DRAFT2019 February 26

Mathematical Methods in Physics - 231B

– Group Theory –

Eric D’Hoker

Mani L. Bhaumik Institute for Theoretical PhysicsDepartment of Physics and Astronomy

University of California, Los Angeles, CA 90095, USAdhoker@physics.ucla.edu

The purpose of this course is to present an introduction to standard and widely usedmethods of group theory in Physics, including Lie groups and Lie algebras, representationtheory, tensors, spinors, structure theory of solvable and simple Lie algebras, homogeneousand symmetric spaces.

Contents

1 Definition and examples of groups 51.1 A little history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Topological groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Relating Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . . 151.8 Tangent vectors, tangent space, and cotangent space . . . . . . . . . . . . 16

2 Matrix Lie groups and Lie algebras 182.1 The general linear group GL(n) . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Closed subgroups of the general linear group . . . . . . . . . . . . . . . . . 212.3 The orthogonal groups SO(n) and Lie algebras so(n) . . . . . . . . . . . . 212.4 The symplectic group Sp(2n) and Lie algebra sp(2n) . . . . . . . . . . . . 232.5 The unitary group SU(n) and Lie algebra su(n) . . . . . . . . . . . . . . . 252.6 Cartan subalgebras and subgroups and the center . . . . . . . . . . . . . . 262.7 Summary of matrix Lie groups and Lie algebras . . . . . . . . . . . . . . . 262.8 Non-semi-simple matrix groups . . . . . . . . . . . . . . . . . . . . . . . . 302.9 Invariant differential forms, metric, and measure . . . . . . . . . . . . . . . 312.10 Spontaneous symmetry breaking, order parameters . . . . . . . . . . . . . 372.11 Lie supergroups and Lie superalgebras . . . . . . . . . . . . . . . . . . . . 38

3 Representations 413.1 Representations of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Representations of Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Transformation of a vector space under a representation . . . . . . . . . . . 443.4 Direct sum of representations . . . . . . . . . . . . . . . . . . . . . . . . . 453.5 Schur’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.6 Unitary representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.7 Tensor product of representations . . . . . . . . . . . . . . . . . . . . . . . 493.8 Characters of Representations . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Representations of SL(2;C), SU(2), and SO(2, 1) 534.1 Irreducible representations of the Lie group SU(2) . . . . . . . . . . . . . . 534.2 Finite-dimensional representations of sl(2,C) . . . . . . . . . . . . . . . . . 544.3 Infinite-dimensional representations of sl(2;C) . . . . . . . . . . . . . . . . 574.4 Harmonic oscillator representation of so(2, 1;R) . . . . . . . . . . . . . . . 574.5 Unitary representations of so(2, 1;R) . . . . . . . . . . . . . . . . . . . . . 584.6 An example of the continuous series . . . . . . . . . . . . . . . . . . . . . . 60

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5 Tensor representations 625.1 Tensor product representations . . . . . . . . . . . . . . . . . . . . . . . . 625.2 Symmetrization and anti-symmetrization . . . . . . . . . . . . . . . . . . . 635.3 Representations of SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.4 Representations of SU(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6 Spinor representations 686.1 Spinor representations of SO(3;R) . . . . . . . . . . . . . . . . . . . . . . 686.2 The Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.3 Representations of the Clifford algebra . . . . . . . . . . . . . . . . . . . . 706.4 Spinor representations of so(d) . . . . . . . . . . . . . . . . . . . . . . . . . 726.5 Reducibility and Weyl spinor representations . . . . . . . . . . . . . . . . . 736.6 Charge and complex conjugation . . . . . . . . . . . . . . . . . . . . . . . 746.7 Spinor representations of so(d− 1, 1;R) . . . . . . . . . . . . . . . . . . . . 75

7 Roots, weights, and representations 787.1 The Cartan-Killing form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.2 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.3 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.4 Raising and Lowering operators . . . . . . . . . . . . . . . . . . . . . . . . 827.5 Finite-dimensional representations . . . . . . . . . . . . . . . . . . . . . . . 827.6 The example of SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.7 The root lattice of An = sl(n+ 1) . . . . . . . . . . . . . . . . . . . . . . . 867.8 The root lattice of Dn = so(2n) . . . . . . . . . . . . . . . . . . . . . . . . 877.9 The root lattice of Bn = so(2n+ 1) . . . . . . . . . . . . . . . . . . . . . . 897.10 The root lattice of Cn = sp(2n+ 1) . . . . . . . . . . . . . . . . . . . . . . 90

8 Representations of the classical Lie algebras 928.1 Weyl reflections and the Weyl group . . . . . . . . . . . . . . . . . . . . . 928.2 Finite-dimensional irreducible representations of sl(n+ 1;C) . . . . . . . . 938.3 Spinor representation of so(2n+ 1;C) . . . . . . . . . . . . . . . . . . . . . 948.4 Spinor representations of so(2n;C) . . . . . . . . . . . . . . . . . . . . . . 94

9 The structure of semi-simple Lie algebras 969.1 Some properties of simple roots . . . . . . . . . . . . . . . . . . . . . . . . 969.2 Classification of finite-dimensional simple Lie algebras . . . . . . . . . . . . 97

10 Weyl’s character formulas 10210.1 Characters on the maximal torus . . . . . . . . . . . . . . . . . . . . . . . 10310.2 Weyl’s first formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

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Bibliography

Standard texts for physicists

• H. Georgi, Lie Algebras and Particle Physics, Benjamin/Cummings, 1982;

• B.G. Wybourne, Classical Groups for Physicists, Wiley, 1974;

• W. K. Tung, Group Theory in Physics, World Scientific;

• P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory,Springer 1997, Chapters 13 and 14.

Classics and more mathematical

• H. Weyl, Classical Groups, Princeton, 1946.

• C. Chevalley, Theory of Lie Groups, Princeton, 1948;

• E. Cartan, The Theory of Spinors, Hermann, Paris 1966, Dover;

• S. Helgason, Differential Geometry, Lie Groups, Symmetric Spaces, Acad. Press 1978;

• M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing (2005)(no group theory per se, but great for topology and differential geometry);

Very useful but more mathematical

• B.C. Hall, Lie Groups, Lie Algebras, and Representations,Springer Verlag Graduate Texts in Mathematics 222 (2015);

• W. Fulton and J. Harris, Representation Theory, Springer Verlag, 1991;

• D.P. Zelobenko, Compact Lie Groups and Their Representations,Tansl. Math. Monographs, Vol. 40, Am. Math. Soc., 1973;

• N.J. Vilenkin, Special Functions and the Theory of Group Representations,Translations of Mathematical Monographs, Vol 22, Am. Math. Soc., 1968;

• F.D. Murnaghan, The Theory of Group Representations, Johns Hopkins, 1938;

Others

• A.W. Knapp, Lie Groups beyond an Introduction, Birkhauser 2004;

• J. Humphreys, Introduction to Lie Algbras & Representation Theory, Springer, 1980;

• N. Jacobson, Lie Algebras, Wiley, 1962; Dover, 1979;

• I.R. Porteous, Clifford Algebras and the Classical Groups, Cambridge, 1995;

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1 Definition and examples of groups

In this section, we shall begin with a little history, and then present the basic definitions andsome simple examples of groups, subgroups, topological groups, discrete groups, parametricgroups, Lie groups, and Lie algebras.

1.1 A little history

The mathematical concept of a group goes back to Joseph-Louis Lagrange (1736 - 1813)and Niels Abel (1802 - 1829), and was articulated by Evariste Galois (1811 - 1832) in thecontext of subgroups of the group of permutations of the roots of a polynomial. Lie groupsand Lie algebras were introduced by Sophus Lie (1842 - 1899) around 1870.

In physics the concept of a group arises almost invariably in the context of a symmetryor invariance of a physical quantity or of an equation which described a physical quantity.Greek antiquity appreciated the beauty of regular polygons and polyhedra. Galileo andNewton undoubtedly understood the notions of translation and rotation invariance, suchas in Galilean relativity, and used them without invoking the notion of a group.

Poincare (1854 - 1912) observed the invariance of Maxwell’s equations (without sources)under Lorentz and Poincare transformation. It is fair to state that Einstein (1879 - 1955)pioneered elevating the symmetry of the equations to a physical principle in his 1905 theoryof special relativity (Poincare invariance), but even more clearly in his 1915 theory of generalrelativity (invariance under general coordinate transformations). Since then, group theoryhas become a dominant organizing principle of modern physics: Herman Weyl (1885 - 1955)applied group theory to quantum mechanics, Lev Landau (1908 - 1968) based his theoryof second order phase transitions on the group-theoretic symmetry properties of an orderparameter; Murray Gell-Mann (1929 - present) used SU(3) group theory to predict the Ωparticle; and the Standard Model of Particle Physics is built on the Yang-Mills theory forthe gauge group SU(3)c×SU(2)L×U(1)Y , though some of this symmetry is spontaneouslybroken and not manifest at low energies.

One of the biggest theoretical questions in modern theoretical and experimental physicsis to find out what lies beyond the Standard Model. The most popular theoretical specu-lations are grand unification, which is based on extending SU(3)c × SU(2)L × U(1)Y to alarger unifying group, and supersymmetry which is based on extending the Poincare groupto a supergroup, with fermionic parameters. One of the most active areas of research overthe past decade has been the conformal bootstrap program, based on exploiting the prop-erties of the conformal group to study quantum field theories that do not necessarily admita Lagrangian or Hamiltonian description.

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1.2 Groups

A set G equipped with a binary operation ? is a group G? provided G and ? satisfy,

1. Closure: For any pair of elements g1, g2 ∈ G the operation obeys g1 ? g2 ∈ G;

2. Associativity: For any triplet of elements g1, g2, g3 ∈ G the operation obeys(g1 ? g2) ? g3 = g1 ? (g2 ? g3) = g1 ? g2 ? g3;

3. Identity: There exists an element e ∈ G such that e ? g = g ? e = g for all g ∈ G;

4. Inverse: For every g ∈ G, there exists a g−1 ∈ G such that g ? g−1 = g−1 ? g = e.

The group axioms imply that the identity element e is unique, and that for every elementg, the inverse g−1 is unique. The direct product G1 × G2 of two groups G1 and G2 withrespective operations ∗1 and ∗2 is a group with elements (g1, g2) under the operation ∗defined by (g1, g2) ∗ (h1, h2) = (g1 ∗1 h1, g2 ∗2 h2) with g1, h1 ∈ G1 and g2, h2 ∈ G2. A groupG? is commutative or Abelian if the product satisfies g1 ? g2 = g2 ? g1 for all g1, g2 ∈ G.Otherwise, the group is said to be non-commutative or non-Abelian. When no confusion isexpected to arise, one often drops reference to the operation in denoting a group, so thatG∗ is simply denoted G.

1.2.1 Subgroups, invariant subgroups, and simple groups

A subgroup H of a group G is a subset H ⊂ G such that h1 ?h2 ∈ H for all h1, h2 ∈ H andh−1 ∈ H for all h ∈ H. The group G itself and the group consisting of the identity elemente only both trivially form subgroups of the group G. Every subgroup of an Abelian groupis Abelian, but a non-Abelian group may have Abelian and/or non-Abelian subgroups.

A subgroup H of a group G is referred to as an invariant subgroup, or normal subgroup,iff it is invariant under all the inner automorphisms of G by conjugation,

γ ∗H ∗ γ−1 ∈ H for all γ ∈ G (1.1)

An inner automorphism of a group G is any transformation T : G→ G which preserves themultiplication structure of the group, so that T (g1)∗T (g2) = T (g1∗g2) for all g1, g2 ∈ G withT (e) = e and T (g−1) = T (g)−1. The key example of an inner automorphism is conjugation,Tγ(g) = γ ∗ g ∗ γ−1 for γ ∈ G. Inner automorphisms, viewed as maps from G to G, form agroup under the operation of composition of maps, Tγ Tγ′(g) = Tγ∗γ′(g). Every subgroupof an Abelian group G is an invariant subgroup.

The quotient spaceG/H of a groupG and one of its subgroupsH is the set of equivalenceclasses in G under the equivalence relation of right-multiplication by elements in H. Twoelements g1, g2 ∈ Γ are equivalent to one another iff there exists an element h ∈ H such thatg2 = g1 ?h, or equivalently g1 = g2 ∗h−1. The class containing an element g ∈ G is denoted[g], and all the elements of G in the class [g] are obtained from g by right multiplicationby H, and are thus given by g ∗H.

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If H is a normal subgroup of G, then the quotient space G/H is itself a group, andthus a subgroup of G. To prove this assertion we will show that the operation ∗ of Ginduces a multiplication on the classes of the quotient space G/H. Let [g1] and [g2] be theclasses of G/H which contain respectively the elements g1 and g2. The general elements ofG in those classes are respectively g1 ∗ h1 and g2 ∗ h2 where e h1, h2 ∈ H. Now considerthe product of two arbitrary representatives of the classes [g1] and [g2], which is given byg1 ∗ h1 ∗ g2 ∗ h2. When H is a normal subgroup of G, we may use the conjugation relationg−1

2 ∗ h1 ∗ g2 = h ∈ H so that

g1 ∗ h1 ∗ g2 ∗ h2 = g1 ∗ g2 ∗ h ∗ h2 (1.2)

By definition the element g1 ∗ g2 ∗ h ∗ h2 belongs to the class [g1 ∗ g2], so that the operation∗ on G induces an operation ? on the classes. When H is not a normal subgroup of G,we cannot use the relation g−1

2 ∗ h1 ∗ g2 = h ∈ H and then it is generally not possible toconsistently map the product of arbitrary members of two classes to a single class.

A group G is referred to as a simple group iff its only invariant subgroups are the groupG itself and the trivial subgroup of the identity e. Simple groups constitute the buildingblocks of general groups which, under certain conditions, may be decomposed into a directproduct of simple groups and certain Abelian groups.

1.2.2 Fields

A field F is defined to be a set F equipped with two operations, which are usually referredto as addition +, and multiplication ×, and which are such that F+ is an Abelian groupwith identity element 0, F0

× is an Abelian group with identity 1, and the operations arerelated by the property of distributivity for all a, b, c ∈ F,

a× (b+ c) = a× b+ a× c (1.3)

Fields form the basis of virtually all the algebra needed in physics.

1.2.3 Examples

Some of the simplest groups under the operations of addition and multiplication are,

integers Z+

rationals Q+ Q0×

reals R+ R0×

complex C+ C0×

quaternions H+ H0× (1.4)

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with the subgroup inclusions Z+ ⊂ Q+ ⊂ R+ ⊂ C+ ⊂ H+ and Q0× ⊂ R0

× ⊂ C0× ⊂ H0

×where the subscript 0 stands for the removal of 0, the identity element of addition. Theabove-mentioned groups are all Abelian, except for H0

× which is non-Abelian. Combiningaddition and multiplication, Q,R and C are fields, but H is not a field with this definitionsince its multiplication is non-Abelian. In physics the fields of interest are R and C.

The group of integers has truncations to finite groups, defined as follows,

Zn = Z/nZ = 0, 1, · · ·n− 1+ addition mod n

Z0p = Z/pZ0 = 1, · · · p− 1× multiplication mod p for p a prime number

where the function mod acts as follows. For N ∈ Z and n ∈ N, we define q ≡ N (modn)such that N = pn+ q with 0 ≤ q < n. Note that for multiplication it is crucial to have p aprime number, as otherwise the product of two elements may vanish mod p, as for examplein (2× 3)(mod 6) = 0. When p is prime, Zp forms a field which is denoted Fp.

