the elements of the algebra V( G) with right translations. 0

Corollary. If G is a connected Lie group, then the map p defines an isomorphism of the center Z(U(g)) onto the algebra of all invariant (i.e. commuting with all left and right translations) differential operators on G.

2.3. Bialgebras Associated with a Lie Group. We continue to use the notation of the previous subsection. In this subsection we shall show that the multiplication in a Lie group G defines on (Oe)~ a certain natural structure


of a bialgebra. After that will be constructed an isomorphism between the bialgebras U(g) and (Oe);.

In order to introduce the bialgebra structure we make use of considerations of duality. Let 11 be the multiplication in the algebra Oe. A simple calculation shows that

M' M" ~ 0~ . (4) M'+M"=M

Thus I1T defines a comultiplication in (Oe);. Let us denote by O(e,e) the algebra of germs of analytic functions at the

point (e, e) on G x G. This algebra contains Oe 0 Oe as a subalgebra. This allows one to construct a linear map (O(e,e)); ~ (Oe);0(Oe);, which turns out to be an isomorphism (one makes use ofthe basis (~M) in (Oe)~ constructed in the proof of Theorem 2.2). We shall identify (O(e,e))~ with (Oe)~ 0 (Oe)~. The multiplication m : G x G ~ G in a Lie group G induces a homomorphism of algebras 8 = m* : Oe ~ O(e,e)' Thus Oe possesses a structure which is close to that of a bialgebra. Since m*(m) is contained in the maximal ideal of the algebra O(e e) (see 1.1 of Chap. 2), 8T(O(* )) C (Oe)~, i.e. 8T de-, e,e c

fines a multiplication in (Oe);. One can easily verify that ((Oe)~,8T,I1T) is a bialgebra.

Theorem 2.3. The map r' : u f---+ r(u~) (see 1.1) is an isomorphism from the bialgebra U(g) to the bialgebra (Oe)~.

Proof. Let us prove that r' preserves multiplication, i. e. that

r'(uv) = 8T (r'(u) 0 r'(v)) (u,v E U(g)).

For this purpose we make use of the formula

00 1 ¢((exp~)(exp1])) = L l!m! (1]r;1]~¢)(e),

l,m=O

(5)

(6)

which holds for ¢ E Oe for sufficiently small ~,1] E g. It can be easily derived

from formula (3). For x = exp (i~ Xiei) , y = exp (i~ Yiei) we see from

(6) that for sufficiently small Xi, Yi

(-1) ILI+IMI ¢(xy) = L L!M! r(e(M)e(L)) xLyM.

L,M

This is indeed the condition (5) for u = e(L)~,v = e(M)~. It remains to show that r' preserves comultiplication, i. e. that

(r' 0 r')6 = I1T r'.

Since the left and the right hand sides of this equality are algebra homomorphisms it suffices to check that they coincide on the elements of the subspace 9 C U(g). This verification can be easily performed making use of (4). D


The algebra Oe is naturally isomorphic with the algebra of convergent power series K{Xl'.'. ,xn}. We embed it in the algebra Oe = K[[Xl, ... ,xnll of formal power series. The homomorphism /5 = m* associated with the Lie group multiplication extends uniquely to a homomorphism /5 : Oe ~ O(e,e) =

K[[Xl, ... ,Xn,Yll ... ,Ynll. The vector space Oe can be naturally identified with the dual ((Oe)~)* of (Oe)* c.

Corollary 1. The map (T')T : Oe ~ U(g)* is an isomorphism of the algebra Oe onto the algebra U(g)* with multiplication jj.T and moreover, /5 is taken to the map dual to the multiplication in U(g).

Corollary 2. There exists a bijective correspondence between homomorphisms of local analytic Lie groups and homomorphisms of their tangent algebras. Two local analytic Lie groups are isomorphic if and only if their tangent algebras are isomorphic.

This corollary was already proved by different means in Chap. 2 (see Theorems 2.2 and 2.10). Let us note also that with the help of Theorem 2.3 one can prove the existence of a virtual Lie subgroup with a given Lie subalgebra, by a method different from the one used in Chap. 2 (Theorem 2.6), see (Helgason 1964).

§3. The Campbell-Hausdorff Formula

In this section the central role is played by the Campbell-Hausdorff series, i.e. the formal power series log(exeY ) in non-commuting variables x, y. By making use of this series one can explicitly express the operation of multiplication in a Lie group in canonical coordinates in terms of the bracket operation in its tangent algebra, and also prove the existence of a local Lie group with a given tangent algebra. For details see (Postnikov 1982), (Bourbaki 1971), (Serre 1965).

3.1. Free Lie Algebras. The system of generators (Xi)iEI of a Lie algebra 9 over K is called free if for any Lie algebra ~ over K and any family of elements (Yi) E J of the algebra ~, indexed by the same set J, there exists a (unique) homomorphism h : 9 ~ ~ such that h(Xi) = Yi (i E J). A Lie algebra is said to be free if it admits a free system of generators. It is easy to see that two Lie algebras admitting free systems of generators of the same cardinality are isomorphic. Moreover, any Lie algebra with a system of generators (Ui)iEI is isomorphic to a quotient of a free Lie algebra with a free system of generators of cardinality III- A free Lie algebra with a free system of generators of arbitrary cardinality can be constructed as a quotient of the algebra of nonassociative polynomials by a certain ideal which is constructed from identities defining the class of Lie algebras. Below we give a different


construction of a free algebra, which at the same time describes its universal enveloping algebra.

Let V be a vector space over K with a basis (Xi)iEI, where I is an arbitrary set. Consider the Lie algebra L(T(V)), corresponding to the tensor algebra T(V). Let us denote by I(V) = 1((Xi)iEI) the subalgebra ofL(T(V)) generated by the family (Xi)iEI.

Theorem 3.1. The family (Xi)iEI is a free system of generators of the Lie algebra I(V) and T(V) is its universal enveloping algebra (with respect to the identity inclusion I(V) ---+ T(V)). The algebra I(V) possesses the following grading: I(V) = ffip~o Ip(V), where Ip(V) = I(V) n TP(V) is the subspace spanned by the elements of the form

[Xill [Xh,··· [Xip_ll xipj .. . J], is E I. If the cardinality d = III is finite, then the dimension ld(P) satisfies the recurrent relation

L mld(m) = dn .

min

Corollary. If d = IJI, then

1" n ld(n) = - ~J.t(m)dm, n

min where J.t is the Mobius function.

Consider the linear map cJ? : T(V) ---+ I(V) given by the formula

¢(Xil ... Xip) = [XiI' [Xi2'· .. , [Xip_l' Xipj . .. J] (p> 0), cJ?(1) = 0

It is not difficult to verify that cJ? : I(V) ---+ I(V) is a derivation of the algebra I(V). From this it follows that

cI>(u) = pu (u E Ip(V». (7)

3.2. The Campbell-Hausdorff Series. Let K be a field of characteristic o. Let us consider the algebra of noncommutative formal power series in the variables (Xi)iEI. We define it as the complete tensor algebra T(V) =

00

I1 TnV, where V is the K-vector space with basis (Xi)iEI. We shall write n=O elements of this algebra as formal power series

00

u = L Un, Un E Tn(V). n=O

We define a subalgebra [(V) of the Lie algebra L(T(V)) by the formula 00

[(V) = II In(V). n=O

00

Let m = I1 Tn(v). n=l


Theorem 3.2. The formulas 00 n

expx= L;' n. n=O

00 n

log (1 + x) = L ( -1 t+1 ~ n

n=l

define mutally inverse bijective mappings exp : m ---+ 1 + m and log: 1 + m ---+

m.

Consider, in particular, the case when the family(xi)iEI consists of two elements x and y. Then (exp x) (exp y) E 1 + m, so that we have a well defined element loge (exp x) (exp y)) E m.

Theorem 3.3 (Campbell-Hausdorff). We have log((exp x) Ell (exp y)) E i(V).

Proof. Consider the diagonal map Ll : T(V) ---+ T(V) ® T(V) for the universal enveloping algebra T(V) of the Lie algebra [(V) = [(x, y). It can be extended to a homomorphism

! : T(V) ---+ T(V Ell V) = II (TP(V) ® Tq (V)). p,q

From Theorem 2.1 it follows that

i(V) = {u E T(V)I!u = u ® 1 + 1 ® u}.

One easily verifies that u = loge (exp x) (exp y)) satisfies the last condition. o

From Theorem 3.3 and formula (7) follows

Corollary. We have

(8)

Formula (8) was first obtained by E. B. Dynkin (see Dynkin 1950). Let us write its right hand side in the form

00

D(x, y) = L Dp(x, y), (9) p=l

where Dp(x, y) E [(x, Y)p. A direct computation shows that

1 1 D 1(x, y) = x + y, D2 (x, y) = 2[x, yJ, D3(X, y) = 12 [x, [x, y]]+ (10)

1 1 + 12 [y, [y, xJ], D4(X, y) = 24 [x, [y, [x, y]]].

Note that Theorems 3.2 and 3.3 and formula (8) hold for any commutative ring K with unit which is a Q-module.

3.3. Convergence of the Campbell-Hausdorff Series. Let g be a Lie algebra over a field K of characteristic 0. Each term Dp(x,y) of series (9) defines a polynomial mapping Dp : g EB g -t g which is homogeneous of degree p. Hence series (9) can be viewed as a g-valued formal power series on the vector space g EB g with center at the point (0,0).

Theorem 3.4. Let g be a finite-dimensional Lie algebra over K = lR or <C. Then the formal power series D(x, y) on g EB g has a positive radius of convergence and defines in a neighbourhood of zero of g a local analytic Lie group whose tangent algebra coincides with g.

Proof. Convergence can be established by using simple estimates. One can verify directly that the operation

x * y = D(x,y) (11)

defines in g the structure of a local analytic Lie group with identity e = 0. By virtue of (10)

1 x * y = x + y + "2 [x, y] + ... ,

where the dots stand in place of terms of degree> 2. By its definition (see 1.1. Chap. 2) the tangent algebra coincides with g. 0

We remark that the methods described here still do not allow us to prove the existence of a global Lie group with a given tangent algebra.

We shall denote by CH(g) the local Lie group with multiplication (1) constructed from a Lie algebra g in Theorem 3.4. One can verify that its one parameter subgroup with tangent vector e Egis the line t 1---+ te. Hence the exponential map exp : g -t Te(CH(g)) = g is the identity map.

Corollary 1. For any Lie group G the operation of multiplication written in canonical coordinates of the first type has the form (11). In particular any Lie group is analytic.

Proof. Let g be the tangent algebra of G. By Theorem 2.10 of Chap. 2 there exists an isomorphism of local Lie groups h : CH(g) -t G such that deh = id. Since the exponential mapping for CH(g) is the identity, it follows from formula (36) of Chap. 2 that H coincides with the exponential mapping exp for the group G. Therefore exp(e * 'T}) = (expe)(exp'T}) for all sufficiently small e,'T} E G. 0

It is not hard to verify that the correspondence g 1---+ CH(g) defines a functor from the category of finite-dimensional Lie algebras over K (= lR or

<C) to the category of local Lie groups over K. With the help of Corollary 1 we obtain

Corollary 2. The category of local Lie groups over K (= IR and !C) is equivalent to the category of finite-dimensional Lie algebras over K.

Chapter 4 Generalizations of Lie Groups

This chapter contains a brief survey of certain theories generalizing the classical Lie theory. This survey does not aspire to completeness. In particular, we do not touch on the theory of algebraic groups and topological groups are considered only in connection with Hilbert's 5th problem. These subjects, naturally, require separate surveys. The same can be said about "infinite" continuous groups (or Lie pseudo-groups), the study of which was originated already by S. Lie. Beyond the scope of this survey remain the Kac-Moody algebras and the groups corresponding to them, as well as Lie supergroups and superalgebras.

§l. Lie Groups over Complete Valued Fields

In connection with this section see also (Bourbaki 1971), (Lazard 1965), (Serre 1965).

1.1. Valued Fields. Recall (see §7 of Encycl. Math. Sci. 11) that by an absolute value or a valuation on a field K one means a real valued function a -t lal over K, possessing the following properties.

1) lal ~ 0 with lal = 0 if and only if a = 0, 2) labl = lallbl; 3) la + bl ::; lal + Ibl· lt is usually also assumed that the absolute value is nontrivial, i.e. satisfies

the condition: 4) lal ¥- 1 for some a E K \ {o}. A valued field is a field equipped with an absolute value. An absolute value is said to be non-archimedean (or ultrametric) ifin place

of 3) the following stronger condition holds: 3') la + bl ::; sup {Ial, Ibl}. In this case it is also said that the valued field is non-archimedean. The best known examples of complete valued fields are the classical fields

IR and C and the field of p-adic numbers Qp (p a prime number), which is

obtained from the field of rational numbers Q by completion with respect to the so called p-adic absolute value lalp = pn, where p is a fixed number, o < p < 1, and n E Z is defined by the equation: a = pn ~ where b, care integers non-divisible by p. By Ostrovski's theorem (see Bourbaki 1964) ~ and C are the only archimedean complete valued fields.

Let K be a non-archimedean complete valued field. Then A = {a E Kllal ~ I} is a subring of K (known as the valuation ring), and m = {a E Allal < I} the maximal ideal in A. For example, if K = Qp, then A = Zp, the ring of p-adic integers and m = pZp.

1.2. Basic Definitions and Examples. Let K be a complete valued field. One defines in the usual way (see Serre 1965) analytic functions on open sets in K n with values in K, analytic manifolds over K and also (analytic) Lie groups over K. In particular, Lie groups over ~ or C are the usual real or complex Lie groups; Lie groups over Qp are called p-adic Lie groups. All the general concepts defined in 1 of Chap. 1: direct products of Lie groups, Lie subgroups, homomorphisms of Lie groups, linear representations, local Lie groups, transfer verbatim to the case of an arbitrary complete valued field K (subgroups and homomorphisms are assumed to be analytic). A large part of the examples of Lie groups given in that section also transfers to the general case. In particular the following are Lie groups over K:

The additive group K and the vector group Kn. The group of non-singular matrices GLn(K) and (isomorphic to it) the

group GL(V) of non-singular linear transformations of an n-dimensional vector space V.

The group of invertible elements of a finite-dimensional associative algebra with unit over K.

The subgroups of diagonal and triangular matrices in GLn(K) are Lie subgroups. The same holds for the subgroup SLn(K) of matrices with determinant 1.

Let us give some examples, specific to the non-archimedean case. Let K be a complete, non-archimedean valued field and A its valuation ring.

Examples. 1. The additive group A is an open Lie subgroup in K; analogously An is an open Lie subgroup in Kn. Also meA and mn c An are open Lie subgroups.

2. The group of invertible matrices GLn(A) over the ring A is an open Lie subgroup of GLn(A).

We remark also that, just as in the cases K = ~,C, every Lie subgroup is closed in the ambient group.

1.3. Actions ofLie Groups. Just as in §2 of Chap. 1 one can define (analytic) actions of Lie groups over a valued field K on analytic manifolds over K. Theorem 2.1 of Chap. 1 does not transfer to the general case, but continues to hold provided char K = 0 (see Serre 1965). The same applies also to


Theorem 2.2, which follows from Theorem 2.1. In particular, if char K = 0, then the stabilizer of any point under the action of a Lie group G is a Lie subgroup of G and the kernel of any homomorphism of a Lie group G is a normal Lie subgroup of G.

Let G be a Lie group over K and H a Lie subgroup. Then there holds the following analogue of Theorem 3.1 of Chap. 1: the set G / H possesses a natural analytic structure, with respect to which the canonical map p : G --t G / H is a quotient map. Moreover, p is a locally trivial fibre bundle and the canonical action of G on G / H is analytic. If H is normal, then the group G / H possesses a natural structure of a Lie group (see Theorem 3.2 of Chap. 1). Finally, if charK = 0, then the generalizations of Theorems 3.3 and 3.4 of Chap. 1 hold over the field K.

1.4. Standard Lie Groups over a Non-archimedean Field. Let K be a nonarchimedean complete valued field. A Lie group Gover K is said to be standard if G coincides, as manifold, with mn and the multiplication has the form

xy = F(x,y),

where F is a power series in XI, ... , xn, yI, ... , Yn with coefficients in An and the free term equal to zero. Clearly, the identity element E of a standard Lie group is O. Note that any formal power series in ZI, ... ,Zm with coefficients in A converges in mn and its sum is a function with values in m.

Theorem 1.1. Every local Lie group G over a non-archimedean field K contains an open Lie subgroup isomorphic to some standard Lie group.

Proof. We can suppose that G is a neighbourhood of 0 in Kn with e = 0 and that the multiplication in G has the form xy = F(x, y), where F is a convergent power series in XI, ... ,xn, yI, ... ,Yn with coefficients in Kn and zero free term. The transformation x ~ x = AX, where A E K \ {O}, of the space K n takes G into the group G with multiplication xy = F(x, y), which is isomorphic to G. Comparing the coefficients of the series F and F we see that if the value IAI is sufficiently large, then all the coefficients of the power series F lie in An and the power series converges in m2n. Thus, G contains a standard open Lie subgroup. D

Corollary. Every local Lie group over a non-archimedean complete valued field is isomorphic to a restriction of some Lie group.

1.5. Tangent Algebras of Lie Groups. The definition of the Lie functor given in §1 of Chap. 2 can be transferred word for word to the case of a Lie group over an arbitrary complete valued field. Thus to every Lie group Gover K there corresponds a Lie algebra 9 over K of the same dimension (the tangent algebra) and to every homomorphism of Lie groups a homomorphism of Lie algebras. Additionally, to a Lie subgroup of G there corresponds a sub algebra of g, which is an ideal if the subgroup is normal.


Let us suppose now that K is non-archimedean and that char K = o. Then we have the following theorems, whose proof makes use of the CampbellHausdorff formula (see Bourbaki 1971).

Theorem 1.2. Let G be a Lie group over K and I) a subalgebra of its tangent algebra g. Then there exists in G a Lie subgroup H, whose tangent algebra is I). If HI is another Lie subgroup, with the same property, then H and HI coincide in some neighbourhood of the identity.

Theorem 1.3. Let G and H be Lie groups over K and cp : 9 ~ I) a homomorphism of their tangent algebras. Then there exists a homomorphism f : Go --t H defined on some open subgroup Go of the group G and such that def = cpo If h : GI --t H is another homomorphism, defined on an open subgroup GI C G and such that deh = cp, then f = h in some neighbourhood of the identity.

Corollary. If two Lie groups over K have isomorphic tangent algebras, then they contain isomorphic open subgroups.

Remark. If char K > 0, then the statement is false. For example, the additive and multiplicative groups K and GLI(K) of the field K have isomorphic tangent algebras (one-dimensional abelian Lie algebras over K), but do not contain isomorphic open subgroups since the formal groups corresponding to them are not isomorphic (see below the example in 2.2).

Theorem 1.4. For any finite-dimensional Lie algebra 9 over K there exists a Lie group Gover K with tangent algebra g.

Proof. Consider the power series D(x, y) introduced in 3.2 of Chap. 3. As in the archimedean case (see Theorem 3.4 of Chap.3) one proves that this power series has a positive radius of convergence and determines in a neighbourhood of the point 0 of the space 9 a local Lie group CH(g) over K, the tangent algebra of which coincides with g. Then one applies the corollary of Theorem 1.1.

From the construction of the group CH(g) and the corollary of Theorem 1.3 follows

Corollary. Every Lie group with an abelian tangent algebra contains an open subgroup· isomorphic to An.

Let G be some Lie group over K, 9 its tangent algebra and CH(g) the local Lie group defined by means of the Campbell-Hausdorff formula. Theorem 1.3 implies the existence of an isomorphism exp : V --t GI , where V and GI are open Lie subgroups in CH(g) and G respectively, such that de exp = id. The map exp is the exponential mapping; in the non-archimedean case it is defined only in a neighbourhood of zero in the tangent algebra.

We also note the following statement, which can be proved analogously to Theorem 3.6 of Chap. 2.

Theorem 1.5. Any closed subgroup of a p-adic Lie group is a Lie subgroup.

§2. Formal Groups

In connection with this section see (Bourbaki 1971), (Serre 1965).

2.1. Definition and Simplest Properties. Let K be a commutative and associative ring with identity. A formal group (or a formal group law of dimension n over K is a system F = (FikSi::;n of formal power series Fi E K[[Xl, ... , Xn, Yl,··., Ynll such that

1) F(x, 0) = x, F(O, y) = Y; 2) F(u, F(v, w)) = F(F(u, v), w).

Examples. 1. The additive group F = (Fih::;i::;n, where

Fi = Xi + Yi (i = 1, ... , n).

2. The multiplicative group (n = 1):

F = x+y+xy.

3. Let G be an n-dimensional Lie group over a complete valued field K. Let us choose in a neighbourhood of the point e EGan analytic coordinate system such that that the identity e has all coordinates equal to zero. If we identify a point X with the corresponding n-tuple of its coordinates Xl, ... X n "

then the multiplication in G can be written in the form

(XY)i = Fi(x, y) (i = 1, ... , n),

where the Fi are convergent power series. Then F = (Fih::;i::;n is a formal group over K which we shall denote by Fc.

4. If the field K is non-archimedean, then a standard Lie group over K determines a formal group over the valuation ring A and conversely (see 1.4).

5. Suppose that the ring K is an algebra over the field Ql (for example, K is a field of characteristic 0). Let 9 be a Lie algebra over K which is a free K-module of finite rank. If we choose a basis in 9 and write in terms of the coordinates the power series F = D of 3.2 of Chap. 3, we obtain a formal group CH(g) over K.

A homomorphism of a formal group F = (Fih<i<n into a formal group P = (Pj h::;j::;m is a set cJ> = (¢j h::;j::;m, where ¢j-E- K[[Xl' ... ,xnll satisfy the conditions:

¢(O) = 0; F(¢(x), ¢(y)) = ¢(F(x, y)).

Formal groups over K form a category, the morphisms of which are homomorphisms.


2.2. The Tangent Algebra of a Formal Group. For any formal group F = (Fih::;i::;n over a ring K we have the following result.

Theorem 2.1. The series Fi have the form

Fi(x,y)=Xi+Yi+ L cMLxMyL, IMI>O ILI>O

(1)

where CML E K and the summation is performed over all ordered tuples M,L chosen out of n non-negative integers.

There exists a unique set H = (Hih::;i::;n, where Hi E K[[Xl, ... , xnll, such that

we have

F(x,H(x)) = F(H(x),x) = OJ

Hi(X) = -Xi + L CMXM, IMI>o

where CM E K. Terms of degree 2 in formula (1) define a certain bilinear map b : K n x K n -t Kn. Putting

[x,y] = b(x,y) - b(y,x),

we obtain a bilinear operation in K n , which, as it turns out, makes K n into a Lie algebra. This algebra is called the "tangent algebra" of the formal group. For example, in the case where the formal group F = FG is associated with a Lie group G (see 1.1 of Chap. 2) the tangent algebra of the formal group CH(g) (see for Example 5 in 2.1) coincides with g.

It is not hard to verify that the differential at 0 (Le. the linear part) of a homomorphism of formal groups is a homomorphism of their tangent algebras. Thus we obtain a functor from the category of formal groups over a ring K into the category of Lie algebras over K (the Lie functor). If K is a field of characteristic 0, then this functor and the functor 9 -t CH(g) define an equivalence of these categories (see also below 2.3). In particular, a formal group over a field of characteristic 0 is determined up to an isomorphism by its tangent algebra. In the case of a field of prime characteristic this is not true, as is shown by

Example. Let K be a field of characteristic p > o. Then the one-dimensional additive and multiplicative formal groups over K (see Examples 1 and 2 of 2.1) are not isomorphic. At the same time they have the same tangent algebra - the one-dimensional abelian Lie algebra over K.

For other examples see (Manin 1963), (Lazard 1955). (Manin 1963) is devoted to the classification of abelian formal groups over fields of prime characteristic. The theory of formal groups is also studied in the following books: (Dieudonne 1973) and (HazewinkeI1978).


2.3. The Bialgebra Associated with a Formal Group. Analogously to the case of Lie groups (see §2 of Chap. 3), with every formal group one can associate some bialgebra. Let us denote by U the K-submodule in (K[[XI' .. . ,xn]])* consisting of linear forms equal to 0 on some power of the ideal m C K[[xl, ... , x n]], generated by the elements Xl, ... , x n. The module U is free and possesses a basis (~M), dual to the system of monomials (xM):

~ M (xL) = {I for M = L o for M i= L.

Moreover, U* = K[[XI, ... , x n]]. Operations in U arise from consideration of duality. As in 2.3 of Chap. 3, it turns out that the multiplication in the algebra K[[XI, .. . ,xnll defines a comultiplcation D in U. Furthermore, substitution into a power series of the power series Fi defines a map K[[XI, . .. ,xnll ~ K[[XI, ... , Xn , Yl, ... , Yn]], extending the associative multiplication in U. It turns out that U is a bialgebra with respect to these operations.

Let 9 = Kn be the tangent algebra of our formal group. We define a map T : 9 ~ U by the formula

n

T(al, ... ,an ) = L ai~ei, i=l

where ei = (0, ... ,1, ... ,0). One can verify that T is a homomorphism of the algebra 9 into the Lie algebra L(U). Hence T can be extended to an algebra homomorphism U(g) ~ U which we shall also denote by T.

Theorem 2.2. If K is an algebra over Q (for example a field of characteristic 0), then the homomorphism T : U(g) ~ U is an isomorphism of bialgebras.

Corollary. If K is an algebra over Q than the Lie functor and the functor 9 1-+ CH(g) define an equivalence of the category of formal groups over K and the category of Lie algebras over K which are free K -modules of finite rank.

§3. Infinite-Dimensional Lie Groups

The most immediate infinite-dimensional generalizations of Lie groups are Banach (in particular Hilbert) Lie groups, to which case almost all basic statements of the finite-dimensional theory can be transferred. However, Banach Lie groups have a rather restricted domain of applicability and, in particular, very rarely appear as groups of transformations of finite-dimensional manifolds (for example, there are no known examples of transitive and effective actions of infinite-dimensional Banach Lie groups on compact manifolds). Significantly larger (although more difficult to study) is the class of Lie groups modelled on Fnkhet spaces, to which, in particular, belongs the group of all diffeomorphisms of an arbitrary compact COO-manifold. Research into groups

of diffeomorphisms has also lead to the definition ofILB- (or ILH-) Lie groups, i.e. topological groups which can be represented as an inverse limit of Banach (or Hilbert) Lie groups (see Adams 1985, Omori 1974).

In connection with infinite-dimensional Lie groups and their applications in physics see (Adams 1985) and also the article (Milnor J., Remarks on infinite-dimensional Lie groups. Relativite, groupes et topol. II. Les Houches, Ec. d'ete phys. theor. Sess. 40, 1983. Amsterdam e. a., 1984, 1007-1058).

In this survey we do not touch upon the theory of Lie pseudo-groups of transformations (i.e. "infinite transformation groups" in the sense of Lie and Cartan) and the related theory of infinite-dimensional filtered Lie algebras.

3.1. Banach Lie Groups (see Bourbaki 1971, Hamilton 1982). Let K be a complete valued field. The definition of a Banach (Hilbert) Lie group G over K is a word for word restatement of the definition of a usual Lie group, with only this difference that G has to possess the structure of a Banach (respectively Hilbert) manifold (for the details see Bourbaki 1971, where the theory of Lie groups is presented at exactly this level of generality). Any ordinary Lie group over K is, of course, a Banach (and Hilbert) Lie group. We shall now give some infinite-dimensional examples.

Examples. 1. Let M and N be real manifolds of class Coo, with M compact. Then for any k ~ 0 the set Ck(M, N) of all maps M ---+ N of class Ck can be given the structure of a real Banach COO-manifold. In order to describe this structure we choose any Coo Riemannian structure on N and denote by Expy the corresponding exponential mapping at the point yEN. Let <p : M ---+ N be some mapping of class Ck • Consider the vector bundle <p*T(N) of class Ck

over M - the pre-image of the tangent bundle of N under the map <Pi its fibre at a point x E M is the tangent space T</>(x)(N). Then the correspondence v ~ <Pv, where <Pv(x) = Exp</>(x)v(x), maps injectively a sufficiently small neighbourhood of zero in the Banach space of sections of class C k of the bundle <p*T(N) into the set Ck(M, N) with <Po = <p. This is the chart in Ck(M, N) containing the point <p. If N = G is some real Lie group, then in the manifold Ck(M, N) = Ck(M, G) there is a natural group structure. It turns out that Ck(M, G) is a Banach Lie group.

Analogously, one defines Hilbert Lie groups H8(M, G) (s > ~dim M) of all maps M ---+ G of the Sobolev class HS (see Palais 1968). The groups Ck(M, G) and HS(M, G) are sometimes referred to as groups of currents and in the case M = SI as loop groups.

2. Generalizing Example 1, one can define Banach and Hilbert manifolds Ck(E) and H8(E) consisting of sections of class Ck and HS respectively of a certain bundle E ---+ M of class Coo with a compact base M. If E is a bundle of Lie groups (i.e. its fibre is a Lie group G and the structure group is the automorphism group Aut G), then Ck(E) and HS(E) (s > ~dimM) are Banach (respectively Hilbert) Lie groups.


Let us note the following important special case. Let P --> M be a principal bundle of class Coo with structure group G. One can then construct a bundle of Lie groups E --> M with fibre G. As E one takes the bundle associated with P corresponding to the action of G on itself by inner automorphisms (see 3.2 of Chap. 1 in Part II). Sections of the bundle E can be viewed as automorphisms of the bundle P, taking every fibre to itself; they are known as gauge transformations. As a result we obtain Lie groups of gauge transformations Ck(E) and Hk(E) (s > ~dimM. In applications it is usual to consider the natural affine action of the group of gauge transformations on the space of connections of the bundle P.

3. Let M be a compact real COO-manifold. Then the group Diffk M of all diffeomorphisms of class Ck (k ~ 0) is open in Ck(M, M) and, therefore, possesses the structure of a real Banach COO-manifold. However, the multiplication Diffk M x Diffk M --> Diffk M is only a continuous but not a differentiable mapping. Thus Diffk M is not a Banach Lie group (in connection with this see 3.3).

4. An algebra A over a complete valued field K is called a Banach algebra if A is a Banach space over K, whose norm is connected with the multiplication in A by the condition

IlxY11 ~ Ilxllllyll (x, YEA). If A is an associative Banach algebra with unit over K, then the set A x of its invertible elements is a Banach Lie group over K. In particular the group GL (H) of all invertible continuous operators of any Banach space H is a Banach Lie group.

3.2. The Correspondence Between Banach Lie Groups and Banach Lie Algebras (see Bourbaki 1971, Hamilton 1982, Dynkin 1950). Let G be a Banach Lie group over K. The tangent space 9 = Te(G) is a Banach space over K. As in the finite-dimensional case, one can introduce in the space 9 an operation [ , 1 which turns it into a Lie algebra. This Lie algebra is a Banach algebra. If char K = 0, then to every Banach Lie algebra over K there corresponds a local Banach Lie group CH(g) which is defined in a neighbourhood of the point 0 with the help of the Campbell-Hausdorff series.

Theorem 3.1. Let char K = O. The correspondences G f--+ 9 and 9 f--+ CH(g) defined above give an equivalence between the categories of local Banach Lie groups and Banach Lie algebras over K.

If the field K is non-archimedean, then, as in the finite-dimensional case, every Banach Lie algebra over K turns out to be the tangent algebra of some global Banach Lie group. In the archimedean case this statement is false, as is shown by the following

Example (Est 1964). Let G = SU2 X SU2 • Consider the real Banach Lie group O(G) = C1(Sl,G) (see Example 1 of 3.1). Its tangent algebra is the Banach Lie algebra O(g) = Cl(Sl, g), where 9 = SU2 EB SU2' Clearly, O(g) =

n(SU2) EB n(SU2)' Consider the 2-cocycle of the Lie algebra n(SU2) with values in lR given by the formula

ZO(¢, 1/1) = 121' tr (¢'(t)1/1(t)) dt,

where we interpret ¢ and 1/1 as functions of class C1 on lR with period 21l'. Then the bilinear function z on n(g) given by the formula

Z(¢1 EB ¢2, 1/11 EB 1/12) = ZO(¢l, 1/11) + (}ZO(¢2, 1/12)) (¢i,1/1i E n(SU2)),

where () is a fixed real number, is a 2-cocycle of the algebra n(g). It determines a central extension

o ~ lR ~ O(g) ~ n(g) ~ 0

(see Vol. 21 of this series). It is easy to see that O(g) is a Banach Lie algebra over R It turns out that there exists a Banach Lie group with tangent algebra O(g) if and only if () E Q. Thus, if () is irrational, then the local Banach Lie group CH(O(g)) is not a germ of any global Lie group.

We know certain conditions which are sufficient for the existence of global Lie groups with a given tangent algebra. In particular we have the following

Theorem 3.2 (see de la Harpe 1972). If 9 is a Banach Lie algebra with trivial center, then there exists a Banach Lie group with tangent algebra g.

As in the finite-dimensional case we can define the exponential mapping exp : 9 ~ G. We note the following result, which is of interest also in the finite-dimensional case.

Theorem 3.3 (Lazard and Tits 1965/66). Let G be a simply connected Banach Lie group over lR or C, 9 its tangent algebra, IIII a norm turning 9 into a Banach Lie algebra. Then the map exp : 9 ~ G is injective and regular in the open ball {x E 9 Illxll < 21l'}.

3.3. Actions of Banach Lie Groups on Finite-Dimensional Manifolds. The following results, obtained in (Omori and de la Harpe 1972), show that infinite-dimensional real Banach Lie groups only in rare cases can act nontrivially on finite-dimensional manifolds (all actions are assumed to be differentiable) .

Theorem 3.4. If a connected real Banach Lie group G acts effectively and primitively on a finite-dimensional manifold, then dim G < 00 (For the concept of primitivity of an action see 1.4 of Chap. 2 in Part II.)

Theorem 3.5. If the tangent algebra 9 of a real Banach Lie group G does not contain proper closed ideals of finite codimension, then 9 does not contain also proper closed subalgebras of finite codimension. In particular, in this case any action of the group G on a finite-dimensional manifold is trivial.

One can verify that for the group GL(H), where H is a infinite-dimensional real Banach space, the condition of Theorem 3.4 is satisfied. Hence we have


Corollary. If H is an infinite-dimensional real Banach space, then the group GL(H) does not admit a non-trivial action on a finite-dimensional differentiable manifold.

3.4. Lie-Frechet Groups (see Hamilton 1982). Let us recall that by a Frechet space one means a complete Hausdorff locally convex topological vector space whose topology is given by a family of semi-norms (this last condition is equivalent to metrizability). We shall assume that the basic field is lR. - the field of real numbers. Generalizing the classical notion of directional derivative one can develop differential calculus for functions with values in Fnkhet spaces, defined in an open neighbourhood of another Frechet space. In an obvious way one defines differentiable (of class COO) Frechet manifolds and differentiable mappings between them, tangent spaces to Frechet manifolds, differentials of mappings etc. (see Hamilton 1982).

By a Lie-Frechet group we mean a Frechet manifold G equipped with a group structure such that the multiplication (g, h) ~ gh and the inversion mapping 9 ~ g-1 are differentiable. The tangent space Te(G) at a point e of a Lie-Frechet group G can be made into a Lie algebra, for example by using the natural isomorphism between Te(G) and the space of right-invariant vector fields on G, which is a Lie algebra with respect to the operation of the commutator (or Lie bracket). (cf. 1.2 of Chap. 2). This Lie algebra 9 is called the tangent algebra of the group G.

Examples. 1. Let M and N be real Coo-manifolds, with M compact. Then, if we define charts analogously to Example 1 of 3.1, the set COO(M, N) of all Coo-mappings M ---+ N becomes a Frechet manifold. Another Frechet manifold is the set Coo (E) of all Coo -sections of a differentiable fibre bundle E with a compact base. If M is compact and G is a (finite-dimensional) real Lie group, then the group COO(M, G) is again a Lie-Frechet group. Its tangent algebra is the Lie algebra COO(M, g), where 9 is the tangent algebra of the group G. The group of sections COO(E) of any differentiable fibre bundle of Lie groups with a compact base (see Example 2 of 3.1) is a Lie-Frechet group. Its tangent algebra is the Lie algebra Coo ( e) of sections of the corresponding fibre bundle e of Lie algebras. To every principal fibre bundle of class Coo there is associated the group of gauge transformations of class Coo, which is a Lie-Frechet group.

2. Let M be a compact differentiable manifold. Then the group Diff M of all of its diffeomorphisms possesses a natural structure of a Lie-Frechet group (see Leslie 1967, Hamilton 1982). The tangent algebra of this group is the Lie algebra V(M) of all vector fields (of class COO) on M with the bracket operation.

The difficulty of studying Lie-Frechet groups is connected, in part, with the fact that the inverse function theorem, which is well known for Banach spaces and manifolds, does not extend to the case of differentiable mappings of Frechet manifolds. The following example shows that this theorem is not valid for the "exponential mapping" related to the group Diff M of Example 2.


3. (see Hamilton 1982). Let M ba a compact differentiable manifold. Every vector field v E V(M) generates a flow o(t) = exptv on M with velocity v (see 2.1 of Chap. 1 in Part II). The map exp: v I---t expv = 0(1) ofthe space V(M) into Diff M is differentiable and do exp = id. However, its image exp V(M) does not, in general, contain a neighbourhood of the identity diffeomorphism e in Diff M. Indeed, let for example, M = Sl. Let a diffeomorphism f : Sl -+ Sl have the form f = expv, where v E V(Sl). If f does not have fixed points, then vex) =I- 0 for all x E Sl. Therefore v can be taken, by a diffeomorphism of the circle, into a rotation invariant vector field and f is conjugate to a rotation in the group Diff M. At the same time, it is easy to construct diffeomorphisms of the circle, without fixed points and arbitrarily close to the identity diffeomorphism, but not conjugate to a rotation.

Several years ago a new category of Frechet manifolds and differentiable mappings was defined, in which a weaker version of the inverse function theorem holds. These manifolds, the Frechet spaces on which they are modelled and the admissible mappings are called tame. Not entering into details (see Hamilton 1982) we only observe that in a tame Frechet space F, besides topology, we also fix some family of semi-norms II Ilk (k = 0,1,2, ... ) which detetermine it, such that Ilxo II :::; IIx111 :::; IIx211 :::; ... for any x E F and a tame mapping f, beginning from a certain n, satisfies the condition Ilf(x)lln :::; cn (1+ Ilxlln+r ) for a certain r ~ O. A Lie-Frechet group G is called tame if G is a tame manifold and the mappings (g, h) I---t gh and g I---t g-l are tame. It turns out that all manifolds and Lie-Frechet groups considered above in Examples 1, 2 are tame. We note certain results on diffeomorphism groups obtained by this method (Hamilton 1982).

Let M be a compact differentiable manifold, D(M) a tame Frechet manifold consisting of all positive densities I-" of class COO on M, satisfying the condition J I-" = 1. The group Diff M admits a natural differentiable action

M on D(M) for which the stabilizer of a point I-" E D(M) is the group Diff/L(M) of diffeomorphisms preserving the density I-" (or the corresponding measure on M).

Theorem 3.6. The group Diff M acts transitively on D(M). For any point I-" E D(M) the subgroup Diff/LM is a tame Lie subgroup of Diff M and the mapping ¢ I---t ¢(I-") (¢ E Diff M) is a projection of a tame differentiable principal fibre bundle with base D(M) and structure group Diff/LM.

It is also known that the group (Diff M)O acts transitively on the connected components of the manifold of symplectic structures and the manifold of contact structures on a compact manifold M, however, it seems to be unknown if the stabilizers of these actions are Lie subgroups of Diff M. Lie subgroups of Lie-Frechet groups are also discussed in (Leslie 1992).

3.5. ILB- and ILH-Lie Groups. Attempts to create a generalization of the theory of Lie groups to include infinite-dimensional groups of automor-


phisms of various geometric structures on manifolds have lead to the following concept, to the introduction and study of which is devoted the book (Omori 1974). Suppose there is given a sequence of Banach (or Hilbert) spaces E S (s = d, d + 1, ... ), with Es+1 linearly and densely embedded in E S, and let the subspace lE = n ES be given the topology of a projective limit. A

s?d

topological group1G is called a (strong) ILB- (respectively ILH-) Lie group, if there exists a sequence of topological groups GS (s 2: d), satisfying the following conditions: GS is a Coo-manifold modelled on ES; Gs+1 is a dense subgroup of GS and the inclusion is of class Coo, G = nGs coincides (as a topological group) with the projective limit of the inverse system of topological groups G8; the multiplication G x G -+ G and the inversion mapping G -+ G extend to maps Gs+l x GS -+ GS and Gs+l -+ GS of class Cl; right translations in GS are transformations of class Coo and induce a mapping of class C l of the manifold Te (Gs+1) x GS into the tangent bundle of GS; local charts in suitable neighbourhoods of the points e in all the groups GS can be consistently given by means of a local chart on Gd with values in Te(Gd ).

In a neighbourhood of the identity of a ILB- (or ILH-) Lie group there exists a chart with values in the Frechet space 9 = n Te (GS) ~ lE, turning

s?d G into a Lie-Frechet group. In the space 9 one can define an operation turning it into a topological Lie algebra known as the tangent algebra of the group G.

It turns out that in the case we are considering there does hold a certain generalization of Theorem 2.10 of Chap. 2. From this follows

Theorem 3.7 (Omori 1974). If the tangent algebras of simply connected ILB-groups G and H are topologically isomorphic, then G and H are isomorphic as topological groups.

In (Omori 1974) it is also shown that, for any compact COO-manifold M, the group Diff M possesses natural structures of an ILB-Lie group and an ILH-Lie group. Further, the following Lie subgroups of Diff M possess the structure of an ILB-group:

the subgroup of all diffeomorphisms preserving some differentiable fibre bundle whose total space is the manifold M;

the subgroup of all diffeomorphisms preserving a given closed submanifold; the subgroup of all diffeomorphisms, preserving the measure determined

by a Riemannian metric, a symplectic or a contact structure on M.

§4. Lie Groups and Topological Groups

A topological group is a group G with a topology such that the multiplication (g, h) f-+ gh and the inversion mapping g f-+ g-1 are continuous mappings. Clearly, any Lie group (real or complex) is a topological group.

1 For more on this concept see §4.


The same applies to Lie groups over normed fields, to Banach Lie groups and to Lie-Frechet groups. At the same time, it is easy to give examples of topological groups which are not manifolds in any sense and therefore do not admit the structure of a Lie group.

In this section we shall briefly consider the following question: what kind of place do Lie groups occupy among topological groups? Many years of research in this direction were stimulated by Hilbert's 5-th Problem, which was solved in the 50's of our century (for a survey of results connected with this problem see Sklyarenko 1969 and Yang 1976).

4.1. Continuous Homomorphisms of Lie Groups. The following theorem shows that real and p-adic Lie groups form full subcategories of the category of topological groups.

Theorem 4.1. Let f : G ---+ H be a continuous homomorphism of real or p-adic Lie groups G, H. Then f is analytic.

Proof. Consider the graph r = {(g, h) E G x Hlh = f(g)} of the homomorphism f. Clearly r is a closed subgroup of G x H. By (Cartan's) Theorem 3.6 of Chap. 2 and by Theorem 1.5 r is a Lie subgroup of G x H. The corresponding Lie sub algebra is the graph of the homomorphism df : 9 ---+ ~.

Hence the projection (g, h) I---t 9 defines an isomorphism r ---+ G. From this our statement easily follows. 0

Corollary. A real (or p-adic) Lie group G possesses a unique structure of an analytic manifold over lR (respectively !Qp), with respect to which the multiplication in G is analytic.

Proof. Apply Theorem 4.1 to the identity isomorphism from G to itself equipped with another analytic structure. 0

Thus real and p-adic Lie groups can be viewed as topological groups of a special type.

4.2. Hilbert's 5-th Problem. In Hilbert's famous lecture "Mathematical Problems" the following question was proposed. Let us suppose that in the definition of a finite continuous group of transformations of lRn given by S. Lie (see 1.3 of Chap. 1 in Part II) we replace the requirement of differentiability or analiticity of functions by the requirement of their continuity. Can one introduce new (local) coordinates in lRn and new local parameters in the group so that the functions defining the group of transformations become differentiable or analytic? Can this be done under some additional assumptions?

Later, from this Hilbert's 5-th problem emerged the following problem, which amounts precisely to the question of characterizing Lie groups in the class of all topological groups: is every locally Euclidean group (i.e. a topological group which is a topological manifold) a Lie. group? This question was answered positively in 1952.


Theorem 4.2. Every locally Euclidean topological group admits the structure of a differentiable manifold, with respect to which it is a real Lie group.

The proof of this theorem was given by Gleason, Montgomery and Zippin in (Gleason 1952) and (Montgomery and Zippin 1952); their methods were perfected in (Yamabe 1953a), (Yamabe 1953b) and (Kaplansky 1971) (for exposition see also (Montgomery and Zippin 1955) and (Glushkov 1957)). Along with it was obtained also the following characterization of Lie groups: a topological group G is a Lie group if and only if it is locally compact and does not contain small subgroups (i.e. there exists a neighbourhood of the identity e in G which does not contain subgroups distinct from {e}).

L. S. Pontryagin (see Pontryagin 1984) has proved that a compact topological group is a Lie group if and only if it is finite-dimensional and locally connected.

The general Hilbert problem concerning groups of transformations can be (in a global form) formulated as follows: is every continuous action of a locally Euclidean topological group on a topological manifold M differentiable with respect to some differentiable structure on M? The answer to this question, negative in general, turns out to be positive for transitive and effective actions. There holds a more general

Theorem 4.3 (Montgomery and Zippin 1955). If a locally compact topological group G acts transitively and effectively on a finite-dimensional compact and locally connected topological space X, then G possesses a Lie group structure and X a structure of a real analytic manifold such that the action is analytic.

§5. Analytic Loops

The subject of this section will be non-associative generalizations of Lie groups first considered by A.1. Maltsev (1976).

5.1. Basic Definitions and Examples (see Bruck 1958). A set G together with a multiplication operation (a, b) t--+ ab on it is called a loop if the following conditions are satisfied:

a) there exists an element e E G (the unit of the loop G) such that ea = ae = a for all a E G;

b) for any a, bEG each of the equations ax = band ya = b has a unique solution.

Condition b) makes it possible to introduce in a loop G the operations of left division (a, b) t--+ a\b and right division (a,b) t--+ alb, possessing the following properties

(a/b) b = (ab)/b = b(b\a) = b\(ba) = a.

A loop G is called a Bol loop if it satisfies the condition


c) a (b (ac)) = (a (ba)) c for all a, b, c E G, and a Moufang loop if c) is satisfied together with the condition

d) ((ca) b) a = ((ab)a)c for all a, b, c E G.

89

A loop G is a Moufang loop if and only if it satisfies any of the following three identities:

a(b(ac)) = ((ab)a) C; ((ca)b)a = c(a(ba)); (ab)(ca) = a((bc) a).

A loop is called monoassociative if each of its elements generates an associative subloop, and diassociative (or alternative) if every pair of its elements generates an associative subloop. In particular, any Bol loop is monoassociative. Moreover, a Bol loop is diassociative if and only if it is a Moufang loop.

An analytic loop is an analytic manifold (over K = IR or q with a loop structure such that the multiplication operation is analytic. There is a natural definition of a local analytic loop.

Examples. 1. If ((]) is the Cayley algebra (see Encycl. Math. Sc. 11, §19), then the set ((]) \ {O} with the operation of multiplication of Cayley numbers is a diassociative analytic loop.

2. Let M be an analytic manifold with a given analytic linear connection, e E M a fixed point. We shall denote by EXPa the exponential mapping at the point a E M which corresponds to the given connection (see Helgason 1964). In a neighbourhood of the point e in M we define a multiplication by the formula

where Te,a denotes parallel translation of tangent vectors along a geodesic segment from the point e to a. It turns out that this multiplication defines the structure of a local analytic loop in a neighbourhood of the point e in Mj it is called the geodesic loop of the given connection. If the connection is locally symmetric, then the geodesic loop is a Bol loop. Every local analytic Bol loop is a geodesic loop of some linear connection (Sabinin and Mikheev 1985).

5.2. The Tangent Algebra of an Analytic Loop. Let G be a local analytic loop and let Xl, ... ,Xn be a local coordinate system on G in a neighbourhood of the identity e = (0, ... ,0). Let

be functions, analytic at the point (0,0), which express the coordinates of the product z = xy through the coordinates of the factors. In view of condition 2) in the definition of a loop, the functions /Li, just as in the case of a Lie

group, have the property () .{)r/LJ . I = () .{)rf'J . I = o. Hence the X'I.} ••• X1.r Ytl'" Y1.r

(0,0) (0,0) Taylor series of the function /Li at (0,0) has the following form:


j,k j,k,l

+ L C~klXjYkYl + ... (1) j,k,l

where b;kl = bLl' C;kl = 4j l and the dots stand in place of terms of degree ~ 4. Let us recall that a local Lie group is determined, up to isomorphism, by terms of degree 2 in formula (1). For arbitrary loops this property, naturally, does not hold. However, local Bolloops, as we shall see later, are completely determined by terms of degree 2 and 3 in expansion (1).

Let 9 = Te (G). We introduce in 9 a binary and a ternary operations [~,1]] and (~,1],() as follows. Let a(t),j3(t),"((t) be differentiable paths in G satisfying the conditions a(O) = 13(0) = "((0) = e,a'(O) = ~,j3'(O) = 1],"('(0) = (. Then

(j3(t)a(t)) \ (a(t)j3(t)) = t2[~, 1]] + 0(t2),

(a(t)(j3(th(t))) \ ((a(t)j3(t)h(t)) = t3(~, 1], () + 0(t3).

If the vectors ~,1], ( E G are given in terms of their coordinates, with respect to the local coordinate system in G chosen above, then

[~, 1]]i = L U~k~j1]k' j,k

where

(~, 1], ()i = L V}kl~j1]k(l' j,k,!

Thus the operations we have defined are linear with respect to each argument. By the tangent algebra of a local analytic loop G we shall mean the binaryternary algebra 9 with the operations [ , land ( , , ). We note that if G is a local Lie group, then (~, 1], () = 0 for all ~,1], ( E g. Moreover, in the general case

[~,~] = 0 (~E g). (2)

Example (Akivis 1978, see also Sabinin and Mikheev 1985). Let G be a geodesic loop of some linear connection (see Example 2 of 5.1). Then the structure constants U;k and V}kl can be expressed through the curvature tensor R and the torsion tensor T of the connection as follows:

. 1. ujk = -"2Tjk (e),

V}kl = ~(RLk(e) - 'VjT~l(e)).

5.3. The Tangent Algebra of a Diassociative Loop. For diassociative local analytic loops the ternary operation in the tangent algebra can be expressed


in terms of the binary one (see formula (4) below). Therefore in this subsection the tangent algebra will be considered as an ordinary algebra with one multiplication operation [ , l. An algebra is said to be binarily Lie if any two elements of it generate a subalgebra, which is a Lie algebra.

Theorem 5.1. If G is a diassociative local analytic loop over the field k = IR or C, then its tangent algebra 9 is binarily Lie. Moreover, any finitedimensional binarily Lie algebra over IR is isomorphic to the tangent algebra of a unique (up to isomorphism) local analytic diassociative loop.

Note that the theorem on the existence and uniqueness of one-parameter subgroups with a given tangent vector (see 3.1 of Chap. 2) extends to the case of arbitrary monoassociative analytic loops, which makes it possible to define for such a loop G the exponential mapping 9 ~ G and canonical coordinates in a neighbourhood of the identity (Kuz'min 1971). If G is diassociative, then the multiplication in G can be expressed in canonical coordinates through the operation [ , 1 in the algebra 9 with the help of the Campbell-Hausdorff formula (see Maltsev 1976).

Let A be some algebra with a bracket operation [ , l. We set

J(e,"7,() = [[e,"7l,Cl + [[C,el,"7l + [["7,C],el (e,"7,C E A).

The algebra A is called a Maltsev algebra (or a Moufang-Lie algebra, see (Maltsev 1976)), if condition (2) and the condition

J(e, "7, [e, C]) = [J(e, "7, C), el· (3)

are satisfied. Condition (3) can be replaced by the following:

[[e, "7l, [e, Cll = [[[e, "7l, Cl, el + [[["7, Cl, el, el + [[[C, el, el, "7l·

For example, we obtain a Maltsev algebra from any alternative algebra with a multiplication (e,,,.,) f--+ e"., by defining the bracket by [e,,,.,1 = e"., - ".,e. For a survey of results on Maltsev algebras see (Kuz'min 1968).

Theorem 5.2 (Maltsev 1955, Kuz'min 1971). A locally analytic loop is a Moufang loop if and only if its tangent algebra is a Maltsev algebra.

Observe that the ternary operation ( , , ) in the tangent algebra of a diassociative loop can be expressed in terms of the binary form by means of the formula

(4)

5.4. The Tangent Algebra of a Bol Loop. A binary-ternary algebra A with operations [ , land ( , , ) is called a Bol algebra if the following conditions are satisfied:

(e,e,"7) =0;

J(e, "7, C) = 2«(e" "7, C) + ("7,C,e) + (C,e,"7)); [(e,,,.,, C), xl- [(e, "7, X), Cl + ([e, "7l, x, C) - ([e, "7l, c, X) + (e, "7, [X, C]) = 0


(A,X, (~,1],()) = ((A,X,~),1],() + (~,(A,X,1]),() + (~,1],(A,X,()), where (~,1],() = -2(~,1],() + [[~,1]],(j.

Theorem 5.3 (Akivis 1976, Sabinin and Mikheev 1985). The binary-ternary tangent algebra of a local analytic Bol loop is a Bol algebra. Every finitedimensional Bol algebra is isomorphic to the tangent algebra of a unique (up to isomorphism) local analytic Bol loop.

Observe that in canonical coordinates the multiplication in an analytic Bol loop is given by the following formula, in which the dots stand in place of terms of degree 2 4:

111 xy = x + y + 2[x, yj + 12 [[x, yj, yj + 12 [x, [x, y]]-

2 1 - 3(x,y,y) + 3(y,x,x) + ...

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II. Lie Transformation Groups

Introduction

V. V. Gorbatsevich, A. L. Onishchik

Translated from the Russian by A. Kozlowski

Contents

99

Chapter 1. Lie Group Actions on Manifolds 100

§1. Introductory Concepts ....... 100 1.1 Basic Definitions ........ 100 1.2 Some Examples and Special Cases 102 1.3 Local Actions ......... 103 1.4 Orbits and Stabilizers ..... 105 1.5 Representation in the Space of Functions 106

§2. Infinitesimal Study of Actions . . . . . . . . 108 2.1 Flows and Vector Fields ........ 108 2.2 Infinitesimal Description of Actions and Morphisms 111 2.3 Existence Theorems ............... 113 2.4 Groups of Automorphisms of Certain Geometric Structures 114

§3. Fibre Bundles ............. 115 3.1 Fibre Bundles with a Structure Group 115 3.2 Examples of Fibre Bundles ..... 116 3.3 G-bundles ............. 118 3.4 Induced Bundles and the Classification Theorem 119

Chapter 2. Transitive Actions . . 121

§1. Group Models ....... 121 1.1 Definitions and Examples 121 1.2 Basic Problems . . . . . 122

96 V. V. Gorbatsevich, A. L. Onishchik

1.3 The Group of Automorphisms ........... 123 1.4 Primitive Actions . . . . .. ........... 124

§2. Some Facts Concerning Topology of Homogeneous Spaces 125 2.1 Covering Spaces .................. 125 2.2 Real Cohomology of Lie Groups . . . . . . . . . . . 126 2.3 Subgroups with Maximal Exponent in Simple Lie Groups 127 2.4 Some Homotopy Invariants of Homogeneous Spaces 128

§3. Homogeneous Bundles ................ 129 3.1 Invariant Sections and Classification of Homogeneous Bundles 130 3.2 Homogeneous Vector Bundles. The Frobenius Duality 131 3.3 The Linear Isotropy Representation and Invariant Vector

Fields ....... 132 3.4 Invariant A-structures 132 3.5 Invariant Integration 134 3.6 Karpelevich-Mostow Bundles 136

§4. Inclusions Among Transitive Actions 138 4.1 Reductions of Transitive Actions and Factorization of Groups 138 4.2 The Natural Enlargement of an Action ....... 139 4.3 Some Inclusions Among Transitive Actions on Spheres 140 4.4 Factorizations of Lie Groups and Lie Algebras . . . . 141 4.5 Factorizations of Compact Lie Groups . . . . . . . . 143 4.6 Compact Enlargements of Transitive Actions of Simple

Lie Groups . . . . . . . . . . . . . . . . . . . . . 145 4.7 Groups of Isometries of Riemannian Homogeneous Spaces

of Simple Compact Lie Groups ............. 146 4.8 Groups of Automorphisms of Simply Connected Homogeneous

Compact Complex Manifolds . . . 147

Chapter 3. Actions of Compact Lie Groups 149

§1. The General Theory of Compact Lie Transformation Groups 149 1.1 Proper Actions . . . . . . . . . . . . . 149 1.2 Existence of Slices ........... 150 1.3 Two Fiberings of an Equi-orbital G-space 151 1.4 Principal Orbits 152 1.5 Orbit Structure . . . . 153 1.6 Linearization of Actions 154 1. 7 Lifting of Actions . . . 155

§2. Invariants and Almost-Invariants 156 2.1 Applications of Invariant Integration 156 2.2 Finiteness Theorems for Invariants 156 2.3 Finiteness Theorems for Almost Invariants 158

§3. Applications to Homogeneous Spaces of Reductive Groups 158 3.1 Complexification of Homogeneous Spaces ...... 158 3.2 Factorization of Reductive Algebraic Groups and Lie Algebras 159

II. Lie Transformation Groups 97

Chapter 4. Homogeneous Spaces of Nilpotent and Solvable Groups 160

§1. Nilmanifolds ............. 161 1.1 Examples of Nilmanifolds . . . . . 161 1.2 Topology of Arbitrary Nilmanifolds 161 1.3 Structure of Compact Nilmanifolds 162 1.4 Compact Nilmanifolds as Towers of Principal Bundles

with Fibre Tl ...... 163 §2. Solvmanifolds ........... 164

2.1 Examples of Solvmanifolds 164 2.2 Solvmanifolds and Vector Bundles 166 2.3 Compact Solvmanifolds (The Structure Theorem) 167 2.4 The Fundamental Group of a Solvmanifold 168 2.5 The Tangent Bundle of a Compact Solvmanifold 169 2.6 Transitive Actions of Lie Groups on Compact Solvmanifolds 169 2.7 The Case of Discrete Stabilizers . . . . . . . . . . . . 170 2.8 Homogeneous Spaces of Solvable Lie Groups of Type (I) 171 2.9 Complex Compact Solvmanifolds ........... 171

Chapter 5. Compact Homogeneous Spaces 172

§1. Uniform Subgroups . . . . . . . 172 1.1 Algebraic Uniform Subgroups 172 1.2 Tits Bundles . . . . . . . . 174 1.3 Uniform Subgroups of Semi-simple Lie Groups 174 1.4 Connected Uniform Subgroups ....... 175 1.5 Reductions of Transitive Actions of Reductive Lie Groups 177

§2. Transitive Actions on Compact Homogeneous Spaces with Finite Fundamental Groups . . . . . . . 178 2.1 Three Lemmas on Transitive Actions ....... 179 2.2 Radical Enlargements . . . . . . . . . . . . . . . 180 2.3 A Sufficient Condition for the Radical to be Abelian 181 2.4 Passage from Compact Groups to Non-Compact

Semi-simple Groups ............... 182 2.5 Compact Homogeneous Spaces of Rank 1 ..... 184 2.6 Transitive Actions of Non-Compact Lie Groups on Spheres 187 2.7 Existence of Maximal and Largest Enlargements . . . 188

§3. The Natural Bundle .................. 189 3.1 Orbits of the Action of a Maximal Compact Subgroup 189 3.2 Construction of the Natural Bundle and Its Properties 190 3.3 Some Examples of Natural Bundles . . . . . . 191 3.4 On the Uniqueness of the Natural Bundle . . . 193 3.5 The Case of Low Dimension of Fibre and Basis 194

§4. The Structure Bundle .............. 195 4.1 Regular Transitive Actions of Lie Groups 195 4.2 The Structure of the Base of the Natural Bundle 196


4.3 Some Examples of Structure Bundles ..... 197 §5. The Fundamental Group ............. 198

5.1 On the Concept of Commensurability of Groups 198 5.2 Embedding of the Fundamental Group in a Lie Group 199 5.3 Solvable and Semi-simple Components 199 5.4 Cohomological Dimension 200 5.5 The Euler Characteristic 201 5.6 The Number of Ends . . 202

§6. Some Classes of Compact Homogeneous Spaces 202 6.1 Three Components of a Compact Homogeneous Space and

the Case when Two of them Are Trivial 203 6.2 The Case of One Trivial Component 203

§7. Aspherical Compact Homogeneous Spaces . 204 7.1 Group Models of Aspherical Compact Homogeneous Spaces 204 7.2 On the Fundamental Group . . . . . . 205

§8. Semi-simple Compact Homogeneous Spaces 206 8.1 Transitivity of a Semi-simple Subgroup 206 8.2 The Fundamental Group ..... 206 8.3 On the Fibre of the Natural Bundle 207

§9. Solvable Compact Homogeneous Spaces 207 9.1 Properties of the Natural Bundle 207 9.2 Elementary Solvable Homogeneous Spaces 208

§10. Compact Homogeneous Spaces with Discrete Stabilizers 209

Chapter 6. Actions of Lie Groups on Low-dimensional Manifolds 210

§1. Classification of Local Actions .............. 210 1.1 Notes on Local Actions . . . . . . . . . . . . . . . . 210 1.2 Classification of Local Actions of Lie Groups on ]R1, «:1 212 1.3 Classification of Local Actions of Lie Groups on ]R2 and «:2 214

§2. Homogeneous Spaces of Dimension::; 3 217 2.1 One-dimensional Homogeneous Spaces 217 2.2 Two-dimensional Homogeneous Spaces

(Homogeneous Surfaces) ....... 218 2.3 Three-dimensional Manifolds ..... 219

§3. Compact Homogeneous Manifolds of Low Dimension 220 3.1 On Four-dimensional Compact Homogeneous Manifolds 220 3.2 Compact Homogeneous Manifolds of Dimension::; 6 . . 221 3.3 On Compact Homogeneous Manifolds of Dimension;::: 7 222

References 223


Introduction

At the end of the last century S. Lie developed the theory of "continuous transformation groups", which gave rise to the field nowadays known as the theory of Lie groups. The work of S. Lie was to a large extent inspired by the idea of constructing the analogue of Galois theory for differential equations, but further development of the theory made clear its close relationship with other areas of mathematics (particularly with geometry) and also with theoretical physics. The authors of the present work do not attempt to give a survey of all the main results of the theory of Lie transformation groups obtained in over a century of its development. In particular almost entirely beyond the scope of this survey remain the geometry and topology of Lie groups and homogeneous spaces and the, closely connected with topology, theory of continuous actions of compact Lie groups. Special attention was paid to the general theory and to transitive actions of Lie groups, in particular, to results on the classification of transitive actions and the structure of homogeneous spaces.

The notation used mostly corresponds to the notation in Part 1. In addition, the following conventions are used:

DPG - the p-th commutator of the group G, i.e. DPG = (DP-1G, DP-1G), DOG = Gj

NG(H) - the normalizer of the subgroup H in the group Gj MG - the set of fixed points of the group G, acting on the set Mj vg - the set of vectors of the space V, annihilated by some linear repre

sentation of the Lie algebra gj A - the universal covering group of the Lie group SL2(JR).


Chapter 1 Lie Group Actions on Manifolds

§l. Introductory Concepts

1.1. Basic Definitions. Actions of Lie groups were already defined in Part I of the present volume (see §2 of Chap. 1). We shall repeat this definition using somewhat different notation, beginning with an action of an abstract group.

By an action of a group G on a set M we mean a map T : G x M --+ M, satisfying the following properties:

T(e, x) = x,

T(a, T(b, x)) = T(ab, x)

(1)

(2)

for any x E M, bEG. Given an action T, to every a E G there corresponds a bijective transformation Ta : x f-t T( a, x) of the set M and the map t : a f-t Ta is a homomorphism of the group G into the group SM of all permutations (bijective transformations) of the set M. Conversely, any homomorphism t : G f-t S M defines an action of G on M by the formula

T(a, x) = t(a)(x) (a E G, x EM).

Usually (when this does not lead to confusion) an action of a group G on M is denoted as multiplication of elements of the group G by elements of M and written

T(a,x) = ax (a E G,x EM).

The group actions defined above are also referred to as left actions. Along with them, one sometimes considers right actions in the definition of which condition (2) is replaced by the condition

T(b, T(a, x)) = T(ab, x). (2')

We shall also make use of the notation T( a, x) = xa, with it (2') takes the form

(xa)b = x(ab) (x E M, a, bEG).

Clearly, every left action (x, a) f-t ax determines a right action (x, a) f-t xa = a-Ix, and conversely, every right action of G on M determines its left action onM.

For any action of a group G on M we shall denote by MG the set {x E

M I gx = x for all 9 E G} of all invariant elements or fixed points of the action. An action T is called trivial if MG = M, i. e. if Tg = id for any 9 E G.

In this study we shall primarily consider the case when G is a (real) Lie group and M a real Coo-manifold. In addition we shall always implicitly assume that T is a Coo-mapping and refer to M together with a given action


as a differentiable G-space. Another important case is that of an analytic G-space, where G is a Lie group, M - a real analytic manifold and the map T is analytic. Further, we shall speak of a topological G-space if G is a topological group, M a topological space and T is a continuous mapping. We mention also the complex analytic (G - a complex Lie group, M - a complex manifold or even a complex space, T holomorphic) and algebro-geometric (G - an algebraic group, M - a variety over some field, T - a polynomial) situation. In each of these cases the transformations Tg are automorphisms of the space M in the relevant category (i.e. diffeomorphisms, homeomorphisms etc.)

By the kernel (or the non-effectivness kernel) of an action T we mean the kernel Ker t of the corresponding homomorphism t : a f--t Ta. The action T is said to be effective if Ker t = {e}. If M is a differentiable (or even topological) G-space of a Lie group G, then its kernel N is a Lie subgroup of G. We say that T is locally effective, if the subgroup N is discrete.

If an action T has kernel N, then the corresponding homomorphism t : G ~ S M determines an injective homomorphism G / N ~ S M, i.e. an effective action of the group G / N. In the differentiable case one can pass to an action of the group G / NO, which turns out to be locally effective.

If an action T is effective, then the homomorphism t identifies G with the subgroup t( G) c S M, i.e. with some group of transformations of the space M. However, even in the general case an action of a group on some set is often referred to as a transformation group of this set. If a Lie group G acts effectively and differentiably on a manifold M, then G can be identified with the subgroup t( G) of the group Diff M of all diffeomorphisms of the manifold M. Differentiable (and analytic) actions which are not necessarily effective are often called Lie transformation groups (in 2.4 we use this term in a narrower sense).

Suppose there are given two actions of the same group G on sets M and N. By a morphism from the first action to the second we mean a map f : M ~ N possessing the property f (gx) = 9 f (x) (g E G, x E M). A morphism is also called an equivariant mapping. By an isomorphism of two actions of a group G we mean an invertible morphism between them. If we fix a group G, then its actions and their morphisms form a category. Analogously one defines the category of differentiable (analytic) actions of a Lie group G, where one requires that morphism be differentiable (analytic).

It is useful to consider a larger category, whose objects are actions of various groups (or Lie groups). Its morphisms are defined as follows. Let an action T of a group G on M and of G' on M' be given. A mapping f : M ~ M' is called a morphism of T into T' if there exists a homomorphism ¢ : G ~ G' such that f(gx) = ¢(g)f(x) for all g E G, x E M. In the case G = G', ¢ = id we obtain the morphisms introduced above (equivariant mappings). The corresponding notions for differentiable and analytic actions are defined in the obvious way.

The invertible morphisms of the just defined category are called similitudes. If f is a similitude of two effective actions, then the corresponding homomorphism of groups ¢ is an isomorphism, uniquely determined by f. Two actions are called isomorphic (similar) if there exists an isomorphism (a similitude) between them. Note that two similar actions T and T' of the same group G = G' need not be isomorphic (see examples in §2 of Chap. 5 and in §1 of Chap. 6). But if there exists a similitude between T and T' such that the corresponding homomorphism ¢ : G --- G is an inner automorphism of G, then T and T' are isomorphic.

Let T' be an action of a group G' on M and ¢ : G --- G' a homomorphism of groups. Define an action T of G on M by Ta = T¢(a) , a E G. Then the identity mapping id : M --- M is an morphism from T to T'. If G is a subgroup of G' and ¢ is the inclusion mapping, then we say that T' is an enlargement of the action T or that T is a reduction of the action T' to the subgroup G. The morphism id is called in this case an enlargement of actions. We also say that there exists an inclusion between the actions T and T' and write T:S T'. An enlargement (or a reduction) is called proper if G =I- G'.

1.2. Some Examples and Special Cases

Example 1. An action of the additive group of real numbers IR on a differentiable manifold M is often referred to as a flow (or a dynamical system with continuous time) on M. This is one of the most fundamental objects of study in the theory of dynamical systems (see Part II of Encycl. Math. Sc. 1 of this series).

Example 2. Let p : G --- GL(V) be a linear representation of a group G on a vector space V over the field IR or <C. Clearly, the formula gx = p(g)x (g E G, x E V) defines an action of the group G on V. These kind of actions and the corresponding G-spaces are called linear. Intertwining operators for two linear representations are morphisms of the corresponding linear actions, but there can also exist non-linear morphisms. The set VG of fixed points of a linear action is a subspace of the space V. An element v E V is called representative or almost invariant if the elements of the form gv (g E G) generate in V a finite-dimensional subspace. Naturally, this notion is interesting only in the case when V is infinite-dimensional.

If G is a Lie group and V is finite-dimensional, then we always assume that the representation p is differentiable (this is equivalent to its continuity or real analyticity).

Example 3. With the help of linear actions we can construct numerous examples of actions of Lie groups on non-linear manifolds. Let M be a differentiable submanifold of a finite-dimensional vector space V, invariant with respect to a representation p : G --- GL(V) of a Lie group G. Then one defines in the obvious way a differentiable action of G on M. A classical example is provided by the action of the orthogonal group On on the sphere sn-l C IRn ,


given by the equation xi + ... + x; = 1. By a linearization of an action of a Lie group G on a manifold M we shall mean a morphism of this action into some linear action of the group G on a finite-dimensional vector space V, which is an embedding of differentiable manifolds M -+ V. It is natural to ask: which actions admit a linearization? For a consideration of this question see 1.4 and 1.5 of Chap. 3.

Example 4. For any group G there are defined the following actions (ana-lytic if G is a Lie group) 0 f the group on itself:

a) L: (g,x) 1---+ Lg(x) = gx (action by left translations) b) R: (g, x) 1---+ Rg(x) = xg- I (action by right translations) Frequently instead of b) the right action R'(g, x) 1---+ xg is considered.

Clearly, Lg · Rh = Rh' Lg for any g, hE G. Hence we can define the following action B of the group G x G on G :

B((g,h),x) = gxh- I (g,h,x E G),

called the action by two sided translations. In addition we have also the action of G on itself by inner automorphisms:

A: (g, x) 1---+ Ag(x) = gxg- I .

The map s : x 1---+ X-I of the group G into itself is an isomorphism between actions a) and b).

Example 5. Let G be a Lie group and H a subgroup. On the set G / H of left cosets of H in G ther~ exists a natural action of G induced by action a) of the previous example, and also denoted by L:

L(g, xH) = Lg(xH) = gxH.

Analogously one defines the action R by right translations on the set H\ G of right cosets.

An action of a group G on a set M is said to be transitive if for any x, y E M one can find 9 E G such that y = gx. In this case it is also said that M is a homogeneous space of the group G. It is easy to see that the actions Land R of G on G / Hand H\ G are transitive.

Let G be a Lie group and H a Lie subgroup. Then on G / Hand H\ G one can canonically introduce the structure of an analytic G-space. In the differentiable category this was done in Part I of the present volume (see Theorem 3.1 of Chap. 1 of Part I). The construction described there can be easily transferred to the analytic case if we take into account the fact that every differentiable Lie group is analytic and that any subgroup of it is an analytic submanifold (see 3.3 of Chap. 3).

1.3. Local Actions. The notion of an action of a Lie group was the starting point of the investigations of S. Lie which lead to the creation of the theory which nowadays bears his name. S. Lie and his immediate successors adopted in their works a local viewpoint, to which one can pass from our definition in the following way.


Let T be an action of a Lie group G on a differentiable (or analytic) manifold M and let V be an open set in M. The action T, generally speaking, does not send V to itself and therefore does not define an action of G on V. However, we can define a mapping T : W -+ V, where W = {(g, x) E G x V I T(g, x) E V} is an open set in G x V containing {e} x V. Moreover, for any fixed point Xo E V, there exists a neighbourhood U of the identity e in G and a neighbourhood V' of the point Xo in V such that T(U x V') C V. We can assume that in U and V there are given local coordinate systems (charts) al, .. . ,ar and Xl, ... ,Xn respectively, with e having all coordinates zero. Then, in U x V', T can be given by formulas of the form

where Yi are the coordinates of the point Y = ax (a E U,x E V'), with ti Coo -(or analytic) functions, satisfying the following conditions which follow from (1) and (2):

(4)

(5)

Here (5) is satisfied for all x E V', a, b E U for which both sides of this identity are defined, and mj are functions expressing in local coordinates the multiplication in G.

S. Lie considered in his works families of local analytic diffeomorphisms of the space IRn (or en), given by formulas (3) and satisfying conditions (4) and (5). In fact what was studied there was local Lie groups (see 1.5 of Chap. 1 of Part I), consisting of local analytic diffeomorphisms and not actions of abstractly defined Lie groups.

The above considerations lead to the following notion. Let G be a Lie group and M a differentiable manifold. By a local action of G on M we mean a differentiable mapping T : W -+ M, where W is an open set in G x M containing {e} x M, such that conditions (1) and (2) are satisfied for all a, bEG, x E M, for which both sides of identity (2) are defined. Note that local actions of G on M are always identified if they coincide on some neighbourhood of the set {e} x M. Actions defined on the whole of G x M are sometimes called global.

A local action T of a Lie group G on a manifold M generates a local action of the same group G on any open subset V c M. This action is called a localization or a restriction of the action T to the set V.

Analogously to the global case (see 1.1) one defines morphisms of local actions of the same Lie group and similitudes of local actions of two different Lie groups G and G'. Here in the definition of similitude instead of an isomorphism of Lie groups one considers a local isomorphism (see 1.5 of Chap. 1 of Part I).


A local action is said to be globalizable if it is similar to a localization of some global action (the latter is known as a globalization of the original local action).

Example 6. Let G be a Lie group, H a virtual Lie subgroup, (see 2.3 of Chap. 2 of Part I), V eGa sufficiently small submanifold of dimension m = dim G - dim H, transversal to H at the point e E V. Then for a sufficiently small neighbourhood V' of the point e in H the mapping (v, h) I-t vh is an analytic isomorphism of the manifold V x V' onto the neighbourhood V" = VV' of e in G (cf. Part I, the proof of Theorem 3.1 in Chap. 1). Set p(vh) = v (v E V, h E V'). Let W C G x V" be a set on which there is defined a localization of the action L of G on itself to the set V". Then the formula Lg (v) = p(gv) (g, v) E W n (G x V)) defines an analytic local action of G on the manifold V, which can be identified with an open set in JR.m • This action may not be globalizable. For example (see Mostow 1950), the action of the group G = SU2 X SU2 associated with the subgroup H = {diag( eit , e- it ), diag( eiat , e- iat ) It E JR.}, where a is a fixed irrational number (here m = 5).

1.4. Orbits and Stabilizers. Let T be an action of a group G on a set M. Then one can define an equivalence relation on M by

x rv y ¢::=} X = gy for some 9 E G.

The equivalence classes are called the orbits of the action. Every point x E M is contained in a unique orbit

G (x) = {gx I 9 E G}.

We shall denote the set of all orbits by MIG. Clearly, G acts transitively on every orbit. The whole action is transitive

if it has only one orbit. For a fixed point x E M consider the mapping T X : G -+ M given by the

formula TX(g) = gx. (6)

Clearly, TX maps G onto the orbit G(x) and is a morphism of the action of G on itself by left translations into the action T. The set

Gx = (Tx)-l(X) = {g E G I gx = x}

is a subgroup of G called the stabilizer (or the isotropy group) of the point x. Clearly, TX(gh) = TX(g) for all h E Gx. Hence we can define a map fx : GIGx -+ M such that

fX(gG x ) = TX(g) = gx.

It is easy to prove

Theorem 1.1. Let T be an action of a group G on a set M. For any point x E M the map fx : GIGx -+ M maps GIGx bijectively onto the orbit G(x) and is a morphism of the action L into the action T.


An analogous theorem holds in the category of differentiable G-spaces. Let T be an action of a Lie group G on a differentiable manifold M. According to Theorem 2.1 of Chap. 1 of Part I, for any x E M the mapping TX is differentiable and has constant rank, Gx is a Lie subgroup of G and its tangent space Te(Gx ) at the point e coincides with the kernel of the mapping dxTx : G/Gx --+ Tx(M). From this and from Theorem 1.1 follows

Theorem 1.2. Let T be an action of a Lie group G on a differentiable manifold M. Then for any x E M the mapping TX : G/Gx --+ M is an injective immersion. Thus, the orbit G(x) is an immersed submanifold of M of dimension dimG - dimGx , where Tx(G(x)) = deTX(Te(G)).

If the action T is transitive, then TX : G / G x --+ M is a diffeomorphism of manifolds (see Theorem 3.3 of Chap. 1 in Part I). Thus every differentiable homogeneous space M of a Lie group G is isomorphic to a G-space G / H, where H = G x is the stabilizer of an arbitrary point x E M. In particular, M is an analytic G-space.

An action of a group G on M is said to be free if Gx = {e} for any point x E M. An action which is at the same time transitive and free is called simply transitive.

Corollary. Any simply transitive action of a Lie group G is isomorphic to the action of the group G on itself by left translations (see Example 4 of 1.2).

Suppose we are again given an arbitrary action of a group G on M. It is easy to see that

Ggx = gGxg- 1 (g E G, x EM).

Thus the stabilizers of two points of the same orbit are conjugate in G. We say that two orbits G(x) and G(y) (x,y E M) have the same orbit type if the subgroups Gx and Gy are conjugate in G. The relation on the set of orbits M / G defined in this way is an equivalence relation. The simplest examples of actions with a single orbit type are transitive and free actions.

For any point x E M the stabilizer Gx admits a linear representation t on the tangent space Tx (M) given by the formula

The representation t is called the linear isotropy representation and its image the linear isotropy group at the point x.

Let f be a morphism of a G-space M into a G' space M' and ¢ : G --+ G' the corresponding group homomorphism. Then ¢(Gx ) C Gf(x} for any x E

M. If f is injective, then Gx = ¢-l(Gf(x})'

1.5. Representation in the Space of Functions. We shall denote by F(M) the algebra of all real CCX>-functions on a differentiable manifold M. Every differentiable mapping of manifolds f : M --+ N determines an algebra homomorphism f* : F(N) --+ F(M), operating according to the formula

/*¢>=¢>ofj

and f is fully determined by the homomorphism !*. For two differentiable mappings f : M -7 Nand 9 : P -7 M we have

(f 0 g)* = g* 0/* With every action T of a Lie group G on a manifold M we can associate a linear representation PT of the group G on the space of functions F(M) given by the formula

PT(g) = T;-l The operators of this representation are automorphisms of the algebra F(M). The representation PT uniquely determines the action T of G on M. Thus, studying of the action T can be, in principle, replaced by studying of the linear action PT.

The set F(M)G of functions which are invariant under PT is a subalgebra of F(M). Clearly, a function ¢> E F(M) is invariant if and only if it is constant on every orbit of the action of G on M. In particular, any invariant function ¢> E F(M)G has the form ¢> = 7r*'ljJ where 'ljJ is a function on MIG and 7r :

M -7 MIG is the natural mapping. If G acts transitively, then F(M)G = lR., however the converse assertion is not true. For example, the standard linear action of the group GLn(lR.) on lR.n (n ~ 1) has two orbits lR.n \ {O} and {O}, however any continuous function on lR.n invariant with respect to this action is constant. The representative functions on M, i.e. the almost invariant elements (see Example 2 of 1.2) of the G-space F(M) form a subalgebra F(M)G of the algebra F(M).

Example 7. Consider the action of a Lie group G on itself by left translations. The corresponding algebra of representative functions F(M)G coincides with the linear envelope of the matrix elements of all finite-dimensional linear re presentations of the group G. The same algebra of representative functions arises from the action B of G x G on G by two sided translations (see Example 4 of 1.2).

Representative functions on a manifold M are closely connected with morphisms of the action of G on M into linear representations. Let Fj M -7 N be a morphism of two actions of a Lie group G. Then the homomorphism f* : F(N) -7 F(M) is an intertwining operator for the representation PT of G on F(N) and F(M), whence it follows that !*(F(N)G) C F(M)G. In particular, if N is a finite-dimensional vector space with a linear action of G, then for any polynomial function ¢> E F(N) we have f*¢> E F(M)G. Moreover, if V C F(M)G is a finite-dimensional subspace invariant under PT, then the mapping f : M -7 V* given by the formula

f(x)(¢» = </J(x) (x E M,</J E V)

is a morphism of actions of the Lie group G, where in V* we are considering the representation conjugate to the reduction of the representation PT to V. These considerations lead to the following assertion (see Onishchik 1976).

Lemma 1.1. If an action of a Lie group G on a manifold M is linearizable, then the subalgebra F(M)c is dense in the algebra F(M) with the Coo_ topology. For compact manifolds M the converse holds.

As we shall see in Chap. 3, for compact Lie groups G the subalgebra F(M)c is always dense in F(M). Now we shall show that, for a compact manifold M and a group G with an effective action on M, denseness of the subalgebra of representative functions implies that the group is very nearly compact.

We shall call a linear action p of a Lie group G on a finite-dimensional vector space V (over JR. or q compact if the closure p( G) of its image in the group GL(V) is compact. This is equivalent to the existence in V of a G-invariant inner (or hermitian inner) product.

Let us now denote by Nc the normal Lie subgroup of G given by the intersection of all kernels of all possible compact linear actions of G.

Lemma 1.2. Let p be an action of of a Lie group G on a manifold M which is a differentiable G-space. Then the natural action PT of G on any finite-dimensional G-invariant sub-space in F(M) is compact. In particular, F(M)c C F(M)NG.

Proof. We give the algebra F(M) a norm by the formula 11<1>11 = max 1(x) I· "'EM

It is easy to see that it is invariant under PT. In any invariant finite-dimen-sional subspace V c F(M) the representation PT induces a homomorphism of the group G into the compact Lie group of linear transformations of the space V preserving the norm II II. 0

Corollary. Suppose that under the assumptions of Lemma 1.2 the representative functions separate the points of the manifold M. Then the subgroup Nc is contained in the kernel of the action. If the group G is connected and its action is locally effective, then G 3:! K x JR.m, where K is a compact Lie group.

§2. Infinitesimal Study of Actions

In this section we shall give, in contemporary form, an account of the main ideas of S. Lie, who proposed (Lie and Engel 1888, 1890, 1893, see Hermann 1975) to study continuous groups of transformations with the help of corresponding infinitesimal transformations, i.e. vector fields. Lie algebras first appeared in this circle of ideas.

2.1. Flows and Vector Fields. Let M be a differentiable manifold. We shall denote by tJ(M) the space (and F(M)-module) of all COO-vector fields on M. Let T be a local flow (Le. local action of the group JR.) on M. Consider a vector field v E tJ(M), whose value v(x) at a point x E M is the "velocity" of the

flow at this point, i.e. a tangent vector to the curve s f-t T( s, x) at the point s = o. The field v is called the infinitesimal transformation corresponding to the flow T. As is well known, a vector field on M can be viewed as a derivation of the algebra F(M). With this interpretation, the infinitesimal transformation corresponding to a flow T, acts according to the formula

d (v¢)(x) = ds ¢(T(s, x))[s=o. (7)

The theorems on the existence and the uniqueness of solutions of a system of ordinary differential equations lead to the following result (see Warner 1983).

Theorem 2.1. Two local flows on a manifold M coincide in a neighbourhood of the set {O} x M if and only if the corresponding infinitesimal transformations coincide. A global flow is completely determined by its infinitesimal transformation. For any field v E tl(M) there exists a local flow on M with v as its corresponding infinitesimal transformation. A point of M is fixed under the flow if and only if v(x) = o.

A vector field v on a manifold M is said to be complete if the corresponding flow on M is global. If M is compact than any vector field on M is complete.

Example 1. Let M be the open interval (0,1) C R The vector field v on M given by the formula v(x) = 1 for all x E M is not complete. The corresponding local flow is defined by the formula

T(s,x) = x + s (x EM,s E JR.)

in the domain W = {O < x + s < 1} C JR. x M. A flow T on a manifold M generates a linear representation s f-t Ts* of

the group JR. on the space F(M). However, it is more natural to consider the representation PT : s f-t T~s (see 1.5). The linear operator Lv tangent to the one parameter group

{PT(S) [s E JR.},

is called the Lie derivative in the direction of the field v and acts by the formula

d (Lv¢)(x) = ds (pT(S)¢)(X)[s=o,

from which it follows that

Lv¢ = -v¢ (¢ E F(M)). (8)

From Taylor's formula we deduce the following formula, which makes it possible to reconstruct Ts* from the operator v:

* S2 2 Sm m m Ts ¢=¢+sv¢+2!v¢+···+m!v ¢+O(S) (¢EF(M). (9)

Next we shall show how to describe morphisms of flows in the infinitesimal language. Let M and N be differentiable manifolds and f : M ---> N a differentiable mapping. We shall say that f projects a vector field v E tl(M)


onto a vector field w E u(N) if w(f(x)) = dfx(v(x)) for any x E M. If f is surjective, then the field w is completely determined by the field v and we shall write w = f*v. In the language of derivations of algebras of functions this relation is expressed as follows

Note that, in general, not every vector field v on M projects onto some vector field on the manifold N or, in other words, is f-projectable. Indeed, a necessary condition is given by the identity dfx(v(x)) = dfy(v(y)) for any X,y E M such that f(x) = f(y). If f : M --* N is a diffeomorphism, then every v E u(M) is f-projectable and f*v is given by the formula

(10)

In particular, if M = N, then we can speak about a vector field invariant with respect to f. An invariant vector field v is determined by the condition f* v = v or, equivalently, the condition

v(f(x)) = dfxv(x) (x EM). (11)

If one considers the field v E u(M) as a derivation of the algebra F(M), then formula (10) takes the form

f*v = r -10 V 0 f*,

and the invariance condition (11) takes the form

f* 0 v = v 0 f*.

(12)

(13)

Theorem 2.2. Let 1 : M ~ N be a differentiable mapping 01 differentiable manifolds and suppose on M and N there are given local flows T and S with infinitesimal transformations u and v respectively. If the mapping 1 is a morphism of flows, then f * u = v and for a surjective mapping f the converse is also true. In particular, a diffeomorphism f : M ~ M is an automorphism of a flow T on M if and only if the vector field v corresponding to the flow is invariant under f.

A function ¢ E F(M) is invariant under a flow T if and only if v¢ = 0, where v is the infinitesimal transformation corresponding to the flow T. We shall next find an analogous condition for a vector field u to be invariant under a flow T, i.e. for it to satisfy the condition (Ts)*u = u for all s E R For this purpose consider the vector field w = fs(Ts)*uls=o. A simple computation shows that

d ds (Ts)*u = (Ts)*w, (14)

and from (7) and (10) it follows that

w = -u 0 v + v 0 u. (15)

If u, v E tl(M), then the operator [u, v] = -u 0 v + v 0 u is a derivation of the algebra F(M) and, therefore, can be viewed as a vector field on M. Thus, in


u(M) we have an operation [ ], known as the commutator or the Lie bracket. With respect to this operation u(M) is a Lie algebra over JR, and by (8)

L[u,vJ = [Lu,Lv] (u,v E u(M)).

Beside this we have

[¢U,v] = ¢[u,v]- (v¢)u (u,v E u(M),¢ E F(M)).

If in a certain local coordinate system Xl, ... , Xn the vector fields u and v n n

take the form u = L: Ui8~.,v = L: Vi8~., then W = [u,v] takes the form i=l ' i=l '

n W = L: Wi 8~., where

i=l '

Wi = t (-Uk OVi + Vk oUi ) . k=1 OXk OXk

From (14) and (15) we obtain

Theorem 2.3. A vector field u E u(M) is invariant with respect to a flow T if and only if [u, v] = 0, where v is the infinitesimal transformation corresponding to T. Two flows Sand T commute (i.e. TaSb = SbTa for all a, b) if and only if their infinitesimal transformations u and v satisfy the condition [u,v] = o.

2.2. Infinitesimal Description of Actions and Morphisms. In this subsection we shall generalize the results of 2.1 to the case of actions of arbitrary Lie groups. Instead of a single vector field corresponding to a flow, in this case to an action of G we assign a homomorphism of the tangent algebra of G into the Lie algebra of vector fields on the manifold on which G acts. In the local case the construction of this homomorphism is due to S. Lie and can be described in the following way.

Let a local action of a Lie group G in the neighbourhood of the point 0 E JR be given by formulas (3). Let

i 8ti Uj(x)=~(O,x) (i=I, ... njj=I, ... ,r).

Uaj n

Then Uj(x) = L: uj(x) 8~k (j = 1, ... , r) are vector fields defined in a k=l

neighbourhood of zero in JRn. S. Lie pointed out that for any j, k = 1, ... , r r

[Uj,Uk] = LC;kU/' 1=1

where C;k E JR, i.e. that the real linear envelope of the fields UI, . •• , Ur is an algebra under the operation [ ]. He called the elements of this algebra the infinitesimal transformations belonging to the given group.


Let us now turn to the modern viewpoint. Let 9 = Te(G) be the tangent algebra of a Lie group G (see Chap. 2 of Part I). To every local action of the Lie group G on a manifold M there is associated a linear action PT of G on the space F(M) (see 1.5). We define the differential of the representation PT as the mapping dPT : 9 ~ tl(M), given by the formula

d ((dPT(v))¢)(x) = ds (pT(d(s))¢)(x)ls=o (¢ E F(M), x EM),

where 0 is any smooth path in the group, having vasa tangent vector at the point e = 0(0). Since dPT can be viewed as the differential of the homomorphism t : G ~ Diff M corresponding to the action T, we shall write dPT = dt. Note that the vector field dt(v) can also be given by the formula

where T X is determined by formula (6). In 1.4 of Chap. 2 of Part I was proved the following

Theorem 2.4. The mapping dt : 9 ~ tl(M) is a homomorphism of Lie algebras.