Another example is the group of permutations Sn of a set An with n elements. Moregenerally, a group G acting on a set A by permuting its elements is a transformation groupacting on A. When the number of elements of A is n <∞ one always has G ⊂ Sn, but Gneed not be the full permutation group. Examples of transformation groups include manyof the transformations we use in physics, including translations on the real line by discretetranslations such as in a crystal or by continuous translations, and similarly for rotations.

A very important class of groups is matrix groups, which are sets of square matriceswith the operation of matrix multiplication. They will be studied in great detail in thiscourse. Here we shall just quote the most general of the matrix groups GL(n) which is thegeneral linear group of n× n matrices. The entries of the matrices may be in Z,Q,R,C oreven quaternions H and are then denoted respectively by,

GL(n,Z) ⊂ GL(n,Q) ⊂ GL(n,R) ⊂ GL(n,C) ⊂ GL(n,H) (1.5)

In each case the determinants form a normal subgroup under multiplication, respectivelygiven by Z2,Q,R,C,H (one often omits the zero superscript and the multiplication sub-script). The quotients of each group GL(n) by its determinant invariant subgroup is againa group referred to as the special linear groups, and denoted by,

SL(n,Z) ⊂ SL(n,Q) ⊂ SL(n,R) ⊂ SL(n,C) ⊂ SL(n,H) (1.6)

The other matrix groups will defined as closed subgroups of one of the above.

Finally, rectangular m × n matrices form a group under addition of matrices. Thisgroup is equivalent to a subgroup of GL(m+n) under multiplication. Denoting two m×nmatrices by M1 and M2, the correspondence is exhibited by the following matrix identity,(

Im M1

0 In

)(Im M2

0 In

)=

(Im M1 +M2

0 In

)(1.7)

where Im and In respectively denote the identity m×m and n× n matrices.

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1.3 Topological groups

The notions of open and closed sets of a set G, of a topological space G, a metric topology,a Cauchy sequence, and continuity were introduced in section 2 of the notes for 231A, andwe shall not repeat the definitions here. A topological group G is a group G equipped witha topology so that G is also a topological space and the group operation is continuous. Forlater convenience we shall recall here a few properties of a topological group G which willbe of fundamental importance in group theory.

1.3.1 Connectedness

A subset S of a topological group G is path-connected (which we shall simply refer to asconnected when no confusion is expected to arise), if any two points x, y ∈ S can be joinedby a continuous function f : [0, 1] → S given by f(s) for s ∈ [0.1], such that f(0) = xand f(1) = y. Note the criterion of continuity and the requirement that f([0, 1]) mustbe entirely contained in S. A set S which is not connected is disconnected. The set ofall disconnected components of S forms a group, denoted by π0(S) and referred to as thezero-th homotopy group of S.

A topological group G where every point is disconnected from every other point is calleddiscrete and we have π0(g) = g for every g ∈ G.

Let G be a topological group and Gid the connected component of G which containsthe identity. Then Gid is an invariant subgroup and every connected component of G iscongruent to G0 with π0(G) = G/Gid.

The set of connected components of the direct product of two topological groups G1, G2

(or more generally of two topological spaces) is the direct product of the sets of components,

π0(G1 ×G2) = π0(G1)× π0(G2) (1.8)

so that the number of disconnected components is multiplicative.

1.3.2 Simply-connectedness

Let S be a connected subset of a topological group G. A closed curve C ⊂ S is given by acontinuous function C : [0, 1] → S such that C(0) = C(1). A subset S is simply-connectedif every closed curve in S can be continuously deformed to a point, while remaining in S.To make this more explicit, a closed curve C can be deformed to a point p provided thereexists a continuous function C from [0, 1]× [0, 1]→ S such that,

C(s, 1) = C(s) C(s, 0) = p s ∈ [0, 1] (1.9)

Subsets which are not simply-connected are also very important, and one can provide ameasure for how much they fail to be simply connected. The key ingredient is the notion of

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homotopy. Two curves C0 and C1 in S which have the same end points p, q are homotopicto one another provided there exists a continuous function C : [0, 1]× [0, 1]→ S such that,

C(s, 1) = C1(s) C(1, t) = C1(1) = C0(1) = p

C(s, 0) = C0(s) C(0, t) = C1(0) = C0(0) = q (1.10)

Homotopy induces an equivalence relation between curves, either with the same end-pointsor between closed curves in S. Thus we may state the condition of simply-connectednessas the fact that all closed curves in S are homotopic to points.

The equivalence classes of closed curves in S under the homotopy equivalence relationare referred to as the elements of the first homotopy group or fundamental group π1(S).The classes form a group under composition of curves by choosing two representatives C1

and C2 which have a point p in common and then composing the curves at the point p.

Under direct product of two connected topological groups G1, G2, the fundamentalgroup behaves as follows,

π1(G1 ×G2) = π1(G1)× π1(G2) (1.11)

For example, the fundamental group of U(1) = S1 = SO(2) is isomorphic to Z (see thecorresponding discussions in the 231A notes). Since S1 is connected, we apply the aboveresult for the torus and we have π1(S1 × S1) = Z2. Since R is simply connected and hasπ1(R) = 0, the fundamental group of the cylinder is π1(S1 × R) = Z.

1.3.3 Compactness

A cover of a subset S of a topological group G is defined to be a class of open sets Onwith n ∈ N such that S ⊂

⋃On and thus covers S. A subspace S of a topological space

is compact if for every cover Onn∈N one can extract a finite sub-cover. The Heine-Boreltheorem states that a subset of Rn, with the topology induced by the Euclidean metric onRn, is compact if and only if it is closed and bounded in the sense of the Euclidean metrictopology. If G1, G2 are two compact spaces, then their direct product G1×G2 is compact.On the other hand, if G1 is not compact then the direct product of G1 with any group isnot compact.

It is important to point out that a given set may be compact under one topology andnon-compact under another. For example, consider the complex plane C = R2. Underthe topology induced by the Euclidean metric, R2 is non-compact since it is unbounded.However, we could equip C with a different metric, such as for example,

ds2 =|dz2

(1 + |z|2)2(1.12)

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With this metric, infinity z = ∞ is a finite distance away from any other point in C, andinfinity reduces to just a single point. Strictly speaking the point z = ∞ does not belongto C, but we can add it to C to produce the set,

C = C ∪ ∞ (1.13)

The set C with the metric ds2 can be covered by just two open sets C = O1 ∪ O2, wherewe can take for example O1 = z ∈ C, |z| < 2 and O2 = z ∈ C, |z| > 1. Thus, theset C with the metric topology induced by ds2 is a compact set, and C is referred to as a(one-point) compactification of C.

1.3.4 Manifolds

A topological space M is a topological manifold provided every point p ∈ M has an openneighborhood which is homeomorphic to an open set in Rn. A map is a homeomorphismprovided it is bijective, continuous, and its inverse map is also continuous. By continuity,the value of n must be constant throughout the manifold, and n is referred to as thedimension of M . Some important aspects of manifolds are as follows.

1. The homeomorphism ϕα from an open neighborhood Uα of a point p ∈ M into Rn

provides‘local coordinates for Uα. The pair (Uα, ϕα) is called a chart (think of ageographical map), and an atlas (Uα, ϕα)α∈S is a set S of charts such that,⋃

α∈S

Uα = M (1.14)

so that all points in M may be described by at least one local coordinate system.

2. The only way the open sets Uα can cover M is by having non-trivial intersections withone another. The intersection Uα ∩ Uβ of two open sets Uα and Uβ is an open set. Ifthe intersection is non-empty we have two different homeomorphisms (i.e. coordinatesets) for the same point p, namely ϕα(p) and ϕβ(p). Their compositions,

ψα,β = ϕα ϕ−1β : Rn → Rn (1.15)

are referred to as transition functions. Since ϕα and ϕβ and their inverses are con-tinuous, the functions ψα,β are automatically continuous.

While it follows from the definition of a topological manifold that the transition func-tions ψα,β are continuous, we may impose conditions on ψα,β which are stronger than merecontinuity. Here are some of the most frequently used extra conditions.

• Differentiable manifold or more precisely Ck differentiable manifold provided the orderk derivatives of the transition functions exist and are continuous;

• Real analytic manifold provided the transition functions are real analytic;

• Complex manifold exists provided n is even, and the transition functions are holo-morphic functions of several complex variables.

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1.3.5 Group manifolds and parametric groups

Requiring a topological group G to be a manifold is yet a further restriction that will turnout to be of great importance. When the group G is a manifold, its dimension dimG isdefined to be the dimension of G as a manifold.

A group G is a parametric group if it can be parametrized (at least locally in theneighborhood of any point) by a system of real coordinates or parameters t = (t1, · · · , tn),

g(t) = g(t1, . . . , tn) (1.16)

so that the group multiplication structure in G is given by,

g(ti) ∗ g(si) = g(fi(t, s)) (1.17)

where the real-valued functions fi(t, s) are continuous. Since the parameters locally describean Euclidean Rn neighborhood, a parametric group G is automatically a manifold endowedwith a group structure.

1.4 Lie Groups

A group G is a Lie group if G is a parametric group and the function fi(s, t) expressingthe composition of two group elements are real analytic in s, t.

Theorem 1 If the functions fi(s, t) admit first derivatives in s, t and those derivatives arecontinuous, then the functions fi are analytic in s, t. (Sophus Lie, 1888)

Theorem 2 Every parametric group is a Lie group. (This gives a solution to Hilbert’s Vth

problem; see D. Montgomery and L. Zippin , 1955 )

These theorems show that the combination of the group structure and the continuityof fi or first order differentiability automatically implies real analyticity.

Continuity in parameters is pervasive in physics, whenever we have a continuous sym-metry, such as rotations or Lorentz transformations, or group transformations. Therefore,our emphasis in these lectures will be on Lie groups, and we shall be brief on discretegroups, such as permutations and crystallographic groups.

1.4.1 Examples of Lie groups

The groups R and C, under addition and multiplication, are Lie groups since the operationson two elements x, y of addition x+y and multiplication xy are real analytic. Similarly, thegroups Rn, Cn, and the additive group of m×n matrices are all Lie groups under addition.The groups Z and Q under addition are not Lie groups, as they are not even manifolds.

12

Under multiplication, the matrix groups GL(n,R), GL(n,C), SL(n,R), SL(n,C) are allLie groups. For example, to see this for the case of GL(n,R) we exhibit the matrix in termsof its entries xij for i, j = 1 · · · , n,

g(x) =

x11 x12 · · · x1n

x21 x22 · · · x2n

· · · · · · · · · · · ·xn1 xn2 · · · xnn

(1.18)

The entries xij may be used as real parameters to promote GL(n,R) into a parametricgroup. Note that the parameters are labelled here by a composite index, but this is equiv-alent to the labelling by a single parameter used in the definition of a parametric group.The composition functions f(x, y) are given by the rules of matrix multiplication,

g(x)g(y) = g(f(x, y)) fij(x, y) =n∑k=1

xik ykj (1.19)

They are continuous functions of x, y, so that GL(n,R) satisfies to axioms of a Lie group,which are in fact real analytic as should be the case for a Lie group. Similarly, GL(n,C),SL(n,R), and SL(n,C) are Lie groups.

1.5 Vector spaces

Next, we will introduce Lie algebras, but to do so we shall first recall the definition of avector space. A vector space V over a field F is a set V , equipped with two operation:

1. under the operation of addition, denoted +, V forms an Abelian group;2. there is also an operation of scalar multiplication of elements of V by elements of F

which satisfies associativity and distributivity axioms, for all α, β ∈ F and all u,v ∈ V ,

(α + β)u = αu + βu

α(u + v) = αu + αv

(αβ)u = α(βu) (1.20)

The zero element 0 of F is related to the zero element 0 of V by 0u = 0, and the unit 1 ofF acting trivially on V by 1u = u. The fields of interest in physics will be R or C givingrespectively a real or a complex vector space. The vector space V will then be isomorphicto Rn, Cn or be an infinite-dimensional generalization, such as a complex function space.

1.6 Lie Algebras

A Lie algebra (defined over a field F which will be either R or C), is a vector space Gendowed with a binary operation [x, y] which is referred to as the Lie bracket and whichsatisfies the following axioms,

13

1. closure: [x, y] ∈ G for all x, y ∈ G;

2. anti-symmetry: [y, x] = −[x, y] for all x, y ∈ G;

3. linearity: [αx+ βy, z] = α[x, z] + β[y, z] for all α, β ∈ F and all x, y, z ∈ G;

4. Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z ∈ G.

When the vector space G has dimension n over F, we denote a basis of G by Xa fora = 1, · · · , n and Xa are referred to as generators of the Lie algebra. Closure implies thatthe Lie bracket of two basis vectors must be a linear combination of the Xa, so that wehave the structure relations of the Lie algebra G,

[Xa, Xb] =n∑c=1

fabcXc (1.21)

The coefficients fabc are referred to as the structure constants of the Lie algebra G. Byanti-symmetry of the Lie bracket, they are anti-symmetric in the first two indices,

fbac = −fabc (1.22)

for all a, b, c = 1, · · ·n. By linearity, the product of any two vectors may be decomposed interms of the structure constants, while the Jacobi identity on the basis vectors,

[Xa, [Xb, Xc]] + [Xb, [Xc, Xa]] + [Xc, [Xa, Xb]] = 0 (1.23)

imposes the Jacobi relations on the structure constants,

n∑d=1

(fadefbcd + fbdefcad + fcdefabd

)= 0 (1.24)

Conversely, a set of coefficients fabc will correspond to structure constants of a Lie algebraprovided fabc is anti-symmetric in the indices a, b and satisfies the Jacobi identity (1.24).

We note that, while the Lie bracket [x, y] in the definition of a Lie algebra has beendenoted by the symbol customarily used for the commutator xy−yx, a Lie algebra structuredoes not assume that products such as xy can be defined for x, y ∈ G: these products mayor may not exist. For the case of matrix algebras, we will in fact have [x, y] = xy − yxbecause the product of square matrices exists independently of their commutator.

1.6.1 Examples of Lie algebras

Any vector space V , over R or C, may be endowed with a vanishing Lie bracket for allpairs of elements of V . The corresponding Lie algebra is then referred to as an Abelian Liealgebra. The vector spaces Rn and Cn where the elements are interpreted as translationsfall in this category.

14

Examples of non-Abelian Lie algebras include the vector space V of n×n matrices wherethe Lie bracket of two elements x, y ∈ V is the commutator [x, y] = xy − yx and operationwhich is defined in terms of the product xy of n × n matrices, is always well-defined, andautomatically satisfies the Jacobi identity.

An example of a non-Abelian Lie algebra is provided by gl(n,R) which is the vector spaceof all n× n matrices with real entries and the Lie bracket is the commutator. This vectorspace has dimension n2 and a basis may be given by n2 matrices Xij with 1 ≤ i, j ≤ n whoseentries are all zero, except for entry (i, j) which is 1. Note that in our general discussionof Lie algebras we used a single index a to label the generators Xa, but here is it muchmore convenient to use the composite label (i, j). To work out the structure constants forgl(n,R) using this notation, we exhibit the entries of the generator Xij as follows,

(Xij)αβ = δiαδjβ 1 ≤ α, β ≤ n (1.25)

The components of the commutator are then obtained by the rules of matrix multiplication,

[Xij, Xk`]αβ =n∑γ=1

((Xij)αγ(Xk`)γβ − (Xk`)αγ(Xij)γβ

)(1.26)

Expressing the components in terms of Kronecker δ, we find,

[Xij, Xk`]αβ =n∑γ=1

(δiαδjγδkγδ`β − δkαδ`γδiγδjβ

)= δjkδiαδ`β − δi`δkαδjβ (1.27)

Recasting the Kronecker δ combinations in terms of components of generators, we arriveat the Lie algebra structure relations of gl(n,R),

[Xij, Xk`] = δjkXi` − δi`Xkj (1.28)

Note that the same basis of generators Xij may be used to span the vector space of n× nmatrices with complex entries. They form the Lie algebra gl(n,C) whose structure relationson the generators Xij coincide with those of gl(n,R).

1.7 Relating Lie groups and Lie algebras

In the subsections above, Lie groups and Lie algebras have been defined independently ofone another. In this subsection, we shall explain how the two are intimately related to oneanother. We begin by stating the results, and will then partially prove them.

Theorem 3 To every Lie group G there corresponds a unique Lie algebra G generated bythe tangent vector fields to G at the identity.

15

The definition and construction of tangent vectors and the tangent space to a point ona manifold will be given in the subsequent subsection. For the special case of a Lie groupG, the tangent space at every point on the Lie group manifold may be obtained from thetangent space at the identity by using left or right translations Lγ,Rγ on G, which aredefined by left and right multiplication,

Lγ(g) = γ g

Rγ(g) = g γ−1 (1.29)

Left translations form a group γ ∈ GL which is isomorphic to G, and right translations forman independent group γ ∈ GR which is also isomorphic to G. Left and right translationscommute with one another, LγRγ′ = Rγ′Lγ for all γ ∈ GL and γ′ ∈ GR and, upon settingγ = γ′ reduces to the inner automorphism discussed earlier. Under right translation by anelement γ = g, the element g is mapped to the identity. Every point in G may be mappedto the identity by such a translation. The tangent space Tg at the point g is thereforeisomorphic to the tangent space Te at the identity in G.

Theorem 4 To every Lie algebra G there corresponds a unique connected and simplyconnected Lie group G obtained by the exponential map applied to the Lie algebra.

The exponential map will be constructed in the next section for matrix groups. Theknowledge of the Lie algebra determines the Lie group up to global properties which cannotbe reconstructed from a local treatment, such as connectedness and simply connectedness.

1.8 Tangent vectors, tangent space, and cotangent space

Let M be a manifold of dimension n. Let Uα be a local open neighborhood of a pointp ∈ M and ϕα : M → Rn the corresponding homeomorphism to Rn. We shall assumethat the manifold is differentiable at least once, so that all transition functions are C1.A differentiable curve γ is a map γ : [−1, 1] → M such that ϕα γ : [−1, 1] → Rn isdifferentiable. We shall assume that the point p lies on γ and set γ(0) = p. The tangentvector to the curve γ at the point p, in the chart Uα is defined by,

d

dtϕα γ(t)

∣∣∣t=0

=d

dtϕα(γ(t))

∣∣∣t=0

(1.30)

Since the image of ϕα is in Rn, the tangent vector defined above is an element of the vectorspace Rn, and we may use the linearity properties of the vector space Rn to add vectorsand multiply them by a scalar. The corresponding vector space is defined to be the tangentspace Tp(M), and has dimension n.

The tangent vectors defined in this manner depend on the chart (Uα, ϕα), on the curveγ, and on the choice of the parameter t. It will be convenient to isolate the dependenceon γ and t by recasting the tangent vectors in terms of vector fields. To do so, consider a

16

differentiable function f : M → R. In the chart (Uα, ϕα) f may be expressed by the localcoordinates xi = ϕiα for i = 1, · · · , n in terms of a function f : Rn → R by,

f(p) = f ϕα(p) = f(x1, · · · , xn) (1.31)

The derivative of f along the curve γ at the point p is obtained by chain rule,

d

dtf(γ(t))

∣∣∣t=0

=m∑i=1

(d

dtϕiα γ(t)

∣∣∣t=0

)∂if ∂if ≡

∂f

∂xi(1.32)

and defines a linear map on the space of differentiable functions f . The coefficients of thederivatives ∂if are the components of the tangent vector to γ at p, but might be replacedwith arbitrary coefficients which no longer refer to the specific curve used,

Dvf =m∑i=1

vi(p)∂if (1.33)

It is a standard abuse of notation, when no confusion is expected to arise, to drop the tildeon f , and to identify the points p ∈M with their local coordinates (x1, · · · , , xn). The mapDv is linear in f , obeys Leibnitz’s rule,

Dv(fg) = fDv(g) + gDv(f) (1.34)

and is referred to as a derivation or vector field. The tangent space Tp(M) at a point p ∈Mis isomorphic to the space of all vector fields. We may view the vector of derivatives ∂ias the basis vectors of Tp(M). Finally, it makes sense to act on a function f by differentvector fields. The commutator of two vector fields is again a vector field,

[Dv, Dw]f = Duf ui =n∑j=1

(vj∂jw

i − wj∂jvi)

(1.35)

The commutator satisfies the Jacobi identity thanks to the Schwarz identity ∂i∂jf = ∂j∂if .

The vector space dual to Tp(M), in the sense of duality in normed vector spaces, is thecotangent space T ∗p (M) at p, and its basis vectors are the differential forms dxi. A generaldifferential 1-form ω may be decomposed in this basis by,

ω =m∑i=1

ωi(p)dxi (1.36)

The total differential d,

d =m∑i=1

dxi∂

∂xi(1.37)

is independent of the coordinates chosen in the chart.

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2 Matrix Lie groups and Lie algebras

In this section, we shall discuss Lie groups and Lie algebras defined in terms of matriceswhose entries are real or complex. The group operation is matrix multiplication of n × nmatrices which gives again an n× n matrix, and is always associative.

The most general such groups were introduced earlier as GL(n,R) and GL(n,C). Fur-ther matrix groups that will be of interest in Physics are subgroups of GL(n) defined bycertain equalities between the group elements. Such subgroups are referred to as closedsubgroups of GL(n) since generally spaces defined by an equality are closed sets. Matrixgroups whose entries are rational numbers or integers, such as GL(n,Q) and GL(n,Z), areoccur in Physics as duality groups, but they are not Lie groups.

We shall exhibit the relations between Lie groups and Lie algebras, introduce Car-tan subgroups and subalgebras, evaluate the dimension and the rank, establish low-rankcoincidences and general inclusions of the standard matrix groups and algebras.

2.1 The general linear group GL(n)

The set of n×n matrices with real or complex entries and non-vanishing determinant formgroups under matrix multiplication denoted by GL(n,R) or GL(n,C). The identity matrixis the unit element, and the condition of non-vanishing determinant guarantees that everymatrix in GL(n) has an inverse given by the inverse matrix. Some basic topological andgroup-theoretic properties are as follows.

• GL(1,R) ≈ R0 is Abelian, non-compact, not connected since the positive and negativecomponents are disconnected, and simply-connected.

• GL(1,C) ≈ C0 is Abelian, non-compact, connected, but not simply connected sincethe complex plane has one point removed.

• For n ≥ 2, the groups SL(n,R) and SL(n,C) are non-Abelian, non-compact sincethey are unbounded in Rn2

or Cn2and connected. While SL(n;C) is simply-connected,

its real counterpart SL(n;R) is not simply connected.

• For n ≥ 2, the groups GL(n,R) = SL(n,R) × R0 and GL(n,C) = SL(n,C) × C0

inherit the properties of their factors, and are non-Abelian, non-compact, the formerbeing not connected and the latter not simply connected.

As was already explained earlier, these matrix multiplication groups are all Lie groups.

2.1.1 The general linear Lie algebra gl(n)

The general linear Lie algebra gl(n,R) is the vector space of all n × n real matrices. TheLie bracket is given by the matrix commutator [x, y] = xy − yx, which gives a real n × n

18

matrix for any x, y ∈ gl(n,R), is clearly anti-symmetric and linear in each argument, andautomatically satisfies the Jacobi identity, since we have,

[[x, y], z] + [[y, z], x] + [[z, x], y] = [xy − yx, z] + [yz − zy, x] + [zx− xz, y] = 0 (2.1)

for all x, y, z ∈ gl(n,R). Thus, gl(n,R) forms a Lie algebra over R. Similarly, the space of alln×n matrices with complex entries gl(n,C) forms a Lie algebra over C. The special linearLie algebra sl(n,R) and sl(n,C) are the subspaces of respectively gl(n,R) and gl(n,C) withvanishing trace, and close under the commutator since tr[x, y] = 0 for all x, y ∈ gl(n).

2.1.2 From GL(n) to gl(n)

From the Lie group GL(n,R) we may construct the corresponding Lie algebra gl(n,R) byexpanding the group elements in a neighborhood of the identity element I ∈ GL(n,R)which corresponds to the parameter values xij = δij for i, j = 1, · · · , n. To organize thisexpansion, we introduce the expansion of the group elements,

gX(α) = I + αX +O(α2) xij = δij + αXij +O(α2) (2.2)

where X is an arbitrary n × n real matrix. Performing this expansion in all directionsaround the identity defines the tangent space Te at the identity, and X ∈ Te. For given X,the group elements gX(α) are labelled by one parameter α and generate a one-parametersubgroup of G. We can now go through the conditions under which GL(n,R) is a group,expand those conditions to linear order in α and deduce their implications on the structureof the tangent space Te.

Closure of the Lie group requires that the product gX(α)gY (β) be in GL(n,R). To firstorder in α and β this requires that αX + βY should be in Te, which means that Te isa vector space. The structure of an algebra on Te is obtained by expanding the productgX(α)gY (β) to terms which are first order in both α and β. To obtain these terms it iseasiest to study the commutator of group elements,

gX(α)gY (β)gX(α)−1gY (β)−1 = (I + αX)(1 + βY )(1− αX)(1− βY ) +O(α2, β2)

= I + αβ(XY − Y X) +O(α2, β2)

= g[X,Y ](αβ) +O(α2, β2) (2.3)

The bracket is defined to be the commutator of the matrices,

[X, Y ] = XY − Y X (2.4)

which automatically satisfies the Jacobi identity. Thus closure in the Lie group requiresthat the commutator [X, Y ] ∈ Te, so that Te with the commutator operation satisfies theaxioms of a Lie algebra, and may be identified with the Lie algebra gl(n,R).

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2.1.3 The exponential map: from gl(n) to GL(n)

Lie’s theorem states that, given a Lie algebra G, there exists a unique connected and simply-connected Lie group G whose Lie algebra is G. Thus, we can ask in the case of matrixgroups such as gl(n) and sl(n) (with real or complex field), how one passes from the Liealgebra to the Lie group. This is done with the help of the exponential map, which formatrix algebras is literally given by the exponential function of matrices.

The Taylor series of an entire function f(z) has infinite radius of convergence near anypoint in C. Thus we may identify the function with its Taylor series at the origin,

f(z) =∞∑k=0

f (k)(0)zk

k!(2.5)

The function f is defined on n× n matrices X by substituting X for z in its Taylor series,

f(X) =∞∑k=0

f (k)(0)Xk

k!(2.6)

where Xk is the k-th power of the matrix X. The result f(X) is an n×n matrix. Generally,matrix sums of this type are difficult to compute, because one generally does not have agood formula for successive powers of X. However, if X is diagonalizable by a matrix g,namely if one has X = gΛg−1 for some diagonal matrix Λ with components Λij = λiδij,then the function may be easily worked out,

f(X) = gf(Λ)g−1 f(Λ)ij = f(λi)δij (2.7)

The calculation may be extended to the case where X cannot be diagonalized, but can onlybe reduced to Jordan normal form.

The exponential map of a matrix X is the exponential function of X, given by eX . Theexponential of 0 ∈ G maps to the identity matrix, while the inverse of eX is given by e−X .A very useful relation between the trace and the determinant is given by,

det (eX) = etr(X) (2.8)

Given an element X of the Lie algebra gl(n) one defines a one-parameter group GX by,

gX(t) = etX (2.9)

For all X and all values of t, gX(t) ∈ GL(n), as the exponential always produces aninvertible matrix. If trX = 0 then det gX(t) = 1 and thus gX(t) ∈ SL(n).

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2.2 Closed subgroups of the general linear group

Subgroups of the general linear groups, either over R or over C, often arise in mathematicsand in physics as the elements of GL(n) which leave a bilinear form on Vn = Rn or Cn, ora Hermitian form on Cn invariant.

A general bilinear form is specified by an n × n matrix Q, which is real or complexaccording to the nature of Vn,

Q(x, y) = xtQy x, y ∈ Vn (2.10)

If Q is degenerate, namely if the rank of Q is strictly less than n, then the form can berestricted to a subspace Vp with p < n on which the restricted form is non-degenerate.Therefore we may assume that Q is non-degenerate without loss of generality. The subsetof linear transformations M ∈ GL(n) which leave Q invariant satisfy,

Q(Mx,My) = Q(x, y) M tQM = Q (2.11)

for all x, y ∈ Vn. They form a group GQ under multiplication since when Mi satisfiesM t

iQMi = Q for i, 1, 2 then their product also satisfies this relation, as may be seen fromthe following manipulations

(M1M2)tQ(M1M2) = M t2M

t1QM1M2 = M t

2QM2 = Q (2.12)

Multiplication is associative since Mi are matrices; the identity matrix is the unit elementof the group; and the inverse of an element M is given by M−1 = Q−1M tQ which makessense because we have assumed that Q is non-degenerate. The quadratic relation impliesthat detM = ±1, so that the group GQ is disconnected and has (at least) two connectedcomponents, of which the identity component satisfies detM = 1.

The bilinear form Q need not be symmetric or anti-symmetric, but may clearly bedecomposed into its symmetric and its anti-symmetric parts. This decomposition will giverise to bilinear forms with definite symmetry but each of which is generically degenerate.The non-degenerate forms with definite symmetry will be studied next.

2.3 The orthogonal groups SO(n) and Lie algebras so(n)

The orthogonal groups, either over the reals or over the complex numbers, are defined toleave a symmetric bilinear form invariant. The real and complex cases behave differentlyand it will be convenient to treat them separately from one another.