From this theorem it follows that dt(g) is a subalgebra in tl(M). Elements of this subalgebra are called the fundamental vector fields of the action T (in the terminology of S. Lie these are just the infinitesimal transformations belonging to the corresponding transformation group).

From Theorem 2.1 follows

Theorem 2.5. Let Tl and T2 be two local actions of a Lie group G on a manifold M. Then we have dh = dt2 if and only if Tl and T2 coincide in some neighbourhood of the set {e} X M. If Tl and T2 are global actions of a connected Lie group G, then from dtl = dt2 follows Tl = T2.

Let N be the kernel of the action T of the group G on M.

Theorem 2.6. The tangent algebra n of the subgroup NeG coincides with Ker dt. In particular, the action T is locally effective if and only if the homomorphism dt is injective.

From Theorem 2.2 we obtain the following infinitesimal description of morphisms.

Theorem 2.7. Let f : M ~ N be a differentiable mapping and suppose there are given actions Tl and T2 of a Lie group G on M and N. If f is a morphism of these actions, then dt2(v) = f*(dtl(V)) for all v E g. If G is connected and f is surjective, then the converse also holds: the identity dt2(v) = f*(dtl(V)) for any v E 9 implies that f is a morphism of actions.

Corollary. A diffeomorphism f : M ~ M is an automorphism of an action T of a connected Lie group G on M if and only if all its fundamental vector fields are invariant under f.


Example 2. Consider the action L of G on itself by left translations (see Example 4 of 1.2). It is easy to see that the automorphisms of this action are all right translations Rg (g E G). Therefore the fundamental vector fields of the action L are right invariant, i.e. they are invariant under all right translations. More precisely, u = dl(v), where v Egis a right invariant vector field on G such that u(e) = v. It is easy to see that a right invariant vector field u is uniquely determined by its value u(e), which can be an arbitrary vector of the space 9 (see Theorem 3.5 of Chapt. 2). From this it follows that dl isomorphically maps 9 onto the Lie algebra of all right invariant vector fields (for a direct proof of this fact see 1.2 of Chap. 2 Part I). Analogously, the fundamental vector fields for the action by right translations are the left invariant vector fields.

2.3. Existence Theorems. We now consider the question of the existence of an action T with a given homomorphism dt. It turns out that for local actions the answer to this question is always positive.

Theorem 2.8 (See Palais 1957a). Let G be a Lie group, M a differentiable manifold and suppose there is given a homomorphism of algebms 7 : 9 --t

tJ(M). Then there exists a differentiable local action T of G on M such that 7 = dt.

The analogous theorem for global actions is false even in the case G = lR (see Example 1). Clearly, all fundamental fields of any global action are complete. It turns out that the condition of completeness happens also to be sufficient for the existence of an action T with a given dt, if one assumes that the group G is simply connected. Furthermore, we have

Theorem 2.9 (See Palais 1957a). Let G be a simply connected Lie group, 9 its tangent algebm and 7 : 9 --t tJ(M) a homomorphism such that the subalgebm 7(g) is genemted (as a Lie algebm) by complete vector fields. Then there exists a differentiable action T of the group G on M such that 7 = dt.

Corollary 1. If a finite-dimensional subalgebm f) C tJ(M) is genemted by complete vector fields, then f) consists entirely of complete fields.

Corollary 2. Under the assumptions of Corollary 1 there exists a unique connected Lie group H contained in Diff M as an abstmct subgroup and possessing the following properties: the natuml action T of the group H on M is differentiable; 1m dt = f).

Corollary 3. Let G be a simply connected Lie group and let the manifold M be compact. Then for any homomorphism 7 : 9 --t tJ(M) there exists an action T of G on M such that 7 = dt.

Corollary 4. Every local action of a simply connected Lie group G on a compact manifold M extends to a global action of G on M.


Note that for a non-compact manifold M complete fields do not, in general, form a subalgebra of u(M).

Example 3. (see Palais 1957a). Let M = lR2 , u(x, y) = (y, 0), v(x, y) = (0, !x2 ). Then u and v are complete fields but the fields u + v and [u, v] are not complete.

2.4. Groups of Automorphisms of Certain Geometric Structures. Many applications of the theory of Lie groups are connected with the fact that the groups of automorphisms of a number of important geometric structures turn out to be Lie groups. More precisely, we shall say that a subgroup G of the group Diff M of all diffeomorphisms of a differentiable manifold M is a Lie transformation group (in the restricted sense) if there exists a Lie group structure on G such that the natural action T of the Lie group G on M is differentiable and that any flow on M, which is contained in G, has the form s f-t T-y(s), where 'Y is a one-parameter subgroup of G. An analogous definition can also be given in the complex-analytic case. In this section we shall formulate certain sufficient conditions for a subgroup G c Diff M to be a Lie transformation group (for more details see Kobayashi 1972).

Theorem 2.10 (Bochner-Montgomery, see Montgomery and Zippin 1955). If a subgroup G c Diff M, where M is a differentiable manifold, is locally compact, then G is a Lie transformation group.

Moreover, Corollary 2 of Theorem 2.9 shows that G c Diff M is a Lie transformation group if the complete vector fields on M, for which the corresponding flows lie in G generate a finite-dimensional subalgebra of the Lie algebra u(M). On this fact is based a number of criteria, which apply in the case G = Aut (M, IT) - the group of automorphisms of a certain A-structure IT on an n-dimensional differentiable manifold M, i.e. a reduction of the structure group of the frame bundle of M to a Lie subgroup A C GLn(lR) (see also Example 6 of 3.2 below).

Let 0 C g[n (lR) be some subalgebra. We shall denote by Ok the space of all symmetric (k + 1 )-linear mappings). : (lRn )k+1 -+ lRn such that for any fixed Vl, ... ,Vn E lRn the transformation v f-t ).(v,VI, ... ,Vk) (v E lRn) belongs to o. The algebra 0 is said to be an algebra of finite order k if Ok-l :I 0 but Ok = o. In this case Ol = 0 for all l ~ k. The subalgebra 0 is called elliptic if 0 does not contain matrices of rank 1. Every subalgebra of finite order is elliptic.

Theorem 2.11. Let IT be an A-structure on an n-dimensional differentiable manifold M, with the tangent algebra 0 of the group A having a finite order k.

k-l Then Aut (M, IT) is a Lie transformation group of dimension::; n+ L dim 0i.

i=l

Since the tangent algebra of an orthogonal group has order 1 we obtain


Corollary (Myers - Steenrod). If 0' is a Riemannian structure on an ndimensional differentiable manifold M, then Aut (M, 0') is a Lie transformation group of dimension :S n(n2+1).

Theorem 2.12. Let 0' be an A-structure on a compact differentiable manifold M and suppose the tangent algebra a of the group A is elliptic. Then Aut (M, 0') is a Lie transformation group.

Corollary 1. If 0' is an almost complex structure on a compact differentiable manifold M, then Aut (M, 0') is a Lie transformation group.

Corollary 2 (Bochner - Montgomery). If M is a compact complex analytic manifold, then the group Aut M of all biholomorphic transformations of the manifold M is a complex Lie group. Its tangent algebra is the Lie algebra of all holomorphic vector fields on M.

Note that Corollary 2 generalizes to compact complex spaces (with singular points) (Kaup 1967, Fujimoto 1968).

§3. Fibre Bundles

3.1. Fibre Bundles with a Structure Group. This subsection is a brief exposition of the basic definitions and constructions of the theory of differentiable fibre bundles (for details see Husemoller 1966, Steenrod 1951, Sulanke and Wintgen 1972).

Let G be a Lie group, F a differentiable G-space and B a differentiable manifold. By a fibre bundle (or just a bundle) with basis B, fibre F and structure group G we mean a collection of the following objects: a differentiable manifold E, a differentiable mapping p : E -+ B, an open covering U = (Ui)iEI of the manifold B, a 1-cocycle Z = (Zij) of the covering U with values in the sheaf :FG of germs of differentiable G-valued functions on B (see Part I of Encycl. Math. Sc. 10, 3.7) and a choice of diffeomorphisms hi: Ui x F -+ p-l(Ui ) c E satisfying the following conditions:

1) (Pi 0 hi)(x,y) = x (x E Ui,y E F)

2) hj(x,y) = hi(x,Zij(X)y) (x E Ui n Uj,y E F).

One naturally defines equivalence of two fibre bundles with given G, Band Fj it reduces to the requirement that the co cycles (Zij) which define them, have to define the same cohomology class in the set Hl(B,:FG). In what follows fibre bundles will be considered up to equivalence class. The manifold E is called the total space of the bundle, p - the projection of the bundle, and the submanifold E z = p-l(X) - the fibre over the point x E B. The fibre bundle will be written:

p F ---+ E ---+ B or p: E -+ B.


Quite analogously one defines topological fibre bundles with a topological structure group.

A fibre bundle is called principal if F = G and G acts on F by left translations. Let G ~ E ~ B be a principal fibre bundle with structure group G. Setting hi(x, y)g = hi(x, yg) (x E Ui , y, 9 E G), we obtain a well defined right action of G on E which takes each fibre to itself. Clearly this action is free and the fibres are its orbits.

Let ¢ : A ~ G be a homomorphism of Lie groups. There is a naturally defined mapping ¢* : Hl(B,FA) ~ Hl(B,FG)' If ¢ defines an inclusion of actions of A and G on a manifold F, then any fibre bundle F ~ E ~ B with structure group A can be viewed as a fibre bundle with structure group G. This operation is known as extension of the structure group and the inverse operation as a reduction. Study and classification of fibre bundles is often done with respect to equivalence in a larger structure group than the given one.

3.2. Examples of Fibre Bundles

Example 1. If Z is the identity cocycle, Le. Zi,j = e for all i, j, then the manifold E can be identified with B x F and p with the projection on the first factor. This kind of fibre bundle is said to be trivial.

Example 2. Let M be an arbitrary n-dimensional differentiable manifold. For any point x E M we shall denote by Ex the manifold of all n-frames (Le. bases) of the tangent space Tx(M) and let E = U Ex. Let (Ui)iEI be a

xEM

covering of M by coordinate neighbourhoods and let x~i), ... ,x~) be a local system of coordinates in Ui . Define a map hi : Ui x GLn{lR) ~ E by the formula:

(

a~l (X)) hi(x,a) = aT a:

ax" (x) Gluing together the manifolds Ui x GLn{lR) with the help of the maps Fij = hi1 ohj (the complex analytic variant of this process is described in 2.7 Part I of Vol. 10), we give E the structure of a differentiable principal fibre bundle with base M and structure group GLn{lR). The right action of this group on E takes the form

(g, (u!, ... ,un)) f---t (Ul,' .. ,un)g (Ui E Tx(M), 9 E GLn{lR)).

As cocycle defining the fibre bundle one can take

Zij = (::~:) f3 l~Ct,f3~n

The principal fibre bundle E = R( M) we have constructed is called the frame bundle of M.


Example 3. Let G be a Lie group, H a Lie subgroup, p : G ---+ G / H the natural mapping. Then p is the projection of a principal analytic fibre bundle with base G / H and structure group H. This follows easily from the existence of a submanifold V c G, which is considered in Example 5 of 1.2. Fibres of this fibre bundle are left cosets gH, and the right action of H on G is given by right translations.

Example 4. If in a fibre bundle F ---+ E ---+ B the fibre F = kn , where k = JR, Cor IHI, and the structure group is the group GLn(k) with the standard action on kn , then the fibre bundle is called a vector bundle over the field k. An example is given by the tangent bundle lRn ---+ T(M) ---+ M of a differentiable manifold M.

Let B be a differentiable manifold and G a Lie group. Suppose we are given a principal fibre bundle 71' : P ---+ B with structure group G. If now we are given an arbitrary differentiable G-space F, then using a co cycle z which defines P we can construct a fibre bundle F ---+ E ---+ B with fibre F. Thus to every principal fibre bundle P with structure group G and a G-space F there corresponds a fibre bundle E with fibre F; it is called a fibre bundle associated with P. The space E can be obtained directly as the fibered product E = PXGF, which is defined as the space of orbits (PxF)/G ofthe manifold P x F with respect to the diagonal action:

g(x,y) = (xg-1,gy) (g E G,x E P,y E F).

The projection of the bundle p : E ---+ B is given by the formula

p(G(x,y)) = 7l'(x) (x E P,y E F)

Example 5. Various classical fibre bundles of geometric objects over a differentiable manifold M are associated with the frame bundle (see Example 2). Among them there is the tangent bundle T( M), the bundle of p-contravariant and q-covariant tensors TP,q (M) and bundles of tensors of various special types. We particularly note the bundle of positive densities D(M), which is associated with the frame bundle and has as its fibre the one-dimensional manifold JR+ = {x E lRl x> O}, on which GLn(lR) acts by the formula

gx = Idetglx (g E GLn(lR),x E ~).

A section of a fibre bundle p : E---+ B is a differentiable mapping s : B ---+ E such that po s = id. We denote by r(E) the set of all sections of a bundle E. Note that sections of the trivial bundle E = B x F can be identified with differentiable mappings B ---+ F. Further, a principal bundle possesses sections if and only if it is equivalent to a trivial bundle. For a vector bundle the set of sections is a vector space over the corresponding field. Sections of the bundle T(M) are vector fields on M, sections of the bundle of positive definite symmetric bilinear forms are Riemannian structures on M, sections of the bundle of positive densities are positive densities on M. Note the following simple criterion for the existence of a section: if the fibre F is homeomorphic to lRn, then r(E) -=I- 0 (see Steenrod 1951).


From this it easily follows, for example, that on any differentiable manifold there exist Riemannian structures and positive densities.

Example 6. Let A be a Lie subgroup of a Lie group G. Considering the action L of G on G / A (see Example 5 of 1.2), we can associate to any principal bundle G -+ E -+ B a certain bundle G / A -+ E -+ B with fibre G / A and structure group G. It turns out (see Husemoller 1966) that there exists a map

0: r(E) -+ H1(B,FA)

such that the sequence

A (j 1 i* 1 r(E) -+ H (B,FA) --+ H (B,Fa),

where i : A -+ G is an inclusion, is exact in the following sense: 1m 0 is the set of all fibre bundles with structure group A which can be obtained from E by reduction of structure group.

Example 7. Let M be an n-dimensional manifold, A a Lie subgroup of GLn(lR). Let us denote by ~A the fibre bundle with base M and fibre GLn(lR)/A associated with the frame bundle R(M). The bundle ~A is known as the bundle of A-structures and its sections A-structures on M. As we saw in Example 6, to every A-structure there corresponds a reduction of the structure group of the bundle T(M) to the subgroup A. Many classical structures on manifolds are A-structures. In particular, for A = On A-structure is a Riemannian structure on M, for

A = {g E GLn(lR) 1 det 9 = ±1}

a positive density, for

A = GL~(lR) = {g E GLn(lR) 1 det 9 > O}

an orientation on M, for

A = {g E GLn(lR) IgJ = Jg}

where J is some complex structure in lRn and n is even - an almost complex structure on M, and for A = {e} an absolute parallelization.

3.3. G-bundles. Let G be a Lie group and B be a differentiable G-space. By a G-bundle with base B we mean a differentiable bundle p : E -+ B together with a given G-action on E such that p is a morphism of actions. If E is a G bundle, then for any point x the stabilizer Gx sends the fibre Ex to itself so that Ex is a Gx-space. Further, on the set of sections r(E) we have the following action of G:

(gs)(x) = g(S(g-lX) (g E G, s E r(E), x E B). (16)

If E is a vector bundle, then we shall assume that the mappings of fibres Ex -+ Egx(x E B) defined by the elements 9 E G are linear.


Example 8. Let M be a differentiable G-space. Then the tangent bundle T(M) has a natural structure of a G-vector bundle:

gv = (dxTg)v (g E G,v E Tx(M)).

The corresponding linear representation of G on the space tl(M) = r(T(M)) have the form g f--T (Tg ). (cf. (10)). Analogous G-structures are defined on the frame bundle, on the tensor bundles TP,q(M), on the bundles of A-structures for any Lie subgroup A C GLn(IR), n = dimM.

A generalization of the situation described in Example 8 gives the following

Lemma 3.1. Let P --t M be a principal bundle with structure group K, possessing the structure of a G-bundle such that the actions of G and K on P commute, i.e. g(xk) = (gx)k for all g E G,k E K,x E P. Then every bundle associated with P has a natural G-structure.

Proof. Let E = P X K F be a bundle associated with P, the fibre of which is a differentiable K -space F. Then the action of G on E considered in Lemma 3.1, is given by the formula

gK(x, y) = K(gx, y) (g E G, x E P, y E F). o

Example 9. If G is a Lie group, Band F differentiable G-spaces, then by the trivial G-bundle with base B and fibre F we mean the trivial fibre bundle E = B x F, with the following G-action:

g(x,y) = (gx,gy) (g E G,x E B,y E F).

Invariant sections of this G-bundle can be identified with morphisms of Gspaces B --t F. If F = lR. and G acts trivially on F, then the space of sections r(E) can be identified with the space of functions F(B) on the base B, and the action (16) with the representation PT considered in 1.5.

An important problem in the theory of G-bundles is to describe the set r(E)G of invariant sections of a given bundle E. Examples of such sections are provided by G-invariant tensor fields, Riemannian structures and invariant densities on differentiable G-spaces (see Example 8). This problem will be considered in §3 of Chap. 2 in the case of a homogeneous space B and in Chap. 3 in the case of compact groups G.

Another interesting problem is the question of the existence of a G-bundle structure on a given fibre bundle E --t B, whose base B is acted upon by a group G, i.e. the problem of lifting the action of G from B to E. For this problem see 1.6 of Chap. 3 (where the group G is assumed to be compact).

3.4. Induced Bundles and the Classification Theorem. In the theory of fibre bundles an important role is played by the following construction. Suppose we are given a differentiable fibre bundle p : E --t B with fibre F and structure group A. Then to any differentiable mapping f : M --t B there corresponds a fibre bundle E = f* E with base M, fibre F and structure group A, which is known as the bundle induced by f. It can be constructed by transferring

the covering and the co cycle defining E to the manifold M. The total space E can be defined by the formula

E = {(x, y) EM x E I f(x) = p(y)},

and the projection p : E -t M by the formula p(x, y) = p(y). One can view the induced bundle as a fibre bundle over M whose fibre over a point x E M is the fibre Ef(x) of the bundle E. Note that the mapping 7 : E -t E,

given by the formula 7(x,y) = y takes Ex = {x} x Ef(x) to Ef(x). If G is some Lie group, E a G-bundle and f a morphism of G-spaces, then the action g(x,y) = (gx,gy) ((x,y) E E) turns E into a G-space, and 7 into a morphism of G-spaces. It turns out that any fibre bundle with structure group A and a given fibre F can be obtained by inducing it from a certain standard (so called universal) bundle, which depends only on A, F and the dimension of the base. Below we shall state the relevant theorem for the case of vector bundles.

Example 10. Let V be a finite-dimensional vector space over one of the fields k = JR., C or 1Hl. Let Gm(V) denote the Grassmann manifold of all possible m-dimensional subspaces of V (1 :S m < dim V). The manifold E = {( 1', x) E Gm (V) x V I x E 1'} is the total space of a differentiable vector bundle over k with' base Gm(V), m-dimensional fibre and projection (,)"x) f--+ 1'. This bundle is sometimes called tautological, because its fibre over point l' E Gm(V) is l' viewed as a vector space. The total space of the associated principal bundle is the manifold of all m-frames in V. We shall write G~(kn) = G~,m' The tautological vector bundle over G~,m can be viewed as a bundle with structure group On (k = JR.), Un (k = q or SPn (k = 1Hl). In this case the total space of the principal bundle is the Stiefel manifold St~ m of all orthonormal m-frames in kn. Note also that the tautological bundle has a natural structure of a GL (V)-bundle, with a transitive action of GL (V) on its base Gm(V).

The following theorem shows that tautological bundles are universal for all vector bundles.

Theorem 3.1 (See Husemoller 1966, Steenrod 1951). Any differentiable vector bundle over k with an m-dimensional fibre and an n-dimensional base B is induced from the canonical bundle by means of some differentiable map B -t G~,m' N 2: n + m + 1, ~ + m, n4"2 + m for k = JR., C, IHl respectively. Under these conditions two differentiable mappings B -t G~ m induce isomorphic vector bundles over B if and only if they are homotopi~.

In 1.5 of Chap.3 we shall consider a generalization of this theorem to G-bundles in the case when G is a compact Lie group.

II. Lie Transformation Groups

Chapter 2 Transitive Actions

§l. Group Models

121

1.1. Definitions and Examples. Recall that any transitive action of a group G on a set M is isomorphic to an action of this group by left translations on the set of all cosets G I H where H = G;]) is the stabilizer of any point x EM. In the differentiable case (see 1.4 of Chap. 1) G I H is an analytic homogeneous space of the Lie group G and the isomorphism is a diffeomorphism. The Gspace G I H is called a group model (or a Klein model) of the homogeneous space M. A group model depends on the choice of the point x so that the subgroup H is defined up to conjugation in G. By means of a group model any property of the homogeneous space M can be expressed in terms of the group G and a subgroup H. We shall now illustrate this method on the simplest examples.

Since all stabilizers of a transitive group G are conjugate, the kernel N of an action of G on M (which coincides with the intersection of all the stabilizers) is the largest normal subgroup of the group G contained in H. In particular, an action is effective if and only if H does not contain nontrivial normal subgroups of G and is locally effective if and only if H does not contain non-trivial connected normal subgroups of G. If we transfer from the given action of G to the effective action of GIN (respectively locally effective action of GINO), then a group model of the new homogeneous space will have the form (GIN)/(HIN) (respectively (GINO)/(HINO)).

Next we give a description of morphisms of homogeneous spaces.

Theorem 1.1. Let f : M --t N be a morphism of G-spaces, where M is homogeneous. Then f is surjective if and only if N is homogeneous. Group models of the homogeneous spaces M and N can be expressed in the form G I Hand G I K respectively, where K :J H, and then the morphism f takes the form gH f--? gK.

Corollary. Homogeneous G-spaces G I HI and G I H2 are isomorphic if and only if the subgroups HI and H2 are conjugate in G.

Conversely, if G :J K :J H are inclusions of subgroups of G, then the natural mapping f : gH f--? gK of the homogeneous space G I H into G I K is a morphism. In the case when G is a Lie group and K and H are Lie subgroups, f is a projection of an analytic G bundle with fibre K I H and structure group K, associated with the principal bundle G --t GIK.

Lemma 1.1. Let G be a Lie group and N :J H two Lie subgroups, where H is normal in N. Then the natural mapping f : G I H --t GIN is the projection of an analytic principal bundle, with structure group NIH, whose right action


on G / H is given by the formula

(gH) (nH) = gnH (g E G, n EN). (1)

Example 1. Let G be one of the classical compact Lie groups On (n ~ 2), Un (n ~ 1) or SPn (n ~ 1), acting linearly on the spaces JRn, Cn and lHIn respectively. Then G acts effectively and transitively on the unit sphere sn-l, s2n-l and s4n-l of the corresponding vector space. Taking as x the vector el of the standard basis we obtain the following group models G / H:

sn-l = On/On-I, s2n-l = Un/Un-I, s4n-l = Sp/SPn_l.

In each of these cases the normalizer Nc(H) is the subgroup of all transformations in G preserving the line (el), so that G /N c (H) is a group model of the projective space over the corresponding field:

IRPn- l = On/Ol x On-I, cpn-l = Un/Ul x Un-I,

lHIPn- l = SPn/SPl X SPn-l.

The corresponding principal bundles sn-l --) IRPn-l, s2n-l --) cpn-l, s4n-l --) lHIPn- l with structure groups 0 1 ~ ~, Ul , SPI respectively are known as Hopf bundles.

Example 2. Generalizing Example 1, consider the natural transitive action of the group G = On, Un, or SPn on the Stiefel manifolds St~ m (k = JR, C, 1HI respectively). Taking as x the frame {el, ... ,em} we obtai~ the following group models G / H j

St~,m = On/On-m, St~,m = Un/Un- m, St~,m = Spn/Spn- m·

The subgroup Nc(H) consists of all transformations preserving the subspace (el, ... , em). Hence G/Nc(H) is a group model of the Grassmann manifold G~,m:

G~,m = On/Om X On-m, G~,m = Un/Um X Un- m

G~,m = Spn/SPm X SPn-m.

The principal bundles St~ m --) G~ m have structure groups Nc(H)/ H ~ Om, Um, SPm respectively and are ass~ciated with the tautological bundles over G~ m (see Example 9 of 3.4 of Chap. 1).

The following criterion for similarity of two transitive actions is easy to prove.

Lemma 1.2. Let Ml = Gd HI and let M2 = G2/ H2 be two homogeneous spaces. The effective actions of Gl and G2 on Ml and M2 are similar if and only if the pairs (GI, HI) and (G2, H2) are isomorphic, i.e. there exists an isomorphism ¢ : Gl --) G2 such that ¢(Hl ) = H2 .

1.2. Basic Problems. We shall consider here some basic problems of the theory of homogeneous spaces and interpret them in the language of group models.


Problem 1. Classify (up to isomorphism or similitude) all transitive actions of a given Lie group G.

By Corollary 1 of Theorem 1.2 of Chap. 1, Corollary 1 of Theorem 1.1 and Lemma 1.2 Problem I is equivalent to the problem of classification of all Lie subgroups of G up to conjugation or up to automorphism of G. Many results have been obtained in the direction of such a classification, in particular in the case of connected subgroups of a connected compact or semi-simple Lie group (see volume "Lie Groups 3").

A differentiable manifold M is called homogeneous if M possesses a transitive action of some Lie group. Thus the notion of a homogeneous manifold differs from the notion of a homogeneous space in that in the first case there is no fixed Lie group action. The same homogeneous manifold may possess effective transitive actions of various Lie groups. For example on the sphere s2n-l (n ~ 2) there are transitive actions of the groups 02n, S02n, Un, and SUn, and on the sphere s4n-l (n ~ 1) of the groups 04n, S04n, U2n , SU2n and SPn (see Example 1).

Problem II. Let M be a homogeneous manifold. Classify (up to isomorphism or similitude) effective actions on M of those Lie groups for which such actions exist.

The solution to Problem II is known for a number of series of homogeneous spaces, for example for spheres (see Chap. 5 and 6). It reduces to the following question. Let M = G I H, where G is a Lie group, H a Lie subgroup, and let A be another Lie group. Find Lie subgroups B c A such that the manifolds M and AlB are diffeomorphic. If in Problem II we consider not only differentiable but also continuous actions, then there arises the problem: when is M homeomorphic to AlB? During investigation of these problems by means of homotopy invariants there arise naturally also the problems of classifying homogeneous spaces up to homotopy equivalence or rational homotopy equivalence.

We also note a special case of Problem II, which consists of describing reductions and enlargements of a given transitive action (for more details see §4).

1.3. The Group of Automorphisms. We shall describe the group Auta M of all automorphisms of a homogeneous G-space M. Let H = G x , where x E M. Then formula (1) (where N = Na(H)) defines a right action of the group Na(H)1 H on M which is free and, in particular, effective.

Theorem 1.2. The image of the right action of the group Na(H)1 H on M = GIH given by formula (1), coincides with the group Auta M. Thus AutaM S:' Na(H)IH. The group AutaM acts on M freely, and any of its orbits (AutaM)( x) coincides with the set Mare of those points y E M for which G y = G x .


A transitive action of a group G on M (or the homogeneous G-space M) is called asystatic if AutcM = {e} and systatic otherwise. It follows from Theorem 1.2 that an action is asystatic if and only if Nc(H) = H or MH = {x}.

Suppose now that M is a smooth homogeneous space of a Lie group G. Then N c (H) is a Lie subgroup of G and the group N c (H) / H is naturally a Lie group. We give AutcM the structure of a Lie group via the isomorphism of Theorem 1.2. By Lemma 1.1 the natural mapping M = G/H ---+ G/Nc(H) is the projection of a principal bundle with structure group AutcM.

In the differentiable case it is convenient to introduce the notion of an asystatic action in a somewhat different way than for abstract groups. Namely, we shall say that a transitive differentiable action of a Lie group G on M is called asystatic if the group Autc M is discrete, or equivalently, if any point x E M possesses a neighbourhood U in M such that MC~ n U = {x}.

For example, the standard action of the group SOn on sn-l (n 2': 2) is asystatic in the differentiable sense and systatic in the abstract sense (see Example 1). The corresponding actions of the groups Un and SPn on s2n-l and s4n-l are systatic even in the differentiable sense. The actions of these groups on projective spaces and Grassmann manifolds (Example 2) are asystatic in the abstract sense.

1.4. Primitive Actions. A transitive action of an abstract group G on a set M is called primitive if the only G-invariant equivalence relations ReM x M are R = {(x, x) I x E M} and R = M x M. In the language of a group model G / H, primitiveness is equivalent to H being a maximal proper subgroup. From Theorem 1.1 it follows that a transitive action is primitive if and only if any of its morphisms into a non-trivial transitive action is an isomorphism.

For Lie groups primitiveness of an action is defined somewhat differently. A transitive action of a Lie group G on a manifold M is called primitive, if on M there is no G-invariant foliation with connected fibres of positive dimension smaller than dim M. This is equivalent to the following condition on the stabilizer H: for any virtual (see Part I) Lie subgroup fI such that H c fI c G we have fIo = HO or fIo = GO. Clearly every action of a Lie group which is primitive as action of an abstract group is also primitive in this sense.

In what follows we shall consider the case when G is a connected Lie group and the action is locally effective. A Lie subgroup H eGis said to be primitive, if G acts primitively on M = G / H. A sub algebra of the tangent algebra 9 of G is called primitive, if it is the tangent algebra of some primitive Lie subgroup of G. Primitive subalgebras can be characterised as follows.

Theorem 1.3 (Golubitsky 1972). Let IJ be a subalgebra of a Lie algebra 9 not containing its ideals of positive dimension, and let H be the corresponding connected virtual Lie subgroup of G. The following conditions are equivalent:

a) the subalgebra IJ is primitive;


b) NG(H) is a maximal Lie subgroup ofG, with NG(H)O = H; c) I) is maximal among subalgebras not equal to 9 subalgebras of 9 which

are invariant under AdNG(H). If the subalgebra I) is primitive then the group G acts primitively on the

homogeneous space GIN G (H).

Clearly every maximal sub algebra of an algebra 9 which does not contain its non-zero ideals is primitive. Under certain conditions the converse statement also holds. Namely, let I) be a primitive subalgebra of a complex algebra g. If 9 is not simple or if 9 is simple and I) is non-reductive, then I) is maximal (Golubitsky 1972). At the same time, any simple complex Lie algebra contains a reductive primitive subalgebra which is not maximal (see Golubitsky 1972, Golubitsky and Rotschild 1971, where all such subalgebras of maximal rank are computed.) The simplest example is as follows.

Example 3. Let 9 = s[n(C), n ~ 2, I) the subalgebra of all diagonal matrices with trace O. Then I) is primitive in g. To the pair (g, I)) there corresponds a primitive action of the group G = SLn(C) on the homogeneous space G ING(H), where H is the subgroup of all diagonal matrices in SLn(C). Clearly the sub algebra I) is not maximal.

The classification of all reductive primitive subalgebras of complex or real simple Lie algebras is given in Komrakov 1991.

§2. Some Facts Concerning Topology of Homogeneous Spaces

In this section we shall give a short survey of facts concerning coverings and real homotopy invariants of Lie groups and homogeneous spaces, which we shall use later.

2.1. Covering Spaces. Below we give a description of all coverings of a given connected homogeneous space in terms of group models.

Let M be a connected homogeneous space of a Lie group G, According to Theorem 4.3 of Chap. 1 of Part I, the connected component of the identity GO acts transitively on M. We shall, therefore, suppose that G is connected and set H = G OJo, where Xo EM. Let 7r : G --> G be the universal covering of the group G. We have a transitive action (g,x) ~ 7r(g)x of the group G on M, with GOJo = H = 7r- I (H). By Theorem 4.8 of Chap. 1 of Part I, the manifold M = if I iIo = M is simply connected. By Lemma 1.1 the natural mapping p : if --> G I iI = M is the projection of a principal bundle with discrete structure group iII iIo, i.e. a covering. Thus p is the universal covering of the manifold M and 7r1 (M) ~ iII iIo.

Theorem 2.1. Let M = G I H = G IiI, where G is the universal covering of a connected Lie group G. Then any covering manifold MI of M is a homogeneous space of the group G with group model G I HI, where iIo C HI C iI


and the covering Ml = G / HI .....-) G / il = M coincides with the mapping gH1 f--+ gil.

Proof. The proof follows from the fact that any covering Ml .....-) M is the projection of a bundle associated with the universal covering p : if .....-) M, the fibre of which is a homogeneous space of the group 11"1 (M) ~ il / ilo. The action of G on Ml exists in view of Lemma 3.1 of Chap. 1. 0

2.2. Real Cohomology of Lie Groups. Let M be a manifold and A an associative commutative ring with unit. By H*(M; A) = ffip::::o HP (M, A) we denote the graded cohomology algebra of of the manifold M with values in A. Every continuous mapping f : M .....-) N generates a homomorphism of graded algebras f* : H* (N, A) .....-) H* (M, A).

Let E = ffip::::oEp be a graded vector space over some field, where the dimensions bp = dim Ep are finite. We define the Poincare series of E by the formula:

P(E, t) = L bptp. P::::O

In particular we have the Poincare series ofthe space H*(M,JR), where M is a manifold with finite Betti numbers bp(M) = dimHP(M,JR); we shall briefly denote it by P(M, t) (it is called the Poincare polynomial of the manifold M).

Let G be a connected Lie group. Then (see Borel 1953) the real cohomology algebra H*(G;JR) of the manifold G is the exterior algebra A(6,.·· '~r)' where ~i are elements of odd degree 2mi + 1. The numbers mi (i = 1, ... ,r) are known as the exponents of G; they completely determine the algebra H*(G; JR). The largest exponent is denoted by m(G). Ifr = 0, i.e. H*(G, JR) ~ JR, we set m( G) = -1. The Poincare polynomial of G has the form

r

P( G, t) = II (1 + em; +1 ).

i=1

We set H+(G,JR) = ffip>oHP(G,JR) and denote by (H+(G,JR))2 the set of elements of the form ~:=1 UiVi, where Ui, Vi are elements of positive degrees. Then E(G) = H+(G, JR)/(H+(G; JR))2 is a graded vector space with a homo-geneous basis ~i = ~i + (H+(G, JR))2 (i = 1, ... , r). We have

P(E( G), t) = em1 +1 + ... + t2mr+1.

If G = G1 X G2 is a direct product of two Lie groups, then H*(G,JR) H*(Gl, JR) ® H*(G2, JR), whence E(G) ~ E(G1 ) ffi E(G2). Note that a finite covering G .....-) G generates an isomorphism H*(G,JR) ~ H*(G,JR) and that the cohomology of a connected Lie group G coincides with the cohomology of a maximal compact subgroup. From the above it follows that it suffices to know the exponents of simple compact connected Lie groups, which need only be considered up to local isomorphism. These exponents are given in Table 1 (see Bourbaki 1968).


Table 1

G r ml···,mr m(G)

SUn (n ~ 2) n-1 1,2, ... , n - 1 n-1

S02n+1 (n ~ 1) n 1,3, ... ,2n - 1 2n-1 S02n (n ~ 3) n 1,3, ... ,2n - 3 2n-3

SPn (n ~ 1) n 1,3, ... ,2n - 1 2n-1 E6 6 1,4,5,7,8,11 11 E7 7 1,5,7,9,11,13,17 17 Es 8 1,7,11,13,17,19,23,29 29 F4 4 1,5,7,11 11 G2 2 1,5 5 S02 1 0 0

If G is a compact Lie group or a reductive algebraic group, then the exponents ml, ... ,mr of G possesses the following properties:

1. The number of exponents r = dim E{ G) coincides with the rank rk G of the group G i.e. with the dimension of a maximal torus T.

2. The numbers mi + 1 are the degrees of free generators of the polynomial algebra on the tangent space of the torus T, which are invariant under the Weyl group G.

3. If G is simple and non-commutative, then the number m{ G)+ 1 coincides with the Coxeter number of G, i.e. the order of a product of reflections in simple roots; moreover m{ G) = d:::'f - 2.

4. The number dimE{Gh (i.e. the number of exponents equal to 0) is equal to the dimension dim Z (G) of the centre of the group G.

5. The number dimE{Gh (i.e. the number of exponents equal to 1) is equal to the number of distinct non-commutative simple summands in the factorization of G into simple components ..

Note also the following monotoneity property of the number m{G) (see Onishchik 1962, Onishchik 1979):

Lemma 2.1. If H is a connected virtual Lie subgroup of a connected Lie group G, then m{H) :s; m{G).

2.3. Subgroups with Maximal Exponent in Simple Lie Groups. We shall call a connected virtual Lie subgroup H of a connected Lie group G a subgroup of maximal exponent if m{H) = m{G). Below we shall enumerate all such subgroups of simple non-commutative complex or compact Lie groups G.

Theorem 2.2 (Onishchik 1962). Let G be a connected simple non-commutative complex or compact Lie group, H a proper connected virtual Lie subgroup (respectively real or complex) of maximum exponent. In the compact case the pairs H c G are exhausted by the following list, which contains one representative of a group G with a given simple tangent algebra, and H is

given up to conjugation in G:

SPn C SU2n (n> 1), G2 c S07, S02n-l C S02n (n> 3),

In the complex case the pairs H C G are obtained by complexification of the compact pairs listed above, i.e. are exhausted by the following list (under the same assumptions on G and H):

SP2n(C) C SL2n (C) (n> 1), G2(C) c S07(C), S02n-l(C)

C S02n(C) (n > 3), Spin7(C) C SOs(C), G2(C) c SOs(C),

F4 (C) c E6(C).

Corollary. Every connected subgroup H of maximal exponent in a connected simple complex or compact non-commutative Lie group G is simple, closed in G and coincides with the connected component of its normalizer. We have rkH < rkG. If we exclude the case G2(C) c SOs(C), then the tangent algebra of such a subgroup is maximal in g.

2.4. Some Homotopy Invariants of Homogeneous Spaces. We shall next consider some invariants of the real homotopy type of a homogeneous space, which can be easily expressed in the language of group models. Invariants of this kind are particularly useful in solving problems of classification of homogeneous spaces (see 1. 2). We shall denote by r s (M) the rank r k 11"8 ( M) ofthe s-th homotopy group of a manifold M (for s = 1 this number is defined only when 1I"1(M) is abelian).

Let h : G1 ---+ G2 be a homomorphism of Lie groups. Then the corresponding homomorphism of cohomology algebras h* : H*(G2,1R) ---+ H*(Gl,IR) induces a homomorphism of graded vector spaces E(G2) ---+ E(G1 ) which we shall also denote by h * .

Theorem 2.3 (Onishchik 1963). Let H be a connected Lie subgroup of a connected Lie group G, M = G / H and let i : H ---+ G be the inclusion map. Set Eo(G) = Keri*, EO(H) = Cokeri*, where i* : E(G) ---+ E(H). Then

dimEo(Ghk+1 = r2k+1(M) (k ~ 0),

dimEo(Hhk+l = r2k+2(M) (k ~ 0).

Corollary 1. The polynomial

00

P(E(G), t) - P(E(H), t) = ~)r2k+1 - r2k+2)t2k+1 k=O

is a homotopy invariant of the manifold M = G / H.


We define the homotopy characteristic of a manifold M by 00

h(M) = - ~) -1)8 r8 (M). 8=1

By the rank and co rank of a manifold M we mean the numbers defined respectively by the formulas

00

rkM = L r2k+l, k=o

00

corM = Lr2k. k=1

Then h(M) = rkM - cor M.

Corollary 2. We have

In particular,

rkM = dimEo(G), cor M = dimEoOH),

h(M) = dimE(G) - dimE(H) ::::: o.

o :::; cor M :::; rk M,

0:::; h(M) :::; rkM.

From Corollary 2 it follows that if G and H are compact, then

rkG - rkH = h(M)

is a homotopy invariant of the manifold M = G / H.

Lemma 2.2. Let G be a connected compact Lie group and H a Lie subgroup, then rk (G / H) = 0 if and only if the manifold G / H is diffeomorphic to ]R.n.

Let X(G/H) = P(M, -1) be the Euler characteristic of the manifold M. We have the following assertion (see Samelson 1958):

Lemma 2.3. Let G be a connected compact Lie group and H a Lie subgroup ofG. Then X(G/H)::::: 0 with X(G/H) > 0 if and only ifrkG = rkH.

Note that X(M) ::::: 0 for any compact homogeneous manifold M (see Corollary 1 of Theorem 1.2 of Chap. 5).

It is known that X(M) = 0 for a compact differentiable manifold M if and only if on M there is a non-vanishing COO-vector field. Hence we have the following

Lemma 2.4 (Hermann 1965). If M is a compact differentiable manifold and X(M) #- 0, then any transitive action of a Lie group on M is asystatic.

§3. Homogeneous Bundles

Let G be a Lie group. By a homogeneous bundle with respect to G (or a homogeneous G-bundle) we mean a G-bundle the base of which is a homogeneous space of G. Many important geometric structures on homogeneous


spaces are sections of homogeneous bundles. In particular we note that homogeneous vector bundles play an important role in the theory of representations of Lie groups.

3.1. Invariant Sections and Classification of Homogeneous Bundles. Let M be a homogeneous space of a Lie group G, x E M and H = G.,. Let P : E --+ M be a G-bundle with base M and let F = E.,. Then F is a differentiable H-space. Consider the set r(E) of all differentiable sections of a bundle E and its subset r(E)G of sections invariant under the action given by formula (16) of Chap. 1, i.e. morphisms of G-spaces s : M --+ E.

Let r., : r(E) --+ F be the mapping given by the formula rz(s) = s(x). Clearly, r z is H-equivariant, whence it follows that r z takes r(E)G to FH. Since the action of G on M is transitive it follows that the mapping is invertible. The inverse mapping sends an element U E FH into a section s E r(E)G given by the formula

s(gx) = gu (g E G).

Thus we have

Theorem 3.1. For any homogeneous bundle F --+ E --+ M the mapping r z : r(E)G --+ FH is bijective.

Analogously one describes morphisms of a homogeneous bundle E into p

another homogeneous bundle F' ~ E' ~ M', i.e. morphisms of G-spaces h : E --+ E' such that p' 0 h = p. The morphism h is called an isomorphism if it has an inverse morphism. Any morphism h : E --+ E' defines a morphism of H -spaces hz : F --+ F' = E~.

Theorem 3.2. The correspondence h f--+ hz is a bijection between the set of morphisms of homogeneous bundles E --+ E' and and the set of morphisms of H -spaces F --+ F'. Moreover, h is an isomorphism if and only if hz is an isomorphism for each x.

From this theorem we can deduce the following theorem on classification of homogeneous bundles with base M = G / H.

Theorem 3.3. By assigning to each G-bundle E --+ M the differentiable H -space F = Ez we obtain a one to one correspondence between isomorphism classes of homogeneous bundles with base M and isomorphism classes of differentiable H -spaces.

Proof. The existence of a homogeneous bundle E to which a given Hspace F corresponds can be proven as follows. The group G can be viewed as a principal bundle with base M and structure group H (see Example 3 of 3.2 of Chap. 1), which is homogeneous with respect to the action by left translations. By Lemma 3.1 of Chap. 1 the associated bundle E = G XH F is also homogeneous. 0


From Theorem 3.3 it follows that every homogeneous bundle over G / H admits H as a structure group.

Note that the classification Theorem 3.3 is finer than the usual classification of bundles in the sense of differential topology, since it takes into account the action of G. In particular, there exist trivial bundles which are not isomorphic as G-bundles. According to Theorem 3.3, two trivial homogeneous spaces (Le. trivial G-bundles over M, see Example 8 of 3.3 of Chap. 1) are isomorphic if and only if the reductions to H of the actions of G on F which define them, are isomorphic.

Theorem 3.4. A homogeneous G-bundle F -+ E -+ M is isomorphic to a trivial bundle if and only if the action of the group of H on F = Ex admits an enlargement to an action of G.

3.2. Homogeneous Vector Bundles. The Frobenius Duality. A homogeneous vector bundle is a G-vector bundle over a homogeneous space G / H. Recall, (see 3.3 of Chap. 1), that the action G is linear on fibres. In particular, the action of the group H on the space F = Ex is linear. We shall also suppose that morphisms of homogeneous vector bundles are linear on fibres. Morphisms of homogeneous vector bundles E -+ E' form a vector space HomG(E, E'). Let us formulate analogues of Theorem 3.1-3.3 for homogeneous vector spaces (the analogue of Theorem 3.4 is, of course, also valid).

Theorem 3.5. For any homogeneous vector bundle F -+ E -+ M the map rx : r(E)G -+ FH is an isomorphism of vector spaces. If F' -+ E' -+ M is another homogeneous vector bundle, then HomG(E', E) ~ HomH(F', F).

Theorem 3.6. Assigning to each homogeneous vector bundle E -+ M the linear H -space F = Ex we obtain a one to one correspondence between isomorphism classes of homogeneous vector bundle with base M and isomorphism classes of finite-dimensional linear representations of the Lie group H.

Thus to each linear representation p : H -+ GL (F) on a finite-dimensional vector space F there corresponds a homogeneous vector bundle Ep with base M = G/H. We denote by R: G -+ GL(r(Ep)) the linear representation of the group G on the space of differentiable sections of the bundle E given by formula (16) of Chap. 1 (and, generally speaking, infinite-dimensional). The representation R is called the representation of G induced by the representation p of the subgroup H. Study of induced representations is based on the following classical result.