2.3.1 Classification over R

We define a real-valued symmetric non-degenerate bilinear form on the vector space Vn =Rn in terms of an n× n real symmetric matrix Q,

Q(x, y) = xtQy x, y ∈ Rn (2.13)

21

Equivalently, one may consider the associated quadratic form Q(x, x). Since Q is realsymmetric it may be diagonalized by an orthogonal matrix M , which satisfies M tM = Isuch that Q = M tDM and D is a diagonal matrix with entries diδij. In terms of the newcoordinates x′, y′, the bilinear form thus reads,

Q(x, y) =n∑i=1

dixiyi (2.14)

Since we have assumed that the quadratic form is non-degenerate, none of the eigenvalues divanish, and we shall denote the number of positive and negative eigenvalues respectively byp and q with p+ q = n. The pair (p, q) is referred to as the signature of the quadratic form.The signature is unchanged under real linear transformations on the coordinates xi, andis thus an intrinsic characteristic of the quadratic form. By also rescaling the coordinates,one arrives at the canonical bilinear form of signature (p, q),

Q(x, y) = xtIp,qy Ip,q =

(Ip 00 −Iq

)(2.15)

where Ip is the p× p identity matrix. One usually considers the bilinear forms with signa-tures (p, q) and (q, p) to be equivalent to one another. Important special cases are,

1. Euclidean signature (n, 0), corresponding to the Euclidean metric on Rn, given byds2 = Q(dx, dx) = dxtdx;

2. Minkowski signature (n− 1, 1), corresponding to the Minkowski metric on Rn, givenby ds2 = Q(dx, dx) = dxtIn−1,1dx;

3. Conformal signature (n− 2, 2), do be discussed later.

The linear group which leaves a real symmetric non-degenerate bilinear form of signature(p, q) invariant is denoted by O(p, q;R) and is defined by,

M tIp,qM = Ip,q M ∈ O(p, q;R) (2.16)

For n = 2, the groups are Abelian, while for n = p + q ≥ 3, the groups are non-Abelian.The defining relation implies that detM = ±1, so that O(p, q;R) is not connected as amanifold, but rather has two connected components. The component connected to theidentity may be defined as follows,

M tIp,qM = Ip,q detM = 1 M ∈ SO(p, q) (2.17)

For Euclidean signature (n, 0), the groups O(n;R) and SO(n;R) are compact since theyare closed and bounded in Rn2

, while for pq 6= 0 the groups SO(p, q) are non-compact, since

22

they are unbounded in Rn2. None of these groups are simply-connected, a property which

is both delicate and important and will be discussed in detail when we deal with spinors.

The associated Lie algebra is so(n,R), the algebraic pairing is the commutator of ma-trices, and the elements m ∈ so(n,R) satisfy,

mt = −Ip,qmIp,q m =

(S UU t T

)(2.18)

where S and T are anti-symmetric p × p and q × q matrices respectively, and U is anarbitrary p× q matrix.

2.3.2 Classification over C

We define a complex-valued symmetric non-degenerate bilinear form Q on the vector spaceVn = Cn in terms of an n× n complex symmetric matrix Q,

Q(x, y) = xtQy x, y ∈ Cn (2.19)

Unlike in the real case, there is no need in the complex case to introduce signatures, sincea factor of −1 may always be absorbed in the quadratic form by multiplication by a factorof i. Making a general linear transformation on the coordinates, one arrives at the canonicalsymmetric bilinear form over the complex numbers,

Q(x, y) = xty (2.20)

Its invariance group is denoted by O(n,C) and consists of the elements g ∈ O(n,C) suchthat gtg = I. It follows that det g = ±1 so that the group has two connected components;the component connected to the identity is denoted SO(n,C). The corresponding Lie alge-bra so(n,C) is generated by the same elements as the real algebra so(n,R) but considerednow with complex coefficients.

2.4 The symplectic group Sp(2n) and Lie algebra sp(2n)

The symplectic groups, either over the reals or over the complex numbers, are defined toleave a anti-symmetric bilinear form invariant. The real and complex cases behave verysimilarly and will be treated together. A non-degenerate anti-symmetric bilinear formcan only exist in even dimensions, and may be put in canonical form by general lineartransformations on the entries,

Q(x, y) = xtJ y J =

(0 In−In 0

)(2.21)

23

where x, y ∈ R2n for the real case and x, y ∈ C2n in the complex case. The closed subgroupSp(2n) of GL(2n) which leaves Q invariant is defined by,

Q(Mx,My) = Q(x, y) M tJM = J M ∈ Sp(2n) (2.22)

for all x, y ∈ R2n for Sp(2n;R) or x, y ∈ C2n for Sp(2n;C). The associated Lie algebras aresp(2n;R) and sp(2n;C) and we have for either the real or the complex case that m ∈ sp(2n)satisfies mtJ + Jm = 0.

2.4.1 Symplectic structure of Hamiltonian mechanics

Hamiltonian mechanics is based on the following three ingredients:

1. Phase space S2n which is a manifold of even dimension 2n, locally parametrized bygeneralized position qi and momentum pi variables for i = 1, · · · , n;

2. The Hamiltonian H(p, q) is a function on phase space;

3. The Poisson bracket x, y is an anti-symmetric bilinear form on phase space. Whenthis form is non-degenerate, S2n is a symplectic manifold. Evaluating the Poissonbracket on the generalized coordinates,

x = (p1, · · · , pn; q1, · · · , qn) xα, xβ = Jαβ (2.23)

for α, β = 1, · · · , 2n. Equivalently, one defines the symplectic 2-form by,

ω =1

2

2n∑α,β=1

Jαβdxα ∧ dxβ =n∑i=1

dpi ∧ dqi (2.24)

and this form is closed dω = 0. The linear transformations on xα which leave thePoisson bracket or, equivalently, the symplectic form ω invariant form the real sym-plectic group Sp(2n;R). In this way, the symplectic group is the subgroup of allcanonical transformations which act linearly on the canonical variables.

The existence of a continuous symmetry transformation which acts on the canonical vari-ables by infinitesimal transformations δaqi and δapi implies, by Noether’s theorem, theexistence of a conserved charge Qa. The transformations are reproduced by the action ofthe Poisson bracket,

δaqi = Qa, qi δapi = Qa, pi (2.25)

The Poisson bracket of two conserved charges Qa, Qb produces a new conserved charge, sothat the vector space of conserved charges produces a Lie algebra under composition bythe Poisson bracket,

Qa, Qb =d∑c=1

fabcQc (2.26)

24

and fabc are the structure constants of the symmetry algebra. The Poisson bracket auto-matically satisfies the Jacobi identity when acting on smooth functions, and therefore thesymmetry algebra is a Lie algbera. The qualification is analogous to the curl of a gradientvanishing, provided both act on smooth functions.

2.5 The unitary group SU(n) and Lie algebra su(n)

A Hermitian form (sometimes referred to as a sesquilinear form) over Cn is given in termsof a Hermitian matrix Q by,

Q(x, y) = x†Qy x, y ∈ Cn (2.27)

with Q† = Q. As with bilinear forms, we assume without loss of generality that the form isnon-degenerate. Since Q is Hermitian, we may diagonalize it by a unitary transformation,and its eigenvalues di will be real and non-zero, so that the form may be expressed as,

Q(x, y) =n∑i=1

di xi yi (2.28)

One may further rescale the variables xi, yi by a complex number, which rescales di by anarbitrary but positive number. Thus, we encounter again the possibility of having formsof different signatures (p, q) with p + q = n where p and q count the number of positiveand negative eigenvalues respectively. Thus a Hermitian form of signature (p, q) may bereduced to the following canonical form,

Q(x, y) = x†Ip,q y x, y ∈ Cn (2.29)

The forms of signature (p, q) and (q, p) are usually assumed to be equivalent to one another.The group which leaves this form invariant is defined by,

Q(Mx,My) = Q(x, y) M †Ip,qM = Ip,q (2.30)

and denoted by U(p, q). There is no need to indicate that this is over the complex, sinceHermitian assumes complex. The group U(p, q) has a U(1) invariant subgroup, and for thisreason one defines the more frequently used group,

SU(p, q) = M ∈ U(p, q), detM = 1 (2.31)

The most familiar case is when q = 0, in which case the group is denoted by SU(n).

The Lie algebra associated with the Lie group SU(n) is denoted by su(n) and is thespace of traceless anti-Hermitian n×n matrices m ∈ su(n). More generally, the Lie algebra

25

associated with SU(p, q) is denoted by su(p, q) and consists of n × n complex matrices m

which satisfy the relation,

m†Ip,q + Ip,qm = 0 m =

(m1 m3

m†3 m2

)(2.32)

where m1 and m2 are anti-hermitian matrices of dimension p × p and q × q respectively,and m3 is an arbitrary p× q matrix.

2.6 Cartan subalgebras and subgroups and the center

A crucial role is played by the Cartan subgroup of a Lie group, and the Cartan subalgebraof a Lie algebra. We shall limit the definition here to semi-simple Lie groups (the productof simple Lie groups), and semi-simple LIe algebras (the direct sum of simple LIe algebras),which all the matrix groups are aside from some Abelian cases. The Cartan subalgebraof a semi-simple Lie algebra G with Lie group G is defined to be its maximal Abeliansubalgebra, and is unique up to conjugation by an arbitrary element of the Lie group G.The dimension of the Cartan subalgebra is defined to be the rank of G and G. The Cartansubgroup of G may similarly be defined as the maximal Abelian subgroup of G, or maximaltorus, which is unique up to conjugation.

The center Z(G) of a group G is the subgroup of elements in G which commute with allelements g ∈ G. The center Z(G) is a normal subgroup of G so that G/Z(G) is a group.

2.7 Summary of matrix Lie groups and Lie algebras

In the table below we give a summary of simple Lie groups, their group relation, Cartan typeof their Lie algebra, rank, and dimension. The unit determinant condition is understoodfor all groups but has been exhibited only for SL groups for brevity.

The above Lie groups are all connected. The groups SU(n), SO(n;R), and USp(2n)are compact; all others are non-compact. The groups SL(n;C) and SU(n) are simplyconnected, while π1(SO(n)) = Z2 for n ≥ 3 and π1(Sp(2n)) = Z2 for all n ≥ 1, so thesegroups are not simply connected. The group USp(2n) = Sp(2n;C)∩U(2n) is the maximalcompact subgroup of Sp(2n;C), and is sometimes denoted Sp(n).

2.7.1 Isomorphisms between matrix groups of low rank

For complex groups we have the following identifications,

• SO(3;C) = SL(2;C)/Z2

• SO(4;C) = SL(2;C)× SL(2;C)/Z2

• SO(5,C) = Sp(4;C)

26

type group group relation rank dimension center

An−1 SL(n;C) detM = 1 n− 1 2n2 − 2 ZnSL(n;R) detM = 1 n− 1 n2 − 1 1 or Z2

SU(n) M †M = I n− 1 n2 − 1 ZnSU(p, q) M †Ip,qM = Ip,q, p+ q = n n− 1 n2 − 1 Zn

Bn SO(2n+ 1;C) M tM = I n 2n(2n+ 1) 1SO(2n+ 1;R) M tM = I n n(2n+ 1) 1SO(p, q;R) M tIp,qM = Ip,q , p+ q = 2n+ 1 n n(2n+ 1) 1

Cn Sp(2n;C) M tJM = J n 2n(2n+ 1) Z2

Sp(2n;R) M tJM = J n n(2n+ 1) Z2

USp(2n) M tJM = J , M †M = I n n(2n+ 1) Z2

Dn SO(2n;C) M tM = I n 2n(2n− 1) Z2

SO(2n;R) M tM = I n n(2n− 1) Z2

SO(p, q;R) M tIp,qM = Ip,q, p+ q = 2n n n(2n− 1) Z2

• SO(6;C) = SL(4;C)

For compact groups we have the following identifications at low ranks,

• SO(3;R) = SU(2)/Z2 = USp(2)/Z2

• SO(4;R) = SU(2)× SU(2)/Z2

• SO(5;R) = USp(4)/Z2

• SO(6;R) = SU(4)/Z2

For real non-compact groups we have the following identifications at low ranks,

• SO(1, 2;R) = SL(2;R) = Sp(2;R)

• SO(1, 3;R) = SL(2,C)/Z2

• SO(2, 2;R) = SO(2, 1;R)× SO(2, 1;R)

• SO(2, 3;R) = Sp(4;R)/Z2

• SO(2, 4;R) = SU(2, 2)/Z2

All these identifications at low rank involve the orthogonal groups and are somehow relatedto the existence of the Clifford algebra and spinors. Many if not all play one role or anotherin physics. Let us establish the most frequently used relations.

2.7.2 Pauli matrices

We begin by introducing the Pauli matrices σi for i = 1, 2, 3 which are 2 × 2 Hermitianmatrices satisfying the following relations,

tr(σi) = 0 σi, σj = 2δijI2 (2.33)

27

It is customary to use a specific representation for these matrices, given by,

σ1 =

(0 11 0

)σ2 =

(0 −ii 0

)σ1 =

(1 00 −1

)(2.34)

2.7.3 Proving SO(3;C) = SL(2;C)/Z2

There is a simple one-to-one map between complex 3-vectors z = (z1, z2, z3) with zi ∈ Cand traceless complex 2× 2 matrices,

M(z) =

(z3 z1 − iz2

z1 + iz2 −z3

)=

3∑i=1

ziσi (2.35)

The trace ofM(x) vanishes while its determinant is the complex quadratic form−detM(z) =z2

1 + z22 + z2

3 . Under conjugation gM(z)g−1 by an arbitrary invertible matrix g, the tracecontinues to vanish so that the result must be a matrix M(z′) of the same type,

M(z′) = gM(z)g−1 (2.36)

Since multiplying g by a complex scalar, including the central element −I ∈ SL(2,C), doesnot affect M(z′) we restrict g ∈ SL(2,C)/Z2. Since detM(z′) = detM(z), the action of gon x must be a complex orthogonal transformation, so that SO(3;C) = SL(2;C)/Z2.

2.7.4 Proving SO(3;R) = SU(2)/Z2

To prove the identification SO(3;R) = SU(2)/Z2, we define the group U(2) by the factthat g ∈ U(2) leaves the following quadratic form on z ∈ C2 invariant under z → gz,

z†z = 1 z =

(z1

z2

)z1, z2 ∈ C (2.37)

Now consider the real vector xi for i = 1, 2, 3 given by the following Hermitian forms,

xi = z†σiz x1 = z1z2 + z2z1 x2 = iz2z1 − iz1z2 x3 = z1z1 − z2z2 (2.38)

For z†z = 1, the vector vi has unit length,∑3

i=1 vivi = 1. The invariant subgroup U(1)

of U(2) merely multiplies z by a phase and does not act on vi. This leaves only SU(2),but its center element −I ∈ SU(2) also does not act on vi. Hence the action is only bySU(2)/Z2. Now this action leaves the length of the three-vector vi invariant, and thus actsby an orthogonal transformation on vi, so that SU(2)/Z2 = SO(3;R).

28

2.7.5 Proving SO(4;R) = (SU(2)× SU(2)) /Z2

There is a one-to-one map between real 4-vectors x = (x1, x2, x3, x4) with xi ∈ R andcomplex matrices of the form,

M(x) =

(x4 + ix3 ix1 + x2

ix1 − x2 x4 − ix3

)= x4I2 + i

3∑i=1

xiσi (2.39)

which span the space defined by the conjugation relation M + M † = I2 tr(M). One mayalso think of these matrices as being isomorphic to quaternions. The following relationgives the Euclidean quadratic form in R4,

M(x)†M(x) = I2

(x2

1 + x22 + x2

3 + x24

)(2.40)

Under independent left and right multiplications by elements (gL, gR) ∈ SU(2)L×SU(2)R,

M(x′) = gLM(x)g−1R gL ∈ SU(2)L, gR ∈ SU(2)R (2.41)

we have M(x′)†M(x′) = M(x)†M(x) and hence this transformation performs a rotation onthe vector x. However, the transformation (gL, gR) = (−I2,−I2) does not act on M(x),and hence we have the identification SO(4;R) = (SU(2)× SU(2)) /Z2.

2.7.6 Proving SO(1, 3;R) = SL(2;C)/Z2

There is a one-to-one map between real 4-vectors x = (x0, x1, x2, x3) with xi ∈ R andHermitian 2× 2 matrices,

M(x) =

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)= x0I2 +

3∑i=1

xiσi (2.42)

The determinant gives the Minkowski quadratic form on R4 detM(x) = x20 − x2

1 − x22 − x2

3

which is invariant under SO(1, 3;R). Under the following action of g ∈ SL(2;C),

M(x′) = g†M(x)g (2.43)

the determinant, and hence the Euclidean quadratic form is preserved. The center element−I2 ∈ SL(2;C) does not act, and hence we can identify SO(1, 3;R) = SL(2;C)/Z2.

2.7.7 Maximal compact subgroups

Compact Lie groups play a central role in Mathematics and in Physics, and the compactsubgroups of a non-compact group also play a central role. The maximal compact subgroupof a non-compact group is the largest such subgroup, in the sense of largest dimension and

29

group max compact subgroup

SL(n;C) SU(n)SL(n;R) SO(n;R)SU(p, q) SU(p)× SU(q)× U(1)SO(n;C) SO(n;R)SO(p, q;R) SO(p;R)× SO(q;R)Sp(2n;C) USp(2n)Sp(2n;R) U(n)SO(p, q;R) SO(p;R)× SO(q;R)

rank. These groups may generally be obtained as intersection groups with the largestcompact group in a given dimension, which is U(n). Thus the maximal compact subgroupsare given in the Table below. For example, the maximal compact subgroup of Sp(2n;R) isobtained by computing its intersection with U(2n). Let M ∈ Sp(2n;R) ∩ U(2n), so thatthe matrix is real and unitary, which means that it belongs also to SO(2n;R) and we haveM tM = I2n as well as M tJM = J , which upon combining both shows that MJ = JM . Inn× n block decomposition, we have,

J =

(0 In−In 0

)M =

(A BC D

)(2.44)

The commutation of M with J implies C = −B and D = −A, while M tM = I2n impliesAtA + BtB = In and AtB = BtA. Identifying J with the complex number i we introducethe complex n×n matrix m = A+ iB, and we see that m†m = In using the above relationsbetween A and B, so that m ∈ U(n).

2.8 Non-semi-simple matrix groups

The Lie groups considered so far are,

1. Abelian groups , such as for example Rn, Cn, or U(1) = SO(2,R);

2. Simple groups, such as for example SL(n), SU(n) for n ≥ 2, SO(n) for n ≥ 5, andSp(2n) for n ≥ 1, along with their various different real forms;

3. Semi-simple groups, such as direct products of the above simple groups;

4. Reductive groups, which direct products of semi-simple groups and Abelian groups,such as for example the gauge group SU(3)c×SU(2)L×U(1)Y of the Standard Model.

But there are plenty of other matrix groups that play an important role in Mathematicsand in Physics. Here, we shall not attempt to give a full classification, but illustrate theideas with the examples of the group of Euclidean motions and the Poincare group, neitherof which belongs in any of the above four categories.

30

2.8.1 The group of Euclidean motions and the Poincare group

The Euclidean metric or distance function in arbitrary dimension n is given by,

ds2 =n∑i=1

(dxi)2 s2 =n∑i=1

(xi − yi)2 (2.45)

In either formulation, the Euclidean metric is invariant under translations xi → xi+ai, yi →yi + ai by an arbitrary translation vector a = (a1, · · · , an)t which form an Abelian groupRn. The metric is also invariant under rotations x → Rx and y → Ry for an arbitraryrotation R ∈ SO(n,R) represented by an n × n orthogonal matrix acting on the columnvector x = (x1, · · · , xn)t, which forms a non-Abelian group. We can assemble these twotransformations into a single package,

M(R, a)x = Rx+ a (2.46)

The composition of two transformations is given as follows,

M(R1, a1)M(R2, a2)x = R1R2x+R1a2 + a1

M(R1, a1)M(R2, a2) = M(R1R2, R1a2 + a1) (2.47)

From this formula, we see that the transformations close, are associative, have unit elementM(I, 0) and inverse M(R, a)−1 = M(R−1,−R−1a), so they form a Lie group which isreferred to the group of Euclidean motions, and denoted ISO(n;R). The compositionlaw reveals that translations and rotations act on x in an intertwined manner so that thegroup is not the direct product of SO(n;R) by Rn, but rather the product is referred to assemi-direct product and denoted by SO(n;R) nRn.

A matrix representation of the group may be obtained by considering (n+ 1)× (n+ 1)matrices M acting on a vector x = (x1, · · · , xn, 1)t ∈ Rn+1. Using a block decompositionof M into blocks of size n×n, n× 1, 1×n and 1× 1, we have the following representation,

M(R, a) =

(R a0 1

)(2.48)

One verifies that M(R, a)x = (Rx+ a, 1) so that M(R, a) acts on x as M(R, a) does.

The Poincare group is obtained by flipping the sign of one term in the metric, and thePoincare group ISO(1, n− 1;R) = SO(1, n;R)nRn. Further generalizations are manifest.

2.9 Invariant differential forms, metric, and measure

A very important notion is that of invariant differential forms on a Lie group G, andtheir relation with the Lie algebra G of G. To develop these objects for arbitrary Liegroups requires a bit of differential geometry, so we shall content ourselves here by defininginvariant differential forms for matrix Lie groups.

31

2.9.1 Invariant differential forms on a Lie group

For a matrix Lie group of dimension d, the group elements g(t1, . . . , td) are representedby square matrices n × n for some value of n, and are parametrized by local coordinates(t1, · · · , td). The partial derivatives of g with respect to ti are again n × n matrices, andthe total differential is defined by,

dg =d∑i=1

∂g

∂tidti (2.49)

Generally, the matrix dg is not an element of the Lie algebra, but we can define differentialforms that do take value in the Lie algebra by introducing the following combinations,

ωL(g) = g−1 dg ωR(g) = dg g−1 (2.50)

It is manifest that ωL is invariant under left multiplication of g by an arbitrary constant(i.e. t-independent) element h ∈ G, since d(hg) = hdg so that ω(hg) = ωL(g). Similarly,ωR(gh) = ωR(g) is invariant under right multiplication by a constant element h ∈ G.

Now the claim is that both ωL and ωR take values in the Lie algebra G which wasdefined to be the tangent space to G at the identity element of G. Ig we consider ωL andωR at the identity element, then the statement is trivial by the definition of the tangentspace to a manifold given earlier. Next, consider ωL (an analogous argument holds for ωR)at a point g0 ∈ G which is not the identity. But now we can use the invariance of ωL underleft multiplication by g0 to pull the form back to the identity,

ωL(g) = ωL(g−10 g) (2.51)

but from the right side it is clear that at the point g = g0, the form is evaluated atthe identity element and thus again belongs to the Lie algebra. The dimension of thespace of left-invariant forms (and of right-invariant forms) is n. Choosing a basis Xa witha = 1, · · · , d for G, we may decompose the forms as follows,

ωL =d∑a=1

ωaLXa ωR =d∑a=1

ωaRXa (2.52)

We shall show later on that ωK and ωR satisfy the Maurer-Cartan equations which willprove that every Lie group admits a flat connection, which has torsion.

2.9.2 Invariant metric on a Lie group

For a matrix group G, one may use the invariant differential forms ωL or ωR to constructthe following metric ds2 on G,

ds2 = −tr(g−1dg g−1 dg

)= −tr (ωLωL) = −tr (ωRωR) (2.53)

32

Expressing the metric ds2 in terms of local coordinates got G, we find,

ds2 =d∑

i,j=1

γij(t)dtidtj γij(t) = −tr

(g−1(t)

∂g(t)

∂tig−1(t)

∂g(t)

∂tj

)(2.54)

The metric ds2 is invariant under both left and right multiplication of g, and is thusinvariant under the group G×G. The signature of the metric will depend on the form ofG, and is dictated by the signature of the quadratic form tr(XaXb). One shows that for acompact Lie group G, the signature of −tr(XaXb) is Euclidean (all positive signs), so thatthe metric is then a positive Riemannian metric.

For example, in the case of the orthogonal group SO(n,R) the Lie algebra so(n,R)consists of real n× n anti-symmetric matrices. We take a basis Xαβ with 1 ≤ α < β ≤ n,whose components are given by (Xαβ)ij = δαiδβj − δαjδβi and evaluate the traces,

−tr (XαβXγδ) = 2δαγδβδ − 2δβγδαδ (2.55)

The trace is non-zero only if (α, β) = (γ, δ) or (α, β) = (δ, γ), but the latter is excludedsince it would require β = γ < α = δ, which is excluded by the ordering α < β. Thus, thebasis vectors Xαβ with 1 ≤ α < β ≤ d are all mutually orthogonal under the trace pairing,and −tr(Xαβ)2 = 2. Therefore the invariant metric is positive definite and Riemannian.

The Lie algebra of the unitary group U(n) consists of all n×n anti-Hermitian matrices.We take a basis consisting of the real anti-symmetric matrices Xαβ introduced for so(n)together with the imaginary symmetric matrices iYαβ with 1 ≤ α ≤ β ≤ n for whichthe components are given by (Yαβ)ij = δαiδβj + δβiδαj. It is immediate to show thattr(XαβYγδ) = 0 and that,

tr (YαβYγδ) = 2δαγδβδ + 2δβγδαδ (2.56)

The trace is zero unless (α, β) = (γ, δ) or/and (α, β) = (δ, γ) in which cases it is positive.Thus, the full basis consisting of the union of Xαβ and Yαβ is orthogonal and non-degenerate, so that the the invariant metric is positive definite and Riemannian.

Now let’s consider the non-compact group SO(p, q;R) with pq 6= 0. For the compactsub-algebra SO(p) we use the basis Xαβ with 1 ≤ α < β ≤ p while for SO(q) we use Xα′β′

with p+ 1 ≤ α′ < β′ ≤ p+ q, while for the off-diagonal block we use the basis Yαβ′ but thistime without the factor of i since those matrices must be real symmetric. The signature ofthis metric is now 1

2p(p − 1) + 1

2q(q − 1) plus signs and pq minus signs, adding up to the

total dimension 12(p+ q)(p+ q − 1) of SO(p, q;R).

2.9.3 Geodesics on a Lie group

Having the G×G invariant metric ds2 on a Lie group G, we can investigate its geodesics,namely the curves of extremal distance between two given points. Differential geometry

33

tell us to evaluate the Christoffel connection for the metric and then solve the geodesicequation. Here we shall be able to take a short cut, and show that for every element X ofthe Lie algebra G of G, the one-parameter subgroup,

GX =gX(t) = etX , t ∈ R

(2.57)

is the unique geodesic through the identity whose tangent vector at the identity is propor-tional to X. Clearly the curve GX contains the identity for t = 0 and has tangent vectorX. To prove that GX is a geodesic, we fix two points, the identity for t = 0, and the pointeX for t = 1, and show that the length of GX is extremal under all infinitesimal variationsδg(t) of the curve gX which leave the end points unchanged, given by,

g−1(t)δg(t) = ε(t) ∈ G ε(0) = ε(1) = 0 (2.58)

The length of an arbitrary curve g(t) is given by,

`[g] =

∫ 1

0

dt(− tr

(g−1g(t)g−1g(t)

)) 12

(2.59)

where g is the derivative of g(t) with respect to t. For example the length of the curve GX

between the identity and eX is given by `[gX ] = (−tr(X2))12 . To compute the variation in

length due to the deformation ε, we first compute the following variation,

δ(g−1g

)= −g−1δgg−1g + g−1δg (2.60)

Using the fact that we parametrize the variations by the Lie algebra valued function ε(t),we have δg = gε+ gε, which allows us to simplify the above expression, and obtain,

δ(g−1g

)= ε+ [g−1g, ε] = ε+ [X, ε] (2.61)

The length of the ε-deformed curve is given by,

`[g] =

∫ 1

0

dt(− tr

(X + ε+ [X, ε]

)2) 1

2+O(ε3) (2.62)

Expanding the trace, we have,

tr(X + ε+ [X, ε]

)2= trX2 + 2trXε+ tr(ε+ [X, ε])2 (2.63)

where we have used the relation trX[X, ε] = tr(X2ε) − tr(XεX) = 0 due to the cyclicsymmetry of the trace. Expanding the length in powers of ε to second order we find,

`[g] = `[gX ] +

∫ 1

0

dt

(−trXε− 1

2tr(ε+ [X, ε])2 +

1

2

(trXε)2

trX2

)+O(ε3) (2.64)

34

The first order term vanishes since X is independent of t and ε(0) = ε(1) = 0. Thisimmediately shows that the curve GX has extremal length, and is a geodesic.

We may learn more about these geodesics by investigating the contributions to secondorder in ε. The last term in (2.64) receives contributions only from deformations ε whichare parallel to X, but these amount to a reparametrization in t of the curve and are geo-metrically immaterial. Thus, we may assume, without loss of generality, that tr(Xε(t)) = 0for all t, and we are left with,

`[g] = `[gX ]− 1

2

∫ 1

0

dt tr(ε+ [X, ε])2 +O(ε3) (2.65)

For a compact Lie group G, the trace of gives a negative definite quadratic form on G.Since the combination ε + [X, ε] ∈ G, we conclude that for compact G, the geodesics areactually (local) minima of the length functional. For non-compact G, we can draw no suchconclusion. We shall return to this issue when we deal with homogeneous spaces.

2.9.4 Invariant measure on a Lie group

From the metric, as always, we can construct a volume form since from distances we cancompute angles, and from distances and angles we can construct the volume. Since themetric is invariant under G×G, the resulting measure or volume form,

dµG =√

det (γ(t))d∏i=1

dti (2.66)

is also invariant under G×G and is referred to as the Hurwitz-Haar measure or often justthe Haar measure. The Haar measure allows one to integrate functions over the Lie groupwith strong symmetry properties, which will be of fundamental importance when we dealwith representation theory and characters. For compact Lie groups, the volume of G withrespect to the Haar measure is finite,

Vol(G) =

∫G

dµG (2.67)

These volumes may actually computed and carry interesting topological information.

2.9.5 Invariant metric on a coset space G/H

Many interesting manifolds are obtained by taking the coset space G/H of a matrix Liegroup G by one of its Lie subgroups H. For example, the sphere Sn, the hyperbolic upperhalf space Hn, and complex projective space CP n of arbitrary dimension n ≥ 2 may be

35

constructed as coset,

Sn = SO(n+ 1;R)/SO(n;R)

Hn = SO(n, 1;R)/SO(n;R)

CP n = SU(n+ 1)/S(U(n)× U(1)

)(2.68)

The examples of the sphere S2 = CP 1 and the upper half plane H2 are familiar already.

We shall now construct an invariant metric for such spaces. Recall that at the levelof the group, the coset space is the quotient of G by the equivalence relation whereby toelements g, g ∈ G are equivalent to one another, g1 ≈ g2, iff there exists an element h ∈ Hsuch that g = g h. Now consider the left-invariant differential forms g−1dg, which we haveconstructed on G, and observe its transformation under right multiplication by an elementh of G, so that g = g h,

g−1dg = h−1g−1(dg h+ g dh

)= h−1(g−1dg)h+ h−1dh (2.69)

Clearly, the left-invariant differential form g−1dg depends on the element chosen in Gto represent the coset, and is thus not well-defined on G/H. However, we see that theinhomogeneous term h−1dh is always an element of the Lie subalgebra H. To construct aninvariant metric on the coset G/H, we will assume that the Lie algebra G of G admits adecomposition into a direct sum of the Lie algebra H of H plus a complement M,

G = H⊕M (2.70)

such that the mixed trace vanishes, tr(XY ) = 0 for all X ∈ H and Y ∈ M. If the Liegroup is compact, then this decomposition is automatically orthogonal under the traceinner product. But the decomposition extends to non-compact cases as well. In particular,this is the case when G/H is a symmetric space iff H and M satisfy,

[H,H] ⊂ H [H,M] ⊂M [M,M] ⊂ H (2.71)

Assuming the existence of the decomposition, we now take the projection of the differentialform g−1dg onto the space M,

ω(g) = g−1dg∣∣∣M

(2.72)

the relation between the one-form ω(g) at two points g, g = gh ∈ G which belong to thesame equivalence class, and thus correspond to the same point in G/H, is given by,

ω(gh) = h−1ω(g)h (2.73)

36

Therefore, we may now construct a metric ds2 on G/H out of these new differential formsby taking the trace,

ds2(g) = tr(ω(g)ω(g)

)(2.74)

The trace is invariant under the conjugation by h and thus we have ds2(gh) = ds2(g),and this metric is well-defined on the coset G/H. Furthermore, since it is constructed outof the left-invariant forms g−1dg, the metric ds2(g) is automatically invariant under lefttranslations in G.