Theorem 3.7. For any linear G-space V and any linear representation p: H -+ GL(V) we have an isomorphism of vector spaces

Proof. The statement follows from the Theorem 3.5 if we take as E' the trivial bundle M xV. 0


Note that, if we identify Ep with G XH F, then r(Ep) can be identified with the space of differentiable vector-functions ¢ : G -+ F such that ¢(gh) = p(h)-l¢(g). The induced representation R acts on this space by the formula

(R(g)¢)(x) = ¢(g-lx) (g, x E G).

3.3. The Linear Isotropy Representation and Invariant Vector Fields. Let M = G / H be a homogeneous space. As we saw in Example 7 of 3.2 of Chap. 1, the tangent bundle T(M) admits the structure of a G-vector bundle, i.e. homogeneous vector bundle with base M. The corresponding representation of the subgroup H is the linear isotropy representation £ : H -+ GL(T",(M)) (see 1.3 of Chap. 1). We shall describe the representation £ in terms of the group model. Since the map dT: : 9 -+ T",(M) is surjective and its kernel coincides with ~ (see 1.3 of Chap. 1), the space T",(M) can be identified with g/~.

Lemma 3.1. The isotropy representation £ can be identified with a quotient representation of the adjoint representation of the group H on the space g, i. e.

£(h)(x + ~) = (Ad h)x + ~ (h E H, x E g). (3)

Further (d£)(y)(x +~) = [y, x] + ~ (y E ~,x E g). (4)

A homogeneous space M = G / H is called reductive if there exists a subspace meg, such that 9 = m E9 ~ and (Ad H)m = m. For reductivity it is sufficient, for example, that H be a compact or semi-simple group with a finite number of connected components. The tangent space T",(M) to a reductive homogeneous space can be identified with m and the isotropy representation with the corresponding subrepresentation of the representation Ad of the subgroup H (see Helgason 1962).

Let tl(M) be the space of all differentiable vector fields on a manifold M. From Theorem 3.5 and Lemma 3.1 follows

Theorem 3.8. The subspace tl(M)G of all invariant vector fields on M is isomorphic to the space (g/~)H of invariants of the representation £ given by formula (3).

As is shown by the Corollary to Theorem 2.7 of Chap. 1, tl(M)G is a subalgebra of the Lie algebra tl(M) tangent to the Lie group AutGM, which by Theorem 1.2 is isomorphic to NG(H)/ H. The isomorphism of Theorem 3.8 sends the commutator in tl(M)G to the operation (x +~, y +~) I-t [x, y] + ~ (x + ~,y + ~ E (g/~)H).

3.4. Invariant A-structures. Below we retain the notation of 3.3. We shall find it convenient to identify the structure group of the frame bundle R( M) with the group GL(g/~). Let A be a Lie subgroup of GL(g/~). Then the bundle of A-structures EA -+ M is homogeneous (see Example 7 of 3.2 of Chap. 1 and Lemma 3.1). Our purpose is to describe the set of all G-invariant A-structures on M. Let Iso = £(H) c GL (g/~) be the linear isotropy group


of the homogeneous space M. Let

S = {g E GL(g/~) Ig-1 (Iso)g c A}. Clearly the group A acts on S by right translations. From Theorem 3.1 we deduce

Theorem 3.9. G-invariant A-structures on M = G / H are in one to one correspondence with elements of the set S / A. In particular such a structure exists if and only if S -I 0.

Corollary 1. There exists on M = G / HaG-invariant Riemannian structure if and only if the closure of the linear isotropy group in GL(g/~) is compact. Such structures are in one to one correspondence with positive definite quadratic forms on the space g/~ which are invariant under L.

Proof. The proof makes use of the fact that for any action of a compact linear group there exists an inner product invariant under it (this easily follows from Lemma 3.2, see below). 0

Corollary 2. The homogeneous space M = G / H possesses a G-invariant orientation if and only if det L( h) > 0 for any h E H or, equivalently, if ~:!:!~~ > 0 for all h E H. If G is connected, then this condition is both necessary and sufficient for orientability of the manifold M.

Example 1. Let M = G be a Lie group acting on itself by left or right translations. Then H = {e}. From Corollary 1 if follows that G has left invariant (i.e. invariant under all left translations) and also right invariant Riemannian structures; they are in a bijective correspondence with inner products in the tangent algebra g.

Consider now the action of G x G on G by left and right translations. In this case L(g,g) = Adg (g E G), and the subgroup Iso coincides with the adjoint linear group Ad G. Corollary 1 shows that on any compact Lie group G there exists a bi-invariant (i.e. invariant under both left and right translations) Riemannian structure. Such a Riemannian structure is necessarily invariant also under the transformation 9 ~ g-l of G. If, in addition, G is simple, then the group Ad G is irreducible, whence it follows that all (Ad G)-invariant inner products in 9 are proportional. Thus, a bi-invariant Riemannian structure on a simple compact Lie group G is unique up to a positive scalar factor.

Note that for a homogeneous space which admits an invariant Riemannian structure the linear group Iso does not necessarily have to be compact. This is demonstrated by

Example 2. Consider the following action of the group G = SUn X IR. on the complex Stiefel manifold M = St~,2 :

(g,s)(x,y) = (e27risg(x),e27ri8sg(y)) (g E SUn,s E IR.,(x,y) E St~,2)'

where () is a fixed irrational number. One can easily verify that Iso is a virtual linear Lie group isomorphic to SUn- 2 X IR. and the subgroup Iso is compact.


On the other hand, it is known that, if G is a Lie group of all automorphisms of a Riemannian structure on some manifold M, then the subgroup G;x (and therefore, also the linear isotropy group) is compact for any point x E M (see Ch. 3, 1.1, Example 3).

Example 3. The Euclidean space En is a homogeneous space of its group of motions E(n), and moreover Iso = On. The latter is also true for the linear isotropy group of the sphere sn considered as a homogeneous space of the group On+1 (see Example 3 of 1.2 of Chap. 1). The same holds for the Lobachevski space An, which is defined as the connected component of the hyperboloid

SO,n = {x E IRn +11 X~ - X~ - ••• - x~+1 = I}.

given by the inequality Xl > 0 and is considered as a homogeneous space of the group G = Ol,n (see 4.2 of Chap. 1 of Part I). According to Corollary 1 each of these spaces has a unique (up to a positive scalar factor) G-invariant Riemannian structure. The spaces En (n 2: 1), sn (n 2: 2), An (n 2: 2) can be characterized as the simply-connected homogeneous spaces M of real Lie groups, satisfying one of the following conditions:

there exists on M an invariant Riemannian structure a with constant sectional curvature;

the linear isotropy group coincides with the full orthogonal group of the tangent space (with respect to some Euclidean metric).

Moreover, under these conditions G = Aut (M, a) (see Wolf 1972).

3.5. Invariant Integration. Another consequence of Theorem 3.9 is the following

Theorem 3.10 There exists a G-invariant positive density on M = G / H if and only if det £(h) = ±1 for any h E H or, equivalently, if IdetgAd hi = Idet~Adhl for any h E H. A G-invariant density is unique up to a constant factor.

Example 4. On any Lie group G there exists a left-invariant and a rightinvariant positive density (each of them unique up to a positive constant factor). In order that a left-invariant density be right-invariant, or equivalently, in order that there exist a bi-invariant positive density on G it is necessary and sufficient that det Adg = ±1 for any 9 E G. Lie groups with these properties are called unimodular. Any of the following conditions is sufficient for unimodularity of a group G: the group Ad G is compact; the group G is semi-simple; the group G is connected and nilpotent.

For any positive density on a manifold M there is an associated measure on M. The integral with respect to this measure of a compactly supported continuous function ¢ on M J M ¢( x) dx is defined as follows. Suppose that the support Supp ¢ is contained in a coordinate neighbourhood with local coordinates Xl,"" xn and let p be the coordinate of our density in this coordinate system. Then


s In general, let Supp ¢l c U Ui , where Ui are coordinate neighbourhoods and

i=l s

let {ei I i = 1, ... , s} be a partition of unity on U Ui corresponding to the i=l

covering (Ui ). Then

1M ¢lex) dx = t 1M ei(x)¢l(x) dx.

If M is a smooth G-space and the density is G-invariant, then integration possesses the following property:

1M ¢l(gx) dx = 1M ¢lex) dx (g E G). (5)

Integration possessing property (5) can also be defined for functions ¢l with values in a Fnkhet space.

Let G be a Lie group. A Frechet space S is called a topological G-module if there is given on S a structure of a topological linear G-space (or, equivalently, there is given a linear representation of G on S such that the mapping (g, s) I-t

gs of the space G x S into S is continuous). The following lemma is frequently used in constructing invariants.

Lemma 3.2. Consider a left-invariant positive density on G. If S is a topological G-module and for some s E S the integral So = fG (gs) dg exists, then So E SG. In particular when G is compact and fG dg = 1, the operator I : s I-t So linearly and continuously transforms S onto SG, with I I SO = id.

An example of a topological G-module is provided by the space r(E) of sections of a G-vector bundle with the COO-topology and the action of G given by formula (16) of Chap.l. A special case is the space F(M) of differentiable functions on a differentiable G-space with representation PT described in 1.4 of Chap. 1.

In computing integrals on Lie groups and homogeneous spaces the following lemma is often useful

Lemma 3.3 (see Helgason 1962). Suppose that a homogeneous space G / H has a G-invariant positive density and let ¢l be a function on G which is integrable with respect to the left invariant positive density on G. Then after a suitable normalization of densities

[ ¢leg) dg = [ 1jJ(x) dx, lG lG/H where

1jJ(gH) = L ¢l(gh) dh

(this integral is taken with respect to the left-invariant positive density on H.)


The statement of Lemma 3.3 can be reformulated as the following double integration formula:

fa ¢(g) dg = fa/H (L ¢(gh) dh) d(gH).

3.6. Karpelevich-Mostow Bundles. This subsection is devoted to a theorem which makes it possible to represent a homogeneous space G / H, where G and H have a finite number of connected components, as the total space of a homogeneous vector bundle, whose base is a homogeneous space of a maximal compact subgroup of the group G (the Karpelevich-Mostow bundle). This theorem generalizes the following property of Lie groups: any Lie group G with a finite number of connected components is diffeomorphic to the manifold K x ]Rn, where K is a maximal compact subgroup of the group G (see "Lie groups and Lie algebras - 3", Encycl. Math. Sc. 41, Ch. 4, 3.4).

Let G be a Lie group, A and B two analytic submanifolds of G. We shall write G = A x B if the map (a, b) r-t ab of the manifold A x B into G is an analytic diffeomorphism. This relationship generalizes to any number of factors. The construction of the bundle sought is based on the following lemma (see Mostow 1955b).

Lemma 3.4. Let G be a Lie group, Hand K a Lie subgroups, L = K n H, E and F analytic submanifolds in G, with G = K x F x E, H = L x E, e E F, and let gFg-1 = F for all gEL. Then the mapping L(k,f) r-t kfH is a K -invariant analytic diffeomorphism of the manifold K XL F onto G / H.

Let now G be a Lie group with a finite number of connected components, H a Lie subgroup with the same property, and K J L two maximal compact subgroups of these groups containing one another. It turns out (see Mostow 1955b, Mostow 1962b) that there exist submanifolds E,F C G satisfying the conditions of Lemma 3.4 which are analytically diffeomorphic to Euclidean spaces. From this we obtain

Theorem 3.11. Let H be a Lie subgroup of a Lie group G, where G and H have a finite number of connected components and K J L are maximal compact subgroups of Hand G respectively. Then G / H as an analytic Kspace is isomorphic to some homogeneous vector bundle with base K / L.

In the case when G and H are connected and semi-simple, this theorem was proved independently by F.1. Karpelevich (1956) and Mostow (1955b), and in the general case by Mostow (1962b). In the semi-simple case the proof of the existence of submanifolds E, F follows from the existence of compatible Cartan decompositions for a semi-simple real Lie algebra and a semi-simple subalgebra (Karpelevich 1953, see Mostow 1955a). In order to grasp the scope of applicability of Theorem 3.11 we note that, if a Lie group G with a finite number of connected components acts transitively on a connected manifold M = G / H with a finite fundamental group, then H has a finite number of connected components. Another important case when the conditions of the


theorem are satisfied is the case of an algebraic group G (over the field lR or C) and an algebraic subgroup H.

Corollary 1. With the assumptions of Theorem 3.11 the manifold G I H has the homotopy type of the compact homogeneous manifold K I L.

Corollary 2. If 7r 8 (G I H) = 0 for all s 2: 0, then the manifold G I H is analytically diffeomorphic to lRn .

Corollary 3 (Montgomery's Theorem). If under the assumptions of Theorem 11 the homogeneous space G I H is compact, then the maximal compact subgroup K ofG acts transitively on GIH, i.e. GIH = KIL.

Going over to the universal covering group (see 2.1) we derive the following statement.

Corollary 4. If a connected Lie group G acts transitively on a compact manifold M with a finite fundamental group 7r1(M), then the commutator of any maximal compact subgroup of G and a Levi subgroup (a maximal connected semi-simple subgroup) of G acts transitively on M.

Corollary 5. If G and H are connected and dim G I H - dim K I L = 1, then G I H is analytically diffeomorphic to K I L x lR.

Example 5 (Samelson, see Mostow 1955b). Consider the projective space ]Rpn with the standard projective action of the group GLn+1 (lR). Let Xo E

]Rpn and M = ]Rpn - {xo}. Then M is a homogeneous space of the group G = GLn+1 (lR)xo, with M = G I H, where H = GLn+l (lR)xo n GLn+1 (lR)Xll Xl E M. It is easy to see that K I L is naturally isomorphic to the hyperplane ]Rpn-l C ]Rpn and the projection of the Karpelevich-Mostow bundle M --; KI L sends each point x E M to the intersection point of the projective line passing through x and Xo with ]Rpn-l. The fibre of this bundle is onedimensional but it is non-trivial.

Example 6. In (Mostow 1955b) there is also an example of a simplyconnected homogeneous space of a semi-simple Lie group for which the Karpelevich-Mostow bundle is non-trivial. In this example GIH = SL3(C)/SL3(lR), KI L = SU3/S03.

Example 7. The conditions imposed on G and H in Theorem 3.11 are genuinely needed. Let, for example, G = SL2(lR) and H be any uniform discrete subgroup of G. Then the action of the maximal compact subgroup K = S02 C G on G I H is non-transitive, so that Corollary 3 is in this case false.


§4. Inclusions Among 'fransitive Actions

In this section we consider the problem of describing all inclusions between transitive (effective) actions of connected Lie groups on a given homogeneous manifold. It can be viewed as a part of the problem of classification of all transitive actions (see 1.2). The problem of describing enlargements arises naturally, for example, in the following situation. Suppose that a Lie group G acts on a manifold M, on which there is given a certain G-invariant geometric structure a such that the group Aut (M, a) of all automorphisms of this structure is a Lie group (see 2.4 of Chap. 1). Then the natural action of the group Aut (M, a) is an enlargement of the given action of G. Hence knowledge of all enlargements of the group G helps to determine the group Aut (M, a).

Studying inclusions between transitive actions is equivalent to studying factorizations of Lie groups as products of two Lie subgroups. We remark that of interest to physics is the problem of unification of two Lie groups, i.e. of constructing Lie groups which decompose as product of the given groups (see Sternheimer 1968, Barut and Rq,czka 1977). We shall return to the consideration of inclusions in Chapters 3 and 5.

4.1. Reductions of Transitive Actions and Factorization of Groups. It is clear that enlargements of a transitive action are always transitive. Next we shall explain when the same holds for a reduction of a transitive action of a group to a subgroup.

Lemma 4.1. Let G be some group, A and H two subgroups. Then the fol-lowing conditions are equivalent:

a) the reduction of the natural action of G on G / H to A is transitive, b) G = AH; c) G = HA; d) the reduction of the natural action of G on G / A to the subgroup H is

transitive.

Condition b) means that every element g E G can be represented in the form g = ah, where a E A, h E H. In this case we shall say that G decomposes as product of the subgroups A and H or that the triple (G, A, H) is a factorization of the group G.

Corollary 1. A triple (G, A, H), where A and H are subgroups of G, is a factorization if and only if the triple (G, H, A) is one.

Corollary 2. If (G, A, H) is a factorization, then· so is the triple (G, aAa-1 , bHb-1 ), where a, b are any elements of G.

Corollary 3. The reduction of the natural action of G on G / H to a subgroup A eGis simply transitive if and only if G = AH (or G = H A) and An H = {e}.


Consider the action of the group G x G on G by two sided translations (see Example 4 of 1.2 of Chap. 1). The stabilizer of the point e EGis the diagonal Gd C G x G, which is isomorphic to G. Applying Lemma 4.1 to the subgroup A x H c G x G we obtain

Corollary 4. The following conditions are equivalent: a)G=AH; b) G x G = (A X H)Gd; c) G x G = Gd(A x H).

Suppose we have an inclusion between transitive actions T' :::; T (see 1.1 of Chap. 1) of groups G' and G on a set M. Clearly, Auta,M :J AutaM. Hence if the action T' is asystatic, then so is the action T. Analogously, an enlargement of a primitive action is primitive.

4.2. The Natural Enlargement of an Action. For any action T of a group G on M we denote by SimaM the group ofits autosimilitudes (Le. of similitudes of T onto itself). Clearly, SimaM contains AutaM as a subgroup. For any g E G the transformation Tg is an autosimilitude of T (the corresponding automorphism of G being the inner automorphism Ag), so that we have a homomorphism t : G -+ SimaM. Its image t(G) is a normal subgroup of SimaM commuting with AutaM elementwise. If T is effective we may identify G with t(G).

We denote by N(T) the natural action of the group SimaM on M. If T is effective, we may consider N(T) as an enlargement of the action T; this enlargement will be called natural; it is proper if and only if AutaM i=- {id}.

Next we describe the group SimaM in the case when the action T is transitive. Write H = Gzo ' where Xo E M. Let Aut (G, H) be the group of automorphisms of G mapping H onto itself. Any 0: E Aut (G, H) defines an autosimilitude Sa by

sa(gxO) = o:(g)xo, g E G.

Then A = {sa 10: E Aut(G, H)} is a subgroup of SimaM fixing the point Xo.

Lemma 4.2. The subgroup A coincides with (SimaM)",o' The action T is transitive if and only if SimaM = t(G)A.

Corollary. For the action L of a group G on M = G by left translations, the group SimaM coincides with HolG = l(G)(AutG) of the group G.

Using Lemma 4.2 and some properties of semi-simple Lie groups (see Lie groups and Lie algebras - 3, Encycl. Math. Sc. 41, Ch. 3, §3) we obtain

Theorem 4.1. Let M = G / H, where G is a connected semi-simple Lie group acting effectively on M. Then SimaM has a finite number of components and (SimaM)O = G (AutaM)O (locally direct product). If G is compact, then SimaM is compact too, and the natural homomorphism Aut (G, H) -+ A is an isomorphism.


Now we mention some applications of Lemma 4.2 and Theorem 4.2. Suppose that we have an enlargement T' of an action T of a group G on M, such that the corresponding group G' contains G as a normal subgroup. If T' is effective, then G' is identified with a subgroup of SimcM, so that T ::; T' ::; N(T). If, in addition, T is transitive, then we get a description of G' in terms of the group G and its automorphisms. More precisely, we have

Corollary 1. Let G be a normal subgroup of G', T a transitive action of G on M, and T' an effective enlargement ofT to G'. Then G' can be identified with a subgroup ofSimcM of the form G' = GB, where B is a subgroup of A. If G is connected and semi-simple, then G'o = GC, where C c (AutcM)o.

A transitive action of a group G on M is called irreducible if no proper normal subgroup of G acts on M transitively. Similarly, a transitive action of a connected Lie group G is called irreducible if no proper connected virtual Lie subgroup of G acts transitively. Corollary 1 permits us to describe transitive actions in terms of irreducible ones.

Since the connected component of the identity of a Lie group G is normal in G, we get

Corollary 2. Let T be a transitive effective action of a Lie group G on a connected manifold M. Identifying G with t(G) we have G C SimcoM and G = GO B, where B is a subgroup of A = (Simco M) Xo •

4.3. Some Inclusions Among Transitive Actions on Spheres. We now consider transitive actions of classical compact Lie groups on spheres, described in 1.1 in Example 1.

Viewing the space en as lR.2n we obtain an inclusion Un C 02n, with both groups acting transitively on sn-I. Since sn-I (n 2 0) is connected, the subgroup SOn C On is transitive on sn-I. It is easy to show that the subgroup SUn C Un is transitive on s2n-l, so that we also have an enlargement of actions S02n =:J SUn on s2n-l. Further, considering the space IHIn as e 2n , we obtain an inclusion SPn C SU2n , with both groups acting transitively on s4n-l. Since AutsPn s4n-1 '::::: SPI' we also obtain a natural enlargement SPn C SPn X SPI' which in this case is locally effective. This enlargement is not a reduction of actions of groups U2n and SU2n but it turns out that there exists a homomorphism ¢ : SPn X SPI --+ SP4n' which defines an enlargement of actions on s4n-I. This homomorphism is given by the formula

¢(g,q)u = guq-I (g E SPn,q E SPI'U E W).

Thus we obtain the following scheme for enlargement of actions on sn-I, each of which can take place if the indices in the symbols denoting classical groups are integers:

SPI! --+ SUI! --+ UI! 4 2 2 ~

1 1 On

SPI! X SPI --+ SOn / 4


We note the following factorizations of Lie groups, which follow, by Lemma 4.1, from the above enlargements:

S02n = SUn· S02n-l,

S04n = SPn . S04n-b

SU2n = SPn . SU2n- 1 ,

4.4. Factorizations of Lie Groups and Lie Algebras. Lemma 4.1 shows that description of inclusions between transitive actions of Lie groups is equivalent to description offactorizations of the form (G, A, H), where G is a Lie group, A a virtual Lie subgroup and H a Lie subgroup. It is natural to consider the following, somewhat more general, notion. A factorization of a Lie group G is defined to be a triple (G, G', Gil), where G' and Gil are virtual Lie subgroups of G with G = G'G". We note some general properties of factorizations of Lie groups. First of all, by Corollary 1 of Lemma 4.1, the triple (G, G', Gil) is a factorization if and only the same holds for the triple (G,G",G'). From now on, two factorizations differing only in the order of summands will be considered identical. Further, for every triple (G, G', Gil), where G', Gil are virtual Lie subgroups of a Lie group G, there is a corresponding triple of Lie algebras (g, g', gil), where 9 ::J g', gil are the tangent algebras of the Lie groups G, G', Gil. Two triples of the form (G, G', Gil) are called locally isomorphic if the corresponding triples of tangent algebras are isomorphic.

Lemma 4.3 (Onishchik 1969). Let (G, G', Gil) and (G, G', Gil) be two locally isomorphic triples of Lie groups, with G and G' connected. Then G = G' Gil if and only if G = G' Gil .

This lemma shows that in studying factorizations (G,G',G") of a connected Lie group G we can assume that G', Gil are connected, and also that we can replace G by any connected Lie group locally isomorphic to it. In this case the property of decomposability of a triple (G, G', Gil) depends only on the corresponding triple of tangent algebras. By a factorization of a Lie algebm 9 we mean a triple (g, g', gil) where g' and gil are subalgebras of 9 and 9 = g' + gil.

Lemma 4.4. Let G be a Lie group, G', Gil a virtual Lie subgroups and (g, g', gil) the corresponding triple of tangent algebms. Then the following conditions are equivalent:

a) 9 = g' + gil ; b) 9 $ 9 = (g' $ gil) + gd, where gd is the diagonal in 9 $ g; c) the set G'G" is open in G. If Gil is closed in G then these conditions

are equivalent to the condition d) the orbit G'(xo), where Xo = Gil E GIG", is open in GIG".

In particular we see that a factorization of a Lie group determines a factorization of its tangent algebra. The converse statement is, in general, not true (see EXanlple 1 below).


We shall call a factorization (g, g', gil) global if G = G' Gil, where G is a connected Lie group with tangent algebra g. Note the following criterion of globality of factorizations.

Lemma 4.5. A triple of algebras (g, g', gil) is a global factorization if and only if for any inner automorphism 0: of the algebra 9 we have 9 = o:(g') + gil .

We also note that a factorization of a Lie algebra is global if the following conditions are satisfied: G' is compact, Gil is closed in G. From this we deduce:

Theorem 4.2 (Onishchik 1962). Any factorization of a compact Lie algebra 9 is global.

In (Malyshev 1975) it is shown that this property continues to hold for almost compact Lie algebras, i.e. semi-direct sums 9 e- a where 9 is compact and a is a real commutative Lie algebra, with a given orthogonal linear representation of the algebra g. For globality conditions for factorizations of reductive Lie algebras see 3.2 of Chap. 3. We also note the following result.

Theorem 4.3 (Malyshev 1978). Any factorization of a nilpotent Lie algebra (over IR or i.C) is global. If every factorization of a complex Lie algebra 9 as a sum of complex subalgebras is global, then 9 is nilpotent.

The following simple lemma establishes a connection between factorizations of real and complex Lie algebras.

Lemma 4.6. A triple (g, g', gil) of real Lie algebras is a factorization if and only if the same holds for the triple (g(C), g' (C), gil (C)).

For Lie groups only the "only if" part of the statement is valid.

Lemma 4.7 (Onishchik 1969). Let (G, G' ,Gil) be a factorization, where G', Gil are virtual complex Lie subgroups of a Lie group G and let H, H' and H" be real forms of the groups G, G' and Gil, with H connected and H :J H', H" . Then H = H'H".

Example 1. Consider the standard action of the group GL2(1R) on ]Rpl.

Since the subgroup O2 :J GL2(1R) is transitive on ]Rpl, we have the factorization

where B is the subgroup of all upper triangular matrices in GL2(1R). The group GL2(1R) and its subgroups O2, B are real forms of the group GL2(C) and its algebraic subgroups 02(C) and B(C) (the subgroup of all upper triangular matrices). However, the group 02(C) is not transitive on !Cpl, so that GL2(C) =f. 02B(C). Thus the converse to Lemma 4.7 is not valid. From this we can see that the factorization of Lie algebras g(2(C) = 02(C) + b(C), obtained from the global factorization g(2(1R) = 02 + b by complexification, is not itself global.


In the classification of factorizations we make use of topological methods. Their application is based on the following theorem (see Onishchik 1962, Onishchik 1969).

Theorem 4.4. Let G be connected Lie group, G', Gil two connected virtual Lie subgroups, such that the subgroup H = G' n Gil has a finite number of connected components. If G = G' Gil then there is an isomorphism of graded vector spaces

E(G) E6 E(Ho) ~ E(G') E6 E(G"). (6)

If G is compact the converse also holds.

Proof. The proof of equality (6) follows from Corollary 1 of Theorem 2.3 applied to the simply connected homogeneous covering space of the homogeneous space G' x Gil / H d , which is diffeomorphic to G. The proof of the converse uses the equality dimG = (2ml + 1) ... (2mr + 1), where ml, ... , mr are the exponents of the connected Lie group G. 0

The condition that H has a finite number of components in Theorem 4.4 is satisfied, in particular, in the following cases: G is an algebraic group (real or complex), Gil an algebraic subgroup; 7rl (G) is finite. Without this condition the theorem is false.

Corollary 1. Let (G, G', Gil) be a factorization satisfying the conditions of the theorem. If K, K' and K" are maximal compact subgroups of G, G' and Gil such that K' c K, then K = K' K" and K' n K" is a maximal compact subgroup of the group H = G' n Gil .

Corollary 2. Let (G, G', Gil) be a factorization satisfying the conditions of the theorem. Then either G' or Gil is a subgroup of maximal exponent in G.

4.5. Factorizations of Compact Lie Groups. In this subsection we present the classification of factorizations of simple compact Lie groups obtained in (Onishchik 1962). This work also provides a description of factorizations of arbitrary compact Lie groups, which we shall omit due to its complexity. Besides this, we shall give a description offactorizations (G, G', Gil) in which the subgroup G' n Gil is discrete.

In the class of all factorizations of compact connected Lie groups we consider the equivalence relation generated by the local isomorphism and by permuting factors. The factorization (G, G, {e}) will be called trivial. In order to simplify the table we shall consider only irreducible factorizations (G, G', Gil), i.e. we shall assume that the action of G' x Gil by two sided translations on G is irreducible. This means that none of the subgroups G', Gil can be replaced in the factorization by its proper connected normal Lie subgroup. Taking into account these remarks we have

Theorem 4.5 (Onishchik 1962). Every non-trivial irreducible factorization of a connected simple compact Lie group as a product of connected virtual


Lie subgroups is equivalent to one of the factorizations (G, G', Gil) listed in Table 2 (all the subgroups in the table are considered as subgroups of G in the standard way.)

Table 2

G G' G" G'nG" G G' G" G'nG"

SU2n SPn SU2n~1 SPn~l

S04n S04n~1 SPn SPn~l (n 2: 2) (n 2: 2)

S07 G2 S06 SU3

S016 S015 Sping Spin7 S05 SU2

S07 G2

S02n S02n~1 SUn SUn~l SOS Spin7 S06 SU3 (n 2: 4)

S05 SU2

Proof. The proof of Theorem 4.5 is obtained according to the following scheme. Applying Corollary 2 of Theorem 4.4, we see that G' is a subgroup of maximal exponent in G. Hence G' is one of the subgroups listed in Theorem 2.2. Making use of Theorem 4.3 and the theory of linear representations of semi-simple Lie groups, we can determine all possible subgroups Gil (it turns out that the cases G = S08, G' = G2 and G = E6, G' = F 4 cannot be realized) . 0

Corollary 1. For any non-trivial irreducible factorization (G, G', Gil) of a connected simple compact Lie group G as product of virtual Lie subgroups G', Gil, the subgroups G', Gil, G'nG" are compact, simple and non-commutative, with G' = Nc(G')o.

In order to obtain all factorizations from Table 2, it suffices to compute the subgroup Nc(G")O for all subgroups Gil in this table. It is easy to see that NC(G")O = G"G~, where either G~ = {e} or G~ or G~ is a connected compact normal subgroup of rank l.

Corollary 2. In any factorization of a simple compact Lie group G as product of connected virtual Lie subgroups G', Gil, both subgroups G' and Gil are compact, one of them is simple while the other is either simple or is a locally prime product of two simple subgroups, one of which has rank 1.

Note that almost all factorizations in Table 2 are connected with inclusions between transitive actions of simple compact Lie groups on spheres. Indeed, the series of factorizations of S02n, SU2n , S04n coincides with the series of factorizations obtained in 4.4, and the factorizations

S07 = G2 . S06, S016 = S015 . Sping , S08 = Spin7 . S07

show that


Besides this, we see from the table that

St~,2 = G2/SU2, St:,2 = SpindSU3 , St:,3 = Spin7 /SU2 .

Next we shall consider a result obtained (in the semi-simple case) in (Sternheimer 1968), which can also be deduced from the general classification of factorizations (see Onishchik 1969). We formulate it in the language of Lie algebras in the form proposed in (Koszul 1978).

Theorem 4.6. Let (g, g', g") be a factorization of a compact Lie algebra such that g' n g" = O. Then

a) There exists a direct sum factorization 9 = a' EEl a" such that the corresponding projections 9 ----) a' and 9 ----) a" define isomorphisms of g' onto a' and of g" onto a". In particular, 9 ~ g' EEl g" .

b) The ideals a' and a" may be chosen in such a way that 3(a') = 7r(3(9')), 3(a") = 7r(3(9")), where 7r : 9 ----) 3(9) is the projection relative to the factorization 9 = 3(9) EEl [g, gJ.

4.6. Compact Enlargements of Transitive Actions of Simple Lie Groups. Here we shall consider actions of connected compact Lie groups which are proper extensions of transitive actions of connected simple compact Lie groups. As was shown in 4.2, to describe such enlargements it suffices to assume that they are irreducible. To begin with we shall display two classes of irreducible enlargements, which can be easily determined with the help of Theorem 4.4.

Let (C', C, H') be a non-trivial factorization of a connected simple compact Lie group, where C, H' are Lie subgroups, C is connected and simple, H = G n H'. By Lemma 4.1, G/ H = G' / H'. Thus we have a proper irreducible enlargement of the action of G on G I H to an action of G'. An enlargement of this sort is said to be of type I.

Embed a connected simple non-abelian compact Lie group G into C x G by the diagonal mapping g f---+ (g,g). Let (C,A',A") be a non-trivial factorization, H = A' n A". Then GI H = (G x C)/(A' x A") = GIA' x CIA" (see Corollary 4 of Lemma 4.1). The corresponding enlargement (of C to C x G) is said to be of type II.

Consider an enlargement of type II. It turns out that the actions of C on CIA' and CIA" admit no enlargement of type II. Further, only one of these actions may admit an enlargement of type I. If such an enlargement C c C' exists, we obtain an enlargement of C to C' x C. Such a composition of enlargements of types II and I is said to be an enlargement of type III.

Theorem 4.7 (Onishchik 1962). Any proper enlargement of a transitive action of a connected simple compact Lie group, which is an irreducible effective action of a connected compact Lie group, is an enlargement of type I, II or III.


A homogeneous space M of a connected simple compact Lie group G is called exceptional if the action of G on M admits an enlargement of type I or II. Using Table 2 it is easy to enumerate all the exceptional homogeneous spaces.

Corollary. Let M be a non-exceptional homogeneous space of a connected simple compact Lie group G and suppose the action of G on M is asystatic. Then this action does not admit any proper compact connected enlargement. The exceptional asystatic homogeneous spaces are (up to local similarity) the following ones:

S02n-t!Un- 1, S04n-t!SPn-1 SPll S04n-t!SPn_1 Ub S04n/SPn_1 SP1 SPn/SPn-1 U1 = cp2n-1, Spin7/G2 = S7, Sping /Spin7 = S15,

S015/Spin7' S016/Spin7' Spins/G2 = S7 x S7, G2 /SU3 = S6.

4.7. Groups of Isometries of Riemannian Homogeneous Spaces of Simple Compact Lie Groups. Let M be a homogeneous space of a compact connected Lie group G, which acts on M effectively. By Corollary 1 of Theorem 3.8 there exist on M Riemannian structures invariant under the action of G. For any such structure a, the corresponding group of isometries Aut (M, a) is a Lie transformation group of M (see Chap. 1, Corollary of Theorem 2.11). This group is compact (see 3.4). Thus, we have a compact enlargement of the action of G on M. Using Theorem 4.2, its corollaries, Theorem 4.7 and its corollary we obtain the following statements.

Theorem 4.8. Let a be an invariant Riemannian structure on a homogeneous space M = G / H, where G is a simple compact connected Lie group. If G / H is non-exceptional, then

Aut (M, a)o = GC (locally direct product),

where C c (AutcM)o. If we exclude the case when M = G and a is biinvariant then Aut (M,a) C SimcM and

Aut (M, a) = GB,

where B c A ~ Aut (G,H).

Corollary 1. If, in addition, the action of G on M is asystatic, then

Aut (M,a)O = G,Aut(M,a) = GB,

where B cAe Aut (G, H).

Corollary of Lemma 4.2 implies

Corollary 2 (see D'Atri and Ziller 1979). Let a be a left-invariant (but not bi-invariant) Riemannian structure on a connected simple compact Lie group G. Then G C Aut (M,a) C HolG i. e., Aut (M,a) = I(G)B,B c AutG.

Now we apply these results to determine the group of isometries of the so called natural Riemannian structure. Let H be a Lie subgroup of a connected simple compact Lie group G and let beg be their tangent algebras. Consider the inner product on 9 invariant with respect to Ad G (which is unique up to a positive factor). Then 9 = bEBm where m = bl... Restricting the inner product to m we obtain, by Corollary 1 of Theorem 3.8, an invariant Riemannian structure 0"0 on M = G / H. It is called the natural Riemannian structure. It follows from Lemma 4.2 that 0"0 is invariant under SimcM. Using Theorem 4.8 and investigating thoroughly the exceptional homogeneous spaces, we come to the following result.

Theorem 4.9. Let G be connected, simple and compact, M = G / H simply connected and 0"0 the natural Riemannian structure on M. Then

Aut (M, 0"0)0 = G(AutcM)o locally direct product

Aut (M, 0"0) = SimcM = G· Aut (G, H),

except for the following cases:

a) M = S6 = G2/SU3 , Aut (S6, 0"0) = 0 7 ; b) M = S7 = SpindG2 , Aut (S7, 0"0) = Os c) M = S7 X 8 7 = SpindG2 Aut (S7 x S7, 0"0) = (Os x Os) )<l (s), where

s is the transposition of factors; d) M = G with the action by left translations, where 0"0 is the bi-invariant

Riemannian structure, Aut (G, 0"0) = (HoI G) )<l (s), s : g -+ g-1 for g E G.

A homogeneous space M = G / H is called isotropy-irreducible if its isotropy representation is irreducible (over JR). On such a homogeneous space, any invariant Riemannian structure differs from the natural one by a positive scalar factor. The classification of isotropy-irreducible homogeneous spaces was obtained, in the compact case, in (Manturov 1966) and, in the general case, in (Wolf 1968), independently. As the action of G on an isotropy-irreducible space G / H must be asystatic, we get

Corollary (Wolf 1968). Let 0" be an invariant Riemannian structure on an isotropy-irreducible simply connected homogeneous space M = G / H, G being connected, simple and compact. Then

Aut (M, 0")0 = G,

Aut (M, 0") = SimcM = G· Aut (G, H),

except for the cases a), b) of Theorem 4.g.

4.8. Groups of Automorphisms of Simply Connected Homogeneous Compact Complex Manifolds. Let M be a connected compact complex analytic manifold. By the classical theorem of Bochner and Montgomery (Corollary 2 of Theorem 2.12 of Chapter 1) the group Aut M of all bi-holomorphic transformation of the manifold M is a complex Lie group and its action on M is analytic. The manifold M is called homogeneous if this action is transitive. In


this sub-section we shall only describe the group Aut M in the case when M is simply connected, referring the reader to Part IV of Encycl. Math. Sc. 10 for a survey of the general theory.

Simply connected compact complex manifolds were studied in (Wang 1954). The most important class of these are flag manifolds, i.e. manifolds of the form M = G / P, where G is a connected linear complex algebraic group and P a parabolic subgroup (which is semi-simple when the action of G is locally effective). They can also be characterized as simply connected homogeneous compact complex manifolds which admit the structure of an algebraic variety (automatically projective). For flag manifolds M the group Aut M coincides with the group of all bi-regular (polynomial) automorphisms and is semisimple.

Let G = GI ... Gr be a factorization of a group G into simple factors. Then P = PI ... PTl where Pi C Gi, and M = MI X ... x Mr, where Mi = Gd Pi' We have AutO M = AutO MI x ... x AutO Mr and each AutO Mi is simple.

Theorem 4.10 (Onishchik1962). Let M = G/P where G is a connected simple complex Lie group which acts effectively on M and P a parabolic subgroup. Then AutO M = G, with the exception of the following cases:

1) M = Cp2n- l , G = PSP2n(C) ~ SP2n(C)/{±E} (projective action), AutO = PGL2n (C).

2) M = Q5 is the quadric in CP6 given by the equation Zo 2 + ZI2 + ... + Z6 2 = 0, G = G2 C S07 (projective action), AutO M = S07'

3) M is the manifold of all n-dimensional totally isotropic subspaces in C2n+l, equipped with a non-singular bilinear form, G = S02n+l, AutO M = PS02n .

Proof. The proof of the theorem reduces to finding all factorizations Q = GH, where Q is a simply connected complex Lie group, G a simple Lie subgroup, and H a parabolic subgroup. By Corollary 2 of Theorem 4.4 and the Corollary of Theorem 2.2 we have m(G) = m(Q), i.e. the pair G C Q belongs to the list of Theorem 2.2. The final listing of possible cases is done by passing to factorizations of maximal compact subgroups (Corollary 1 of Theorem 4.4) and using Theorem 4.5. 0

We describe now the full group Aut M for a flag manifold M. We can assume that M = G / P, where G = AutO M and P is a parabolic subgroup. Let T C B be a maximal torus and suppose the Borel subgroup of G is contained in P. Denote by r the group of automorphisms of the system of simple roots of the group G (corresponding to the pair T C B). As a subgroup of Aut (G, P) the group r acts on M by similitudes (see 4.2). It is clear from Lemma 4.2 that SimcM C Aut M, and Corollary 2 of Theorem 4.1 implies the converse inclusion. More precisely, the following statement is true.

Theorem 4.11 (Kantor 1974, OnisCik 1981). The group Aut M coincides with SimcM, where G = (Aut M)O, and admits the semi-direct product decomposition Aut M = G Xl r.


Now let M be an arbitrary simply connected homogeneous compact complex manifold. By Corollary 4 of Theorem 3.10, M = GIH, where G is a connected semi-simple complex Lie group and H a connected complex Lie subgroup. As was shown in (Tits 1962) (see also §1 of Chap. 5) the subgroup P = N c (H) is parabolic, Le., M = G I P is a flag manifold.

Theorem 4.12 (Onishchik 1962). Assume that G = (Aut M)o. Then AutcM = PI H is a compact complex torus. We have (Aut M)O = G·AutcM (locally direct product) and Aut M = SimcM = G . Aut (G, H).

Chapter 3 Actions of Compact Lie Groups

The theory of compact Lie transformation groups is a large and well researched area of mathematics, in which topological methods are widely used and which requires a separate survey. In this chapter we only touch upon certain problems of this theory. In §1 we shall survey the classical results of Gleason, Montgomery, Mostow, Palais, Samelson and other authors concerning the structure of of orbits of actions of compact Lie groups. We emphasize the differentiable aspects of the theory and a number of results can be formulated in a more general setting - for proper actions of not necessarily compact Lie groups. §2 is devoted to properties of differentiable invariants and almost-invariants of compact Lie groups. In §3 we consider complexifications of homogeneous spaces of compact Lie groups and factorizations of reductive algebraic groups. Surveys of many questions not considered here can be found in (Bourbaki 1982), (Bredon 1972), (Hsiang 1975), (Janich 1968), (Palais 1960), (Schultz 1984).

§l. The General Theory of Compact Lie Transformation Groups

1.1. Proper Actions. A continuous action of a Lie group G on a Hausdorff topological space M is called proper if the mapping () : G x M ----+ M x M given by the formula (}(g, x) = (x, gx) is proper (Le. the set (}-l(Z) is compact for any compact subset Z eM). Proofs of the following properties of proper actions can 'be found in (Bourbaki 1960):

If H is a Lie subgroup of G then the reduction of a proper action of G to H is also proper

Stabilizers of a proper action are compact. The orbit space MIG of a proper action of G on M is Hausdorff and the

orbits are closed in M.


An action of a Lie group G on M is proper if and only if for any x, y E M, x #- y there exist neighbourhoods V", :3 x and Vy :3 Y such that the set {g E GI gV", n Vy #- O} is relatively compact in G.

Example 1. An action of a compact Lie group is always proper.

Example 2. Transitive actions are proper if and only if their stabilizers are compact.

Example 3. If a is a Riemannian structure on a manifold M the natural action of the Lie group Aut (M, a) (see the corollary of Theorem 2.11 of Chap. 1) on M is proper. Conversely, if there is a proper action of a Lie group G on a manifold M, then M possesses a G-invariant Riemannian structure, which can be assumed complete (Alekseevskij 1979).

1.2. Existence of Slices. Let G be a Lie group, M a differentiable G-space, x E M. A submanifold ScM is called a slice through the point x if it satisfies the following conditions:

a) GS is open in M; b) G",(S) = S; c) the differentiable G",-space S is isomorphic to an open ball in some

Euclidean space with an orthogonal linear action of G",; d) the mapping f.L : G xG. S --> M, given by the formula f.L(G",(g, s)) =

gs (g E G, s E S), is a diffeomorphism of the manifold G xG. S onto GS. We note that the mapping f.L (defined whenever condition b) is satisfied) is

always a morphism of G-spaces. Thus if S is a slice through a point x, then f.L is an isomorphism ofthe differentiable G-space G xG. S onto the G-invariant neighbourhood GS of the point x.

Theorem 1.1 (see Bourbaki 1982). For a proper differentiable action of a Lie group G on a manifold M every point x E M has a slice.

In the case when G is compact this result is due to Koszul. Montgomery and Yang generalized it to the case of topological G-spaces (see Bredon 1972, Hsiang 1975, Palais 1960).

Note that Theorem 1.1 contains the following assertion: if T is a proper differentiable action of a Lie group G on a manifold M, then for any x E M the orbit G(x) is a closed submanifold of M and the mapping T'" : GIG", --> G(x) of 1.4 of Chap. I is a diffeomorphism (for a compact Lie group G see Theorem 2.3 of Chap. 1 of Part I).

Proof. Let us sketch the proof of the theorem in the case when G is compact. Making use of left-invariant integration on G (see, for example, Lemma 3.2 of Chap. 2), one can prove that M has a G-invariant Riemannian structure. For any point y E G(x) we set Ny = Ty(G(x))1- (the orthogonal complement in Ty(M) with respect to the Riemannian structure). Then the normal bundle N over the orbit G(x), whose total space has the form


N = U Ny c T(M), is a G-vector bundle and hence is isomorphic to yEG(x)

G xGz N x (see 3.1 of Chap. 2). Let Be = U Y E G(x)Be,y, where Be,y is the open ball of radius e > 0 with center 0 in Ny. Then, Be c:::' G XGz Be,x and for sufficiently small e the exponential mapping Exp arising from our Riemannian structure, maps Be diffeomorphically and G-invariantly onto a neighbourhood of the orbit G(x). The slice at the point x is given by the submanifold S = ExpBe,x' D

By considering a slice S through x one can reduce investigation a G-space M in an invariant neighbourhood of the point x to investigation of the Gx -

space S. Of course, this is of interest only in the case when Gx -# G, Le. when x ~ MG (in the case x E MG one can take as S a small G-invariant neighbourhood ofthe point x). Theorem 1.1 makes it possible to prove certain theorems on actions of compact Lie groups by induction on the dimension of the group, and its number of connected components.