We leave it up to the reader to show that the round metric on Sn indeed correspondsto this coset metric, and similarly for the other spaces. It is straightforward to check thatgeodesics on G/H are generated by one-parameter subgroups on G with a tangent vectorat the identity given by Y ∈M.

2.10 Spontaneous symmetry breaking, order parameters

Consider a statistical mechanics system with a very large number (strictly speaking aninfinite number) of degrees of freedom or a quantum field theory. Suppose its microscopicdynamics is governed by a Hamiltonian or by a Lagrangian which is invariant under asymmetry represented by the action of a group G on the dynamical degrees of freedom. Afamiliar example is given by translations or rotations in space, which are certainly symme-tries of the microscopic Hamiltonian or Lagrangian of Physics. In quantum mechanics, theground state of the system is the state of lowest energy, and we shall often adopt the sameterminology for a classical statistical mechanical system at very low temperatures (thinkof it as the ~→ 0 limit of a quantum system).

Now the ground state may or may not be invariant under G. Taking again the exampleof rotation invariance, the ground state of a gas is statistically rotation invariant, andthe same is true for a liquid. But the magnetic field of a magnet is not invariant underall rotation, but only under the rotations that leave the magnetic field invariant. Thisis an example of spontaneous symmetry breaking, a phenomenon which occurs wheneverthe ground state fails to be invariant under all the symmetries of the microscopic system,which are the symmetries of the Hamiltonian or Lagrangian. Thermal fluctuations mayalter the symmetry and in the case of a magnet produce a second order phase transition toan unmagnetized (approximately) rotation symmetric state above the Curie temperature.Quantum fluctuations also contribute and may induce a quantum phase transition at verylow temperatures.

Lev Landau (1908 - 1968) proposed a theory of second order phase transitions basedon an order parameter which reflects the change in symmetry of the system. The orderparameter is a space-time dependent field (the scalar analogue of electric or magnetic fields),governed by an effective Lagrangian which summarizes the dynamics near the ground state.

37

For every continuous symmetry that is spontaneously broken, Goldstone’s theorem assuresus that the spectrum will contain a massless scalar particle, the Nambu-Goldstone boson.Near the ground state energy, the only excitations in the spectrum will be the Goldstonebosons, for which we can write down an effective Lagrangian which must be invariant underPoincare symmetry, the symmetry G which and whose ground states must be invariantunder a subgroup H of G. The Goldstone field g(x) takes values in G for every point x ofspace-time which we will just take to be flat Rn. The Lagrangian must be invariant underG, but take values only in G/H, since a rotation of the ground state by H is immaterial. Itis immediate to deduce such a candidate Lagrangian in terms of the invariant differentialform constructed on G/H,

L =1

2tr(ωM ωM

)ωM = g−1∂µg

∣∣∣M

(2.75)

These Lagrangians are special cases of what are referred to as non-linear sigma models,and they play a crucial role in many areas of theoretical physics, including particle physicsand condensed matter and play a role in differential geometry as well.

2.11 Lie supergroups and Lie superalgebras

Fermions obey Fermi-Dirac statistics which requires Fermi fields to obey anti-commutationrelations. In the functional integral, Fermi fields are represented by anti-commuting vari-ables referred to as Grassmann variables. (Perturbative Yang-Mills theory in a covariantgauge requires the ghost fields which are also represented by Grassmann variables.) By theColeman-Mandula theorem (1967), there are no Lie group transformations that exchangebosons and fermions consistently with the physical principle of quantum mechanics and rel-ativity. It was discovered by Golfand and Lichtmann (1969), Ramond (1970), Gervais andSakita (1971), that algebras using anti-commuting variables exist that can exchange bosonsand fermions consistently. A full implementation in field theory was pioneered by Wess andZumino in 1974, and goes under the name of supersymmetry. The mathematical structuresrealizing supersymmetry are a generalization of Lie groups in which the parameters live ina Grassmann algebra, and a generalization of Lie algebras with both commutators as wellas anti-commutators.

2.11.1 Grassmann algebra

We postulate an arbitrary number of anti-commuting variables ξi, i = 1, · · ·N with thefollowing properties,

ξi, ξj = ξiξj + ξjξi = 0 for all i, j = 1, · · · , N (2.76)

In particular this means that the square of each variable vanishes. These variables generatea vector space either over the real or over the complex numbers in which linear combinations

38

of ξi may be formed. We may supplement this vector space by a copy of RM or CM withcommuting variables xI with I = 1, · · ·M , and the combined vector space is denoted eitherRM |N or CM |N . One introduces a binary grading by which xI has even grading and each ξi

has odd grading, and one only allows linear combinations that preserve the grading. Thus,it is illegal to form xI + ξi. The grading is preserved under multiplication. This restrictionis required by the physical super-selection rule that one cannot form a linear combinationof a fermion and a boson state.

The vector spaces RM |N and CM |N generate a Grassmann algebra which is obtained bytaking linear combinations of products of an even number of ξi and linear combinations ofproducts of odd numbers of ξi subject to the grading. Thus, we can take linear combinationssuch as 2 + ξ1ξ3 + 6ξ1ξ2ξ5ξ7 with even grading, and πξ1 + 19ξ7ξ8ξ9 with odd grading.

The Grassmann algebra may appear a bit abstract at first and it may not be immediatelyclear that such objects actually exist. A convenient construction in terms of matrices isobtained by starting from γµ which satisfy the Clifford-Dirac algebra,

γiγj = 2δijI i, j = 1, · · · , d (2.77)

For d = 3, one realizes the γi by the Pauli matrices γi = σi, and we shall see later how toconstruct the matrices γi explicitly for arbitrary d. Now take d = 2N even and form thefollowing combinations,

ξj = γ2j−1 + iγ2j j = 1, · · · , N (2.78)

Using (2.77), one readily verifies that ξi, ξj = 0 for all i, j = 1, · · · , N .

2.11.2 Super groups

We shall denote the elements of the vector spaces Rm|n and Cm|n by graded columns,

X =

(xξ

)(2.79)

where xI are even with I = 1, · · · ,m and ξi are odd with i = 1, · · · , n. A general lineartransformation M takes X to a vector Y = MX in the same vector space, and thus withthe same grading. Using block matrix notation,

Y =

(yυ

)M =

(A BC D

) y = Ax+Bξυ = Cx+Dξ

(2.80)

where A,B,C,D are matrices of dimension m×m, m× n, n×m and n× n respectively.To preserve the grading, A,D must be even, and B,C must be odd. The product of twomatrices M1,M2 is defined as the block-wise product,

Mk =

(Ak Bk

Ck Dk

)M1M2 =

(A1A2 +B1C2 A1B2 +B1D2

C1A2 +D1C2 C1B2 +D1D2

)(2.81)

39

The product closes and is associative. The identity matrix has A = D = I and B = C = 0.The inverse of M exists when the matrices A and D are invertible, and is then given by,

M−1 =

(A′ B′

C ′ D′

) A′ = (A−BD−1C)−1

D′ = (D − CA−1B)−1

B′ = −A−1BD′

C ′ = −D−1CA′(2.82)

The space of invertible matrices M forms a group GL(m|n;R) or GL(m|n;C) either overR or over C. However, GL(m|n) is not a Lie group in the sense defined earlier, since someof its parameters are not real or complex numbers, but rather Grassmann numbers. Notethat the even matrices are not just real or complex numbers, but rather even elements ofthe Grassmann algebra. For this reason it is referred to as a Lie super algebra.

2.11.3 The Lie super groups SU(m|n) and OSp(m|2n)

Closed subalgebras of GL(m|n) are defined by requiring invariance of a non-degenerateeven quadratic form on Cm|n or Rm|n. The quadratic form is the sum of quadratic formsin the even variables xi and the odd variables ξi, the latter being very restrictive. In thecomplex case, it may be diagonalized to the following canonical form,

QC = x†x+ ξ†ξ (2.83)

The corresponding Lie supergroup is U(m|n) with maximal even subgroup SU(m) ×SU(n) × U(1). In the real case, non-degeneracy requires their number to be even 2nand the canonical form is given by,

QR = xtx+ ξtJξ (2.84)

and the Lie supergroup is OSp(m|2n) with maximal even subgroup SO(m)× Sp(2n).

2.11.4 Lie super algebras

A Lie super algebra G is a graded algebra with even subspace G0 and odd subspace G1, onwhich one defines a graded Lie bracket, [·, ·] or ·, · depending on the arguments,

1. closure: [G0,G0] ∈ G0, [G0,G1], [G1,G0] ∈ G1, and G1,G1 ∈ G0;

2. (anti-)symmetry: [x, y] = −[y, x] and x, y = y, x;3. linearity: [αx+ βy, z] = α[x, z] + β[y, z], αx+ βy, z = αx, z+ βy, z;4. Jacobi identity: if at most one of x, y, z ∈ G1 we have the usual Jacobi identity,

with x ∈ G0 and y, z ∈ G1 we have [x, y, z] + y, [z, x] − z, [x, y] = 0, and withx, y, z ∈ G1 we have [x, y, z] + [y, z, x] + [z, x, y] = 0.

40

3 Representations

A group is an abstract mathematical concept. The transformation of physical quantities isrealized through a representation of the group on physical variables. When the transforma-tions are linear, we are dealing with a linear representation (or representation for short),in terms of square matrices or in terms of linear operators. The concept of a representa-tion is of fundamental importance in group theory, and provides a tool to classify groups.Representations of Lie groups will be closely related with representations of Lie algebras,which we shall study in detail. Non-linear representations or non-linear realizations of agroup also enter physics and will be discussed in a later section.

3.1 Representations of groups

A (linear) representation R of a group G with operation ∗ is a map R : G → GL(N) interms of which the group operation ∗ of G maps to matrix multiplication in GL(N),

1. R(g ∗ g′) = R(g)R(g′) for all g, g′ ∈ G;

2. R(e) = IN ;

3. R(g−1) = R(g)−1 for every g ∈ G.

If the map is R : G→ GL(N,R) one refers to R as a real representation, while if the mapis R : G → GL(N,C) one refers to R as a complex representation. A special case whichis important in physics and in mathematics is when the map R is to the unitary groupR : G→ U(N) ⊂ GL(N ;C), in which case R is referred to as unitary representation.

The dimension of the representation R is defined to be N in either case, though it mustbe stressed that this definition refers to the dimension over R when R is real, and over Cwhen R is complex. For N =∞, the representation is said to be infinite-dimensional.

Two representations R,R′ of the same group G are referred to as being equivalentrepresentations if their dimensions are equal N = N ′ and there exists a matrix A ∈ GL(N)(which is thus invertible) such that

R′(g) = AR(g)A−1 (3.1)

for all g ∈ G. Clearly, the matrix A may be viewed as effecting a change of basis in thevector space on which G acts so that we shall not be interested in distinguishing betweenequivalent representations, and consider only the equivalence classes of representations.

3.1.1 Examples of representations

We shall spend a lot of time constructing and examining representations of various groupslater on. Here we wish to present just a few examples to clarify the definition.

41

1. The trivial representation assigns to every group element g ∈ G the value R(g) = 1.

2. The multiplicative group C0× has a two-dimensional representation R in terms of real

matrices, mapping a complex number z = a+ ib with a, b ∈ R to the matrix,

R(z) =

(a −bb a

)R(i) =

(0 −11 0

)(3.2)

3. The additive group G+ of M ×N matrices admits a representation R in GL(M +N)by (M +N)× (M +N) matrices of the following form for x, y ∈ G+,

R(x) =

(1 x0 1

)R(x)R(y) = R(x+ y) (3.3)

4. The group of permutations SN has a representation by N × N matrices. To apermutation σ ∈ SN , we assign a matrix R(σ) whose entries as given by,

R(σ)ij = δi,σ(j) i, j = 1, · · · , N (3.4)

This matrix has exactly one entry on each row which equals 1, all others being zero.

5. Another representation of the permutation group is given by the signature of thepermutation, denoted by R(σ) = (−)σ, and takes only the values ±1. In quantummechanics, a system of N identical bosons is invariant under permutations, while asystem of N identical fermions transforms with the signature representation.

6. Matrix groups, introduced in the preceding section, are defined in terms of subgroupsof the general linear groups GL(N ;R) or GL(N,C) and therefore naturally exhibitedin terms of a linear representation, which is referred to as the defining representation.For GL(N), SL(N), SO(N), and SU(N) the defining representation has dimensionN while the dimension of the defining representation of Sp(2N) is 2N .

A non-linear representation of a group G arises when G acts not on a vector space, butrather on a manifold such as a coset space G/H of G by one of its subgroups H. A simpleexample of a non-linear representation of SL(2,R) acting on the real line x ∈ R is givenby the Mobius transformation,

g : x→ ax+ b

cx+ dg =

(a bc d

)∈ SL(2,R) (3.5)

Non-linear representations are ubiquitous in problems where the group action is on variablesand fields which satisfy non-linear differential equations, such as Einstein’s equations ofgeneral relativity or Yang-Mills theory.

42

3.2 Representations of Lie algebras

Consider a Lie algebra G of dimension n with generators Xa, structure constants fabc, for

a, b, c = 1, · · · , n, and structure relations,

[Xa, Xb] =n∑c=1

fabcXc (3.6)

A representation ρ of a Lie algebra G is a linear map ρ : G → gl(N) where gl(N) is thespace of N ×N real or complex matrices (see the previous section), such that ρ preservesthe Lie algebra bilinear product,

ρ([x, y]) = [ρ(x), ρ(y)] for all x, y ∈ G (3.7)

Note that, by an abuse of notation, [x, y] on the left side stands for the Lie bracket of the Liealgebra G, while [ρ(x), ρ(y)] stands for the commutator of matrices in gl(N). The dimensionof the representation ρ is defined to be N (over R or C as in the case of representations ofgroups, see the preceding subsection). Taking Xa to be a basis of the Lie algebra G witha = 1, · · · , n = dimG, we see that the linearity of ρ implies that ρ(Xa) satisfy the samestructure relations as Xa itself,

[ρ(Xa), ρ(Xb)] =n∑c=1

fabc ρ(Xc) (3.8)

As introduced above, the dimension n of the Lie algebra G is independent of the dimensionN of the representation ρ.

3.2.1 Examples of representations of Lie algebras

Here we just give some immediate examples of Lie algebras to illustrate their definition andreserve for later a more systematic discussion.