Example 4. Consider the standard linear action of the group G = On on M =]Rn. As a slice at a point x -# 0 we can take the ray S = {cx I c > o}.

Example 5. Let G = M = Un with G acting on itself by inner automorphisms. Let T be the maximal torus in G consisting of all diagonal matrices, Treg C T the set of regular (that is possessing mutually distinct eigenvalues) diagonal matrices. If x E T reg , then Gx = T, and as slice S at x we can take a small neighbourhood of the point x E T lying in T reg . In the case x E T \ Treg

a slice looks more complicated. For example, let n = 3 and x = diag(1, 1, eo), where leol = 1, eo -# 1. Then Gx = U2 X UI consists of matrices of the form

where A E U2 , lei = 1, and a slice through x is a subset of Gx of the form S = V x W, where V is a small neighbourhood of the unit matrix in U2 ,

invariant under inner automorphisms and W CUI a neighbourhood of the point eo not containing eigenvalues of matrices from V.

1.3. Two Fiberings of an Equi-orbital G-space. In this subsection we assume that G is compact and that M is a differentiable G-space which is equi-orbital (Le. all of its orbits are of the same orbit type (see 1.3 of Chap. 1). From Theorem 1.1 we deduce

Theorem 1.2 (see Bredon 1972). If G is compact and all orbits of the Gspace M are isomorphic to G / H, then the natural mapping M ~ M / G is the projection of a differentiable bundle with fibre G / H and structure group AutG(G/H) c:::' NG(H)/H. Conversely, every such bundle can be obtained from an equi-orbital G-space.


Corollary. If a compact Lie group G acts freely on M then the natural mapping M ~ MIG is a projection of a principal bundle of structure group G.

Borel (see Borel 1960, Bredon 1972) revealed another way of fibering an equi-orbital G-space M. Let Qx = NG(Gx) (x EM). Clearly Qx sends MG z

to itself.

Theorem 1.3. If G is compact and the G-space M is equi-orbital, then for any fixed Xo EM the mapping J1,: G xQzo MGzo ~ M, which sends Qxo(g, x) into gx is an isomorphism of differentiable G-spaces.

Example 6. If the group G acts transitively then the Borel bundle MGzo

~ M ~ GIQxo' obtained from Theorem 1.3 coincides with the principal bundle with structure group AutGGIGxo considered in 1.3 of Chap. 2.

As an example of an application of the above fibre bundles we note the following result, the proof of which makes use of techniques of algebraic topology.

Theorem 1.4 (see Bredon 1972). Let a compact Lie group G act differentiably on a sphere sn. If dim G > ° and all the orbits have the same orbit type, then the action is either transitive or free. In this second case the group G is isomorphic to either UI or SU2 or the normalizer NSV2 (Ud.

1.4. Principal Orbits. Let G be a Lie group, M some topological or differentiable G-space. Recall that two orbits G(x) and G(y) have the same orbit type if the subgroups G:x and G y are conjugate in G, i.e. if G(x) and G(y) are isomorphic as G-spaces (see 1.1 of Chap. 2). Denote by 5)(G) the set of isomorphism classes of homogeneous differentiable G-spaces (which can also be interpreted as the set of conjugacy classes of Lie subgroups of G). Then there is a map T : MIG ~ 5)(G), which assigns to each orbit its orbit type. We introduce the following partial order in 5)(G): MI ~ M2 if the homogeneous space MI admits a morphism into M2 , i.e. if the stabilizer HI of some point Xl E MI is conjugate to a subgroup of the stabilizer H2 of a point X2 E M2 (see Theorem 1.1 of Chap. 2). Clearly in 5)(G) there is a largest element (a simply transitive action) and a least element (the action on a singleton set).

Suppose that the action of G on M admits a slice through a point x E M (see 1.2). From property c) of a slice it follows that the mapping T is semicontinuous from below at the point G(x), i.e. that x possesses a G-equivariant neighbourhood U in M such that T(G(y)) ~ T(G(X)) for all y E U. From Theorem 1.1 we deduce

Theorem 1.5 (see Bourbaki 1982). Suppose we are given a proper differentiable action of a Lie group G on a manifold M, such that MIG is connected. Then the mapping T has a largest value on MIG i. e. there exists an orbit G(xo), Xo EM such that for any point x E M the subgroup Gxo is conjugate to some subgroup lying in Gx . The union Mo of all orbits of the largest orbit


type is open and dense in M, and Mo/G is connected. The set MG z intersects every orbit.

Note that the theorem remains valid for topological actions of compact Lie groups on manifolds (see the article of Montgomery in Borel 1960).

The largest orbit type is also called the principal orbit type and the corresponding orbits and stabilizers principal orbits and principal stabilizers. Clearly, principal orbits have the largest possible dimension. Orbits of lower dimension are called singular, and non-principal orbits of maximal dimension exceptional orbits.

From the definition of a principal orbit it follows that the normal bundle of a principal orbit of a differentiable action is trivial. Hence a principal orbit G(x) possesses an invariant neighbourhood isomorphic to G(x) x B.,x (see 1.1).

Example 7. For the action of Example 4, all orbits, with the exception of the fixed point 0, are principal - they consists of all the possible spheres with centre at O.

Example 8. Let G be a connected compact Lie group, T a maximal torus, r = dim T = rk G. Consider the action of G on itself by inner automorphisms. Then dimGx = dimZ(x) ~ r for all x E G. An element x EGis called regular if dimGx = r and singular if dimGx > r. Clearly, the set B of all singular elements coincides with the union of all singular orbits. Note (see Helgason 1962), that dim B S dim G - 3, where the equality holds for all semi-simple groups G (cf. Theorem 1.8 of the following sub-section). An orbit G(x) is principal if and only if Gx is connected or, equivalently, is a maximal torus of G; the principal orbit type is G/T. It is known (see Dynkin and Onishchik 1955), that exceptional orbits exist if and only if the commutator subgroup of G is not simply connected. For example, for G = Un all orbits of regular elements are principal (see Example 5), and in G = 803 the element x = diag(l, -1, -1) is regular, but it has an exceptional orbit isomorphic to JRP2 •

Example 9. In the notation of Example 8, consider the adjoint linear action of G on its tangent algebra g. An elements x Egis called regular if the centralizer of the subgroup {exp tx I t E lR} in G has dimension r. Principal orbits are precisely the orbits ofregular elements, they have orbit type G/T. There are no exceptional orbits.

1.5. Orbit Structure. Let G be a compact Lie group and M a differentiable G-space. For any Lie subgroup H c H consider the set

M(H) = {x E M I T(G(X)) = G/H}

- the union of all orbits of orbit type G / H. With the help of Theorem 1.3 we can prove

Theorem 1.7 (see Bredon 1972). The set M(H) is a submanifold in M, the natural mapping M(H) --t M(H)jG is a projection of a fibre bundle with fibre GjH and structure group Nc(H)jH. Ifx E M(H) then T(G(X)):::; GjH.

Next, we assume that MjG is connected, and put

B(H) = {x E M(H) IdimG(x) < dimGjH},

E(H) = {x E M(H) IdimG(x) = dimGjH,T(G(x)) < GjH}.

If G j H is a principal orbit type, then B(H) and E(H) are unions of all singular orbits and all exceptional orbits respectively.

Theorem 1.8 (see Bredon 1972). The sets B(H) and E(H) are closed in M. We have

dimB(H) :::; dimM(H) - 2, dimE(H):::; dimM(H) - 1.

If H1 (M,Zz,) = 0, GjH is a principal orbit type and GjH is connected, then dimE(H) :::; dimM - 2.

We shall now examine the question of the number of orbit types. From Theorem 1.1 follows

Theorem 1.9 (see Bredon 1972, Palais 1960). Any compact subset of M intersects only a finite number of orbits of different orbit type. If M is compact or M is an Euclidean space with a linear action of G, then the number of orbit types is finite. In the general case it is at most countable.

The following deep theorem is due to Mann (see (Borel 1960), (Bredon 1972)).

Theorem 1.10. Let M be an orientable manifold whose homology groups Hi(M, Z) are finitely generated. Then the number of orbit types of any action of a compact Lie group on M is finite.

1.6. Linearization of Actions

Theorem 1.11. Let G be a compact Lie group. A differentiable G-space M admits a differentiable equivariant embedding into a finite-dimensional linear G-space if and only if G has only a finite number of orbit types in M.

Proof. In the case of a compact manifold M the shortest proof of Theorem 1.11 is given in (Mostow 1957) and consists of applying Lemma 1.1 and Theorem 2.1 of §2. 0

The first proof of Theorem 1.11 for compact manifolds M was given in (Palais 1957b). In (Mostow 1957) its topological variant was proved, under the assumption that M is a finite-dimensional separable metric space. In the above formulation its proof is sketched in (Bourbaki 1982, Exercise 8 of §9), where, as a matter of fact, a more general situation is considered: G is a Lie group, which admits a faithful linear representation and the action is proper.


Let G be a Lie group, V a finite-dimensional linear G-space over R Let us denote by Q(V) the class of differentiable spaces M with the following property: any point x E M possesses a G-invariant neighbourhood, which admits a differentiable equivariant embedding into V t \ {O}, where V t is the direct sum of t copies of the G-module V, for some t > O.

Theorem 1.12 (Wasserman 1969). Let G be a compact Lie group and V a finite-dimensional linear G-space. If M E Q(V) then M is isomorphic to a closed G-invariant submanifold in V t for some t > O.

Theorems 1.11 and 1.12 can be viewed as generalizations of classical theorems about embedding into Euclidean spaces, which can be obtained from them in the case of trivial action. Next we shall consider the analogue of Theorem 3.12 for G-vector bundles, which generalizes Theorem 3.1 of Chap. 1.

Let again V be a finite-dimensional G-module, where G is a Lie group. Let us denote by 8(V) the class of differentiable G-vector bundles]Rm -+ E -+ M such that the Gx-module Ex for any x E M is isomorphic to a submodule of the Gx-module vm.

Theorem 1.13 (Wasserman 1969). Let G be a compact Lie group, V a finite-dimensional real G-module, ]Rm -+ E -+ M a vector bundle from the class 8(V). Then for any t 2 m + dim M + 1 the bundle E is induced by the tautological bundle over Gm(Vt ) with the help of a certain morphism of Gspaces M -+ Gm(Vt ). Here two morphisms M -+ Gm(vt) induce isomorphic vector G -bundles if and only if they are homotopic in the class of morphisms of G-bundles.

Corollary. Let G be a compact Lie group, ]Rm -+ E -+ M a differentiable G-vector bundle, with M having only a finite number of orbit types and for any subgroup H C G among the H-modules E"" x E M, H = Gx , there is only a finite number of non-isomorphic pairs. Then there exists a finitedimensional G-module V such that E is induced by the tautological bundle over Gm(V) with the help of a morphism M -+ Gm(V).

1.7. Lifting of Actions. Let p : E -+ B be a differentiable principal bundle with structure group A and suppose there is given a differentiable action of a Lie group G on B. A lifting of this action is an action of G on E, such that g(xa) = (gx)a for all g E G, x E E, a E A and p is a morphism of actions. In particular, existence of a lifting turns E into a G-bundle. Suppose the group A is abelian, let a be its tangent algebra and let q, be the space of differentiable mappings B -+ a. To any action of G on B there corresponds some linear representation of its tangent algebra 9 on the space q, (see 2.2 of Chap. 1). The problem of lifting of the action leads to consideration of a certain obstruction which is a cohomology class in H2(g, q,). Making use of properties of cohomology of semi-simple Lie groups and passing from abelian to solvable structure group, we obtain the following result.

156 v. V. Gorbatsevich, A. L. Onishchik

Theorem 1.14 (Palais and Stewart 1961b). Let E -7 B be a differentiable principal fibre bundle, whose structure group is a connected solvable Lie group, and let G be a simply connected compact Lie group. Then any action of G on B admits a lifting.

§2. Invariants and Almost-Invariants

2.1. Applications of Invariant Integration. Let G be a compact Lie group, E -7 M a differentiable G-vector bundle over the field lR. or C. As was noted in 3.5 of Chap. 2, we can apply Lemma 3.2 of Chap. 2 to the space f(E) of Coo -sections of E, i.e. there exists a linear continuous averaging operator I : f(E) -7 f(E) with respect to the group G, which projects f(E) onto the subspace f(E)G of invariant sections. As above, we denote by f(E)G the space of almost-invariant sections. With the help of the operator I we can prove

Lemma 2.1. Let N be a closed submanifold of M, which is invariant under to G. Then the restriction mapping f(E)G -7 f(E I N)G and f(E)G -7

f(E I N)G are surjective.

Corollary. Orbits of a differentiable action of a compact Lie group G on M are separated by the elements of the algebra F(M)G. In particular, an action is transitive if and only if all invariant functions are constant.

The second important application application of invariant integration over a group is

Theorem 2.1. If G is a compact Lie group and S a topological G-module, then SG = S. In particular, f(E)G = f(E) in the COO-topology for any differentiable G-vector bundle E.

Proof. In the case when S = F( G) and the structure of a G-module is defined by the action of G on itself by left translations, this theorem coincides with the classical Peter-Weyl Theorem (see Pontryagin 1973, Chevalley 1946). In the general case, consider the continuous bilinear mapping (¢, s) f--t ¢ * s of the space F(G) x S into S, given by the formula

¢ *S = i ¢(g)(gs) dg

where the integration is done with respect to a left invariant density. One can check that any fixed s E S lies in the closure of the set {¢ * s I ¢ E F( G)} and that ¢ * S E SG, where ¢ E F(G)G. 0

2.2. Finiteness Theorems for Invariants. For algebraic invariants of a compact linear group there is a well known principle of finite generation, first formulated by Hilbert. It turns out that the analogous statements hold also for COO-invariants.

As is well known, the subalgebra of invariants JR[V]G of the algebra of polynomial functions JR[V] on G has a finite system of generators </>1,"" </>r. The mapping </> : V ---+ JRr given by the formula </>( v) = (</>1 (v), ... , </>r (v)), is constant on orbits and hence induces a mapping ¢ : VjG ---+ JRr . The image Q = </>(V) = ¢(V j G) is a closed semialgebraic set in JRn .

Theorem 2.2. (Schwarz 1975, Mather 1977). The mapping ¢ : VjG ---+ Q is a homomorphism. There exists a continuous linear mapping a: F(V)G ---+

F(JRr) such that </>* 0 a = id. In particular, F(V)G = </>* F(JRr ). Thus, every invariant function of class coo on V has the form f (</>1, ... , </>r), where f is a Coo -function on JRr.

Using Theorem 1.11 we obtain

Corollary. Let G be a compact Lie group and M a compact differentiable G-space. There exist functions </>1,"" </>r E F(M)G such that the mapping

induces a homeomorphic mapping of the space M j G onto the closed set Q = 1m</> c JRn, with F(M)G = </>* F(JRr ).

Example 1. Consider the linear action of the group G = Z2 on M = JR, in which the generator g E G acts by the formula gx = -x. Then F(M)G is the algebra of all Coo-functions on R According to Theorem 2.2 every even function has the form f(x 2 ), where f E F(JR). The algebra of all even periodic COO-functions with period 211', can be identified with the algebra F(Sl )Z2, where Z2 acts on the circle Sl = {(x, y) E JR2 I x2 + y2 = I} by the formula g(x, y) = (x, -y). This algebra consists of functions of the form f(x) = f(cos t), where f E F(JR) and t is the angle parameter on Sl. Suppose now that, in addition to a linear action of a compact Lie group G on V, we are given a linear action of the same group on a finite-dimensional vector space W over the field K = JR or C. Consider the trivial G-bundle E = V x W with base V and fibre W (see Example 9 of §3 of Chap. 1). The space r(E) contains the subspace P(V, W) of all G-invariant polynomial mappings V ---+ W. Its invariants form a finitely generated module P(V, W)G over the algebra JR[V]G.

Theorem 2.3 (Onishchik 1976). In the JR[V]G -module P(V, W)G there is a system of generators U1, . .• ,Un, which generates the module r(E)G over the algebra F(V)G ® K of invariant K -valued functions.

Corollary. Let G be a compact Lie group, E ---+ M a differentiable G-vector bundle over the field K with compact base M. Then the module r(E)G over the algebra F(M)G ® K of invariant K -valued functions is finitely generated.

Example 2. Consider the trivial ~-bundle E = Sl X JR with base Sl and fibre JR, on which ~ acts as in Example 1. Then r(E)Z2 can be identified with the space of all odd periodic Coo functions on JR with period 211'. This is a module over the algebra of even functions with generator u = sin t. Note


that the module of odd continuous periodic functions is not finitely generated (Onishchik 1976).

2.3. Finiteness Theorems for Almost Invariants. From Lemma 2.1 and Theorem 1.11 we deduce

Theorem 2.4 (Onishchik 1976). Let G be a compact Lie group and M a compact differentiable G-space. Then the algebm F(M)G has a finite system of genemtors over F(M)G.

Note that as generators of the algebra F(M)G we can take a coordinate system in the Euclidean space in which M is embedded by linearization of the action.

Corollary. If M is a homogeneous space of a compact Lie group G, then the algebm F(M)G is finitely genemted over R

Let now E -t M be a differentiable G-vector bundle over the field K = JR or C. Clearly, the space of almost-invariant sections r(E)G is a module over the algebra F(M)G ® K.

Theorem 2.5 (Onishchik 1976). If G is compact and E satisfies the conditions of Theorem 1.13, then r(E)G is a finitely genemted projective (see Vol. 11 of the Encycl. Math. Bc., §21) module over F(M)G ® K.

§3. Applications to Homogeneous Spaces of Reductive Groups

3.1. Complexification of Homogeneous Spaces. Let G be a compact Lie group, F(G, C)G = F(G)G ® C is the algebra of complex valued almost invariant functions on the group G, which acts on itself by left translations. According to the corollary of Theorem 2.4. the algebra F(G,C)a is finitely generated over the field C. As is well known, every finitely generated associative algebra A with unit and without nilpotent elements over the field C is isomorphic to the algebra of polynomial functions over some (uniquely determined) affine algebraic variety Spec A, the elements of which are algebra homomorphisms a : A -t C such that a(l) = 1. Let G(C) = SpecF(G,C)a. Then G(C) is a reductive linear algebraic group (see Chevalley 1946). Assigning to each element x E G the element ax E G(C), given by the formula

(1)

we obtain an embedding x f---+ ax ofG into G(C). The image of this embedding is the real form of the group G(C), distinguished by the conjugation a f---+ a, where

a(¢) = a(4»)). (2)

In this way G acquires the structure of a linear algebraic group over the field JR, with G(C) as its complexification. To every linear representation

p of G on a finite-dimensional vector space V over IR there corresponds a polynomial linear representation p(C) : G(C) -) GL(V(C)) on the complex space V(C) = V0C, and moreover p(C) is exact if p is exact. Every reductive linear complex algebraic group Q is isomorphic to the group G(C), where G is a maximal compact subgroup of Q.

Suppose now that we are given a transitive action of a compact Lie group G on a differentiable manifold M. The algebra F(M, C)G = F(M)G 0C of complex almost invariant functions on M is finitely generated by the corollary to Theorem 2.4. Consider the affine algebraic variety M(C) = Spec F(M, C)G. Just as for the group G, one defines an embedding x f-+ ax of the manifold Minto M(C), where ax is given by the formula (1), and also the complex conjugation a f-+ a in M(C), given by the formula (2).

We now define an action of G(C) on M(C). For this purpose we note that the natural linear action of G on F(M, C)G extends to a linear action of G(C). The required action on M(C) is defined by the formula

h~)(</» = ~h-1</» h E G(C),~ E M(C),</> E F(m,C)G))' (3)

Theorem 3.1 (Onishchik 1960, Iwahori and Sugiura 1966, Onishchik 1976). Formula (3) defines a polynomial transitive action of the algebraic group G(C) on M(C). For any point x E M we have G(C)x = Gx(C). The submanifold Me M(C) coincides with the set {~ E M(C) Ie = o·

Thus every homogeneous space M = G / H, where G ~ H are compact Lie groups, is a real form of the affine algebraic complex homogeneous space

M(C) = G(C)/ H(C).

Conversely, any homogeneous space Q / R, where Q ~ R are reductive complex linear algebraic groups, can be identified with M (C), where M = G / H, G ~ H are maximal compact subgroups of the groups Q ~ R.

We note that the complexifications of homogeneous spaces of compact Lie groups described above exhaust all affine complex algebraic homogeneous spaces (and also all homogeneous complex Stein manifolds) of reductive groups; see Part IV, §5 of Vol. 10 of the Encycl. Math. Sc.

3.2. Factorization of Reductive Algebraic Groups and Lie Algebras. From Theorem 3.1 and Corollary 1 of Theorem 4.3 of Chap. 2 we deduce the following statement

Theorem 3.2 (Onishchik 1969). Let (G, G', G") be a factorization of a compact Lie group G as product of its Lie subgroups G', and G". Then the triple (G(C), G' (C), G" (C)) is a factorization with G'(C) n G"(C) = (G' n G")(C). Conversely, if (Q, Q', Q") is a factorization of a reductive algebraic group Q over C as product of reductive algebraic subgroups Q', Q"" then there exists an element g E Q such that the factorization (Q, Q' , gQ" g -1) can be obtained by complexification of some factorization of a maximal compact subgroup of Q.


Corollary. Suppose that (G, G', Gil) is a factorization of a connected reductive complex algebraic group G as product of connected virtual Lie subgroups and that the subgroup G' n Gil has a finite number of connected components. Then G = G~G~, where G~, G~ are maximal reductive algebraic subgroups of the groups G', Gil respectively.

The condition on G' n Gil is satisfied, in particular, in the case when G is semi-simple. With the help of Theorem 3.2 and its corollary the problem of describing factorizations of complex Lie groups as products of complex Lie subgroups reduces to the problem of describing factorizations of compact Lie groups (for the latter see 4.6 of Chap. 2).

A subalgebra ~ of a complex Lie algebra 9 is called reductive in 9 if ~ is reductive as a Lie algebra (Le. rad ~ coincides with the centre of ~) and for any z E rad ~ the operator ad z in 9 is semi-simple. Analogously one defines a reductive subalgebra ~ c 9 in the real case; here one demands that ad z be semi-simple in g(C) for all z E rad~.

By making use, in the complex case, of Theorem 3.1, and of Lemma 4.5 of Chap. 2 in the real one, we can prove the following theorem, which generalizes Theorem 4.1 of Chap. 2.

Theorem 3.3 (Onishchik 1969). Every factorization of a reductive (complex or real) Lie algebra 9 as sum of two (complex or real) subalgebras reductive in 9 is global.

In (Onishchik 1969) there is given also a description of all factorizations of semi-simple real Lie algebras g, as sums of two subalgebras which are reductive in g.

Chapter 4 Homogeneous Spaces of Nilpotent

and Solvable Groups

Homogeneous spaces of nilpotent and solvable Lie groups are called nilmanifolds and solvmanifolds respectively. In addition it is assumed that the transitive Lie group is connected. At this time no comprehensive classification of nil- and solvmanifolds is known, as there is no classification of nilpotent and solvable Lie algebras. In this chapter we give a survey of results, which reduce the study of arbitrary nil- and solvmanifolds to the compact case, and also of results about topological construction of compact nil- and solvmanifolds and about transitive actions of Lie groups on them.


§l. Nilmanifolds

1.1. Examples of Nihnanifolds. The simplest example of a nilmanifold is a connected abelian Lie group, acting on itself by left translations. In particular, the torus Tn is a compact nilmanifold. We note that numerous works are devoted to attempts to generalize classical properties of tori to various classes of nilmanifolds. For example, one studies harmonic analysis, dynamical systems on nilmanifolds, the theory of approximations of number systems, etc.

Example 1. The simplest non-trivial examples of compact nilmanifolds are Iwasawa manifolds Nn(lR)/Nn(Z) and Nn(C)/Nn(Z[i]), where Z[i] is the ring of Gaussian integers and for any ring R with unity Nn(R) denotes the group of all unipotent upper triangular matrices whose entries are elements of the ring R. Generalizations ofIwasawa manifolds N3 (lR)/N3 (Z) are compact nilmanifolds N3(lR)/rp,q,Tl where the subgroup rp,q,r, with fixed p,q,r E N consists of all matrices of the form

k p

1

o

~) p·q·r ~ , where k,l,m E Z.

1.2. Topology of Arbitrary Nihnanifolds. In studying nilmanifolds one often makes use of the fact that any simply-connected nilpotent Lie group G possesses a real algebraic group structure, which is determined by a faithful unipotent linear representation ¢ : G ~ Nn(lR) and does not depend on the choice of this representation (see Auslander 1973).

The most fundamental results of this and the following subsection are due to A.1. Maltsev (Maltsev 1949, see also Auslander 1973).

Theorem 1.1. An arbitrary nilmanifold M is diffeomorphic to M* x lRn , where n ~ 0 and M* is some compact nilmanifold.

Proof. Let M = G / H, where G is a simply connected nilpotent Lie group (see 2.1 of Chap. 2). Consider the algebraic (Le. in the Zariski topology) closure a H of H in G, with respect to the above mentioned algebraic structure. One can show that the manifold M* = a H / H is compact and G /a H is diffeomorphic to lRn. The natural fibre bundle

M* = aH/H ~ G/H = M ~ G;aH

is smoothly trivial (being a bundle with contractible base), so that M is diffeomorphic to M* x lRn. 0

Remark. By means of the method used in the sketch of the proof of Theorem 1.1 given above, one can show that for any nilmanifold M there exist n, lEN such that M x lRl is diffeomorphic to the nilmanifold Nn(lR)/r, where r is some discrete subgroup in Nn(lR) (isomorphic to 71'1 (M)) and


1 = dim Nn(lR) - dim M, i. e., as it is sometimes said, M is stably diffeomorphic to the manifold Nn(lR)/r.

1.3. Structure of Compact Nihuanifolds

Theorem 1.2. Let M = NIH be a compact homogeneous space of a nilpotent Lie group N. If the action of N on M is locally effective, then the subgroup H is discrete.

Proof. We can suppose that the Lie group N is simply-connected. Let us consider the natural algebraic structure on it. From the compactness of M it easily follows that a H = N, hence HO is normal in N. From the local effectiveness of the action of N on M we see that HO = {e}. D

Theorem 1.2 reduces the study of compact nilmanifolds to the study of lattices in nilpotent Lie group (which are considered in Part I of Encycl. Math. Sc. 21).

Let r be a lattice in a nilpotent simply-connected Lie group N. Then ar = N and moreover, it turns out that, as an algebraic group, N is defined over Ql, and the lattice r is commensurable with N(Z), i.e. r n N(Z) is a subgroup of finite index in r and in N(Z) (this means that r is an arithmetical subgroup in N). These facts together with other properties of lattices in nilpotent Lie groups lead to the following result.

Theorem 1.3. Let M = N Ir be a compact nilmanifold, where r is a lattice in a simply-connected nilpotent Lie group N (by Theorem 1.2 any compact nilmanifold can be represented in the form N Ir). Then (i) The pair (N, r) is determined by the group 71"1 (M) uniquely up to isomorphism (understood in the obvious sense). In particular, the manifold M is determined by its fundamental group uniquely up to a diffeomorphism}.

(ii) The groups 7I"dM) for all compact (and, by Theorem 1.1 also noncompact) nilmanifolds M are precisely all finitely generated torsion-free nilpotent groups.

From Theorem 1.3 (i) it follows, in particular, that the compact nilmanifold M* in Theorem 1.1 is determined by the manifold M (and even by its fundamental group) uniquely up to diffeomorphism, for 71"1 (M) is isomorphic to 71"1 (M*).

We now consider some corollaries of the above results.

Corollary 1. Let M = NIH be a nilmanifold such that the group 71"1 (M) is abelian. Then 71"1 (M) ~ zm for some m 2: 0 and the manifold M is diffeomorphic to r x IRn - m (where n = dim M), i. e. to some connected abelian Lie group.

Note that the Lie group N acting transitively on M may be non-abelian.

Corollary 2. Let M be a nilmanifold such that 71"1 (M) is a (nilpotent) group of rank 3 (for rank or Hirsch number see 3.1 of Chap. 4 of Part I in Encycl.


Math. Sc. 21). Then M is diffeomorphic to 11'3 X lRn - 3 or M* x lRn - 3 , where M* = N3(lR)/f1,1,r (r E N) is a three-dimensional compact nilmanifold, described in Example 1.

Proof. The point is that any finitely generated nilpotent group of rank 3 without torsion (as is the case with 7rl (M)) is isomorphic to 7i} or the group

{(I k !!:!)

f1,1,r= ~ ~ ~ k'l,mn:}

for some r E N (in particular, any group f p,q,r from Example 1 is isomorphic to f1,1,r). Hence the assertion of Corollary 2 follows from Theorem 1.3. D

Note that if M is a nilmanifold and 7rl (M) is a group of rank ::; 2 then 7rl (M) is abelian and Corollary 2 is applicable.

Corollary 3. Any nilmanifold is parallelizable.

Proof. Let M be a compact nilmanifold; it then has the form M = N If, where N is a nilpotent group and f is a lattice in N. A parallelization of such M is induced by the natural left-invariant parallelization of the Lie group N. By Theorem 1.1 any nilmanifold is parallelizable. D

Let f be some lattice in a nilpotent simply-connected Lie group N. The exponential mapping exp : n --7 N (where n is the tangent algebra of the Lie group N) is a diffeomorphism. The subset D = exp-l (f) is discrete in n and one can show (beginning with the uniformity of f in N), that the linear envelope (D)'Jt of this subset coincides with n. The rational linear envelope (D)Q determines a Q-structure on the space n and one can easily check that (D)Q is a Lie algebra over Q. The Q-structure on n thus obtained corresponds (under exp) to the above mentioned Q-structure on the Lie group N, i.e. exp( (D)Q) = N(Q).

Theorem 1.4. Let M = N If be a compact nil-manifold, where f is a lattice in the simply-connected nilpotent Lie group N. Then the lattice f naturally defines a Q-structure on the tangent algebra n of the Lie group N, and this Q determines the manifold M uniquely up to a finite covering.

Proof. The lattice f is a subgroup of the group N(Q) of rational points and is commensurable with the subgroup of integral points N (Z) of the group N. Moreover, all lattices in N which lie in N(Q), are comensurable with one another and hence the homogeneous spaces corresponding to them coincide, up to passage to a finite covering, with M = N If. D

The problem of Q-structures on nilpotent Lie groups and algebras is considered in greater detail in Part I of Encycl. Math. Sc. 21.

1.4. Compact Nilmanifolds as Towers of Principal Bundles with Fibre 11'. Let M = N If be a compact nilmanifold where f is a lattice in a simply-


connected nilpotent Lie group N. If Z(N) is the center of the group N, then r n Z(N) is a lattice in the abelian simply-connected Lie group Z(N) (see Raghunatan 1972 or Part I of Encycl. Math. Sc. 21). Choose an element 'Y Ern Z(N), 'Y i= e and let C be the (unique) one-parameter subgroup of N which passes through 'Y. Then C C Z(N). Consider the natural bundle

r . C Ir -4 N Ir -4 N Ir . C.

This is a principal fibre bundle with structure group

r . C Ir ~ C Ir n C ~ 1I' ~ S02.

The base N Ir·c is also a compact nilmanifold, and the same construction can be applied to it. As a result we see that the manifold N Ir can be obtained from a point by way of constructing a sequence of principal bundles with fibres 1I' or, as is sometimes said, is a tower of such principal bundles.

Conversely, one can prove (for one special case see Palais and Stewart 1961a), that the space corresponding to a tower of principal bundles with fibre 1I' is diffeomorphic to some compact nilmanifold. Note that the length of the tower (the number of stages) is equal to the rank of the (nilpotent) group 7fl(M), and is also equal to dimM.

§2. Solvrnanifolds

An arbitrary solvmanifold has the form M = G I H, where G is a connected solvable Lie group and H a closed subgroup. Sometimes solvmanifolds are referred to as solvable manifolds, but we shall not make use of this term as it will be used in Chap. 5 for objects of a considerably larger class.

The class of solvmanifolds is much larger than its subclass of nilmanifolds. The study of solvmanifolds (even the compact ones) meets with noticeably greater obstacles than the study of nilmanifolds, however during the last thirty years a large number of significant results were obtained in this direction, many of them close to being definitive (see the survey Auslander 1973).

2.1. Examples of Solvmanifolds. The group of proper motions of the Euclidean plane E(2)O is solvable and factors as a semi-direct product E(2)O = S02 IX: ]R2. We shall show that the Mobius band Mb is diffeomorphic to ]Rp2 \ {xo}, where Xo E ]Rp2 is an arbitrary point. The duality between points and lines on the projective plane leads to the diffeomorphism of the manifold ]Rp2 \ {xo} and the manifold of all straight lines in the Euclidean space E2 (to a straight line in E2 given by the equation ax + by + c = 0, corresponds, under this diffeomorphism, the point [a : b : c] E ]Rp2, and as the point Xo one takes [0 : 0 : 1]). Thus Mb can be identified with the manifold of all straight lines on the plane E2, which is endowed with a natural transitive action of the group E(2)o. Considering motions of the plane E2


as projective transformations taking the line at infinity to itself, we identify E(2)0 with the subgroup of GL3 (IR), consisting of all matrices of the form

(

COS 2rrc - sin 2rrc a) sin 2rrc cos 2rrc b , where a, b, c E IR

o 0 1

Then a group model of the solvmanifold Mb takes the form E(2)0 H, where H is the subgroup of matrices of the form

(±1 0 a) o ±1 0 , o 0 1

in particular, HO ~ IR, H / HO ~ ~. The manifold Mb is the total space of a bundle IRl ----t Mb ----t 'lI'. However

Mb is not diffeomorphic to IRl x'lI', hence it is not a nilmanifold (see Theorem 1.1). From this it follows that Theorem 1.1 does not transfer to solvmanifolds (however, Theorems 2.1 and 2.2 below serve as analogous results).

Example 2. The action of E(2)0 on Mb described in the previous example is easily seen to be asystatic. However, it is not primitive since it leaves invariant, for instance, the following equivalence relation between lines in the plane: h rv l2 if the lines hand l2 are parallel and the distance between them is an integer. The set of equivalence classes is diffeomorphic, as can be easily checked, to the Klein bottle K2 and is a homogeneous space of the group E(2)0. A group model of this space has the form E(2)0 / iI, where the subgroup iI consists of all matrices of the form

o a) ±1 n , o 1

where a E IR,n E Z.

Since the manifold K2 is non-orientable it is not parallelizable and hence is not a nilmanifold (see Corollary 3 of Theorem 1.3).

Example 3. Besides the action of itself, the torus T' for n ~ 2 admits also transitive and effective actions of non-abelian solvable Lie groups. In order to construct such actions, consider some torus T (Le. a compact connected abelian subgroup) in the group SOk and form the (solvable) Lie group G = T ~ IRk, embedded in the natural way in the group of proper motions E( k)O = SOk ~ IRk of the Euclidean space Ek. Let H be a closed subgroup of IRk such that the group S = IRk / H is compact. The manifold S is a torus, hence so is the manifold G / H which is diffeomorphic to T x S. If H does not contain nontrivial subspaces invariant under T then the action of G on G / H is locally effective.

In the simplest case k = 2 we have G = E(2)0. Writing elements of this group as matrices, just as in the previous examples we take as H the subgroup of matrices of the form


o ~ n, where a E R, n E Z.

Then the group G acts locally effectively on the manifold G / H which is diffeomorphic to ']['2. From the previous example we see that there exists a G-invariant double covering ']['2 --t K2.

2.2. Solvmanifolds and Vector Bundles. As we saw from the example of the Mobius band, a solvmanifold may not factor as a direct product of ]Rn

and a compact solvmanifold. However, we have the following

Theorem 2.1 (Mostow 1954). For any solvmanifold M there exists a solvmanifold M' which is a finite covering of M and is diffeomorphic to M* x ]Rn,

where M* is some compact solvmanifold.

Further, there holds the following result, put forward as a conjecture by Mostow in the early fifties.

Theorem 2.2 (see Auslander 1973). Any solvmanifold M is diffeomorphic to the total space of some vector bundle

~ : ]Rk --t M --t M*

over a compact solvmanifold M*. The structure group of this bundle may be reduced to ~, where l :::; ~.

Note that by no means every total space of a vector bundle over a compact solvmanifold is itself a solvmanifold. For example, let M* = ']['2 and consider the vector bundle ~ : ]R2 --t M --t ']['2, whose characteristic class C1 E H2 (']['2, Z) ~ Z is non-trivial. It is easy to verify that for any finite covering M*' --t M* the induced vector bundle (over M*') is non-trivial. From this, by Theorem 2.1 (or 2.2), follows that the total space M of the bundle ~ cannot be a solvmanifold.

We shall now consider in detail vector bundles corresponding to solvmanifolds, and also the conditions for total spaces of these bundles to be diffeomorphic.

Theorem 2.3 (Auslander and Szczarba 1975a). Let M* be a compact solvmanifold. The total space of a vector bundle ~ over M* is a solvmanifold if and only if ~ is equivalent to the direct sum Al EB .•• EB Ak of line bundles Ai, for which the first Stiefel- Whitney class WI (Ai) is a reduction mod 2 of elements of infinite order in HI (M* , Z).

As a matter of fact, Theorem 2.3 is a consequence of a more general result (Auslander and Szczarba 1975a), which relates the study of solvmanifolds with the study of flat (i.e. having a discrete structure group) bundles over a torus. Namely, for any compact solvmanifold M* one can construct in a canonical way a bundle M* ---- 1r' over the torus 1r' (where r = b1 (M*) is

p


the first Betti number of M*). Then the vector bundles over M* whose total spaces are solvmanifolds are exactly the bundles induced from flat bundles over T r with the help of the mapping p. Further, with certain restrictions on M* the solvmanifolds thus obtained are determined uniquely up to diffeomorphism by the Stiefel-Whitney classes WI and W2 of the corresponding vector bundles over Tr. These restrictions are satisfied, for example, by M* = ']["" . In this case there is a complete description of the corresponding solvmanifolds up to diffeomorphism.

Example 4. Consider the factorization 1r' = 1I' x ... x 1I', and let Pi : Tn ~ Tl be the projection on the i-th factor. Let x denote the generator in Hl(lI', Z2) ~ ~ and set Xi = p;(x) E Hl(1r', ~).

To every element of HI there corresponds (up to equivalence) exactly one line bundle over 1r', for which this element coincides with the first StiefelWhitney class. Let ~i (1 ::; i ::; n) be line bundles over 1r' for which WI (~i) = Xi. It is not difficult to give a geometric description of ~i' For example, for n = 1 we obtain a line bundle ~~ over 1I', whose total space is diffeomorphic to Mb and for n > 1 the bundle 6 can be induced from ~~ by the map p. We derive now the following vector bundles over 1r':

ak = 6 EB ... EB 6k (1::; 2k ::; n),

13k = ak EB (6 ® ... ® 6k) (1::; 2k ::; n),

'Yk = 13k EB 6k+l (1::; 2k ::; n - 1).

Let X k, Yk, Z k be total spaces of the bundles ak, 13k, 'Yk respectively. From Theorem 2.3 it follows that Xk, Yk, Zk are solvmanifolds. For example, the manifold Xk is diffeomorphic to

Mb x ... x Mb xr-2k • , ' v 2k

The fundamental group of each of the manifolds X k, Yk, Zk is isomorphic to zn. It turns out that the following theorem holds.

Theorem 2.4 (Auslander and Szczarba 1975a). Let M be a solvmanifold with an abelian fundamental group. Then M is diffeomorphic to W x ]Rl,

where W is one of the solvmanifolds X k, Yk, Zk.

The number l 2: 0 and the solvmanifold Ware determined by M uniquely. There is also a result analogous to Theorem 2.4 in the case when 11"1 (M) is

a free nilpotent group (for example when 1I"1(M) ~ N3(Z)), (Auslander and Szczarba 1975b).

2.3. Compact Solvmanifolds (The Structure Theorem). We now turn to consideration of arbitrary compact solvmanifolds. One of the basic results here is the following, sometimes referred to as Mostow's structure theorem.

Theorem 2.5 (Mostow 1954, see also Auslander 1973). Let M = R/ H be a compact homogeneous space of a solvable Lie group R and suppose that the


action of R on M is locally effective. If N is a nilradical in R then HO c N and H n N is a uniform subgroup of N.

With the notation of this theorem, consider the natural bundle

H . NIH -+ RI H -+ RI H . N.

The fibre is the com pact nilmanifold H . NIH = NIH n N. Moreover, the Lie group A = RIN is abelian, hence the base ofthis fibre bundle is diffeomorphic to the torus Air, where r = HI H n N. The bundle we have just constructed, known as the Mostow bundle, is not in general principal. For example, let K2 = E(2)O IiI be the Klein bottle (see for Example 2). The Mostow bundle for K2 has, as one can easily check, the form l' -+ K2 -+ 1'. If it were principal then the group 11"1 (K2) would be nilpotent, which is not the case.

Considering the towers of principal bundles with fibre l' (see 1.4) corresponding to the fibre and base of the Mostow bundle, we see that any compact solvmanifold M is obtained from a point by constructing a sequence of bundles with fibre l' (not necessarily principal). The converse does not hold, i.e. not every tower of bundles with fibre l' is diffeomorphic to a compact solvmanifold.

The fibre and base of the Mostow bundle, and also their dimensions, depend on the choice of the Lie group which acts transitively on M.

2.4. The Fundamental Group of a Solvmanifold. Note that solvmanifolds are aspherical (i.e. 1I"i(M) = 0 for i 2': 2). Hence the fundamental group 1I"1(M) plays a singular role in the study of them.

Theorem 2.6 (Mostow 1954, see also Auslander 1973). A compact solvmanifold is determined by its fundamental group uniquely up to diffeomorphism.

Note that, unlike in the case of compact nilmanifolds (see Theorem 1.3 (i)), for a compact solvmanifold M = RI H the pair (R, H) is not always determined uniquely by the group 1I"1(M) (for M = T' see Example 2.3).

Corollary 1. For each solvmanifold M the compact solvmanifold M*, over which, by Theorem 2.2, M fibres, is uniquely determined by the group 11"1 (M) up to diffeomorphism.

Corollary 2. If M is a compact solvmanifold and the group 11"1 (M) is abelian then M is diffeomorphic to a torus. If the group 11"1 (M) is nilpotent then M is diffeomorphic to some nilmanifold.

The following theorem, which generalizes Corollary 2, gives a topological classification of compact solvmanifolds among all compact homogeneous spaces.

Theorem 2.7 (Gorbatsevich 1977c). Let M be a compact aspherical homogeneous manifold. Then

(i) If the group 11"1 (M) is nilpotent, then M is diffeomorphic to some nilmanifold, which is unique up to diffeomorphism.


(ii) If the group 7r1(M) is solvable then M is homeomorphic to some solvmanifold, which is unique up to diffeomorphism.

The proof of this theorem is based on the study of transitive actions of Lie groups on compact aspherical manifolds.

The basic assertion of part (ii) of Theorem 2.7 holds, it seems, in the stronger form: "M is diffeomorphic to to some solvmanifold", but so far this has not been proved.

We shall now consider in detail the construction of fundamental group of a solvmanifold.

Let M be an arbitrary compact solvmanifold, and L ----+ M ----+ T8 its Mostow bundle (here L is a compact nilmanifold). The homotopy sequence of this fibre bundle gives us the following exact sequence

{e} ----+ ~ ----+ 7r1(M) ----+ 'J'.,8 ----+ {e}, (*)

where ~ = 7r1 (L) is a finitely generated nilpotent torsion-free group (Theorem 1.3 (ii)).

The groups 7r which can appear in an exact sequence of the form (*), where s 2: 0 and ~ is some finitely generated nilpotent torsion-free group, are called Wang groups; they are considered in greater detail in Part I of Encycl. Math. Sci. 21. For us the following result is important.

Theorem 2.8 (Wang 1956, Auslander 1973). A group 7r is isomorphic to the fundamental group of a solvmanifold if and only if 7r is a Wang group.

Wang groups are clearly polycyclic. The converse is, in general, not true (see Raghunathan 1972), hence there exist polycyclic groups without torsion, which are non-isomorphic to the fundamental group of any solvmanifold.

2.5. The Tangent Bundle of a Compact Solvmanifold. The following theorem proves that the tangent bundle T(M) ----+ M of a compact solvmanifold M is very close to being trivial. Nevertheless, in general, it is not trivial (for dim M 2: 5), since the Stiefel-Whitney classes of M may be non-zero (Auslander and Szczarba 1962).

Theorem 2.9 (Auslander and Szczarba 1962). Let M be a compact solvmanifold. Then

(i) The complexification of the tangent bundle of M is a trivial bundle. (ii) The manifold M is linearly parallelizable, i.e. there exist on it n (=

dim M) smooth fields of tangent lines, which are linearly independent at every point.

Corollary (Auslander and Szczarba 1962). All Pontryagin classes Pi(M) of an arbitrary solvmanifold vanish.

2.6. Transitive Actions of Lie Groups on Compact Solvmanifolds. The problem of describing all transitive actions of Lie groups on a given compact solvmanifold was solved by Auslander (Auslander 1973). For an arbitrary


such M he presented an explicit, though rather complicated, construction of of these actions. The construction is based on the notion of a semi-simple splitting of solvable Lie groups (including non-connected ones), which is considered in §3 of Chap.2 Part II in Encycl. Math. Sc. 21 (for more details see Auslander 1973). Of a number of concrete applications of Auslander's approach we note the following assertion.

Theorem 2.10 (Johnson 1972). Let M be a compact solvmanifold, then there exists transitive and effective actions on M of solvable Lie groups of arbitrarily large dimension.

For the case M = 1r" the statement of the theorem follows easily from the construction given in Example 3.

Another approach to describing transitive actions of solvable Lie groups on compact manifolds was developed by M. V. Milovanov (Milovanov 1980). It is based on a description of all simply-connected homogeneous solvable Lie groups R, containing a given group H as uniform (see 1.1 of Chap. 5) subgroup (note that 7rl(R/H) ~ H/HO). One also makes use of the concept of algebraic splitting (for which see also Mostow 1971, Raghunathan 1972), a variant of semi-simple splitting. With the help of this description one proves, in particular,

Theorem 2.11 (Milovanov 1980). Let H be some solvable Lie group (generally speaking, non-connected). Then there exists at most a finite number of mutually non-isomorphic, connected simply-connected solvable Lie groups R, which contain uniform subgroups isomorphic to the given H.

In (Milovanov 1980) there is given an explicit construction which allows for a given H to construct all possible Lie groups R.

2.7. The Case of Discrete Stabilizers. The study of compact solvmanifolds of the form M = R/r, where r is a discrete subgroup, is, in effect, the study of lattices in solvable Lie groups. For a survey of results about these lattices see Part I of Volume 21 of the Encycl. Math. Sc. where, in particular, a construction of certain Lie groups containing a lattice isomorphic to a given group is described, as well as a criterion for existence of at least one lattice in a given solvable Lie group, etc.

As is shown by the example of the Klein bottle (Example 2), there exist compact solvmanifolds which cannot be represented in the form R/r, where r is a discrete group (since r is orient able and K2 is not). Nevertheless, there holds

Theorem 2.12 (Auslander 1973). Let M be a compact solvmanifold. Then there exists a solvmanifold M', which is a finite covering of M and has the form M' = R' /r', where r' is a lattice in some simply-connected solvable Lie group R'.

This theorem admits a generalization to arbitrary aspherical compact homogeneous manifolds (see Theorem 7.2 in Chap. 5)

2.8. Homogeneous Spaces of Solvable Lie Groups of Type (I). A connected Lie group R is called a group of type (I), if for any g E R the adjoint linear operator Ad g in the tangent algebra t of the Lie group R, has only eigenvalues of modulus 1. Tangent algebras of such groups are characterized by the fact that all eigenvalues of the linear operators ad x, x E t, are pure imaginary numbers. For more details concerning these groups see Part 1 of Encycl. Math. Sc. 21.

Let R = K I>< N be a semi-direct product of a nilpotent group N and a torus K. Clearly R is a solvable Lie group of type (I). Conversely, one can show that any connected solvable Lie group of type (I) can be embedded as a subgroup into a Lie group of this type.

If H is a uniform subgroup in a solvable Lie group R of type (I), then we can check that the fundamental group of the solvmanifold R/ H is almost nilpotent (Le. contains a nilpotent subgroup of finite index). Moreover, we have

Theorem 2.13 (Auslander 1973). (i) Let M be a homogeneous space of a solvable Lie group R of type (I). Then the group 1fl (M) is almost nilpotent.

(ii) Let M be a compact homogeneous space of a solvable Lie group R' and suppose the group 1fl (M) is almost nilpotent. If the action of the group R' on M is locally effective then R' is type (I).

As an example consider again the Klein bottle K2; it is a homogeneous space of the Lie group E(2)O = 1[' I>< ]R2 of type (I). The group 1fl (K2) is not nilpotent; it contains a subgroup of index 2 isomorphic to Z EB Z.

2.9. Complex Compact Solvmanifolds. Complex homogeneous spaces of solvable complex Lie groups are called complex solvmanifolds. It turns out that the properties of compact complex solvmanifolds are close to properties of compact real solvmanifolds.

Theorem 2.14 (Barth and Otte 1969). Let M = R/ H be a compact complex homogeneous space of a solvable complex Lie group R whose action on M is locally effective. Then

(i) The subgroup H is discrete. (ii) H n (R, R) is a lattice in (R, R). (iii) The subgroup H is nilpotent if and only if the Lie group R is nilpotent.

Statement (ii) of this theorem is not explicitly formulated in (Barth and Otte 1969), but can be easily deduced from Lemma 2.4 given there.

We also note that a compact complex homogeneous space M is a solvmanifold if and only if it is aspherical, i.e. 1fi(M) = 0 for i > 1 (this can be deduced from (Barth and Otte 1969) and (Gorbatsevich 1983)).


Chapter 5 Compact Homogeneous Spaces

In this chapter we shall study homogeneous spaces G / H, under the assumption of their compactness. This assumption is trivially satisfied if the group G is compact, but it imposes very strong restrictions on the subgroup H in the case when G is not compact. The problem of describing the Lie subgroups H c G such that G / H is compact (uniform Lie subgroups) is considered in §1. In connection with this problem we consider the real analogue of a remarkable fibre bundle introduced by Tits in the case of complex compact homogeneous spaces (Tits 1962, see also Part IV of Encycl. Math. Sc. 10). In §2 we give a survey of results on classification of transitive actions on compact manifolds with a finite fundamental group. The remaining part of this chapter is devoted to the study of compact homogeneous spaces of general type, based on certain natural fiberings of these spaces, which make it possible to clarify their topological structure. The majority of results of this chapter hold, sometimes with obvious changes, for a wider class of homogeneous spaces than the compact ones, namely: for plesi-compact homogeneous spaces. A homogeneous space M = G / H of a Lie group G is called plesicompact (Gorbatsevich 1988), if there exists a uniform subgroup P of G, containing H and such that the space P/H has a finite P-invariant measure.

§l. Uniform Subgroups

1.1. Algebraic Uniform Subgroups. A subgroup H of a Lie group G is called uniform (or cocompact) if there exists a compact subset Q c G such that G = Q H. A Lie subgroup H eGis uniform if and only if G / H is compact.

It is known that an algebraic subgroup H of a complex linear algebraic group is uniform if and only if H is a parabolic subgroup, i.e. contains a maximal connected solvable subgroup of the group G. The analogous statement in the real case has the following form (Borel and Tits 1965): an algebraic subgroup H of a real linear algebraic group G is uniform if and only if H contains a maximal connected subgroup of the group G, the elements of which can be, in some base, simultaneously represented by triangular matrices.

When studying uniform subgroups of an arbitrary real Lie group it is convenient to introduce the following analogue of the concept of an algebraic subgroup. A subgroup H of a Lie group G is called Ad-algebraic if H = Ad-1 (H*), where H* is a certain (real) algebraic subgroup ofGL(g). Clearly, an Ad-algebraic if H = Ad-1 (H*), where H* is a certain (real) algebraic subgroup of the group GL(g). Clearly, an Ad-algebraic subgroup of G is a Lie subgroup. If G is a real algebraic group then any Ad-algebraic subgroup


of it is an algebraic subgroup in the usual sense, containing the centre of G. The intersection of any family of subgroups of a Lie group G is an Adalgebraic subgroup. From this it follows that for every subgroup H of a Lie group G there exists the smallest Ad-algebraic subgroup H a , containing H.

Example 1. Let H be a virtual Lie subgroup of the Lie group G. Then NG(HO) is an Ad-algebraic subgroup, so that Ha c NG(HO). Indeed, NG(HO) = {g E G I (Adg)1J = IJ}, where IJ egis the tangent algebra of the subgroup H.

A subgroup A eGis called triangular if all operators Ad a (a E A) can be simultaneously represented by triangular matrices in some basis of the tangent algebra 9 of G. It is known that all maximal connected triangular subgroups of of a Lie group are conjugate to each other (see Vinberg 1961, Mostow 1961). A subgroup of G is called a t-subgroup if it contains a maximal connected triangular subgroup of G. Analogously, (with the help of the adjoint representation ad) one defines triangular subalgebras and t-subalgebras of a Lie algebra g. Moreover, connected Lie t-subgroups and Lie t-subalgebras are in a natural correspondence.

Theorem 1.1. Every Ad-algebraic uniform subgroup H of a Lie group G is at-subgroup.

Proof. From the fact that H is Ad-algebraic it follows that there exists a linear representation G ----t GL(V), such that one of the orbits of the corresponding projective action of G on P(V) is isomorphic to G / H. A maximal connected triangular subgroup A C G has a fixed point in this compact orbit (Vinberg 1961) and hence is conjugate to subgroup of H. D

Corollary 1. If H is a uniform subgroup of a Lie group G, then Ha is a t-subgroup.

Corollary 2. If H is a uniform virtual Lie subgroup of G, then NG(HO) is at-subgroup.

In the radical Rad G of a Lie group G there is a largest connected triangular subgroup T, which is thus normal in G. From the triviality of the subgroup T it follows that G is semi-simple.

Corollary 3. Let H be a uniform Lie subgroup of a Lie group G, where the action of G on G / H is locally effective. If (Ha)O = HO then G is semi-simple. In particular, this holds if NG(HO)O = HO, i.e. if the action of the group G on G / HO is asystatic.

Corollary 4. Let H be a uniform virtual Lie subgroup of a Lie group G, possessing a finite number of connected components. Then there exists in G a maximal connected triangular subgroup A, such that (A, A) c HO.

Corollary 5. Let H be a uniform subgroup of a Lie group G, having a finite number of conneted components, with G acting on G / H locally effectively.


Then the largest connected triangular subgroup T of the group Rad G is isomorphic to IRP and coincides with the nilpotent radical of the group G . We have D2RadG = {e}.

1.2. Tits Bundles. Let H be a virtual Lie subgroup of a connected Lie group G. Since H C NG(HO), the set F = HNG(HO)O is a Lie subgroup of G, with H c F and FO = NG(HO)o. If H is a Lie subgroup of G then the homogeneous space G / H admits a natural fibre bundle structure (Tits bundle)

F/H -t G/H -t G/F. (1)

Its fibre F / H = NG(HO)O /(NG(HO)O n H) is connected and can be represented in the form F/H = N/r, where N = NG(HO)o/Ho, and r = (NG(HO)O n H)/ HO is a disq:ete subgroup of the connected Lie group N. Thus F / H is a parallelizable connected homogeneous manifold.

Theorem 1.2. Let M = G / H be a compact homogeneous space of a connected Lie group G, F = HNG(HO)o. Then in the fibre bundle (1) the fibre F / H is a connected compact parallelizable manifold, and the base G / F admits a transitive action of a compact Lie group. The Li~ group N = NG(HO)O is unimodular.

Proof. The fact that N is unimodular follows from the compactness of the manifold N /r = F / H, see Proposition 1.3 of Chap. 1 of Part I of Encycl. Math. Sc. 21. 0

Corollary 1. Let M be a connected compact homogeneous manifold. Then X(M) ~ 0. If x(M) > 0, then the fibre of the Tits bundle is trivial and 1l'1 (M) is finite.

Proof. The proof makes use of the following property of the Euler characteristic: if F -t E -t B is a fibre bundle whose total space is a compact manifold E and whose fibre F is connected, then X(E) = X(B)X(F) (see Serre 1951).

Corollary 2. If M = G / H is compact and X(M) -# 0, then HO = NG(HO)O and H is a t-subgroup of G. If the action is locally effective then the group G is semi-simple.

As can be seen from simple examples, neither the base nor the fibre of the Tits bundle can, in general, be determined uniquely by the homogeneous manifold M alone, but it depends on the choice of the Lie group G acting transitively on M.

1.3. Uniform Subgroups of Semi-simple Lie Groups. Let G be a connected semi-simple Lie group. We identify its tangent algebra 9 with the linear Lie algebra adg C g[(g) with the help of the faithful linear representation ad. This linear Lie algebra is algebraic (over the field IR), since it coincides with


the tangent algebra of the algebraic group Aut 9 (see Encycl. Math. Sc. 41, Ch. 1, 3.3). A subalgebra IJ egis algebraic if and only if IJ is the the tangent algebra of some Ad-algebraic subgroup H of G. For an arbitrary sub algebra IJ C 9 we denote by lJa a smallest algebraic subalgebra of the algebra 9 containing IJ.

Let H be a uniform virtual Lie subgroup of the group G. Then, by the Corollary 1 of Theorem LIlJa is a t-subalgebra in g, and by Theorem 1.2 the adjoint representation of the Lie algebra lJa /IJ has trace O. From this follows

Theorem 1.3 (Goto and Wang 1978). If H is a uniform virtual Lie subgroup of a connected semi-simple Lie group G, then IJ contains the unipotent radical n = {x E rad lJa I ad x is nilpotent} of the algebraic t-subalgebra lJa.

We note that under the assumptions of Theorem 1.3 a H = RN, where N is the connected normal Lie subgroup with tangent algebra nand R is a reductive (Le. with its radical contained in the centre) subgroup. From Theorem 1.3 it follows that H = (H n R) . N, where H n R is uniform in R. This statement is of interest only in the case when lJa f=. g, and for this it is necessary that the subgroup H should not be discrete. For a survey of results on discrete uniform subgroups see Part I of Vol. 21 of the Encyclopaedia of Mathematical Sciences.

We note also the following result, which can be deduced from Theorem 1.3.

Theorem 1.4 (Moore 1984). Let H be a non-discrete uniform Lie subgroup of a real semi-simple linear algebraic group G, not containing compact simple normal Lie subgroups. If HO is nilpotent, then HO coincides with the unipotent radical of some (uniquely determined) parabolic subgroup P of G. Moreover, HO is abelian if and only if G is locally isomorphic to the product of the groups SOl,n and P is the product of their minimal parabolic subgroups.

1.4. Connected Uniform Subgroups. This subsection is devoted to the description of connected uniform virtual Lie subgroups of semi-simple Lie groups. Let us first note the following lemma, which applies to arbitrary Lie groups.

Lemma 1.1 (Goto and Wang 1978). Let H be a connected virtual Lie subgroup and K a maximal compact subgroup of a connected Lie group G. The following conditions are equivalent: H is uniform in G; G = HK; 9 = IJ + t.

Let 9 be a real Lie algebra. To every subalgebra IJ C 9 there corresponds a connected virtual Lie subgroup H* of the group Int 9 of inner automorphisms (see 4.1 of Chap. 2 of Part I), the tangent algebra of which is the subalgebra ad IJ C g. We call the subalgebra IJ uniform in g, if the subgroup H* is uniform in Int g.

Lemma 1.2. Let G be a connected Lie group and H a connected virtual Lie subgroup of G. If the subgroup H is uniform in G, then the subalgebra IJ is


uniform in g. If G is a semi-simple Lie group with a finite center, then the converse also holds.

Corollary. A subalgebra ~ of a semi-simple real Lie algebra 9 is uniform in 9 if and only if 9 = ~ + t where t is a maximal compact subalgebra of 9 {i.e. the tangent algebra of a maximal compact subgroup of the group Int g.

As can be seen from Lemma 1.2, the first step in describing connected uniform subgroups of semi-simple Lie groups should be a description of uniform subalgebras of semi-simple Lie algebras. We shall give this description in terms of t-subalgebras, referring the reader to (Onishchik 1967) for the classification of the latter.

Every algebraic sub algebra ~ of a semi-simple Lie algebra 9 has the form ~ = t EB n, where t is a reductive algebraic subalgebra and n is the unipotent radical of ~. FUrther, t = .5 + 3, where .5 = [t, t] is a semi-simple ideal and 3 is the centre of the algebra t. Represent .5 in the form .5 = .5C EB .5n , where .5 C

is a compact (i.e. corresponding to a compact Lie group) ideal and .5n does not contain any non-zero compact ideals. We also have 3 = 3c EB 3n , where 3n

(respectively 3C ) is the subalgebra of all z E 3 such that ad z has only real (respectively purely imaginary) eigenvalues. We set t C =.5c EB t.

Theorem 1.5 (Onishchik 1967). Let ~o = tEBn be an algebraic t-subalgebra of a real semi-simple Lie algebra 9 and, with the notation introduced earlier, let m be a subalgebra of.5c EB 3 such that .5c EB 3 = t C + m. Then ~ = ~ EB.5n EB n is a uniform subalgebra in g. Any uniform subalgebra in 9 can be obtained by this method from some algebraic t-subalgebra.

From Theorem 1.5 it follows that every uniform sub algebra of a semisimple Lie algebra can be obtained from some t-subalgebra by means of a deformation of the centre of its reductive part.

Corollary 1. A subalgebra of a split real semi-simple Lie algebra is uniform if and only if it is parabolic.

Corollary 2. Let 9 be a complex semi-simple Lie algebra and ~ a real {complex} subalgebra of g. The subalgebra ~ is uniform in 9 if and only if it has the form ~ = m EB .5 EB n, where ~o = 3 EB .5 EB n is some parabolic subalgebra of the algebra 9 and m is a real {respectively complex} subalgebra ofJ such that 3 = 3c + m.

In view of Lemma 1.2, Theorem 1.5 gives a description of connected uniform virtual Lie subgroups of connected semi-simple Lie groups with finite centre. In particular, from Corollary 2 follows a description of connected uniform complex Lie subgroups in complex semi-simple Lie groups, first obtained in (Wang 1954).

If a semi-simple Lie group has an infinite centre, then to a uniform subalgebra of its Lie algebra there may correspond a non-uniform subgroup.


Example 2. Let SLn(lR) = A be the universal covering group of SLn(lR). According to Corollary 1, any proper uniform subalgebra of the the algebra S(2 (JR) is conjugate to the sub algebra of all triangular matrices. The corresponding triangular subgroup of the group SLn (JR) is uniform. At the same time, in A there exist no proper connected uniform virtual Lie subgroups, since the maximal compact subgroup of this group is trivial (see Lemma 1.1). The same property holds in any group An, where n > O.

If a connected semi-simple Lie group G has infinite centre, then the maximal compact subalgebra e of its tangent algebra 9 is not semi-simple. The description of the connected uniform subgroups of G is closely connected with the question: which uniform subalgebras ~ c 9 satisfy the condition 9 = ~ + fe, e]? For simple groups G this question is considered in (Gorbatsevich 1974), where, in particular, all uniform subalgebras ~ satisfying the above condition and such that ~ n fe, e] = 0 are found. A proof is given of the following

Theorem 1.6. Let G be a connected Lie group and Ko its maximal compact subgroup. Then in any connected uniform Lie subgroup U of G there exists a connected Lie subgroup H C U, such that G = KoH and Ko n H = {e}.

1.5. Reductions of Transitive Actions of Reductive Lie Groups. Let G be a connected reductive Lie group, which acts transitively on a compact manifold G / H. In this subsection we will be concerned with describing virtual Lie subgroups G' c G, which act transitively on M. Consider a more general situation, when G = G' H, where G' and H are virtual Lie subgroups, and H is uniform. A subgroup A eGis called compact in G if the corresponding subgroup Ad A c GL(g) has a compact closure. The following theorem was proved in (Onishchik 1977).

Theorem 1.7. Let (G, G' ,H) be a factorization, where G is a connected reductive Lie group, G', H its virtual Lie subgroups, with H uniform and not containing non-compact connected normal subgroups of G. Then the group G' is reductive and its radical is compact in G. If G is simple and non abelian and G'o does not contain connected normal subgroups G1 -I G'o such that G = G1H, then G' is also simple and non abelian.

Corollary 1. Let M = G / H be a compact homogeneous space, where G is reductive and acts on M locally effectively. If a virtual Lie subgroup G' C G acts transitively on M, then G' is reductive and its radical is compact in G. If G is simple and non abelian and G' acts transitively and irreducibly on M, then G' is simple and non abelian.

Corollary 2. Let (G, G', Gil) be a factorization of a simple Lie group G as the product of virtual Lie subgroups G' and Gil. Then either G' or Gil is reductive in G and its radical is compact in G.


Proof. If, for example, Rad Gil is non compact in G, then Gil is contained in a parabolic subgroup P of the group G. From Theorem 1.7 it follows that RadG' is compact in G. D

A factorization (G, G', Gil) of a Lie group G is called maximal, if G' and Gil are maximal connected Lie subgroups of G. For a semi-simple group G, a maximal connected subgroup is either reductive (and its radical is compact in G), or it is parabolic. The classification of all maximal decompositions of non-compact simple Lie groups is given in (Onishchik 1969). The works (Nazaryan 1975a), (Nazaryan 1975b), (Nazaryan 1981) are devoted to the classification of arbitrary factorizations of these groups.

§2. Thansitive Actions on Compact Homogeneous Spaces with Finite Fundamental Groups

Compact homogeneous spaces with finite fundamental groups (in particular simply connected ones) make up the most well researched class of compact homogeneous spaces. In this section we shall describe results on classification of transitive actions of Lie groups, and also on inclusions between transitive actions.

Let M = G / H be a compact homogeneous space, with G connected and 71'1 (M) finite. Let us denote by S a Levi subgroup of G, by Ko a maximal compact subgroup of S and by K a maximal compact subgroup of G containing Ko. According to Corollary 4 of Theorem 3.11 of Chapter 2, the subgroups (K, K) and S act transitively on M. Hence the subgroup Ko also acts transitively. Therefore, the problem of describing of transitive actions on M falls naturally into the following stages:

a) Describing transitive actions of connected compact (or connected compact semi-simple) Lie groups on M.

b) For any connected semi-simple Lie group S, describing enlargements of transitive actions of a maximal compact subgroup Ko on M to actions of S.

c) For any connected semi-simple Lie group S, describing enlargements of its transitive actions on M to actions of connected Lie groups, containing S as a Levi subgroup (enlargements of this type we shall call radical).

Moreover, all these actions can be assumed to be locally effective. It turns out that solutions to problems b) and c) can be obtained in the general case. A much more difficult problem, whose solution is known only for certain classes of manifolds, is presented by a). Here we shall give its solution for simply connected compact manifolds of rank 1, which will enable us to give a description of transitive actions of Lie groups on spheres. The basic results on transitive actions of compact Lie groups on manifolds of other classes can be found in (Onishchik 1963), (Hsiang and Su 1968), (Onishchik 1970), (Scheerer 1971), (Schneider 1973), (Schneider 1975), (Hsiang 1975), (Mkhitaryan 1981), (Shchetinin 1988), (Shchetinin 1990) (see also the sur-


veys (Alekseevskij 1974), (Alekseevskij 1979), (Vinberg 1963)). A survey of these results will be given in one of the following volumes of the present edition. We note that the first work devoted to this subject was the work of Montgomery and Samelson (Montgomery and Samelson 1943) on transitive actions of compact Lie groups on spheres, which to a substantial extent determined the direction and methods of future research.

2.1. Three Lemmas on Transitive Actions. In this subsection we shall give three general lemmas on compact homogeneous spaces G / H, where G is connected and H has a finite number of connected components. (the latter condition is satisfied if the group 7r1 (G / H) is finite). What these lemmas have in common is that both allow us to consider properties of transitive actions in terms of the topological structure of the manifold G / H. We note that by Theorem 1.2 of Chap. 2 the group of automorphisms of an arbitrary compact homogeneous space is a compact Lie group.

Lemma 2.1 (see Onishchik 1966). Let M = G / H be a compact homogeneous space of a connected Lie group G, with H having a finite number of connected components. Then

rk AutcM ~ h( G / HO),

where h is the homotopy characteristic (see 2.4 of Chap. 2).

For any reductive Lie group G let us denote by l (G) the number of its simple factors.

Lemma 2.2 (see Onishchik 1968). Let M = G/H be a compact homogeneous space of a connected reductive Lie group G, with H having a finite number of connected components. If G acts irreducibly on M, then l(G) ~ rk G / HO. In the general case we have a locally direct decomposition G = GOG1, where Go acts transitively and irreducibly, so that l(Go) ::; rkG/Ho and G1 is a Lie group with a compact tangent algebra, with rk G1 ~ h( G / HO). In particular l(G) ~ 2rkG/Ho

In the following lemma we denote by rk Z (G) the rank of the centre of G (viewed as a finitely generated abelian group).

Lemma 2.3 (OnisCik 1988). Let M = G/H be a compact homogeneous space of a connected semi-simple Lie group G, where G acts effectively on M and H has a finite number of connected components. Then

rkZ(G) ~ max{O,h(G/HO) -I}. (4)

Corollary. If under the assumptions of Lemma 2.3 h( G / HO) ~ 1, then G has a finite centre.

Example 1. Let M = SUn, n 2: 3. Then h(M) = n -1. Using the results of (Gorbatsevich 1974) (see Theorem 1.6 above) one can construct an effective action on M of the universal covering G = SU1,n of the group SU1,n' The


group G has an infinite centre of rank 1. In particular, the bound (4) IS

attained in the case n = 3.

2.2. Radical Enlargements. This subsection is devoted to problem c) formulated at the beginning of this section. Let again M = G / H be a compact homogeneous space with a finite fundamental group, with G connected and acting on M locally effectively. Let T be the largest connected triangular subgroup of the group Rad G, which by Corollary 2 of Theorem 1.1 is contained in Nc(HO), and in view of Corollary 5 of the same theorem is isomorphic to ]RP and coincides with the nilpotent radical of the group G. The tangent algebra of the group G can be naturally identified with a subalgebra 9 C tJ(M) and to the subgroup T corresponds a nilpotent radical t C 9. If the subgroup H is connected, then T C Nc(H) and T acts on M by automorphisms of the homogeneous space (see formula (1) of Chapter 2). To this action corresponds a commutative subalgebra a C u(M)).

Lemma 2.4. Suppose that H is connected. Then t C CPa, where cP is some finite-dimensional G-submodule in

F(M)U = {¢ E F(M) I v¢ = 0 for all v E a}.

In particular, t is a compact G-module.

Proof. The second statement of the lemma follows from Lemma 1.2 of Chap. 1. D

From this lemma one can deduce the following theorem, which shows that our assumptions impose fairly strong restrictions on the structure of G.

Theorem 2.1 (Vishik 1973, see also Onishchik 1977). Let G be a connected Lie group, which transitively and locally effectively acts on a compact manifold M = G / H, where 7fl (M) is finite. Then

G = G1 ~ N,

where G1 is a connected reductive Lie subgroup, N a vector Lie group, the action of G1 on N by inner automorphisms is compact and NCI = O. The tangent algebra of the subgroup N coincides with [9, rad 9J·

Proof. By passing to a finite covering G / HO of the manifold M, we can assume that the group H is connected. From Lemma 2.4 it follows that the nilpotent radical of the Lie algebra 9 is a compact G-module. Let s be a Levi subalgebra of the Lie algebra 9. One can show that the s-invariant complement c to the ideal n = [9, rad 9J in rad 9 is an abelian sub algebra. Putting 91 = s EB c, we obtain a semi-direct factorization 9 = 91 e- n. The corresponding decomposition of G is the one we are seeking. D

Corollary 1. Suppose that under the assumptions of Theorem 2.1 G =

SnSeC, where C = Rad G1 and Se (correspondingly Sn) is the product of all compact (non-compact) simple connected factors of the group (G1 ,Gd. Then the decomposition G = SnGO, where Go = (SeC) ~ N, is locally direct.


Corollary 2. In the notation of Corollary 1 we have

Corollary 3. If the subgroup Bn C G acts transitively on M then the group G is reductive.

Let us now consider how this result can be applied to the solution of problem c). Suppose that M = B/F is a compact homogeneous space of a connected semi-simple Lie group B, acting locally effectively on M, 1Tl(M) is finite and suppose that we have a locally effective action of a group G with B as a Levi subgroup, which extends the given action of B. The structure of the group G is described by Theorem 2.1, and moreover, one can assume that B C G1 • The enlargement B C G1 can be constructed easily with the help of 4.3 of Chap. 2 (it it a reduction of the natural enlargement of the group B). The enlargement G1 C G can be described with the help of Lemma 2.1 (here one has to pass to a finite covering of the manifold M). Conversely, if a is any commutative subalgebra in tl(M)G" then F(M)na is a G1-invariant commutative sub algebra in tl(M) Any finite-dimensional G1-submodule n C F(M)na, such that nG1 = 0, defines an action of the Lie group G = G1 ~ n on M, which is an enlargement of the given action of the group G1 • For the description of radical enlargements in terms of stabilizers see (Onishchik 1966), (Vishik 1973).

Let B = BnBe be a locally direct factorization, where Be is the largest connected compact normal Lie subgroup in Band Bn a connected normal Lie subgroup without compact factors. According to Corollary 3 of Theorem 2.1, if the subgroup Bn is transitive on M then the group G = G1 is reductive. In the contrary case, making use of Corollary 3 and Theorem 2.1 of Chap. 3 we obtain

Corollary 4. If M = B / F, the subgroup Bn does not act transitively on M and the action of B on M is asystatic, then there exists a radical enlargement of the action of the group B on M with a radical of arbitrarily large dimension.

2.3. A Sufficient Condition for the Radical to be Abelian

Theorem 2.2 (Onishchik 1966). Let M = K/L where K is a connected semi-simple Lie group, and let rkK AutK K / LO :S 1. If the action of K on M is a reduction of a locally effective action of some Lie group G :J K, then Rad G is abelian.

Note that in the case when rkAutKK/Lo = 0, i.e. when NK(LO)O = LO, the group G is semi-simple (see Corollary 3 of Theorem 1.1). Using Lemma 2.1 we obtain

Corollary. Let M be a compact manifold with a finite fundamental group, Sf its universal covering space, G a Lie group which acts transitively and

locally effectively on M. If h(M) = 0 then G is semi-simple and if h(M) = 1 then Rad G is abelian.

Example 2 (Onishchik 1966). Let M = S2n-l, n 2: 2. Since h(M) = 1, every Lie group acting transitively on M has an abelian radical. Such, in particular, are the radical enlargements of the standard action of SUn on s2n-l. As we saw in 1.1 of Chap.2 (Example 1.1), this action is systatic, with AutsunS2n-l consisting of transformations )"'E,)... E CX (using the standard inclusion s2n-l C cn), and its tangent algebra is the one-dimensional subalgebra a = CVo, where vo(z) = iz (z E S2n-l). The algebra F(s2n-l)o is naturally isomorphic to the function algebra F(cpn-l). For any finitedimensional SUn-submodule <[> C F(cpn-l) the space r = <[>a = <[>vo is a commutative sub algebra in l1(S2n-l) invariant under SUn. We obtain a locally effective action of the group SUn ~ r with abelian radical r on s2n-l. By Lemma 2.1 any radical enlargement of the action of SUn has this form. Here the radical r can have arbitrarily large dimension (cf. Corollary 4 of Theorem 2.1).

The following example shows that the conditions imposed on M in the corollary to Theorem 2.2 are essential.

Example 3 (Onishchik 1966). Let M = SU3 . Then h(M) = 2. Let K = U3 ,

L ~ U1 . be the subgroup of matrices of the form diag (I,E, 1), where lEI = 1. Then K = SU3 · L, SU3 n L = {e}, so that K/L = SU3 . Consider the group G = U 3 I>< C3 , where the representation of U 3 in C3 is the standard one, and the subgroup H, consisting of pairs of the form

(d' (1 27riRez1 ) ( )) lag ,E,e , Zl,Z2,Z3 ,

where lEI = 1, (Zl,Z2,Z3) E C3 . It is easy to see that G = KH, KnH = L and that H does not contain non-trivial normal subgroups of G. Thus there exists an effective action on SU3 of the group G, which is an enlargement of the given action of K = U 3. Moreover, Rad G = Z (U 3) I>< C3 is non-abelian.

2.4. Passage from Compact Groups to Non-Compact Semi-simple Groups. In this subsection we consider problem b), for simplicity limiting ourselves only to the case of connected stabilizers. Let G be a connected semi-simple Lie group and K a connected Lie subgroup corresponding to a maximal compact subalgebra t of its tangent algebra g. A connected Lie subgroup L C K will be called a G-subgroup if the standard action of K on K / L extends to an action of G. These subgroups possess a simple description. Namely, let G = K P be a Cartan decomposition of G. For any point pEP denote by Lp the identity component of the subgroup

Let Mp be the largest connected normal Lie subgroup in

Zc(p) = {g E G I gpg-l = p},


contained in K. Then we have a locally direct decomposition Lp = MpLp * where Lp * is a connected normal Lie subgroup of Lp.

Theorem 2.3 (Onishchik 1967). A subgroup L c K is a G-subgroup if and only if it has the form

L = MLp*,

where pEP and M is a connected Lie subgroup of Mp.

We can also describe (in terms of Lie algebras) all possible enlargements of the standard action of K on K / L to an action of the group G.

Corollary 1. If the tangent algebra of G is split then G -subgroups in K are subgroups of the form Lp, where pEP.

Corollary 2. Let G = K(C) be a connected semi-simple Lie group. A subgroup L c K is a G-subgroup if and only ifD1Z K (k)0 C L c ZK(k)O, where ZK(k) = {g E K I gkg- 1 = k} for some element k E K.

This corollary gives rise to the description of simply connected compact complex homogeneous spaces obtained in (Wang 1954) (cf. Corollary 2 of Theorem 1.5).

Corollary 3. Suppose that a G-subgroup L does not contain connected normal subgroups of the group K and is maximal among connected Lie subgroups of K possessing this property. Then L = L p, for some pEP.

We note that for pEP, the subgroup ZK(p) can be interpreted as the stabilizer of the point p with respect to the action (k, x) f-+ kxk- 1 (k E

K, x E P), of K on P by inner automorphisms. Thus K / Lp = K / Z K (p)O is a homogeneous space, which covers the orbit K(p) C P. Instead of K(p) one can consider the homogeneous space (Ad) K(s) isomorphic to it, where 8 E p, exp s = p and 9 = t EB P is a Cartan decomposition of the Lie algebra 9 of G.

Example 4. Suppose that the real rank of G is equal to 1, i.e. that dim II = 1 in the notation of 1.3. In this case the group K acts transitively (by means of the adjoint representation) on the set of all one-dimensional subspaces of p (Helgason 1962). Hence K acts transitively on the sphere SN-1 = {s E

pi (8, s) = I}, where N = dimp, and ( , ) is the Killing form on g. Making use of the classification of simple groups of real rank 1 we obtain transitive actions of the following non-compact simple groups on spheres:

1) G = SO~,n on sn-l (K = SOn), 2) G = SU1,n on s2n-1 (K = Un), 3) G = SP1 n on s4n-l (K = SPn x SPn), 4) G = FU'on S15 (K = Sping).

Moreover, the action of K extends to an action of G in a unique way; the stabilizer H c G coincides with the unique (up to conjugation) proper connected parabolic subgroup of G.

Let us give the classical geometric description of the action of the group 01,n on sn-l, of which the above action of SO~,n is the reduction. The group 01,n naturally acts on the cone -x5 + xi + ... + x; = 0 in lRn+1 , which leads to its transitive action on the quadric Q C IRPn , given by the same equation in homogeneous coordinates. Clearly, Q is contained in an affine open set (xo =I 0) and in non-homogeneous coordinates ~i = :~ (i = 1, ... , n) is given by the equation ~r+" .+~~ = 1. Thus Q = sn-l. Analogously, (with the help of projective spaces over C or JHI) one constructs actions of U l,n and SPl,n on s2n-l and s4n-l respectively and (making use of the octal projective space) of the group FII on S15.

Example 5. Let G~ k be the manifold of all oriented k-dimensional subspaces of the space lRn (the double covering of the Grassmann manifold G~,k)' The natural action of the group SLn(lR) on G~,k and its reduction to the

subgroup SOn are transitive, with G~,k = SOn/SOk x SOn-k. Clearly, the

homogeneous SOn-spaces G~,1 and G~,n-l are isomorphic to the sphere sn-l with the standard action of the group SOn. At the same time, for n 2: 3 the actions of SLn(lR) on these manifolds are non-isomorphic (although they are similar). The corresponding stabilizers are two non-conjugate maximal connected parabolic subgroups of SLn(lR), which are taken into each other by an inner automorphism ofthis group. Thus on sn-l (n 2: 3) there exist two similar, but non-isomorphic actions ofthe group SLn(lR), which are enlargements of the standard action of the group SOn-

2.5. Compact Homogeneous Spaces of Rank 1. In this subsection we will give a solution to problem a) for simply connected compact homogeneous spaces of rank 1 (in the sense of 2.4 of Chap. 2). In view of Lemma 2.2 we can restrict ourselves to the case when the compact group which acts transitively, is simple and non-abelian. Therefore we shall start with listing (up to a local isomorphism) all pairs of connected compact Lie groups (G, H) such that HcGandrkG/H=1.

Theorem 2.4 (Onishchik 1963). All connected Lie subgroups H of connected simple non-abelian compact Lie groups G, such that rk G / H = 1, are given in Table 3. The groups G are considered up to local isomorphism, and subgroups H up to conjugation in G.

In Table 3 we denote by <P2k+1 the irreducible representation of 803 in lR2k+1. The factorization 804 = 8U28U2 is a locally direct factorization of 804 into simple factors. The last column shows only the non-trivial values of Aut~(G/H).

The following theorem gives the homotopy classification of homogeneous spaces of rank 1.

Theorem 2.5 (Onishchik 1963). Let Ml and M2 be two simply connected homogeneous spaces of rank 1 of simple compact connected Lie groups. We have


Table 3

G H G/H h(G/H) Aut~(G/H)

SUn-1 s2n-1 1 U1 SUn (n ~ 2)

S(U1 X Un-I) cpn-I 0

SU3 S03 SU3/S03 1

S02n-1 St~n+1,2 1 S02

S02n+1 S02n-1 X S02

-R 0 (n ~ 2) G2n+ I ,2

S02n s2n 0

Sping Spin7 S15 1

Spin7 G2 S7 1

S05 CP5(S03) S05/CP5(S03) 1

SPn_1 s4n-1 1 SP1

SPn (n ~ 2) SPn-1 X U1 cp2n-1 0

SPn-1 X SP1 Hpn-1 0

S02n (n ~ 3) S02n-1 s2n-1 1

F4 Sping OP2 0

SU3 S6 0

SU2 St~2 1 SU2

SU2·S02 -R G7,2 0

S04 = SU2· SU2 G2/S04 0 G2

SU2 G/SU2 1 SU2

SU2·S02 G2/SU2· S02 0

2CP3(S03) G2/S03 1

CP7(S03) G2/CP7(S03) 1

1l"k(M1 ) ~ 1l"k(M2) for all k if and only if Ml and M2 are denoted in the same way in Table 3.

Corollary. If two simply connected compact homogeneous manifolds Ml and M2 of rank 1 have isomorphic homotopy groups (for example, if Ml and M2 are homotopy equivalent) then Ml and M2 are diffeomorphic.

From Theorems 2.4, 2.5 and Lemma 1.2 we can deduce the classification of transitive actions of connected compact Lie groups on simply connected manifolds of rank 1 (the groups acting are considered up to local isomorphism).


Theorem 2.6 (Onishchik 1963). Let M be a simply connected homogeneous manifold of rank 1. Any locally effective transitive action of a connected compact Lie group on M is locally similar to one of the following actions:

1) In the case M = sn (n;:::: 2), the standard linear action of SOn+! or its reduction to one of the following subgroups:

SUm, Um, (n = 2m - 1); SPm' SPm U1, SPmSPI (n = 4m - 1);

Spin9(n = 15); Spin7(n = 7); G2 (n = 6).

2) In the case M = cpn (n;:::: 2), the standard projective action of SUn+! or (for n = 2m - 1) its reduction to the subgroup SPm.

3) In the case M = St~n+I,2 (n > 1, n # 3), the standard action of the group S02n+! or the action of the group S02n+! X S02, which is its natural enlargement. In the case M = M = St~,2 the standard action of the group S07, its reduction to the subgroup G2, the natural enlargements of these actions to the groups S07 x S02, G2 X SU2, the reduction of the last action to the subgroup G2 x S02.

4) In the case M = G2/SU2 the standard action of the group G2 , its natural enlargement to the group G2 x SU 2, the enlargement of the last action to G2 x S02.

5) In the case when M is not diffeomorphic to any of the manifolds listed above, the natural action shown in Table 3.

The classification of transitive actions of compact Lie groups on spheres, Part 1 of Theorem 2.6 was obtained in (Montgomery and Samelson 1943), (Borel 1950). Note that the corollary of Theorem 2.5 does not transfer to the case of manifolds of rank 2. Moreover, there exist simply connected compact homogeneous manifolds of rank 2, which are homeomorphic but not diffeomorphic.

Example 6 (Kamerich 1977). Let G = SPn X SP2, H = SPn-1 X SPI X SPI be a subgroup embedded in G in the following way:

(a,b,c)~ ((~ ~),(~ ~)) (aESPn_llb,CESPI).

Then rk G / H = 2. The manifold G / H is homotopy equivalent to the manifold S4 X s4n-1 if and only if n is divisible by 24. Further, the first rational Pontryagin class of G / H is non trivial, from whence it follows that G / H is not homeomorphic to S4 X s4n-l. In particular, for n = 24 the manifold G / H is homotopy equivalent but not homeomorphic to S4 x S95.

Example 7 (Kreck and Stolz, 1988). Suppose that G = SU3 X SU2, p, q are integers, H = SU2 X U1 the subgroup embedded in G in the following way:


Set Mp,q = G / H. One can classify Mp,q up to homeomorphy or diffeomorphy. In particular, it turns out that there exist 28 classes homeomorphic but not diffeomorphic manifolds M k ,4; the representatives of these classes are M 32s+l ,4 (s = 0,1, ... ,27).

We note that in both examples we have h( G / H) = 1, and that for compact homogeneous spaces of positive Euler characteristic homotopy equivalence implies diffeomorphism (Shchetinin 1990).

2.6. Transitive Actions of Non-Compact Lie Groups on Spheres. The results described above allow one also to describe all transitive actions of non compact Lie groups on compact manifolds of rank 1 (see Onishchik 1968). We will limit ourselves to describing actions on spheres. Let G = S Rad G be a Levi factorization of a connected Lie group G, which acts transitively and effectively on sn, n ~ 2. From Lemma 2.2 it follows that in S there exists a simple normal Lie subgroup So, acting transitively on sn, with either S = So or S = SoSU2, or S = SOS03' If S acts asystatically, then G = So (if n is even, then h(sn) = 0 hence G = So). If So is non compact, then by Corollary 3 of Theorem 2.1 the group G is reductive. If, on the other hand, So is compact and acts systatically. This is possible only for linear actions of the group SUm on s2m-l and the group SPm on s4m-l, then Rad G is abelian and can have arbitrarily large dimension (the corollary to Theorem 2.2 and Corollary 4 to Theorem 2.1). We note also that by the corollary to Lemma 2.3 S is a group with finite centre. By the above it suffices to describe transitive actions of simple non-compact Lie groups and the automorphism groups of these actions. Making use of Theorem 2.3, we come to the following result.

Theorem 2.7. The list in Table 4 below contains all connected simple non compact Lie groups G which admit effective actions on spheres sn (n ;::: 2). The groups SO~,n' SUl,n, SPl,n, FII each possess a unique (up to isomorphism) transitive action on the corresponding sphere, described in Example 4, the group SL3(lR) has a unique transitive action on S3 = SU2, arising from the factorization SL3(lR) = SU2 . D, where D is a maximal connected triangular subgroup and the group SLn(lR) has two transitive actions on sn-l, described in Example 5. The reductions of these actions to the subgroups SL ¥ (C), SPn (lR) (n even), Sp ¥, SL'i (lHI) (n = 4m) define transitive actions of these subgroups on sn-l (for SL¥ (C) and SP¥ (C) we obtain two nonisomorphir; but similar actions). For the groups SL¥(C) and Sp¥(C) there exist also one parametric families of pairwise non-isomorphic and non-similar transitive actions, obtained from the above by means of a deformation.

In Table 4 K denotes a maximal compact subgroup of G, SL3(lR) the connected covering group of SL3(lR), the rest of the notation is the same as in Table 3.


Table 4

G K G/H Aut~(G/H)

SO~,2n SOn sn-l

SUl,n S(UI X Un) s2n-l

SPl,n SPn X SPI s4n-l

FII Sping S15

SLn (IR) SOn sn-l

SLn(1C) SUn s2n-l Ul

SP2n(IR) Un s2n-l

SP2n(1C) SPn s4n-l Ul SLn(lHI) SPn s4n-l SPI SL3 (IR) SU2 S3

2.7. Existence of Maximal and Largest Enlargements. Let M = G/H be a compact homogeneous space of a connected Lie group G acting locally effectively on M. We shall now pose the following question: does ther~ exist an enlargement of the natural action of the group G on M, which is maximal or largest among all enlargements with respect to the order ::;, introduced in 1.1 of Chap. I? As above we shall assume that the group 1l'1 (M) is finite. Let S = SnSe be a factorization of a Levi subgroup S of the group G as the product of a compact normal subgroup Se and a normal subgroup Sn without compact factors.

Theorem 2.8 (Onishchik 1977).1) IfG acts asystatically on G/Ho, then there exists only a finite number of pairwise locally non-isomorphic Lie groups, possessing transitive locally effective actions on M = G / H, which are enlargements of the natural action of G. In particular there exist maximal enlargements of this action.

2) If the subgroup Sn acts transitively on M, then the natural action of G on M possesses the largest locally effective enlargement.

3) If G acts systatically on M and the subgroup Sn is not transitive on M, then for any N > 0 there exists a Lie group G* of the form G* = SRad G* , where Rad G c Rad G* and dim Rad G* > N, with a locally effective action on M, which is an enlargement of the natural action of G. In particular, the latter action does not possess the largest enlargement.