1. The trivial representation assigns to every x ∈ G the zero element, ρ(x) = 0.

2. The defining representation for each matrix Lie group SL(N), SU(N), SO(N),Sp(2N) induces the defining representation with the same dimension of the corre-sponding Lie algebras sl(N), su(N), so(N) and sp(2N).

3. Every Lie algebra G possesses the adjoint representation whose representation matri-ces are given by the structure constants fabc with a, b, c = 1, · · · , n = dimG,

(Fa)bc = −fabc (3.9)

The Jacobi identity of the Lie algebra G given in (1.24) may be recast in terms ofmatrix multiplication for the matrices Fa, effected by the summation over the index

43

d in (1.24). To see this, we recast the Jacobi identity in the following form by usinganti-symmetry of the structure constants in their first two indices,

n∑d=1

(fac

dfbde − fbcdfade

)= −

n∑d=1

fabdfdc

e (3.10)

Expressing the four f on the left side and the second f on the right side in terms ofF , but leaving the first f on the right side, we find,

[Fa, Fb]ce =

n∑d=1

fabd(Fd)c

e (3.11)

Thus the matrices Fa form a representation of the Lie algebra G. This representationhas exactly the same dimension as the Lie algebra, namely n, and is referred to as theadjoint representation of G. For an Abelian Lie algebra, the adjoint representation isalways equivalent to the trivial representation.

In a basis-independent formulation, the adjoint representation of an element x ∈ Gis denoted Adx and acts on an arbitrary element z of G (which makes sense since itsdimension is exactly the same as that of the Lie algebra) by,

Adx(z) = [x, z] (3.12)

The fact that this map is a representation of G again follows from the Jacobi identity,

[Adx,Ady] (z) = [x, [y, z]]− [y, [x, z]] = [x, [y, z]] + [y, [z, x]]

= [[x, y], z] = Ad[x,y](z) (3.13)

which was used in going from the first to the last line.

3.3 Transformation of a vector space under a representation

The general linear group GL(N ;C) naturally transforms the vector space V = CN into V bylinear transformations, and analogously for GL(N ;R) on V = RN . Therefore, an arbitrarygroup G may also transform the vector space V through an N -dimensional representationR : G → GL(N) of G. By the same token, the vector space V transforms under anN -dimensional representation ρ of a Lie algebra G, and we have,

R(g) : V → V v ∈ V → R(g)v ∈ V g ∈ Gρ(x) : V → V v ∈ V → ρ(x)v ∈ V x ∈ G (3.14)

where we have represented the elements v of the vector space V as column vectors in agiven basis of V , and R(g)v and ρ(x)v are the products of N × N square matrices by a

44

column vector on the right which gives again a column vector. The vector space V and itselements are then said to transform under the representation R of the group G and underthe representation ρ(x) of the Lie algebra G.1 In physics, the vector spaces V describeN physical quantities, such as position in space and time transformed by translations,rotations, and Lorentz transformation, or a set of electromagnetic fields, or states in theHilbert space of a quantum mechanical system with N degrees of freedom.

3.4 Direct sum of representations

In this subsection, we shall define the direct sum of representations, and the reducibility ofa general representation into a direct sum of representations. We shall write things out forrepresentations of a group G, the transcription to representations of a Lie algebra beingimmediate. First, we need to review the direct sum of vector spaces.

The direct sum of two vector spaces V1 and V2 of respective dimensions N1, N2 (eitherover R or C) is a vector space V1 ⊕ V2 of dimension N1 + N2 consisting of ordered pairs(v1, v2) with v1 ∈ V1 and v2 ∈ V2 such that addition on V1 ⊕ V2 is defined by,

(v1, v2) + (v′1, v′2) = (v1 + v′1, v2 + v′2) (3.15)

In terms of bases e11, · · · , e

N11 for V1 and e1

2, · · · , eN22 for V2 a basis of V1⊕V2 may be chosen

to be e11, · · · , e

N11 , e1

2, · · · , eN22 thereby clearly giving a vector space of dimension N1 + N2.

In this basis, we may represent the vectors in V1 and V2 by column matrices,

v1 =

N1∑i=1

vi1ei1 =

v1

1

v21

· · ·vN1

1

v2 =

N2∑i=1

vi2ei2 =

v1

2

v22

· · ·vN2

2

v1 ⊕ v2 =

v11

· · ·vN1

1

v12

· · ·vN2

2

(3.16)

This matrix notation, in a given basis, will be used throughout.

The direct sum of two representations R1 and R2 of the group G, of respective dimen-sions N1, N2, is a representation R1⊕R2 of dimension N1 +N2 obtained by taking the directsum of the separate transformations of G on each vector space, and may be represented inthe above basis in terms of the following block-matrix decomposition,

(R1 ⊕R2)(g) =

(R1(g) 0

0 R2(g)

)(3.17)

1In the mathematics literature, V is referred to as the representation space of R (or module) while inthe physics literature, by a confusing abuse of notation, often refers to V as the representation as well.Here, we shall try to stick to the terminology that the vector space and its elements transform under therepresentation R.

45

The operation of direct sum on the space of representations is commutative.

Given a representation R : G → GL(N) acting on a vector space V of dimension N ,a vector v ∈ V is said to be invariant under a subgroup H ⊂ G if R(g)v = v for allg ∈ H. Similarly, a vector space W ⊂ V is said to be invariant under a subgroup H ⊂ Gif R(g)w ∈ W for all w ∈ W and all g ∈ H. Clearly, V and 0 are trivial invariantsubspaces. The direct sum R1 ⊕R2 of two representations of G, as constructed above, hastwo non-trivial invariant subspaces, namely V1 and V2.

A representation R : G → GL(N) is said to be reducible if it admits at least onenon-trivial invariant subspace under G. A representation R : G → GL(N) is said to becompletely reducible if every invariant subspace V1 has an invariant complement V2, sothat V = V1 ⊕ V2. The difference between a reducible representation R′ and a completelyreducible representation R, both with invariant subspace V2, may be illustrated by usingthe matrix notation, in which case they respectively take the following form,

R(g) =

(R1(g) 0

0 R2(g)

)R′(g) =

(R1(g) S3(g)

0 S2(g)

)(3.18)

The vector space V1 in the upper block is invariant under both R(g) and R′(g) for all g ∈ G.Its complement V2 is also invariant under R(g) but not under R′(g), since the action ofR′(g) on V2 will have a component along V1 generated by the block matrix S3(g).

A representation R : G → GL(N) acting on the vector space V of dimension N isirreducible if its only invariant subspaces are V and 0. Irreducible representations formthe building blocks for general representations, so that the knowledge of all irreduciblerepresentations will suffice to classify all representations.

3.5 Schur’s Lemma

Let R : G→ GL(M ;C) and S : G→ GL(N ;C) be two irreducible complex representationsof the group G, such that,

AR(g) = S(g)A for all g ∈ G (3.19)

where A is an N ×M matrix which is independent of g. If R and S are inequivalent thenA = 0. If R and S are equivalent, then A is unique up to a multiplicative constant. Inparticular, if R = S, then A = λI with λ ∈ C.

The last assertion means that any N × N matrix which commutes for all g ∈ G withan irreducible representation R(g) of dimension N is proportional to the identity matrix.

To prove Shur’s Lemma, denote by X and Y the vector spaces, of respective dimensionsM and N , which are transformed under R and S, so that R(g) : X → X and S(g) : Y → Y ,

46

and the matrix A is a map from X → Y . Now define the following subspaces,

X0 = Ker (A) = x ∈ X such that Ax = 0Y0 = Im (A) = y ∈ Y such that y = Ax for some x ∈ X

It follows from the conjugation relation that X0 is a subspace of X invariant under R sincewe have AR(g)x = S(g)Ax = 0 for every x ∈ X0 and thus R(g)X0 ⊂ X0. Similarly, Y0

is a subspace of Y invariant under S. To see this, take an element y ∈ Y0 which may beexpressed as y = Ax, apply the conjugacy relation to x so that AR(g)x = S(g)Ax = S(g)y,and observe that this implies S(g)y ∈ Y0. Since the representations R and S have bothbeen assumed to be irreducible, the subspaces X0 and Y0 must both be trivial subspaces,namely either 0 or respectively X or Y . We now examine these different cases.

The cases X0 = Y0 = 0 and X0 = X, Y0 = Y are ruled out linear algebra. The caseX0 = X and Y0 = 0 corresponds to A = 0. Finally, the case X0 = 0 and Y0 = Ycorresponds to the matrix A being invertible and the representations R and S are thenequivalent. The uniqueness up to multiplication by a constant may be seen as follows.Suppose there were a second A′ satisfying A′R(g) = S(g)A′, then take the combination(A′ − λA)R(g) = S(g)(A′ − λA) for λ ∈ C. Now let λ be a root of det (A′ − λA) = 0 (notethat it is here that we need to be in C). For that value of λ, the matrix A′ − λA is notinvertible, but then by the first part of Shur’s lemma, it must vanish. In the case whereS = R, we can clearly take A = I but since A must be unique up to multiplication by aconstant, this is in fact the only possibility, which gives rise to the last assertion.

An important application is that all irreducible representations R : G → GL(N ;C) ofan Abelian group G are one-dimensional. Note that there is no analogous result for realrepresentations R : G → GL(N ;R) since for example the Abelian group of multiplicationgiven by elements of the form,

R(a, b) =

(a b−b a

)(3.20)

for a, b ∈ R is reducible over C to a± ib but not reducible over R.

3.6 Unitary representations

Unitary representations will be important because they preserve the Hermitian inner prod-uct in complex vector spaces, and Hilbert spaces of quantum states or of functions. Thereis also a very important immediate result, valid for representations of finite or infinitedimension, which simplifies the study of unitary representations considerably.

Consider a complex vector space V . A Hermitian inner product is a map V × V → C,denoted by (u,v) such that for all u,v,w ∈ V and all α, β ∈ C we have,

(v,u) = (u,v)∗

(u, αv + βw) = α (u,v) + β (u,w) (3.21)

47

One defines ‖u‖2 = (u,u), which is a norm on V provided it is positive ‖u‖2 ≥ 0 forall u ∈ V ; definite so that ‖u‖ = 0 implies u = 0; and satisfies the triangle inequality‖u + v‖ ≤ ‖u‖+ ‖v‖ for all u,v ∈ V . The resulting normed vector space is automaticallya metric topological space. If V is complete then V is a Hilbert space.

Theorem 5 Every unitary representation is completely reducible.

To prove this result, let (x, y) be the inner product on V with respect to which therepresentation R(g) is unitary, so that we have,

(R(g)x,R(g)y) = (x, y) for all x, y ∈ V (3.22)

Let V1 be an invariant subspace of the representation R(g) so that R(g)V1 = V1. Theorthogonal complement V2 = (V1)⊥ with respect to the inner product (x, y) is defined by,

V2 = v ∈ V such that (v, x) = 0 for all x ∈ V1 (3.23)

Now let y ∈ V2 then its transform R(g)y satisfies,

(R(g)y, x) = (y,R(g)−1x) (3.24)

for all x ∈ V and thus all x ∈ V1, and all g ∈ G. Since V1 is an invariant subspace of R,the image R(g)−1 is in V1, so that the second inner product vanishes by virtue of y ∈ V2.Therefore we have (R(g)y, x) = 0 for all x ∈ V1 so that R(g)y ∈ V2. Therefore, V2 is aninvariant subspace under R(g) and the representation R(g) is completely reducible into therepresentations on the subspaces.

Theorem 6 Every finite-dimensional representation of a finite or compact group G isequivalent to a unitary representation.

To prove this theorem for the case of a finite group G, we consider a representationR : G → GL(N ;C) on a vector space V of dimension N < ∞ with a non-degenerateHermitian inner product (x, y). An arbitrary representation R(g) may not be unitary withrespect to this inner product. We form a new inner product by summing over the group,

〈x, y〉 =∑h∈G

(R(h)x,R(h)y) (3.25)

The sum is over a finite number of elements and thus convergent. Now transform both x, yby the representation R(g),

〈R(g)x,R(g)y〉 =∑h∈G

(R(h)R(g)x,R(h)R(g)y) (3.26)

48

and use the fact that R is a representation, so that R(h)R(g) = R(h∗g). But the sum overh is the same as the sum over h ∗ g for any h ∈ G, so that we find,

〈R(g)x,R(g)y〉 = 〈x, y〉 (3.27)

for all x, y ∈ V and all g ∈ G. This means that R is unitary with respect to the newHermitian inner product 〈x, y〉.

The only change required for a compact group G is to replace the summation over Gby an integral over the compact Lie group G, using the Hurwitz-Haar measure dµ = dµG,

〈x, y〉 =

∫G

dµ(h)(R(h)x,R(h)y) (3.28)

Since G is compact and the integrand continuous in h, the integral is convergent for allx, y. Transforming x, y under R(g), we find,

〈R(g)x,R(g)y〉 =

∫G

dµ(h)(R(h ∗ g)x,R(h ∗ g)y) (3.29)

where we have again used the fact that R is a representation so that R(h)R(g) = R(h ∗ g).Changing from integration variable h to h′ = h ∗ g, we use the translation invariance of themeasure dµ(h′∗g−1) = dµ(h′) and recover 〈R(g)x,R(g)y〉 = 〈x, y〉 so that the representationR(g) is unitary with respect to the inner product 〈x, y〉.

Corollary 1 It follows from combining Theorems 3.6 and 3.6 that every finite-dimensionalrepresentation of a finite or of a compact group is completely reducible.

This result is particularly helpful as it means that for these cases it will suffice tostudy just irreducible representations, all others being then given by taking direct sums ofirreducible representations.

3.7 Tensor product of representations

To define the tensor product of representations, we need to first define the tensor productof vector spaces V1 and V2 of respective dimensions N1 and N2. The tensor product of twovectors v1 ∈ V1, v2 ∈ V2 is given by a bilinear map, denoted by,

(v1, v2)→ v1 ⊗ v2 (3.30)

and all linear combinations of such tensor products define a vector space V1⊗V2 of dimensionN1N2. In terms of bases e1

1, · · · , eN11 for V1 and e1

2, · · · , eN22 for V2 a basis of V1⊗ V2 may be

chosen to be ei1 ⊗ ej2 where i = 1, · · ·N1 and j = 1, · · · , N2, thereby clearly giving a vector

49

space of dimension N1 + N2. Using the decomposition of the two vectors onto these basisvectors as given in (3.16), the tensor product v1 ⊗ v2 takes the form,

v1 ⊗ v2 =

N1∑i=1

N2∑j=1

vi1vj2 e

i1 ⊗ e

j2 (3.31)

A general element v of the tensor product space V1 ⊗ V2 is given by a general linearcombination of the basis vectors ei1 ⊗ e

j2 given as follows,

v =

N1∑i=1

N2∑j=1

vij ei1 ⊗ ej2 (3.32)

where vij are arbitrary coefficients in the field underlying the vector space V .