Proof. Statement 1) can be derived from the semi-simplicity of the asystatically acting group (Corollary 1 of Theorem 1.2). Statement 3) is a slight strengthening of Theorem 2.4. The proof of statement 2) is based on the following idea. Suppose that a locally effective action of some Lie group G* on M is an enlargement of the natural action T of G. Then the tangent algebra g* of the group G* can be identified with the subalgebra of the algebra tJ(M), which is invariant under all (Tg)* (g E G) (see 2.1 of Chap. 1). Thus g* is contained in the space tJ(M)c of all representative vector fields, which is a sub algebra of tJ(M). If this sub algebra is finite-dimensional, then to it

corresponds the largest enlargement of the action T. At the same time we have

Lemma 2.5 (Dao Van Tra 1975). Let E ---> M be a homogeneous vector bundle, where M is a compact homogeneous space of a semi-simple Lie group G without compact simple factors and such that IT1 (M) is finite. Then dimr(E)G < 00.

In (Dao Van Tra 1981) there is a description of the largest enlargements of transitive locally effective actions of semi-simple Lie groups G without compact factors, with connected parabolic stabilizer H. If G = G1 ... Gr is a factorization of G into simple factors, then H = H1 ... Hr , where Hi is a proper parabolic subgroup in G i (i = 1, ... ,r), whence G / H = Gd H1 x ... x Gr/ Hr. It turns out, that any enlargement of an action of G is obtained by multiplying certain enlargements of actions of Gi on Gd Hi. Therefore the description of enlargements reduces to the case when G is simple.

Theorem 2.9 (Dao Van Tra 1981). Let G be a connected simple noncompact Lie group and H a connected parabolic stabilizer, H -I- G. The natural action of the group G on M = G / H is maximal in all cases, except the following (G is given up to a local isomorphism and by G* we denote the group which gives the largest locally effective extension of the action):

1) G = SP2n(lR) (n ~ 1), M = S2n-l, G* = SL2n(lR). 2) G C S03,4 is a non compact real form of the group G2(C), M is the

component of the manifold of isotropic lines in the pseudo-Euclidean space of signature (3,4), diffeomorphic to S2 x S3, G* = soL.

3) G = SO~_1,n (n ~ 3), M is the component oj the manifold of totally isotropic (n - 1) -dimensional subspaces in the pseudo-Eucllidean space of signature (n - 1, n), diffeomorphic to SOn, G* = SO~ n.

4) G = 8P2n(C): (n ~ 2), M = cp2n-1, G* = 8L2~(C). 5) G = G2(C), M is a quadric in CP6 , G* = 807 (C). 6) G = S02n-1 (C) (n ~ 3), M is the manifold of totally isotropic (n -1)

dimensional subspaces in c2n-1, G* = S02n(C).

Note that the enlargements 4), 5), 6) coincide with the enlargements of Theorem 4.7 of Chapter 2. Thus, in the complex case G* coincides with AutO M. Pairs (G, G*) from enlargements 1) - 3) are split real forms of the corresponding pairs from 4) - 6).

§3. The Natural Bundle

3.1. Orbits of the Action of a Maximal Compact Subgroup. Let M G / H be a compact homogeneous space of a connected Lie group G and K a maximal compact subgroup of G. The group K has a natural action on M, its stabilizers have the form KgH = K n gHg-l, g E G and they are conjugate to the subgroups of the form g-1 Kg n H. For the point eH E G/ H the orbit


of the group K has the form KI L, where L = K n H; orbits of the other points may generally speaking, have a different orbit type and even may be non-diffeomorphic to the orbit KIL. For example, if M = G/r, where r is a discrete subgroup of G, then the subgroups of the form gKg~l n r are finite subgroups of r and it can happen that among them there are non-isomorphic ones (see also Example 2 of 3.2). However, there holds

Theorem 3.1 (Gorbatsevich 1981b). Let M = GIH be a compact homogeneous space of a connected Lie group G, and let K be a maximal compact subgroup of G. Then all the subgroups K~ (x E M) are conjugate in G. In particular, all orbits of the natural action of K on M have the same dimension.

Proof. Consider the subgroup P = (NG(HO))o. First, making use of Theorem 1.1 one shows that subgroups of the form g~l Kg n P are maximal compact subgroups of the Lie group P. Whence we deduce that subgroups of the form g~l Kg n HO (conjugate with subgroups of the form KgH , g E G are maximal compact in HO. But then all these subgroups are conjugate in HO. Hence all the subgroups K~: (x E M) are conjugate in G. In particular, all the subgroups have the same dimension.

For any compact homogeneous space M = G I H the orbits of K may, in spite of having the same dimension, have a different orbit type. This situation, however, can be corrected, by replacing M by a suitable finite covering.

Theorem 3.2 (Gorbatsevich 1981b). Let M = GIH be a compact homogeneous space of a connected Lie group G, and let K be a maximal compact subgroup of G. Then there exists a subgroup H' of finite index in H such that all orbits of the natural action of the group K on the space M' = G I H' (finite covering of M) have the same orbit type. If the fundamental group 71"1 (M) has no torsion than one can take M' = M.

Proof. The construction of the required subgroup H' is rather complicated, it makes use of a lemma of Selberg, according to which in any finitely generated matrix group r there exists a subgroup of finite index which is torsion free (see Part I of Encycl. Math. Sc. 21). As r one can take some quotient group HIHo. D

3.2. Construction of the Natural Bundle and Its Properties. Let M = G I H be a compact homogeneous space of a Lie group G, which we shall now assume to be simply connected. If K is a maximal compact subgroup of G then K is semi-simple and also simply connected. Let M' = G I H' be the space from Theorem 3.2, which is a finite covering of M. According to the slice theorem the natural mapping M' ----> K\M' (projection on the orbit space of K) is the projection mapping of a locally trivial fibre bundle with fibre K I L', where L' = K n H' and structure group Q = NK(L')I L' (Theorem 1.3 of Chap. 3). This fibre bundle is known as the natural bundle (note that the total space is not M but M').

The fibre Me = K I L' of this fibre bundle is a homogeneous space of the compact semi-simple Lie group K, the group 7rl(Me) is finite (though not always trivial), i.e. Me is almost simply connected.

The base space Ma = K I L' of the natural bundle can be represented as the space of double cosets K\ G I H'. The universal covering manifold Ma of Ma is diffeomorphic to ]Rn, where n = dim Ma. From this it follows, in particular, that the manifold Ma is aspherical, i.e. has the homotopy type of K(7rl(Ma), 1). From this and from the finite-dimensionality of Ma it follows that 7rl (M) is torsion free.

Theorem 3.3 (Gorbatsevich 1981b). The universal covering M of a compact homogeneous space M is diffeomorphic to Me X Ma = Me x]Rn (here Me = K I L' is a simply connected compact homogeneous space.)

In particular, we find that the Karpelevich-Mostow bundle for M (see 3.6 of Chap. 2) becomes trivial after a suitable enlargement of the structure group.

3.3. Some Examples of Natural Bundles

Example 1. Let M = G If, where f is a uniform torsion free lattice in a connected Lie group G. The natural action of a maximal compact subgroup KeG on M is free, hence we obtain a principal fibre bundle

K -+ M -+ K\Glf

with structure group K. If G is simply connected, then the subgroup K is semi-simple and we obtain a natural bundle for M, with Me = K, Ma = K\Glf.

For example, in the case G = 8L2 (<<=) we have Me = K = 8U2 . 8ince 8L2

is a (double) covering of SO~ 3, the manifold K\ G can be identified with the

Lobachevski space A 3 = SO~ \SO~,3 and Ma = A 3 If. As follows from the theory of obstructions to the existence of a section, the natural bundle is in this case trivial.

Example 2. Let G = SO~ 3 be the group of proper motions of the Lobachevski space A 3 and f the uniform lattice in G, generated by reflections in A 3

(on the subject of such lattices see volume 29 of the present series). We will show that the compact homogeneous space M = G If cannot be fibred over aspherical spaces with simply connected fibre. Indeed, from the exact homotopy sequence of such a fibration it would follow that the set Tors f of elements of finite order in f is generated by elements of order 2 (reflections). Hence we must have Tors f = f, which is impossible, since the group SO~ 3 = G is non compact while the manifold M = G If is compact. '

In particular, it follows from the above that the manifold M itself has no natural bundle, so that the passage to a suitable finite covering of M' over M in the construction of the natural bundle is here unavoidable.


Example 3. As is known, a compact orient able surface Fg of genus g for g 2: 2 is not a homogeneous manifold (see 7.2 below). Let us show that the manifold Fg x S3 is homogeneous.

Let G = SL3(JR). Consider in G the connected subgroup

Further, consider the group SL2(JR), with the natural inclusion in G:

The group SL2(JR) acts transitively on the Lobachevski plane A2 (or on the upper half-plane by means of linear fractional transformations). It is known that for g 2: 2 the manifold Fg is diffeomorphic to A 2 /r, where r is some torsion free lattice in SL2(JR). Then H = rHo is a closed uniform subgroup in G = SL3(JR). Consider the compact homogeneous space M = G/H. It is not hard to check that in this case there exists a natural bundle for M itself and that it has the form IRP3 --+ M --+ Fg • With the help of obstruction theory, one can easily show that the bundle thus obtained is trivial, so that M is diffeomorphic to IRP3 x Fg. The manifold S3 x Fg is a double covering of M and is therefore homogeneous (see Theorem 2.2 of Chap. 2).

Example 4. Let G = SOn X N3, n 2: 2 (see 1.1 of Chap. 4). Consider in SOn the subgroup S02 X SOn-2, which preserves the plane spanned by the first two vectors of the standard basis in JRn . Further, let Z '::::' JR be the centre of the group N3(JR); it consists of matrices of the form

(1 0 Z) o 1 0 ,zEJR 001

Let us denote by C the Lie subgroup of G, isomorphic to JR, contained in S02 x Z, and consisting of pairs

cos 27rt - sin 27rt sin 27rt cos 27rt ;0 D

1 0

1 1 , tEIR. 0

1

Then H = N3(Z) . (C X SOn-2) is a uniform group in G. The homogeneous space M = G/H fibres naturally as SOn/SOn-2 --+ M --+ N3(JR)/ZN3(Z). We have Me = St~ 2' Ma = ']]'2. This bundle is non-trivial for n = 4, since

then M is not diffe~morphic to St~.2 x ']]'2, (Gorbatsevich 1981a).


3.4. On the Uniqueness of the Natural Bundle. After constructing the natural bundle there arises the question of its uniqueness. In the construction of the natural bundle for M = G / H there is involved a Lie group G, and a priori it is not clear whether by passing to another Lie group, which acts transitively on M, we obtain an entirely different fibre and base. Since the construction of the natural bundle makes use of passage to a finite covering, which, in general is not canonical, it is clear that one can expect uniqueness of the fibre Me and the base Ma to hold at most up to a finite covering.

Let us begin with the problem of uniqueness of the fibre Me. The universal covering manifold M of M, is diffeomorphic to Me x]Rn (Theorem 3.3). Hence the manifold Me is homotopy (and even tangential homotopy 1) equivalent to the manifold M. Thus, the (tangential) homotopy type of the manifold Me is uniquely determined by M.

Let us now consider the base of the natural bundle. The group 11"1 (Ma) is determined by the group 1I"1(M) uniquely up to commensurability (see §5 below). The manifold Ma is aspherical and hence it is determined by its fundamental group uniquely up to homotopy equivalence.

Suppose now that the group 11"1 (M) has no torsion. Then M itself has a natural bundle, and its projection M ----) Ma can be viewed as the first level of the Postnikov system of M. In particular, p acquires a homotopy invariant meaning. The above, combined with certain additional considerations leads to the following result.

Theorem 3.5 (Gorbatsevich 1981b). Let M be a compact homogeneous manifold. Then

(i) The tangential homotopy type of the manifold Me, which is the universal covering of the fibre Me of the natural bundle is uniquely determined by the manifold M.

(ii) The homotopy type of the base Ma of the natural bundle is determined by M uniquely up to finite covering (for a stronger statement see Theorem 4.4 below).

(iii) If the group 11"1 (M) has no torsion, then the natural bundle for M is homotopically simple (i. e. the group 11"1 (Ma) acts trivially on the groups 1I"i(Me) for i ~ 2)) and is determined by M uniquely up to fibre homotopy equivalence (for which see Husemoller 1966).

Note that for any compact homogeneous space M there exists a finite covering M' for which the natural bundle is homotopically simple.

The tangential homotopy type of a compact homogeneous manifold (such as Me) generally speaking does not determine a smooth type (the corresponding examples in dimension 7 have been constructed recently by Kreck and

1 Smooth manifolds M and M' are said to be tangentially homotopy equivalent, if there exists a smooth mapping I : M ---> M', which is a homotopy equivalence and such that the fibre bundle j*(T(M')) over M, induced by the tangent bundle T(M') over M' by means of the mapping I, is stably equivalent to the fibre bundle T(M), (i.e. j*(T(M')) E9 () = T(M) E9 (), where () is some trivial vector bundle over M).


Stolz, see 2.5). From this it is easy to see that the smooth type of the manifold Me is not uniquely determined by M. One can, however, show that for dimMa ~ 2 the manifold Me is determined uniquely by M up to diffeomorphism (Gorbatsevich 1981b).

Note that in a number of cases homotopy equivalence of manifolds implies that they are diffeomorphic (see, for example, Shchetinin 1988, 1990).

3.5. The Case or Low Dimension or Fibre and Basis. To begin with, let us assume that dimMe ~ 4. Note that the case dim Me = 1 is impossible, since the group 1l'1 (Me) is finite.

Theorem 3.6 (Gorbatsevich 1981b). Let M be a compact homogeneous manifold and let dim Me = 2 or 4. Then for a suitable M', which is a finite covering of M, the natural bundle is trivial, and Me is diffeomorphic to S2, S4, S2 X S2 or ICp2, and Ma is an aspherical homogeneous space.

The proof of this theorem is based on the fact that for dim Me = 2 or 4 we necessarily have X(Me) =f:. 0, from whence we can deduce that the structure group Q of the natural bundle (see Theorem 3.4) is finite and for a suitable M' reduces to {e }. The statements concerning Me and Ma have slightly more complicated proofs.

In order to study the case dim Me = 3 we make use of another fibre bundle with total space M, the Borel bundle.

Let M = G I K be a compact homogeneous space of a simply connected Lie group G, with K a maximal compact subgroup of G. Replacing M with a suitable finite covering M', we can suppose that the action of K on M is equi-orbital. In this case we have the Borel bundle ML -+ M -+ KIN, where N = NK(L) is the normalizer of the subgroup L = K n H in K (see 1.3 of Chap. 3). It is possible to prove that the fibre ML is often a homogeneous space (Gorbatsevich 1981d). Unlike in the case of the natural bundle, the fibre and the basis of the Borel bundle (and even their dimensions) depend essentially on the choice of the Lie group acting transitively on M. With the help of the Borel bundle one can prove:

Theorem 3.7 (Gorbatsevich 1983a). Let M be a compact homogeneous manifold and dim Me = 3. Then Mc is diffeomorphic to S3 and a suitable finite covering M' of M is diffeomorphic to one of the following:

(i) S3 X Ma, where Ma is the base of the natural bundle for M'. (ii) The total space of the fibre bundle over S2 associated with the Hopf

bundle

(see Example 1 of 1.1 of Chap. 2) and having as fibre some compact homogeneous space.

(iii) G/r where r is a uniform lattice in some Lie group G.


Note that the fibre bundle which appears in (ii) is a Borel bundle for M'. The construction of G in (iii) is described in (Gorbatsevich 1983).

Let us now consider the case when dim Ma :::; 2.

Theorem 3.8 (Gorbatsevich 1981b). Let M be a compact homogeneous manifold. Then

(i) If dim Ma = 1 then Ma = 81 and a suitable finite covering manifold of M is diffeomorphic to Me X 81 •

(ii) If dim Ma = 2, then Ma is diffeomorphic to T2, K2 or Fg where g 2: 2.

If Mg = T2, then the corresponding manifolds M can be described precisely (Gorbatsevich 1981a, Gorbatsevich 1982, 1985); the case Ma = K2

reduces to the fact that K2 has the torus 1['2 as a double covering. For the case Ma = Fg see §8 below and (Gorbatsevich 1981e, 1986b). One can also classify all Ma for dimMa = 3 (see Gorbatsevich 1981b and, partially, §3 of Chap. 6).

§4. The Structure Bundle

As is well known, it is convenient to describe the structue of a Lie group with the help of a Levi decomposition, separating it into the solvable part (radical) and the semi-simple Levi subgroup. Below we will show that the analogous device can also be used in the study of the base Ma of the natural bundle for a compact homogeneous manifold M. The analogue of the Levi decomposition is the structure bundle for Ma.

4.1. Regular Transitive Actions of Lie Groups. In order to construct the structure bundle we shall find it useful to have certain general results on Lie groups acting transitively on compact manifolds. Let M = G / H be a homogeneous space of a connected Lie group G and S a Levi subgroup, i.e. a maximal connected semi-simple Lie subgroup of G. A transitive action of G on M is called regular if Nc(HO)S = G. This condition is equivalent to transitivity of the natural action of S on G/Nc(HO) (see Lemma 4.1 in Chap. 2). It does not depend on the choice of the subgroup S, since all Levi subgroups are conjugate in G.

An arbitrary transitive action is not, in general, regular. For example, let M = R/ H be a compact solvmanifold, where R is a solvable Lie group acting effectively on M, and HO =I=- {e} (for example, see 2.1 of Chap. 4). Here the action of R on M is not regular. On the other hand, any transitive action of a semi-simple Lie group is, clearly, regular. The following theorem shows that there are sufficiently many regular actions.

Theorem 4.1 (Gorbatsevich 1981c). Let M = G/H be a compact homogeneous space of a simply connected Lie group G. Then on some finite covering manifold of M, there exists a regular transitive action of some Lie group G' .


In addition, one can assume that G and G' are subgroups of some Lie group G and have isomorphic (and, with a suitable choice, even the same) Levi subgroups S' c:::' S.

Proof. The proof of this theorem is based on making use of the construction of the splitting of a simply connected Lie group (see "Lie Groups and Lie Algebras ~ 3", Encycl. Math. Sc. 41, and also Gorbatsevich 1981c, Auslander 1973). First one constructs the group G, and then, with the help of a suitable change of the radical of G we obtain the group G'. The stabilizer H C G is not changed by this (up to passage to subgroup of finite index). In the case when M is a compact solvmanifold, the statement of Theorem 4.1 was actually proved in (Auslander 1973), the general case is considerably more complex.

4.2. The Structure of the Base of the Natural Bundle. With the help of passage to a regular transitive action (based on Theorem 4.1) one can prove the following statement, which is the basis for the study of the base Ma of the natural bundle.

Theorem 4.2 (Gorbatsevich 1981b). Let M be a compact homogeneous manifold. Then some finite covering manifold M' of M admits a natural bundle, whose base has the form Ma = K'\ G' /r', where r' is a uniform lattice in some connected Lie group G' and K' is a maximal compact subgroup in G'.

By Theorem 4.2 the study of the manifold Ma is closely connected with the study of uniform lattices in Lie groups. The proof of the following result is based on making use of this relation.

Theorem 4.3 (Gorbatsevich 1979). Let M be a compact homogeneous manifold. Then some manifold M', which is a finite covering of M, the base Ma of the natural bundle admits a bundle Mr ---+ Ma ---+ MS) where the fibre Mr is a solvmanifold and the base Ms is diffeomorphic to a locally symmetric Riemannian space of negative curvature.

Proof. We shall give a sketch of the proof. First we consider an important special case: let M = G /r, where r is a uniform torsion free lattice in the simply connected Lie group G. Let G = S . R be a Levi decomposition (R is the radical, S a Levi subgroup). Suppose that the centre Z (S) is finite and S possesses no compact quotient groups. Then the subgroup r n R is a lattice in R. For our M, as is easy to show, there exists a natural bundle and it has the form K ---+ G/r = M ---+ K\G/r = M a, where Me = K is a maximal compact subgroup of G. For the base of the natural bundle Ma = K\G/r, we have, in turn, the following natural bundle

R/r n R = K n r· R\r· R/r ---+ Ma = K\G/r

---+ K\G /r· R = Ks \s/r.,

where Ks is a maximal compact subgroup of S, and r s = r jr n R is a lattice in S. Put Mr = Rjr n R, Ms = Ks \Sjr s, then Mr is a solvmanifold and M is locally symmetric, since it has as a covering the locally symmetric space Ks \S. We obtain (in the special case under consideration) the required bundle Mr --+ Ma --+ Ms. For arbitrary compact homogeneous spaces with a discrete stabilizer the construction of the bundle is analogous to the one above, though more complicated. The case of an arbitrary compact homogeneous manifold reduces to the one already considered, with the help of Theorem 4.2. 0

The bundle Mr --+ Ma --+ M s, obtained in Theorem 4.3, is called the structure bundle. It can be viewed as an analogue of the Levi decomposition for Lie groups; the manifold Mr plays the role of the solvable part of the base Ma, and Ms that of the semi-simple part (for a more precise statement see §5 and 6 below).

Theorem 4.4 (Gorbatsevich 1981c). Let M be a compact homogeneous manifold. Then

(i) The base Ma of the natuml bundle, and also the manifolds Mr and Ms are determined by M up to a finite covering.

(ii) For a given Ma the structure bundle is determined up to fibre homotopy equivalence.

Statement (ii) of Theorem 4.4 is a consequence of the fact that the manifolds Ma, Mr, Ms are aspherical and therefore are determined by their fundamental groups up to homotopy equivalence, and the groups 11"1 (Mr), 1I"1(Ms) are determined by 11"1 (Ma) up to isomorphism (for more details see §5 below).

4.3. Some Examples of Structure Bundles

Example 1. Consider a homogeneous space M = S jr, where S is a semisimple simply connected Lie group whose centre Z(S) is finite, and r is a uniform lattice in S. Assume, moreover, that the group r jr n Z(S) has no torsion. (Such lattices can easily be constructed with the help of Selberg's lemma - see Part I of Encycl. Math. Sc. 21.) The base Ma of the natural bundle for such M has the form Ma = K\Sjr, where Ms = Ma, and Mr degenerates to a point.

Example 2. Let now S be a simply connected simple Lie group with infinite centre, hence Z(S) ~ Z + <P, where <P is some finite group (Helgason 1962). Consider the homogeneous space M = S jr, where r is a uniform lattice in S, and suppose that r :J Z (S) and the group r j Z (S) has no torsion. Then Ma = K\Sjr and the structure bundle over Ma has the form

kjK. Z(S) = Mr = 11' --+ Ma = K\Sjr --+ Ms = k\sjr,

where k is a maximal connected compactly embedded subgroup of S, containing K. Geometrically, this structure bundle can be described in the following way. The manifold k\S is a hermitian symmetric space of negative curvature, and Ms = k\S jr is hermitian locally symmetric. Consider the

canonical line bundle over the complex manifold Ms (see Wells 1973). The corresponding principal bundle over Ms with structure group 11' coincides with the structure bundle 11' ----; Ma ----; Ms constructed above.

For example, let S = SL2(lR) and M = Sjf, and let us assume that f :J Z(S) and the group f* = f jZ(S) has no torsion. Then

Ma = M, Mr = 11' and Ms = S02 \PSL2 (lR) jf* ,

is diffeomorphic to some closed oriented surface of genus 9 :2: 2. The structure bundle for this M is isomorphic (with the above assumptions on r) to the fibre bundle

11' ----; TI (Fg) ----; Fg

of unit tangent vectors over Fg (Auslander et al. 1963). In particular, TI(Fg) is a compact homogeneous space of the group SL2(lR). Let us note, that the structure bundle is non-trivial, since its characteristic class c E H2 (Fg, Z) is not equal to zero (Auslander et al. 1963).

§5. The Fundamental Group

From the results given in §3 and 4 (see also Chap. 4), it is clear that the fundamental group plays a major role in the study of compact homogeneous spaces. In this section we consider its properties in detail.

5.1. On the Concept of Commensurability of Groups. In the study of fundamental groups of homogeneous spaces and some related manifolds it turns out to be convenient to consider these groups not only up to isomorphism, but to make use of a stronger equivalence relation.

We shall call two groups IT and IT' commensurable, if they contain isomorphic subgroups of finite index (sometimes commensurability is understood in a weaker sense, see 1.2 of Chap. 1 of Part I of Encycl. Math. Sc. 21). We shall call two groups IT and IT' weakly commensurable (and denote this by IT ~ IT') if there exist subgroups of finite index ITI C IT, ITI' C IT' and finite normal subgroups If> <lITI , If>' <lITI ', such that the groups ITl/1f> and ITl/If>' are isomorphic. For example, if If> is a finite normal subgroup of IT, then ITjlf> ~ IT. Both relations - commensurability and weak commensurability - are equivalence relations on the class of all groups. For torsion free groups commensurability and weak commensurability are, clearly, equivalent.

Let Me ----; M' ----; Ma be the natural bundle for a compact homogeneous space M (here M' is some finite covering manifold of M). From the exact homotopy sequence of this fibre bundle it follows that 7fdMa) ~ 7fdM') ~ 7f1 (M). Thus, the group 7f1 (Ma) is determined by 7f1 (M) up to weak commensurability. Since 7f1 (Ma) is torsion free it is in fact determined by 7fd M) up to commensurability. Note that this statement is used in an essential way in the proof of Theorem 4.4.


5.2. Embedding of the Fundamental Group in a Lie Group. The following theorem (as well as Theorem 4.2) is the foundation on which one can build proofs of the basic results on the fundamental group of compact homogeneous manifolds. This allows one to reduce almost completely the study of the group 11"1 (M) to the study of lattices in Lie groups.

Theorem 5.1 (Gorbatsevich 1979). Let M be a compact homogeneous manifold. Then in the group 1I"1(M) there exists a subgroup 11"' of finite index, isomorphic to a uniform lattice in some connected Lie group F.

Proof. We give a sketch of the construction of F. Let G be a connected Lie group acting transitively on M. By Theorem 4.1 we shall suppose that the action of G on M is proper. If H is the stabilizer of this action then it turns out that one can take as F the group (NG(HD)D / HD X ]Rn for some n 2: 0 (uniquely determined by G and H). D

Note that for an arbitrary (not necessarily compact) homogeneous manifold M there exists a subgroup 11"' C 11"1 (M) of finite index, isomorphic to a discrete subgroup of a connected Lie group.

An arbitrary connected Lie group is locally isomorphic to some linear Lie group. Hence by Theorem 5.1 the properties of the group 1I"1(M) should be rather close to properties of linear groups. For example, the Tits alternative holds for 11"1 (M) (for linear groups see Merzlyakov 1980): the group 11"1 (M) is either commensurable with a solvable group or it contains a free abelian subgroup. Another example is provided by Theorem 5.2 below.

5.3. Solvable and Semi-simple Components. Let II be an arbitrary group. The radical of II is the largest solvable normal subgroup of II (when one exists).

Theorem 5.2 (Gorbatsevich 1981a). Let M be a compact homogeneous manifold. Then in 11"1 (M) there exists a radical.

Proof. We give a sketch of the proof. In view of the compactness of M the group 11"1 (M) is finitely generated. Hence, if 11"1 (M) is isomorphic to a linear group, the existence of the radical is a classical result (Auslander 1973). In the general case from Theorem 5.1 one can easily deduce that 1I"1(M) is commensurable with a group with a radical. But then one can show that 11"1 (M) also must have a radical. . D

Consider now the structure bundle Mr -+ Ma -+ Ms for a compact homogeneous manifold M. From the homotopy exact sequence of this fibre bundle we obtain the sequence

from where it is clear that the group 11"1 (Ma) can be viewed as an extension of the group 11"1 (Mr) by 11"1 (Ms). From Borel's density theorem for lattices in semi-simple Lie groups (see Part I of Encycl. Math. Sc. 21) it follows that the


radical of the group 7fd Ms) is trivial. Therefore 7fd Mr) is isomorphic to the group r(7f1 (Ma)), and hence 7f(Ms) C::' 7f1(Ma)/r(7f1(Ma)). Thus, the groups 7fdMr) and 7f1(Ms) are determined by the group 7fdMa) uniquely up to isomorphism. They are also determined by the group 7f( M) up to commensurability. The group 7f1 (Mr) is called the solvable part and the group 7f1 (Ms) the the semi-simple part of the group 7f1 (M). They determine the smooth manifolds Mr and Ms uniquely up to a finite covering (see Theorem 4.4).

In some cases exact sequence (*) splits (with a suitable choice of the base Ma of the natural bundle), i.e. the group 7f1 (Ma) is a semi-direct product of 7f1(Ms) and 7f1(Mr)). This is the case, for example, when the Lie group G is complex. But in general sequence (*) does not always split, even after passage to a finite covering of M (or Ma). For example, consider M = SL2 (IR.)/r, where r is a uniform torsion free lattice. Then exact sequence (*) has the form

(see Example 2 in 4.3). We see that 7fdM) is an extension of the group Z by the group 7f1 (Fg), and it is known that the characteristic class of this extension is non-trivial (Auslander et al. 1963). Hence, in the case we are considering, the exact sequence (*) does not split.

5.4. Cohomological Dimension. The cohomological dimension of a group 7f1 (M) is an important invariant. To study it we need certain auxiliary notions, which, in slightly lesser generality, were considered by Serre (1971).

A group II is called virtually torsion free, if it is weakly commensurable with a torsion free group (in Serre's definition commensurability is required).

If II is a finitely generated linear group, then it contains a torsion free subgroup of finite index, i.e. II is virtually torsion free in the sense of Serre (see Part I of Encycl. Math. Sc. 21). If, however, II is a finitely generated subgroup of a connected Lie group, then II is not necessarily virtually torsion free in the sense of Serre, but it is such in our sense. Moreover, if M is a compact homogeneous manifold, then 7f1 (M) ;:::0 III (Ma), so that 7f1 (M) is virtually torsion free.

A group II is called a group of type (FL), if the group Z, viewed as a trivial II module, has a finite free resolution. For such groups one defines the cohomological dimension cd to be the minimal length of a resolution of the above kind. All groups of type (FL) are torsion free. Hence it is useful to introduce the notion of virtual cohomological dimension vcd II (also slightly more general than its namesake due to Serre (1971)), which is finite even for some torsion groups.

We shall call II a group of type (VFL) (Le. virtual (FL) type), if it is weakly commensurable with some group II' of type (FL). For such a group II we put vcd II = cd II' (one can show that vcd II is well defined). Properties of the invariant vcd II are close to those given in (Serre 1971) in a slightly less general situation.

If II is a polycyclic group, then vcd II is equal to the rank (or the Hirsch number) of the group II, for whose definition see Part I Encycl. Math. Sc. 21 or (Auslander 1973). The role ofvirtual cohomological dimension in the study of compact homogeneous manifolds, can be seen, for example, from the following result.

Theorem 5.3. Let M be a compact homogeneous manifold. Then (i) The group 7I"1(M) is of type (VFL). (ii) vcd7l"1(M) = vcd7l"1(Ma ) = dimMa and vcd71"1 (M) ~ dimM. (iii) vcd 71"1 (M) = dim M if and only if, when M is aspherical. In the

contrary case vcd 71"1 (M) ~ dim M - 2.

Note that by statement (ii) the number dimMa depends only on the group 71"1 (M).

The group 71"1 (Ma) (and also 71"1 (M)) is torsion free and is clearly a Poincare Duality group (Le. a PD-group, see Part I of Encycl. Math. Sc. 21).

5.5. The Euler Characteristic. If II is a group of type (FL), then its Euler characteristic is defined by the formula

00

X(II) = ~) _1)ibi , i==O

where bi = dim Hi (II, Q) is the i-th Betti number. Suppose now the group II satisfies the weak (VFL) condition. By definition, II contains a subgroup II' of finite index, which has a finite normal subgroup such that II' /<1> is a group oftype (FL). We define the Euler characteristic by the formula

X(II) = I~~~'I X(II' /<1»,

where the number X(II' / <1» is defined above. The Euler characteristic of a group of type (VFL) is well defined. The number X(II) is rational; its properties are near to the properties of the ordinary Euler characteristic (see, for example, Serre 1971).

Consider now some examples of applications of the Euler characteristic to the study of compact homogeneous manifolds M. The characteristic X(7I"1(M)) is always defined. One can prove that if X(7I"1(M)) -I 0 then the radical of the group 71"1 (M) is finite (Gorbatsevich 1981a) (Le. the group 71"1 (M) is semi-simple, see §6).

Further, let M = sir, where S is a semi-simple Lie group, and r is a torsion free uniform lattice in S. Let the group be such that Z (S) = {e} and a maximal compact subgroup of it is semi-simple. Then the natural bundle for M has the form

K~ M = sir ~ K\s/r = M a ,

where Ma = Ms, and Mr degenerates to a point. It turns out that one can view the bundle as the principal bundle corresponding to the tangent bundle


T(Ma) ---t Ma over Ma, and K as the holonomy group of the locally symmetric Riemannian structure on Ma (Gorbatsevich 1981b). This fact makes it possible to give a sufficient condition for non-triviality of the natural bundle expressed in terms of the Euler characteristic of r.

Theorem 5.4 (Gorbatsevich 1981b). Let M = sir, where r is a uniform torsion free lattice in a semi-simple Lie group S with trivial centre and a semi-simple maximal compact Lie subgroup K.

Then, ifx(r) #- 0 (or, equivalently, X(7rI(M)) #- 0), the natural bundle for M (viewed as a principal bundle with structure group K) is non trivial.

Proof. If the natural bundle is trivial, then the tangent bundle T(Ma) ---t M is also trivial, i.e. the manifold Ma is parallelizable. But then X(Ma) = O. On the other hand, X(Ma) = X(7rI(Ma)), and 7r1(Ma) :::: r. Hence X(r) = 0, which contradicts the assumption. D

Theorem 5.4 is applicable, for example, to any torsion free lattice in the Lie group S = PSOI ,2n (see Gorbatsevich 1981b).

5.6. The Number of Ends. By the number of ends e(II) of an infinite group II we mean the number 1 + dim&::2 HI (II, :32 [II]), where :32 [II] is the group ring of the group II over :32 , viewed naturally as a II-module. In the case of a finite group II we set e(II) = O. The number of ends of a group can also be defined in topological terms. Let X be a cell complex, such that 7r1 (X) :::: II. Then e(II) = e(X), where e(X) is the number of ends of the universal covering of the complex X (Massey 1967). If M is a compact homogeneous manifold, then e(7rI(M)) = 0,1 or 2. Using this fact we can prove

Theorem 5.5. Let M be a compact homogeneous manifold. Then if M decomposes as a connected sum M = MdJ: M2, then either one of MI , M2 is simply connected or 7r1 (Md :::: 7r1 (M2 ) :::: :32 ,

Proof. If the manifolds M I , M2 are not simply connected and the order of the fundamental group of one of them is larger than two, then e( 7r1 (MI # M 2 ))

= 00, which is impossible in view of the above. D

As an illustration of Theorem 5.5 we remark that the manifold M = ]Rp3#]Rp3 is homogeneous; it is diffeomorphic to the manifold S(]Rp2) (see 2.3 of Chap. 6).

§6. Some Classes of Compact Homogeneous Spaces

Natural bundles and structure bundles allow one to distinguish certain natural classes of compact homogeneous spaces. These classes are defined in this section, and are considered in greater detail in 7-9.


6.1. Three Components of a Compact Homogeneous Space and the Case when Two of them Are Trivial. Let us combine the natural and structure bundles for a compact homogeneous manifold M into a single diagram:

Mr 1

Me -t M' -t Ma 1

Ms

Here M' is some finite covering manifold of M. The smooth manifolds Me, Mr and Ms are called the almost simply connected, solvable and semisimple components of the manifold M. Consider the class of compact homogeneous manifold characterized by triviality (Le. degeneration to a point) of some of the components. It is natural to consider first the case when two of the three components are trivial.

Suppose the components Me and Ms are trivial, then M' = Mr - a solvmanifold. It turns out then that the original manifold M is also a solvmanifold. Conversely, for any solvmanifold M, the components Me and Ms are trivial. Therefore, this case leads to the class of compact solvmanifolds (for which see Chap. 4).

If the components Mr and Ms are trivial, then M = Me - a homogeneous space with finite fundamental group, and the same also holds for the original manifold M. By Corollaries 4 and 3 of 3.6 of Chap. 2, any compact homogeneous manifold M with finite fundamental group admits a transitive action of some semi-simple compact Lie group; if M is such a manifold, then Mr and Ms are always trivial. Thus we arrive at the class of compact homogeneous manifolds with finite fundamental group (for which see §2).

The components Mr and Ms cannot be simultaneously trivial, since the manifold Ms is never homogeneous (see Theorem 7.3 below).

6.2. The Case of One Trivial Component. Let us turn to a survey of the cases when for the homogeneous manifold M one of the three components is trivial.

Theorem 6.1 (Gorbatsevich 1983b). Let M be a compact homogeneous manifold. Then the following conditions are equivalent:

(i) The component Me is trivial. (ii) M is aspherical. (iii) VCd7rl(M) = dimM.

Thus, in the case when the component Me is trivial we arrive at the class of aspherical compact homogeneous spaces. A survey of results concerning these is given in § 7.

Theorem 6.2 (Gorbatsevich 1981b, 1986b). Let M be a compact homogeneous manifold. Then the following are equivalent:


(i) The component Mr is trivial. (ii) The radical r(7I"1(M)) of the group 7I"dM) is finite.

We shall call a group II semi-simple if its radical exists and is finite (this definition of semi-simplicity differs somewhat from those accepted in the theory of finite groups and the theory of Lie groups). Compact homogeneous spaces which satisfy one of the conditions (i), (ii) of Theorem 6.3 are called semi-simple; more details about them are given in §8.

Theorem 6.3 (Gorbatsevich 1981a). Let M be a compact homogeneous manifold. Then the following conditions are equivalent:

(i) The component Ms is trivial (ii) In the group 71"1 (M) there exists a solvable subgroup of finite index. (iii) The group 7I"l(M) is noetherian (i.e. it satisfies the maximality con-

dition for subgroups). (iv) The group 71"1 (M) does not contain non-abelian free subgroups.

A homogeneous manifold M satisfying one of the conditions (i) - (iv) of this theorem is called solvable. Solvable compact homogeneous spaces are considered in detail in §9 (for the subclass of solvmanifolds see Chap. 4).

§7. Aspherical Compact Homogeneous Spaces

7.1. Group Models of Aspherical Compact Homogeneous Spaces. Let G be an aspherical Lie group (i.e. Gis aspherical as a topological space). Beginning with the classification of semi-simple real Lie algebras (see Vol. 41), it is easy to show that the unique aspherical simply connected Lie group is the group A = SL2 (lR), the universal covering of SL2 (lR); it is diffeomorphic to lR3 •

Hence a Levi subgroup of an aspherical Lie group is locally isomorphic to the group A x ... x A. Conversely, any connected Lie group with such a subgroup is aspherical. Note that any simply connected aspherical Lie group G is diffeomorphic to lRn , where n = dim G.

If M = G / H is a homogeneous space of an aspherical Lie group G, it is not difficult to show that M is aspherical. It turns out that for compact homogeneous spaces the converse also holds.

Theorem 7.1. Let M = G / H be a compact aspherical homogeneous space of a connected Lie group G, whose action on M is locally effective. Then

(i) The group G is aspherical. (ii) HO c T R, where R is the radical of the Lie group G and T is some

maximal connected triangular subgroup of a Levi subgroup of G.

For further details see (Gorbatsevich 1983b). Note that for non-compact aspherical homogeneous spaces statement (1) of

Theorem 7.1 does not hold. For example, any noncom pact simple connected


Lie group G acts locally effectively on a homogeneous space G I K (where K is a maximal compact subgroup in G), diffeomorphic to ]Rn.

The following result shows that when studying aspherical compact homogeneous spaces one can often restrict oneself to considering the case of discrete stabilizer.

Theorem 7.2 (Gorbatsevich 1981c). Let M = GIH be an aspherical compact homogeneous space. Then there exists a manifold M', which is a finite covering over M and has the form M' = G' Ir', where r' is a uniform lattice in some aspherical Lie group G'.

Proof. In the case when M is a solvmanifold we obtain Theorem 2.12 of Chap. 4. In the general case the proof of Theorem 7.2 is based on passing from the original action of the Lie group G on M to a proper action (see 4.1). In turn, regular actions on aspherical compact manifolds reduce (after passing to a suitable transitive subgroup) to actions with discrete stabilizer. D

7.2. On the Fundamental Group. An aspherical compact homogeneous manifold M coincides with the base of its natural bundle. Hence the manifold M is determined up to a finite covering, by its fundamental group 7r1 (M) (Theorem 4.4). As a matter offact, the group 7r1 (M) clearly determines such M up to diffeomorphism.

When considering the manifold M up to a finite covering, we can, in view of Theorem 7.2, assume that M = Glr, where r is a uniform lattice in a simply connected aspherical Lie group G, then 7r1 (M) ~ r. Hence, the study of aspherical compact homogeneous manifolds up to a finite covering reduces to description up to commensurability of groups, isomorphic to uniform lattices r in simply connected aspherical Lie groups (for more details about groups r see Gorbatsevich 1983b).

From Theorems 7.2 and 5.2 we deduce that, if M is an aspherical compact homogeneous manifold, then the radical r(7r1(M) is infinite, and moreover, there holds the inequality (Gorbatsevich 1983b)

dimM 2': rkr(7r1(M)) 2': ~dimM,

where rk is the rank of a polycyclic group (for this see part I of Encycl. Math. Sc. 21). From this follows

Theorem 7.3 (Gorbatsevich 1983b). Let M be a compact Riemannian locally symmetric space of negative curvature. The manifold M does not admit any transitive action of a Lie group.

Proof. If M is homogeneous, then r(7r1(M)) is infinite. But, on the other hand, r(7r1(M)) = {e} (see 5.3). We obtain a contradiction. D

The simplest (and well known) corollary of Theorem 7.3 is the non homogeneity of an arbitrary closed oriented surface Fg of genus 9 2': 2.


§8. Semi-simple Compact Homogeneous Spaces

The definition of a semi-simple compact homogeneous space was given in 6.2 above.

8.1. Transitivity of a Semi-simple Subgroup. The following theorem shows that any transitive action on a semi-simple compact manifold is an extension of a transitive action of a semi-simple Lie group.

Theorem 8.1 (Gorbatsevich 1981e, 1986b). Let M = G / H be a semi-simple compact homogeneous space. If S is a Levi subgroup in G, then the natural action on M is transitive, i.e. G = SH.

If a certain semi-simple Lie group S acts transitively and locally effectively on a manifold M then dimS::; n(n+ 2), where n = dimM (Hermann 1975). However, M (even in the compact and semi-simple case) can also sometimes act transitively non semi-simple Lie groups, of arbitrarily large dimension (see 1.4).

8.2. The Fundamental Group. Let M be a semi-simple compact homogeneous space, and let Me ---+ M' ---+ Ma be the natural bundle (for a suitable M').

By the semi-simplicity of M we have Ma = Ms. The manifold Ms is determined by the group 11'1 (M) uniquely up to a finite covering.

With the help of Theorem 5.1 we can show that for a compact semi-simple homogeneous manifold M the group 11'1 (M) is weakly commensurable with a lattice in some semi-simple Lie group S. This group S can be chosen so that it has trivial centre and no compact factors, and then it is determined by the group 1I'1(M) uniquely up isomorphism (Gorbatsevich 1986b) (we denote it by S (11'1 (M)). The following theorem gives a description of the group 11'1 (M) up to weak commensurability (and hence of the manifolds Ma up to finite coverings) .

Theorem 8.2 (Gorbatsevich 1986b). (i) Let M be a semi-simple compact homogeneous space. Then the group 11' = 11'1 (M) is weakly commensurable with a uniform lattice in some semi simple Lie group S(1I'), which has trivial centre and no compact factors.

(ii) Let S be a semi-simple Lie group with trivial centre, and without compact factors. A uniform lattice r in S is weakly commensurable with the group 1I'1(M) for some semi-simple compact homogeneous space M if and only if the tangent algebra of the Lie group S contains no ideals isomorphic to

SUp,q for (p, q) -# (2,2), (3, 3), s[n(lR) for n ~ 4,

sO;n(n -# 4),E6III,E7VII.

In (Gorbatsevich 1986b) a description is given, in terms of the group 11'1 (M) and the Satake scheme of the corresponding Lie group S (11'1 (M)), of the cor-


responding semi-simple homogeneous manifolds M. This description is rather complex, and will not be given here. Further, in (Gorbatsevich 1986b) there are exhibited all possible semi-simple Lie groups, which act transitively on each such M. This may be considered as a natural generalization of the results of (Onishchik 1967b), relating to simply connected compact homogeneous manifolds.

If M is a semi-simple compact homogeneous space, then either the group 1I"1(M) is finite (and then vcd11"1 (M) = 0), or vcd1l"1(M) 2: 2 (Gorbatsevich 1986b). Moreover, vcd 11"1 (M) = 2 if and only if Ma is diffeomorphic to some closed orientable surface Fg of genus g 2: 2 (and then 1I"1(M) ~ 11"1 (Fg)).

8.3. On the Fibre of the Natural BlUldle. The base Ma of the natural bundle is almost completely determined by the group 11"1 (M). If the group 11"1 (M) is semi-simple then it also strongly effects the fibre Me of the natural bundle. All possible manifolds Me for semi-simple compact homogeneous spaces are described in (Gorbatsevich 1986b). Here we will give only one partial result.

Theorem 8.3 (Gorbatsevich 1986b). Let Me be the fibre of the natural bundle for a semi-simple compact manifold M. If the group 11"1 (M) is infinite, then either rk (Me) 2: 2, or Me is diffeomorphic to a real Stiefel manifold (and moreover then Ma = Fg for some g 2: 2).

By rk (Me) we denote the rang of the manifold Me (see 2.4 of Chap. 2).

§9. Solvable Compact Homogeneous Spaces

9.1. Properties of the Natural BlUldle. From general results on compact homogeneous manifolds (see §4) follows

Theorem 9.1 (Gorbatsevich 1981a). Let M be a solvable compact homogeneous manifold. Then for some manifold M', which is a finite covering of M, there is a natural bundle Me ---7 M' ---7 Ma, the fibre of which Me is simply connected, and the base Ma is diffeomorphic to a solvmanifold of the form R/r, where r is a lattice in a connected solvable Lie group R.

Note that for a compact solvable homogeneous manifold M the group 11"1 (M) contains a polycyclic subgroup of finite index. On the other hand, in any polycyclic group there is a subgroup of finite index, isomorphic to 11"1 (M) for some solvable compact homogeneous space M (moreover as M one can take a solvmanifold - see Chap. 4).

The following statement gives a sufficient condition for triviality of the natural bundle for a suitable M', which is a finite covering of M (for the extent of the necessity of this condition see Gorbatsevich 1985).

Theorem 9.2 (Gorbatsevich 1981a). Let M = G/H be a solvable compact homogeneous space. If the radical of the Lie group (Nc(HO)O / HO admits an


exact finite-dimensional linear representation, then for some manifold M', which is a finite covering of M, the natural bundle is trivial.

Theorem 9.3 (Gorbatsevich 1982). Let M be a solvable compact homogeneous space of a reductive Lie group G. Then some manifold M' which is a finite covering of M is diffeomorphic to Me X r, where Me is the fibre of the natural bundle for M', and Ma = r is the base of this bundle.

As fibre Me of the natural bundle in the case of solvable homogeneous manifold M can serve any homogeneous space of a semi-simple compact Lie group. As for the base Ma, it can be any compact solvmanifold.

9.2. Elementary Solvable Homogeneous Spaces. Clearly, any compact solvmanifold is solvable. Also solvable are homogeneous spaces of arbitrary compact Lie groups. We have also the following more general assertion:

Theorem 9.4 (Gorbatsevich 1981a). Let G be a connected Lie group, whose Levi subgroup is compact. Then any compact homogeneous space of G is solvable.

It turns out that any solvable compact homogeneous manifold M has a finite covering M', which is a homogeneous space of a Lie group with a compact Levi subgroup. Moreover, as M' one can take the homogeneous space obtained from the following construction, a special case of which was given in Example 4 of Sect 3.3.

Let K be some connected semi-simple compact Lie group with a connected subgroup L. Further, let R be a simply connected solvable Lie group, r a lattice in R, and A a connected Lie subgroup of Z(R) such that An r is a lattice in A. We want to construct a solvable compact homogeneous space M of the Lie group G = K x R, which admits a natural bundle structure with fibre Me = K / L and base Ma = R/r A. Suppose that, in addition, that in K there exists a torus T, which centralizes the subgroup L, and possesses only a finite intersection with it, such that dim T = dim A.

Since the group A is abelian and simply connected there exists an epimorphisms A ---+ T. Choose one of them h : A ---+ T. Its graph G = {(h(a), a) I a E A} is a subgroup in G. Then H = LCr is a Lie subgroup of G and the homogeneous space M = G / H possesses all the properties required by us. Homogeneous spaces obtained by means of this construction (for all possible admissible K, L, T, R, r, h), we shall call elementary (they are solvable and compact).

Theorem 9.5 (Gorbatsevich 1982). Any solvable homogeneous manifold has a finite covering by some elementary solvable homogeneous space.

Proof. First one can show that for a suitable manifold £I', which is a finite covering of M, the structure group Q of the natural bundle Me ---+ £I' ---+ Ma reduces to some torus T c Q. The corresponding principal bundle over Ma with fibre T is determined uniquely up to equivalence by the characteristic


class

C E H2(M',7rl(,i)) c::: H2(7rl(Ma),zn),

where n = dim T. One can further construct an elementary homogeneous space M', for which the corresponding Ma, Me and the class c coincide (up to a transformation induced by passing to a finite covering) with Ma, Me and c obtained above. From this it follows that the manifold M' is a finite covering of M. 0

From the proof of Theorem 9.5 it is clear, that a solvable compact homogeneous manifold is determined uniquely up to a finite covering by the manifold Me = K I L, the solvmanifold Ma and some cohomology class c E H2(7rl (Ma), zn).

For an elementary solvable homogeneous space M, the Levi decomposition of a Lie group which acts transitively on M is a direct sum. In (Akhiezer 1974) it is shown that if M is a complex compact homogeneous space and the group 7rl (M) is nilpotent, then for a simply connected complex Lie group, which acts transitively and locally effectively on M, the Levi decomposition is always a direct sum. In the case of an arbitrary solvable group 7rl (M) this is not always the case.

§10. Compact Homogeneous Spaces with Discrete Stabilizers

As we saw above, in many questions involving arbitrary compact homogeneous spaces, a major role is played by spaces of the form M = G Ir, where r is a uniform lattice in the connected Lie group G. The study of such M possesses also independent significance, as they turn up in the most disparate areas of mathematics. Properties of spaces of the form G Ir are closely connected with properties of lattices in Lie groups, for which see Part I of Encycl. Math. Sc. 21. Here we shall consider only some results about such spaces.

Theorem 10.1 (Mostow 1975). Let M = G/r where r is a uniform lattice in a connected Lie group G, whose Levi subgroup has no compact factors. Then the manifold M is determined uniquely up to a finite covering by its fundamental group.

In other words, if 7rl (G Ir) is isomorphic to 7rl (G' Ir') (where r', G' satisfy the same conditions as r, G), then the manifolds G Ir and G'/r' admit a finite covering by some third manifold. The proof of this theorem in (Mostow 1975) makes use of the following result, which is of independent interest.

Theorem 10.2 (Mostow 1971). Let r be a uniform lattice in a simply connected Lie group G and let G = SR be a Levi decomposition, where the Levi subgroup S has no compact factors. Then there exists a subgroup r c r of finite index, isomorphic to a uniform lattice r' in G, such that r' = (r' n S)(r' n R) is the semi-direct product of the lattices r' n S in Sand


r' n R in R. Moreover, the manifold G jf, which is a finite covering of G jr, is isomorphic to G jr' .

In view of this theorem many problems concerning compact homogeneous spaces of the form G jr can be reduced to the cases when G is solvable or semi-simple (see Part I of Encycl. Math. Sc. 21).

Usually compact homogeneous manifolds of the form G jr admit also transitive actions of Lie groups other than G, and moreover the stabilizer of such an action may happen to be non discrete (for example on M = IRjZ = SI acts transitively G = SL2 (IR), see 1.2 of Chap. 6). However, the following theorem exhibits a class of manifolds, for which the Lie group acting transitively is practically unique.

Theorem 10.3 (Gorbatsevich 1986a). Let M = Sjr, where r is a uniform lattice in a semi-simple Lie group S, without compact factors. If the Lie group G acts transitively and locally effectively on M, then G is locally isomorphic to the group S. Moreover, it is a finite covering of the Lie group S* = SjZ(S).

One can prove that, for a given manifold M among the Lie groups in Theorem 10.3, the number of non-isomorphic groups is finite (even though it can be arbitrarily large).

In (Lukatskij 1971) it is proved, that under the conditions of Theorem 10.3 the natural action of the Lie group S on S jr does not admit proper extensions, i.e. is maximal (this follows from Theorem 10.3).

Chapter 6 Actions of Lie Groups on Low-dimensional Manifolds

§ 1. Classification of Local Actions

This section is devoted to results on classification of analytic local actions of Lie groups on open subsets of the spaces IRn and en for small n. In the cases n = 1,2 the classification was already obtained by Lie (see, for example, Vladimirov 1979, Chebotarev 1940).

1.1. Notes on Local Actions. One can classify local actions of a given Lie group up to isomorphism of these actions (see 1.3). However, beginning already with S. Lie, this classification is usually given up to local similarity. Suppose we are given two local actions: T (of a Lie group G on a manifold M) and T' (of a Lie group G' on a manifold M'). These two actions are called locally similar if there exist open subsets U c M and U' c M' such that the localizations of the action T to U and T' to U' are similar. This


means that there exists a local isomorphism of Lie groups h : G ---+ G' and a diffeomorphism f : U ---+ U' such that T'(h(g),f(x)) = f(T(g,x)) for all g E G, x E U, for which both sides of this equation are defined. In the case when G = G' and h = id, we obtain the notion of locally isomorphic local actions T and T'. If G = G', and the isomorphism h is inner (i.e. has the form g f--+ aga-1 for some a E G), then locally similar actions will also be locally isomorphic (the local isomorphism is given here by the diffeomorphism f : x f--+ T'(a, f(x)), defined in some open subset (; c U).

In the classification of local actions it is convenient to pass to infinitesimal language. A local action T of a Lie group G on a manifold M is uniquely defined by a homomorphism r : 9 ---+ tl(M) (see 2.2 of Chap. 1) of the tangent algebra 9 of the group G into the Lie algebra tl(M) of vector fields (differentiable or analytic depending on the situation considered) on the manifold M. Such a homomorphism r is called an action of the Lie algebra 9 on the manifold M. If U C M is an open subset, then there is a unique restriction (or localization) homomorphism of Lie algebras tl(M) ---+ tl(U). An action r of a Lie algebra 9 on a manifold M and an action r' of a Lie algebra g' on a manifold M' are called locally similar, if there exist an isomorphism h : 9 ---+ g' of Lie algebras and a diffeomorphism f : U ---+ U' of open subsets U c M and U' eM', such that

r'(h(X)) = df(r(X))

for any X E g, here r(X) is being considered (with the help of localization) as some vector field on U, the case of r'(h(X)) is analogous. If 9 = g' and h = id we obtain the notion of locally isomorphic actions of a Lie algebra g.

Theorem 1.1. Local actions T of a Lie group G on a manifold M and T' of a Lie group G on a manifold M' are locally similar if and only if the corresponding actions of Lie algebras r : 9 ---+ tl(M) and r' : g' ---+ tl(M') are locally similar.

There is also an analogous statement for locally isomorphic local actions of Lie groups and local Lie algebras.

An action of a Lie algebra r : 9 ---+ tl(M) on a manifold M is called effective, if Kerr = {O}. In the case when algebras of analytic vector fields on an analytic manifold M are being considered (and from now on we shall assume that this is the case), the property of effectiveness of the action is preserved by localization. In the assumption of effectiveness of action instead of considering a homomorphism 9 ---+ tl(M) it suffices to consider its image, i.e. a subalgebra of tl(M). Clearly, two actions determined by subalgebras 9 C tl(M) and g' C tl(M'), are locally similar if and only if for some open sets U C M and U' C M' there exists a diffeomorphism f : U ---+ U', such that f* maps the restriction of the subalgebra 9 to U to the restriction of the subalgebra g' to U' (in this case we shall say that the sub algebras 9 and g' are locally similar).


Clearly, in a local classification of actions (Up to local similarity or local isomorphy) it suffices to consider domains in ]Rn and en. At present classification results exist only for analytic actions, hence below (in 1.2) we limit ourselves only to the analytic case.

In studying actions of Lie algebras it is convenient (by analogy with the case of actions of Lie groups) to make use of the notions of transitivity and primitivity. An action T : 9 ---+ tJ(M) of a Lie algebra 9 on a manifold M is called transitive, if for any x E M the set of vectors of the form T(X)(X), where X E g, coincides with the tangent space Tx(M). An action T : 9 ---+

tJ(M) is called locally transitive if after restriction to some open subset U c M it becomes transitive.

A transitive action of a Lie algebra is called primitive if the stabilizer algebras gx = {X E 9 I T(X)(X) = O} for all points x E M are maximal subalgebras in the Lie algebra g. Analogously one defines also the notion of local primitivity. For subalgebras of tJ(M) the notions of transitivity and primitivity are defined in the natural way.

To conclude let us consider the connection between classifications of local and global actions of Lie groups. In the general case they differ significantly: not every local action of a Lie group can be globalized (see Example 6 in 1.3 of Chap. 1) and locally similar global actions are not necessarily globally similar. However we have

Theorem 1.2. Any local action of a Lie group on]Rn (or en) is globalizable, ifn::; 4.

In (Mostow 1950) this statement is proved for all locally transitive actions. The general case can be reduced to this special case.

1.2. Classification of Local Actions of Lie Groups on ]RI, el • Following the general approach, indicated in 1.1, we shall begin the classification of local actions of Lie groups on ]Rl by describing up to local similarity all finitedimensional subalgebras of tJ(U), where U is an open subset of Rl. We shall describe canonical subalgebras by displaying a basis, and we shall also give all non zero commutativity relations.

Theorem 1.3 (S. Lie, see Lie and Engel 1893, and also Vladimirov 1979, Chebotarev 1940). Let 9 be a non zero finite-dimensional Lie subalgebra of the Lie algebra of all analytic vector fields on an open subset of IR. Then 9 is locally similar to one of the following mutually locally non similar subalgebras:

(1) gil) = (Xl) r::: ]R, Xl = :x' (2)

As abstract Lie algebra g~l) is the unique real two-dimensional non-abelian Lie algebra.


(3)

The classification up to local similarity of finite-dimensional complex Lie subalgebras in the Lie algebra of complex analytic vector fields on reI is analogous (the subalgebras obtained are the complexifications of the subalgebras

g~l), i = 1,2,3, of Theorem 1.3).

It is easy to verify that the subalgebras gil), g~1) and g~1) are transitive and primitive.

By Theorem 1.2 there exist global actions of Lie groups corresponding to the subalgebras g~l), i = 1,2,3. Let us consider examples of these actions.

Example 1. a) The Lie group Gil) = JRI acts on JRI by parallel translations:

T(a,x) = a+x, where a,x E JRI .

b) The Lie group G~l) = AffoJRI (the connected component of the identity

of the group AffJRI ) acts on JRI by affine transformations. The group G~l) is isomorphic to the matrix group

and its action on JRI has the form

x 1--+ ax + b, x E JRI .

c) The Lie group G~l) = SL2 (JR) of projective transformations of the projective line]Rp1 (diffeomorphic to the circle). This action has the form

ax+b x 1--+ ---

cx+d'

where (~ :) is an element of the group SL2(JR).

All three of the above actions of Lie groups are transitive and primitive, and moreover actions a) and b) are reductions of action c), restricted to JRI . It is not difficult to verify that for any of the subalgebras g~l), i = 1,2,3, every automorphism is induced by a diffeomorphism of the line JRI (and for g~l) it is even inner). Hence we obtain the following corollary of Theorem 1.3.

Corollary. Any local action of each of the groups G~l) (i = 1,2,3) is locally isomorphic to one of the actions described in Example 1.

For global classification of transitive actions of Lie groups on one-dimensional manifolds see §2 below.

1.3. Classification of Local Actions of Lie Groups on ]R2 and «:2. As in the one-dimensional case, we begin by describing finite-dimensional subalgebras of the Lie algebra of analytic vector fields. However, here it will be more convenient to consider the complex analytic case, i.e. to the study subalgebras in tl ((:2) (the classification of Lie su balge bras in tl (]R2) is more cumbersome). Coordinates in «:2 will be denoted by x, y.

Theorem 1.4 (S. Lie, see Lie and Engel 1893, and also Vladimirov 1979, Chebotarev 1940). Let 9 be a non zero finite-dimensional complex subalgebra of the Lie algebra tl((:2) of complex analytic vector fields on «:2. Then 9 is locally similar to one of the following subalgebras:

1. Locally transitive subalgebras (all transitive besides 9i!) ). A) Primitive:

(1) (2) _ (a a a a a a 2 a a a 2 a )

91 - ax' ay'X ax'x ay'y ax'y ay'X ax +xy ay'XY ax +y ay ~ .5l3(C),

(2) _ (~a a (3) 93 - ax' ay' Xay ' y~ x~ -y~). ax' ax ay

(5)

B) Non primitive (here r = dim 9):

s where mi E N, Q;i E «:, i = 1,2, ... , s, :E mi + 1 = r ~ 2

i=O

(2) (a a a a a 9 = - y- e<>l'" - xe<>l'" - xml - 1e<>1"'_

5 ax ' ay' ay' ay' ... ay'

e<>2"'~ xm.-1e<>."'~) ay' ... ay ,

s where mi E N, Q;i E «:, i = 1,2, ... , s, :E mi + 2 = r,

i=O


(2) _ (a a a r-3 a a a ) (8) 98 - ax' ay' x ay' ... x ay' x ax + ayay

for a E C, a =I r - 2,

(2) _ (a a a r-3 a a r-2 a) (9) 99 - ax' ay' xay ' ... x ay' xax +[(r-2)y+x lay'

(2) _ (a a a r-4 a a a ) (10) 910 - ax' ay' x ay' ... x ay' x ax' Yay'

(2) _ ( a a a) (12) 912 - ax' x ax + ay ,

(2) _ (a a r-5 a a a 2 a a ) (14) 914 - ax' ay' ... ,x ay' Yay' x ax' x ax + (r - 5)xyay ,

(2) _ ( a a a 2 a a ) (15) 915 - ax' x ax + ay' x ax + 2x ay ,

(2) _ ( a a 2 a a a 2 a ) (1) (1) (17) 917 - ax' x ax' x ax' ay' y ay' y ay ~ 93 ® C + 93 ® C,

II. Locally intransitive

(18)

(19)

(20)

9 (2) = (~) '" 9(1) .0. C 18 ax - 1 IDI ,

(21)

where ¢Ji (x) are arbitrary analytic functions

(22)

where ¢Ji(X) are arbitrary analytic functions.

One should note that among the Lie algebras g~2)_g~;) there are locally similar ones (among the non primitive algebras) for some special values of the parameters. There exist also other versions of the classification of Lie subalgebras of u(<C2 ), differing in form from the one given above (see Vladimirov 1979, Hermann 1975).

There exists also classification up to local similarity of Lie subalgebras of u(JR.2 ), it is obtained by breaking up some of the classes of subalgebras, locally similar over <C into several classes of subalgebras, locally similar over JR. (the list of subalgebras obtained can be found, for example, in Vladimirov 1979).

Theorem 1.4 gives, in fact, also the classification up to local similarity of local complex analytic actions of Lie groups on <c2 • A classification of these actions up to local isomorphism has not yet been obtained. The following example, due to G. G. Mikhailichenko, shows that on <c2 classifications up to local similarity and up to local isomorphism are different.

Example 2. Consider the subalgebra of u(<c2 )

9 = (Xl, X2, X 3 , X4),

where Xl = tz' X 2 = ty ' X3 = xtz' X3 = yt ' and let G be the corresponding simply connected Lie group. Define on tte elements of the basis an automorphism a of the Lie algebra 9 by:

Xl I--t Xl, X2 I--t X 3 , X3 I--t -X2' X4 I--t -X4 .

The identity inclusion i: 9 ---; U((C2) and the inclusion aoi define two actions on (C2 of the Lie group G. One can easily convince oneself that these actions are not locally isomorphic (although they are, of course, similar).

In view of Theorem 1.1 all Lie subalgebras in Theorem 1.4 correspond to some global actions of Lie groups. Let us give some examples of these actions:

Example 3. a) The subalgebra g~l) for s = 1, a1 = 0, m1 = 1 has the form

and corresponds to the unique action of the Lie group (C2 on itself by translations. Note that the subalgebra 9 can be viewed as the direct sum gP) ®

<C EB g~l) ® (C of two subalgebras - the complexifications of the subalgebras

g~1) C u(JR.1 ).


Yet another global model of 9 is given by the action of the two-dimensional torus ([:2/'Z} on itself by translations.

b) To the subalgebra g~2) corresponds the action of the group SL3(([:) by projective transformations. The subgroups of SL3 (([:) consisting of affine and equiaffine transformations correspond to the subalgebras g~2) and g~2) .

c) To the subalgebras g~2), i = 6,11,17,18,19,20, correspond the actions of the Lie groups obtained by complexification of the actions of the Lie groups {e}, G~1), G~1) and G~1) on the real line and their products.

We remark that there exists a partial classification up to local similarity of local actions on ([:n and for n ~ 3. For example, there is a classification of all local actions of non solvable Lie groups on ([:3 (Kim Sen En and Morozov 1955), and ([:4 (see Zabotin 1958a, Zabotin 1958b), primitive actions are classified for n :S 6 in the works of Lie and his successors.

For the complete classification of primitive subalgebras of simple complex or real Lie algebras we refer to (Komrakov 1990, 1991).

§2. Homogeneous Spaces of Dimension::::; 3

Here we give a description of all homogeneous manifolds of dimension :S 3 and display some transitive actions of Lie groups on these manifolds.

2.1. One-dimensional Homogeneous Spaces. Let M = G I H be a onedimensional homogeneous space of a Lie group G. The one-dimensional manifold M must be diffeomorphic to JR.1 or the the circle S1. We can suppose that the Lie group G is simply connected and its action on M is locally effective. From results of Lie (see §1) it follows that the group G isomorphic to lR.1, AffoJR or the group A-the universal covering group of SL2(JR).

For examples of natural actions of the Lie group JR1 on the manifolds JR1 or S1 see 1.1 of Chap. 4. In 1.2 of that chapter a transitive action of the group G = Aff JR on JR1 is given. Let us consider in more detail transitive actions on JR and S1 of Lie groups locally isomorphic to SL2(JR).

Example 1. Consider the Lie group A and in it some maximal connected triangular subgroup B (all such subgroups are conjugate in A and isomorphic to the group AffoJR). The manifold AlB is diffeomorphic to JR.1, and hence we obtain a transitive action of the Lie group A on JR1, this action is effective.

We shall now display a transitive action of the group A on S1. We have Z(A) ~ z. Put H(k) = (Z(A))k B, where Z(A)k = {zk I z E Z(A)}, and k is some natural number. We obtain closed subgroups H(k) in A, and it is easy to verify that AIH(k) is diffeomorphic to S1 for any k > 0. As a result we obtain a series of transitive actions of the Lie group A on 8 1. All these actions are pairwise non-similar, although they are locally isomorphic. After dividing by the kernels of non-effectivness we obtain transitive and effective actions on S1 ofthe Lie groups A(k) = AI(Z(A))k, which are finite coverings


of the group PSL2(IR). For example, the natural action ofthe group PSL2(IR) on S1 by projective transformations is obtained from this construction for k = 1. The following theorem was, in fact, already known to Cartan.

Theorem 2.1. Let G be a connected Lie group, acting transitively and effectively on a one-dimensional manifold M. Then M is diffeomorphic to IR1 or S1 with

(i) If M = IRl, then G ~ IR1, AffoIR or A. Any effective action of each of these groups on IR is isomorphic either to the action, described in Example 1 of 1.2 (G = IRl, AffoIR), or to the example described in in Example 1 of that section (G = A).

(ii) If M = S1, then either G ~ '][' or G ~ A(k), where kEN. Moreover, each of the groups '][', A(k) admits unique up to isomorphism an effective action on S1 (free in the case G = '][', and described in Example 1 in the case G = A(k)).

2.2. Two-dimensional Homogeneous Spaces (Homogeneous Surfaces). We begin with some examples of two-dimensional homogeneous spaces.

Example 2. The surfaces IR2, IR1 X '][' and ']['2 are diffeomorphic to abelian Lie groups and therefore homogeneous (with respect to the natural actions of these Lie groups on themselves by means of translations).

Example 3. The group S03 of rotations of the three-dimensional Euclidean space E3 is transitive on the unit sphere S2 C E3 and generates a transitive action of the group S03 on IRP2.

Example 4. On the manifold Mb (Mobius band) and K2 (Klein bottle) acts transitively the group EO(2) of motions of the Euclidean plane (see §2 of Chap. 4).

Let us show that on Mb the group SL2(IR) acts transitively. Recall that Mb is diffeomorphic to the space of all straight lines in E2. However, it is clear that the space of all straight lines in the Lobachevski plane A 2 is also diffeomorphic to Mb. The natural transitive action by isometries of the group SL2 (IR) on the plane A 2 = S02 (IR) \SL 2 (IR) defines a transitive action on the space of all lines in A 2 .

Note that all transitive actions described in Examples 2, 3, 4 are minimal (see p. 220).

Theorem 2.2 (Mostow 1950). Any homogeneous two-dimensional manifold is diffeomorphic to

Let us consider briefly the problem of classification of all transitive actions of Lie groups on surfaces. The classification up to similarity of actions can be obtained beginning with the classification of local actions of Lie groups on IR2 (see §1). Every such local action is globalizable (Theorem 1.1), and

it only remains to explain on exactly which surfaces this or that Lie group acts transitively. In (Mostow 1950), precisely by this method, all transitive actions of Lie groups on surfaces were determined (up to similarity). For some examples of such actions see Chapters 4, 5 (see also Examples 2, 3, 4 above), the complete classification is rather cumbersome and will not be given here.

2.3. Three-dimensional Manifolds. A homogeneous manifold is called (homogeneously) decomposable if it is diffeomorphic to a direct product of (homogeneous) manifolds of lower dimension. In order to describe all three-dimensional homogeneous manifolds it suffices to describe all homogeneously indecomposable among them (as homogeneous manifolds of dimension < 3 were already described above). We begin by constructing some such manifolds.

Example 5. Let G be a three-dimensional connected Lie group, and r a discrete subgroup of G. Then M = G Ir is a three-dimensional homogeneous space.

a) If G = SU2, then all discrete subgroups r of G are finite and well known (see Dubrovin et al. 1979). The homogeneous spaces G/r all admit finite coverings by the sphere S3.

b) Let G = A be the universal covering of SL2 (lR). We obtain an aspherical three-dimensional homogeneous space M = Air. One can show that such a manifold M is homogeneously decomposable if and only if the group r is abelian (and in addition isomorphic to {e}, Z or Z2).

c) Let G = lR ~ lR2 be the semi-direct product corresponding to a homomorphism ¢ : lR ---t GL2(lR). Suppose that ¢(1) E SL2(Z) and consider the discrete subgroup r = Z ~..p Z2, where 'IjJ is the restriction of the homomorphism ¢ to the subgroups Z c lR and Z2 C lR2. We obtain a three-dimensional compact solvmanifold M = G/r. We note that there exist compact threedimensional solvmanifolds not diffeomorphic to the one constructed here (see Chap.4).

Example 6. Let L(ru>2) be the total space ofthe (unique) non trivial line bundle over ru>2, and S(ru>2) the total space of the (unique) non trivial fibre bundle over ru>2 with fibre S1 and structure group ~. By L(K2) we denote the total space of the (unique) non-trivial line bundle over K2. These three manifolds are homogeneous (Gorbatsevich 1977a)j they are homogeneously indecomposable.

Theorem 2.3 (Gorbatsevich 1977a). Any homogeneous, homogeneously indecomposable three-dimensional manifold is diffeomorphic to one of the following:

(i) A solvmanifold: L(K2) or a compact one (in particular, N s (lR)/N3 (Z), lR ~ lR2 IZ ~'" Z2 - see Example 5c).

(ii) L(ru>2), S(ru>2). (iii) SU2ID, where D is a finite subgroup of SU2


(iv) Air where r is a non abelian discrete subgroup of A.

The manifolds mentioned in various parts of Theorem 2.3 are not diffeomorphic to one another. Two solvmanifolds in (i) are diffeomorphic if and only if their fundamental groups are isomorphic. The manifolds in (ii) are mutually non diffeomorphic and the manifolds in (iii) are diffeomorphic if and only if the groups D are isomorphic. We note that three-dimensional homogeneous manifolds are homeomorphic if and only if they are diffeomorphic.

From the above results one can, in particular, deduce that a compact three-dimensional manifold M is determined up to diffeomorphism by its fundamental group. For non compact manifolds this is not always the case: ]Rl x S2 and ]R3 are homogeneous and simply connected, however they are not diffeomorphic. The statement is also false for four-dimensional simply connected homogeneous manifolds (for example for S2 x S2 and S4).

The aim of describing arbitrary transitive actions on manifolds M for dim M :::: 3 is at present not attainable. Therefore from among all transitive actions it is convenient to separate and study those which are minimal in the sense defined below.

A transitive action of a connected Lie group G on a manifold M is called minimal, if it is locally effective and if G does not contain subgroups acting transitively on M. Finding all transitive actions of Lie groups on a given manifold M reduces to

a) Finding of all minimal actions on M. b) Finding of all extensions of each of the minimal actions found in a). Taking (Mostow 1950) as the starting point, it is possible to compute all

minimal transitive actions of Lie groups on surfaces. All minimal actions on three-dimensional manifolds are described in (Gorbatsevich 1977a).

§3. Compact Homogeneous Manifolds of Low Dimension

3.1. On Four-dimensional Compact Homogeneous Manifolds. In §2 we considered homogeneous manifold of dimension ::; 3. As the dimension increases the number of homogeneous manifolds, even just the compact ones, grows and their description becomes considerably more complicated. At the present time (in 1992) there are described up to diffeomorphism compact homogeneous manifolds only up to dimension 4 (for partial results in higher dimensions see below). Homogeneously indecomposable compact homogeneous four-dimensional manifolds are the solvmanifolds (the number of which is even), 7 separate manifolds and 4 infinite series, for details see (Gorbatsevich 1977b). Let us consider some concrete examples.

The manifolds S4, JlU>4 and CP2 are homogeneous and homogeneously indecomposable. Also homogeneous are the total spaces of bundles with fibre

S1 over a base of the form SU2/ D, where D is a finite subgroup of SU2. All these spaces admit a finite covering by the manifold S1 x S3.

We note that on some compact four-dimensional manifolds M there exists (up to isomorphism) only one minimal transitive action of a Lie group - for example for M = S4 this is the natural action of the group S05. Some M admit even minimal actions of Lie groups of arbitrarily large dimension (for example the torus 1[4).

3.2. Compact Homogeneous Manifolds of Dimension ::::: 6. For dim M ~ 5 determination of compact homogeneous, homogeneously indecomposable manifolds M has not so far been achieved. However, for dim M ::::: 6 one can describe such manifolds up to a finite covering.

Theorem 3.1 (Gorbatsevich 1980). Let M be a compact homogeneous manifold of dimension ::::: 6 Then there exists a manifold M', which is a finite covering of M, and which decomposes as a direct product of manifolds from the following list {in which all manifolds are homogeneous and homogeneously indecomposable} :

{i} Simply connected manifolds: Sk {2::::: k ::::: 6}, Cpk {k = 2, 3}, SU3/S03, SU3/1'2 {where 1'2 is a maximal torus in SU3}, G~2 {Grassmann manifold}.

{ii} Solvmanifolds of the form R/r where r is' a lattice in some simply connected solvable Lie group R, with dim R = 1,3,4,5 or 6 and the group r does not decompose as a direct product of proper subgroups.

{iii} A/r, A x A/ D, where A = SL2 (lR), rand D are uniform lattices with D indecomposable.

{iv} A ~ Ad lR3 /r, where r is a uniform lattice in the Lie group A ~ Ad lR3 -the semi-direct product of the groups A and lR3 , corresponding to the adjoint representation Ad : A ---) GL3(lR).

{v} S3 x Fg, where Fg is a closed orientable surface of some genus g ~ 2. {vi} SL2 (CC)/r, where r be a uniform lattice.

Moreover, the direct factors of M', which appear above, are determined by the original manifold M uniquely up to a finite covering and the order of listing.

The proof of this theorem is based on the general results on compact homogeneous manifolds given in Chap. 5. In particular, for a suitable M', which is a finite covering of M, one considers the natural bundle and the Borel bundle, and then makes use of topological methods of classifying such bundles. If the fibre and the base degenerate to a point, then one applies other, special methods. For example, if M is aspherical, then one makes use of results of (Gorbatsevich 1983b). If M is simply connected, then for dimM = 6 see (Gorbatsevich 1980) and for dimM < 6 the argument is simple.

From Theorem 3.1 it follows, in particular, that for dim M ::::: 6 for a compact homogeneous manifold M there exists a finite covering M', such that the natural bundle for M' is trivial.


Another corollary of Theorem 3.1 concerns the determination of a compact homogeneous manifold M by its homotopy groups, or, more generally, by its homotopy type.

If Ml and M2 are compact manifolds and Ml is homotopy equivalent to M 2 , than sometimes it is possible to show that Ml is diffeomorphic to M 2 • For example, from Theorem 3.1 it easily follows that this is true if dim Mi ~ 6 and Mi are simply connected (i = 1,2). In the general case, homotopy equivalent compact homogeneous manifolds are not always diffeomorphic (for details see 2.5 of Chap. 5 and also 3.3 below). With the help of Theorem 3.1 one can prove the following statement.

Theorem 3.2 (Gorbatsevich 1980). Let Ml and M2 be compact homogeneous manifolds of dimension ~ 6. Then, if the homotopy groups 7l"i(Md and 7l"i(M2) are isomorphic for 1 ~ i ~ 5, then Ml and M2 are diffeomorphic up to a finite covering.

Let us now consider the question of the relation of decomposability and homogeneous decomposability for compact manifolds of low dimension. One can prove that, if M is a compact homogeneous manifold of dimension ~ 4 and M is diffeomorphic to Ml x M2 (i.e. M is decomposable), then the manifolds Ml and M2 are homogeneous. (Hence, for compact homogeneous manifolds of dimension ~ 4 the properties of decomposability and homogeneous decomposability are equivalent.) In dimensions ;::: 5 this no longer holds. For example, the manifold S3 x Fg is homogeneous (see Theorem 3.1 (v)), but Fg for g ;::: 2 is not homogeneous (see, for example, Theorem 2.2).

A few words concerning the complex case. Let M be a compact complex homogeneous manifold and dimcM ~ 3. Then dimIRM :S 6 and hence up to a finite covering the smooth type of the manifold M is described by Theorem 3.1. In fact, if dimcM = 1, then M is diffeomorphic to 1f2 or S2, and for dimcM = 2,3 the description of all such M is given in (Tits 1962).

3.3. On Compact Homogeneous Manifolds of Dimension ;::: 7. Attempts to classify compact homogeneous manifolds of dimension 7 and above (even up to finite coverings) meet with the following obstacles.

First, there exist already an countably infinite number of mutually non diffeomorphic simply connected homogeneous manifolds M of dimension 7 (from Theorem 3.1 it follows that for dimM ~ 6 the number of such M is finite). For example, the set of mutually non diffeomorphic manifolds of the form SU3/1I', where 1I' is the one-dimensional torus in SU3 is countably infinite. Another countably infinite series consists of manifolds of the form SU3 x SU2/ H (see 2.5 of Chap. 5), where for the stabilizer H = SU2 . T, the semi simple part L = SU2 is embedded in the standard way in SU3 and T is some one-dimensional torus, which normalizes the subgroup L, with L n T discrete. It turns out (Kreck and Stolz 1988), that among manifolds of this form there exist the following:

a) Homotopy equivalent but non-diffeomorphic manifolds.


b) Homeomorphic but not diffeomorphic manifolds (see Example 7 of 2.5 of Chap. 5).

Further, beginning with dimension 7, there exist compact homogeneous manifolds M, such that for any finite covering M' of M, the natural bundle is non trivial (and is not diffeomorphic to Me X Ma, which makes classification very much harder), for examples see 3.3 of Chap. 5.

Finally, the classification of compact solvmanifolds M is very complicated (and at the present time has not been explicitly carried out) already in dimensions 5, 6 and for dimM = 7 the complications appear, at this time, to be insurmountable.

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Auslander, L. 169

Betti, E. 167 Birkhoff, G. 61 Bochner, S. 114, 115 Bol, G. 88, 91 Borel, A. 152

Campbell, J. E. 59, 71 Cartan, E. 47, 87, 218 Cayley, A. 89 Coxeter, H. 127

Dufio, M. 64 Dynkin, E. B. 72

Frechet, M. 84, 135 Frobenius, F. 131

Gelfand, 1. M. 64, 65 Gleason, A. 88 Goldie, A. W. 64 Grassmann, H. G. 120

Hausdorff, F. 59, 71 Heisenberg, W. 50 Helgason, S. 46 Hilbert, D. 74, 87, 88, 156 Hirsch, K. A. 201 Hopf, H. 122

Iwasawa, K. 161

Jacobi, C. 32

Karpelevich, F. 1. 136 Killing, W. 183 Kirillov, A. A. 64, 65 Klein, F. 121, 170 Koszul, J.-L. 150 Kreck, M. 186, 193

Author Index

Levi, E. E. 137 Lie, S. 4, 6, 86, 99, 103, 111, 210, 212, 214 Lobachevski, N. 1. 134

Mobius, A. 164 Maltsev, A. 1. 53, 88, 91, 161 Mann, L. N. 154 Mikhailichenko, G. G. 216 Milovanov, M. B. 170 Montgomery, D. 88, 114, 115, 147, 149,

150, 153, 179 Mostow, G. D. 136, 137, 149, 166, 218 Moufang, Ruth 89, 91 Myers, S. B. 115

Ore, 0. 64 Ostrowski, A. 75

Palais, R. S. 149 Peter, F. 156 Poincare, H. 59, 61 Pontryagin, L. S. 88, 169 Postnikov, N. N. 29, 193

Samelson, H. 137,149,179 . Selberg, A. 190 Serre, J .-P. 200 Steenrod, N. 115 Stein, K. 159 Stiefel, E. 120, 166, 167 Stolz, S. 186, 194

Taylor, B. 68 Tits, J.174, 199

Wang, E. H.-C. 169 Weyl, H. 65, 127, 156 Whitney, H. 166,167 Witt, E. 61

Yang, C.-T. 150

Zippin, L. 88

Subject Index

A-structure 114 Absolute value 74 Action of a group 100

asystatic 124 by inner automorphisms 103 by left translations 103

- - by right translations 103 by two sided translations 103 effective 101 free 106 left 100

- - linear 102 primitive 124 right 100 simply transitive 106 systatic 124

- - transitive 103 trivial 100

- of a Lie algebra 211 locally primitive 212

- - locally transitive 212 - - primitive 212

transitive 212 - of a Lie group 12, 100

affine 12 asystatic 124

- - irreducible 140 - - linear 12

linear compact 108 local 104 local globalizable 105 locally effective 101

- - minimal 220 - - primitive 124

proper 149 regular 195

Algebra, Maltsev 91 - universal enveloping 60 - Banach 82 - Bol91 - Heisenberg 50 - Hopf66 - Lie 32 - - free 70

nilpotent 58 semisimple 57 solvable 56

- Moufang-Lie 91 - Weyl65

Bialgebra 66 Bundle 115 - associated 117 - Borel 194 - frame 116 - homogeneous 129 - Hopf 122 - induced 119 - Karpelevich-Mostow 136 - locally trivial 15 - Mostow 168 - natural 190 - of A-structures 218 - of positive densities 117 - principal 116 - structure 197 - tautological 120 - Tits 174 - trivial 15, 116 - universal 120

Canonical coordinates of the first kind 45

- of the second kind 45 Central series decreasing of a Lie group 58 - - algebra 58 Centralizer of a Lie subgroup 40 - an element of a Lie algebra 36 Centre of a Lie algebra 40 - group 40 Commensurability of groups 198 - weak 198 Commutator (ideal) mutual of ideals 55 - (subalgebra) of a Lie algebra 52 - (subgroup) mutual of normal subgroups

55 - (subgroup) of a Lie group 52

Subject Index

- higher of a Lie algebra 56 - - of a Lie group 56 - of vector fields 111 Component of a homogeneous manifold - almost simply connected 203 - semisimple 203 - solvable 203 Comultiplication 71 Corank of a manifold 129

Decomposable manifold 219 Deformation of a path 40 Dense winding of the torus 14 Derivation of a Lie algebra interior 50 Derivation of an algebra 35 Diagonal map 66 Differential of a homomorphism of Lie

groups 32 - of an action of a Lie group 34 - of the exponential mapping 46 - operator right invariant 67 Dimension, cohomological 200 - Gelfand-Kirillov 65 - virtual 200 Direction vector of a one-parameter

subgroup 44

Element almost invariant 102 - invariant 100 - regular of a compact Lie group 153 - - of a tangent algebra 153 - representative 102 - singular of a compact Lie group 153 Enlargement of an action 102 - natural 139 - of type I 145 - of type II 145 - radical 178 Euler characteristic of a group 201 - of a manifold 129 Exponential mapping 45, 77 Extension of structure group 116

J-projectable field 110 Factorization of a group 138 - of a Lie algebra 148 - of a Lie group 141

global 142 irreducible 143 maximal 178 trivial 143

Fibre bundle 115 - of a bundle 115 Fixed point 100

Flag manifold 148 Flow 102 - local 108 Formal group law 78 Function representative 107 Functor Lie 29 79

G-bundle 118 - homogeneous 129 - trivial 119 G-space 101 - analytic 101 - differentiable 101 - equiorbital 151 - topological 101 Gauge transformation 82 Geodesic loop 89 Globalization of a local action 105 Group formal 78 - isotropy 105 - - linear 106 - Lie 6

abelian 43 abelian complex 43 abelian real 43 aspherical 204 Banach (Hilbert) 81 complex 6 linear 8

- - local 11 nilpotent 58 of transformations 101 of type (I) 171 p-adic 75 semi-simple 204 solvable 56 standard 76 universal covering 26 vector 7

- Lie-Frechet 84 - Lie-Frechet tame 85 - loop 81 - model 121 - of components of a Lie group 22 - of covering transformations 26 - of flows 81 - of interior automorphisms of a Lie

algebra 49 - - of a Lie group 49 - of transformations 101 - semi-simple 214 - topological 86 - virtually torsion free 200 - Wang 169

233

234 Subject Index

Homogeneous manifold 123 Homomorphism of formal groups 78 - of Lie groups 9 - - covering 24 Homotopical characteristic 129

ILB-group (strong) 86 ILH-group (strong) 86 Inclusion of actions 102 - improper 102 - proper 102 Isomorphic actions 101 - local Lie groups 11 Isomorphism of group actions 101 - of homogeneous bundles 130 - of Lie Groups 9 Iwasawa Manifold 161

Jacobi identity 32

Kernel non-effectiveness 101 - of action 101 Klein model 121

Largest nilpotent ideal 58 - normal Lie subgroup 59 Lie bracket 111 - derivative 109 Lifting of an action 155 Linear isotropy representation 106 - representation of a Lie group 9 Linearlization of an action 103 Localization of an action 104 Locally isomorphic Lie algebra actions 211

Lie groups 11 - local actions of Lie groups 211 - triples of Lie groups 141 Locally similar actions of Lie algebras 211 - local actions of Lie groups 211 - subalgebras of the Lie algebra of vector

fields 211 Loop 88 - alternative 89 - analytic 89

Bol88 - disassociative 89 - local 89 - monoassociative 89 - Moufang 89

Maltsev closure 53 Manifold homogeneously decomposable

219 Mapping equivariant 101

Morphism of actions 101 - of homogeneous bundles 130 - of local actions 104 Multiplication 66

Nilmanifold 160 Norm 74 - non-archimedean 74 - ultrametric 74 Normalizer of a subspace in an algebra 37 Normed field 74 - non-archimedean field 74 Number Coxeter 127 - of ends of a group 202

One-paremeter subgroup 44 Orbit 105 - exceptional 153 - principal 153

singular 153

Part of a fundamental group semi-simple 200

- solvable 200 Path 37 Poincare Polynomial 126 - series 126 Primitive element of a bialgebra 66 Product direct of Lie groups 7 - fibred 117 - semi-direct 20 Projection map of a fibre bundle 115 Pseudo orthogonal matrix 23 Pseudo unitary matrix 23

Quotient group Lie 17 - map 15

Radical of a group 199 - of a Lie algebra 57 - of a Lie group 57 Rank of a compact Lie group 127 - of a manifold 129 Reduction of action 102 - of structure group 116 Representation adjoint of a Lie algebra 36 - - of a Lie group 35 - induced 131 Restriction of a local Lie group 11 - of an action 104

Section of a bundle 117 Similar group actions 102 Similitude of group actions 102

- of local actions of Lie groups 104 Skew-field enveloping 64 - Lie 64 Slice 150 Solvmanifold 160 - complex 171 Space Frt!chet 84 - - tame 85 - homogeneous 103

elementary 208 reductive 132 semi-simple 204 solvable 204

- Lobachevski 134 Stabilizer of a point 105 Subalgebra elliptic 114 - of finite order 114 - primitive 124 - reductive 160 - stabilizer 212 - triangular 173 - uniform 175 Subgroup Ad-algebraic 172 - arithmetical 162 - cocompact 172 - - in a Lie group 177 - Lie 7 - - virtual 38 - of maximal exponent 127 - primitive 124 - stabilizer (isotropy) 105 - triangular 173 - uniform 172

Subject Index

Sum semi-direct 51 - of Lie algebras 51

235

System dynamical with continuous time 102

- of generators free 70

t-subalgebra 173 t-subgroup 173 Tangent algebra of a formal group 79 - of a Lie group 30, 76, 86 - of a local analytic loop 90 Theorem Cartan's 47 - Mostow's Structure 167 Topological G-module 135 Torus 7 Total space of a fibre bundle 115 Tower of principal fibrations 164 Transformation inifinitesimal 109 Type orbit 106 - principal 153

Unification 138 Unit of a loop 88 Universal covering 26

Valuation ring 75 Vector field, fundamental 112 - right-invariant 31, 113 - /-projectable 110 - complete 109 - invariant 110 Velocity field of an action 34 - of a path in a Lie group 37

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