The tensor product R1 ⊗ R2 of two representations Ri : G → GL(Ni;C) is a represen-tation of dimension N1N2, namely,

R1 ⊗R2 : G→ GL(N1N2;C) (3.33)

whose transformation on V = V1 ⊗ V2 is defined on the tensor product of two vectors by,

(R1 ⊗R2)(g)(v1 ⊗ v2) = (R1(g)v1)⊗ (R2(g)v2) (3.34)

and in particular on basis vectors of V1 ⊗ V2 by,

(R1 ⊗R2)(ei1 ⊗ ej2) =

N1∑k=1

N2∑`=1

(R1)ik(R2)j`ek1 ⊗ e`2 (3.35)

where we have suppressed the dependence on g. Note that the tensor product is associative,

(R1 ⊗R2)⊗R3 = R1 ⊗ (R2 ⊗R3) for all R1, R2, R3 (3.36)

3.8 Characters of Representations

A group character of a representation R of a group G is a complex-valued function on G

which depends on the representation R and on the conjugacy class of each group element.For a finite-dimensional representation R(g), we define,

χR(g) = trR(g) (3.37)

The trace is taken of an N × N matrix where N = dimR. Invariance under conjugationfollows from the fact that R is a representation and the cyclic property of the trace,

χR(h ∗ g ∗ h−1) = tr(R(h)R(g)R(h)−1

)= trR(g) = χR(g) (3.38)

50

One of the most important properties of group characters is that their behavior underthe operations of direct sum and tensor product of representations. For representationsR1 : G→ GL(M ;C) and R2 : G→ GL(N ;C) we have,

χR1⊕R2(g) = χR1(g) + χR2(g)

χR1⊗R2(g) = χR1(g)χR2(g) (3.39)

To prove the first line is immediate from inspection of (3.17). To prove the second line, werecall from (3.35) that the matrix elements of R1 ⊗ R2 are given by (R1)ik(R2)j`. To takethe trace, we must sum the diagonal elements (R1)ii(R2)jj obtained by setting (k, `) = (i, j)over the composite index ij, which clearly produces the product of the traces.

For the special case of g = e, the identity element of G, we have R(e) = IN so that,

χR(e) = dim(R) (3.40)

The direct sum and tensor product formulas for characters then reduce to the dimensionformulas derived earlier,

dim(R1 ⊕R2) = dim(R1) + dim(R2)

dim(R1 ⊗R2) = dim(R1) dim(R2) (3.41)

For infinite dimensional representations, the definition does not make sense as it stands,since in particular the value of the character would diverge at the identity element. Some-times, the character may still be defined in a useful way.

Theorem 7 The following orthogonality relations hold for finite-dimensional irreduciblerepresentations R1 and R2 of a finite group G,∑

g∈G

χR1(g)χR2(g) = NR1δR1,R2 (3.42)

while for a compact group G with Hurwitz-Haar measure dµ we have,∫G

dµ(g)χR1(g)χR2(g) = NR1δR1,R2 (3.43)

where the overline indicates complex conjugation, δR1,R2 equals 1 when R1 and R2 areequivalent to one another and vanishes otherwise, and NR1 are normalizations.

The proof proceeds from Shur’s lemma. In fact we shall prove a Lemma first.

Lemma 1 Two irreducible representations R1 and R2 of a compact Lie group G (or afinite group upon replacing the integral by a sum over G), satisfy the orthogonality relation,∫

G

dµ(g)R1(g)ij R2(g−1)k

` = C(R1) δR1,R2 δi` δk

j (3.44)

51

where δR1,R2 vanishes when R1 and R2 are inequivalent representations, and equals 1 whenR1 and R2 are equivalent in which case the result has been recorded for when the represen-tations are actually equal.

To prove the lemma, we introduce the the matrix M defined by,∫G

dµ(g)R1(g)ij R2(g−1)k

` = Mikj` (3.45)

Multiplying to the left by R1(h)ai, summing over i to produce the product R1(h)R1(g) =

R1(h ∗ g), changing variables from g to h ∗ g, and using the invariance of the measure givesdµ(g) = dµ(h ∗ g), we find the following relation,∑

a

R1(h)iaMak

j` =

∫G

dµ(g)R1(g)ij R2(g−1 ∗ h)k

` =∑b

MikjbR2(h)b

` (3.46)

For fixed values of the indices j, k, denote the matrix Mikj` = Mi

`. The above relationthen implies that M satisfies R1(h)M =MR2(h) for all h ∈ G. By Shur’s lemma, eitherR1 and R2 are inequivalent, in which case M = 0 for all values of j, k. Or R1 and R2

are equivalent in which case M is unique up to a multiplicative factor by Shur’s lemma.Repeating the same procedure on the indices j, k we conclude the proof of the lemma.

The Theorem follows by taking the trace in the indices i, j and separately in k, `.

Theorem 8 A unitary irreducible representation of a compact group is finite dimensional.

The proof proceeds by using the orthogonality theorem, and we leave it up to the reader.

Representations of Lie algebras

So far we have discussed representations of groups, including Lie groups. The relationbetween a representation ρ of a Lie algebra G and a representation R of its associatedconnected and simply connected Lie group G is given explicitly by the exponential map,

R(g) = eρ(x) g = ex (3.47)

for all x ∈ G. Thus, having a representation of a Lie algebra will easily produce a repre-sentation of the corresponding Lie group.

52

4 Representations of SL(2;C), SU(2), and SO(2, 1)

The group SU(2) is familiar to any physicist. Its elements are 2×2 complex matrices whichmay be parametrized in a variety of ways. As a matrix we have,

g =

(a bc d

)g†g = I2 det g = 1 (4.1)

Equivalently, it may also be parametrized in terms of Euler angles α, β, γ,

g = eiαJ3eiβJ2eiγJ3 Ja =σa2

(4.2)

where σa are the Pauli matrices.

4.1 Irreducible representations of the Lie group SU(2)

We begin by constructing the finite-dimensional irreducible representations of the Lie groupSU(2). The trivial representation is denoted D0. The formulation of SU(2) in terms of2 × 2 matrices provides the defining representation which is often denoted by D 1

2(g) = g,

and which is manifestly irreducible. The direct sum of n defining representations D 12

is arepresentation of dimension 2n which is obviously reducible.

To construct new irreducible representations of SU(2) we take tensor products of thedefining representation with itself and then completely reduce those into irreducible repre-sentations. To do this, we define the vector space V = C2 on which D 1

2acts, and denote

its vectors by vα with components viα for i = 1, 2. The representation D 12

acts as follows,

vα → D 12(g)vα viα →

2∑i′=1

D 12(g)ii′ v

i′

α =2∑

i′=1

gii′ vi′

α (4.3)

The tensor product of two vectors transforms under the tensor product representation,

vα ⊗ vβ → (D 12⊗D 1

2)(g)(vα ⊗ vβ) (4.4)

Writing out the tensor products in components we have,

viαvjβ →

2∑i′,j′=1

D 12(g)ii′ D 1

2(g)jj′ v

i′

α vj′

β =2∑

i′,j′=1

gii′ gjj′ v

i′

α vj′

β (4.5)

Clearly, the tensor product representation D 12⊗D 1

2has dimension 4.

53

Since we are dealing with a finite-dimensional representation of a compact group, weknow by theorem 3.6 that the representation D 1

2⊗D 1

2is completely reducible. Symmetriza-

tion and anti-symmetrization of the tensor product provides a general procedure to reducetensor product representations. In the case of D 1

2⊗D 1

2, we have,

1

2(viαv

jβ ± v

iβv

jα)→

2∑i′,j′=1

gii′ gjj′

1

2(vi′

α vj′

β ± vi′

β vj′

α ) (4.6)

Now, the anti-symmetric combination involves,

viαvjβ ± v

iβv

jα = εij(v1

αv2β ± v1

βv2α) (4.7)

where εij is the anti-symmetric symbol with ε11 = ε22 = 0 and ε12 = −ε21 = 1, andsimilarly for vi

′α v

j′

β ± vi′

β vj′α . But now consider the combination,

2∑i′,j′=1

gii′ gjj′ ε

i′j′ (4.8)

which is anti-symmetric in i, j and thus has a single independent component, obtained bysetting i = 1 and j = 2, and given by g1

1g2

2−g21g

12 = det g = 1. Thus the anti-symmetric

combination is invariant under SU(2) and transforms under the trivial representation D0.The symmetric combination has dimension 3, is irreducible, and is denoted D1.

4.2 Finite-dimensional representations of sl(2,C)

The 3-dimensional Lie algebra with generators Ja for a = 1, 2, 3 and structure relations,

[J1, J2] = iJ3 [J2, J3] = iJ1 [J3, J1] = iJ2 (4.9)

is the Lie algebra sl(2;C) or equivalently so(3;C) provided the vector space generated bythe generators Ja is over C. When the generators −iJa are real we have the algebra so(3;R),and when the generators Ja are Hermitian it is the Lie algebra su(2) familiar from angularmomentum theory. Finally, we may also set J1 = X1, J2 = X2, J3 = iX3,

[X1, X2] = −X3 [X2, X3] = X1 [X3, X1] = X2 (4.10)

and now require X1, X2, X3 to be real matrices, which gives the Lie algebra so(2, 1;R).The Lie algebras su(2), so(3;R) and so(2, 1;R) are referred to as different real forms of thecomplex Lie algebra sl(2;C), and their representation theory is closely related to that ofsl(2;C), with which we will begin.

54

The rank of sl(2;C) is one. We choose the generator of the Cartan subalgebra to beJ3 and assume that J3 is diagonalizable.2 We organize the remaining generators in linearcombinations chosen so that the adjoint map AdJ3 is diagonal,

AdJ3J± = ±J± J± =1√2

(J1 ± iJ2) (4.11)

In this basis, the structure relations become,

[J3, J±] = ±J± [J+, J−] = J3 (4.12)

This organization will generalize to all simple Lie algebras in which case the J± generatorswill correspond to roots.

To construct the most general finite-dimensional irreducible representation ρ of sl(2;C),we shall assume that the representation matrices ρ(Ja) are N ×N , for as yet undeterminedvalues of N <∞, and satisfy the structure relations of the Lie algebra,

[ρ(J3), ρ(J±)] = ±ρ(J±) [ρ(J+), ρ(J−)] = ρ(J3) (4.13)

We denote the N -dimensional vector space on which they act by V and the eigenvalues ofρ(J3) by m. Since there is no assumption of Hermiticity of the generators for sl(2;C), wegenerally may have m ∈ C. The eigenspace of ρ(J3) may have dimension one or higher,and we label the different vectors with the same eigenvalue m by α. The spectrum of ρ(J3)is the set of quantum numbers S(ρ) = (m,α) and has cardinality N .

We shall often use the terminology and notations of quantum mechanics, refer to vectorsas states, and denote by |ρ;m,α〉 the basis vectors of a basis of V in which ρ(J3) is diagonal,

ρ(J3)|ρ;m,α〉 = m|ρ;m,α〉 (m,α) ∈ S(ρ) (4.14)

The generators ρ(J±) map V to V by the definition of a representation, and thereforeρ(J±)|m,α; ρ〉 must be a linear combination of vectors |ρ;m′, α′〉 with (m′, α′) ∈ S(ρ).Actually, we may restrict the form of this linear combination by evaluating,

ρ(J3)ρ(J±)|ρ;m,α〉 =(

[ρ(J3), ρ(J±)] + ρ(J±)ρ(J3))|ρ;m,α〉

= (m± 1)ρ(J±)|ρ;m,α〉 (4.15)

obtained using the structure relations. Thus we have the following decompositions,

ρ(J±)|ρ;m,α〉 =∑β

C±αβ(m, ρ)|ρ;m± 1, β〉 (4.16)

2When the Lie algebra is su(2), the generator J3 is Hermitian and thus automatically diagonalizable.

55

On the right side the sum is over all vectors such that (m±1, β) ∈ S(ρ). The matrix ρ(J+)raises the eigenvalue of ρ(J3) by 1 while ρ(J−) lowers the eigenvalue by 1.

Next we apply ρ(J±) repeatedly and make use of the assumption that the dimension ofthe representation ρ is finite. This implies that the eigenvalue of ρ(J3) cannot get raised orlowered indefinitely since this would produce a representation space of infinite dimension.Thus, there must exist j± with non-zero vectors |j±, α±; ρ〉 and (j±, α±) ∈ S(ρ) such that,

ρ(J+)|ρ; j+, α+〉 = 0

ρ(J−)|ρ; j−, α−〉 = 0 (4.17)

This may occur for one or more values of j± and one or more values of α± for each valueof j±. Repeatedly applying ρ(J−) to any one of the vectors |ρ; j+, α+〉 will produce all thevectors of an irreducible representation of sl(2;C). If several α+ occur then applying ρ(J−)will produce the direct sum of as many irreducible representations as there are α+.

We shall now focus on a single irreducible representation produced from a single j+ = jand a single α. In general, this representation is no longer ρ, but rather one of the irreduciblecomponents of ρ. We shall denote this representation by ρj and drop the label α, labellingthe states simply |ρj;m〉. In mathematics terminology, the quantum number m is referredto as the weight and the corresponding vector |ρj; j〉 as the highest weight vector.

Applying lowering and raising operators we have,

ρj(J±)|ρj;m〉 = N±(ρj;m)|ρj;m− 1〉 N+(ρj, j) = 0 (4.18)

Enforcing the structure relation [ρj(J+), ρj(J−)] = ρj(J3) gives,

N+(ρj,m− 1)N−(ρj,m)−N−(ρj,m+ 1)N+(ρj,m) = m (4.19)

Considering the sum of this relation over m up to m = j,

j∑n=m

(N+(ρj, n− 1)N−(ρj, n)−N−(ρj, n+ 1)N+(ρj, n)

)=

j∑n=m

n (4.20)

The second term in the parentheses is the opposite of the first shifted by n→ n+ 1. Thusthe sum is given by its endpoints and, using the fact that N+(ρj, j) = 0, we obtain,

2N+(ρj,m− 1)N−(ρj,m) = (j +m)(j −m+ 1) (4.21)

The formula reproduces N+(ρj, j) = 0 and shows that the only value of m < j for whichN− can vanish is m = −j, so that N−(ρj,−j) = 0. Given the highest weight j of therepresentation ρj, the allowed values of m are such that j −m ∈ Z+, where Z+ stands for

56

the zero or positive integers. But −j must be such a value, so that 2j ∈ Z+. Therefore thestates in the representation ρj are given for j −m ∈ Z+ and,

|ρj;m〉 0 ≤ j −m ≤ 2j ∈ Z+ (4.22)

and the dimension of the representation is dim(ρj) = 2j + 1. Note that we did not use theassumption that m or j had to be real from the outset: for sl(2;C) this result arises solelyfrom requiring the representations to be finite-dimensional.

Imposing the restrictions of Hermiticity for su(2), or reality for so(3;R) changes nothingto the structure and dimensions of the representations derived above. In the case of com-pact groups we know that all finite-dimensional representations are equivalent to unitaryrepresentations.

4.3 Infinite-dimensional representations of sl(2;C)

Now let us remove the assumption that the representation ρ is finite-dimensional, but stillassume that ρ(J3) is diagonalizable, so that we still have the relations,

N+(ρ;m− 1)N−(ρ;m)−N−(ρ;m+ 1)N+(ρ;m) = m (4.23)

Consider now a representation ρ for which N−(ρ; j) = 0, so that ρ(J−)|ρ; j〉 = 0 for anon-zero state |ρ; j〉. This time applying m raising operators to |ρ; j〉, we find,

−2N+(ρ; j +m)N−(ρ; j +m+ 1) = (m+ 1)(2j +m− 2) (4.24)

If 2j were a negative integer, then we recover the finite-dimensional case already discussed.But if we assume that 2j is not a negative integer, then we can apply an infinite numberof raising operators for which N+(ρ; j +m) 6= 0 and thus |ρ; j +m〉 6= 0. This produces aninfinite-dimensional representation of sl(2;C).

4.4 Harmonic oscillator representation of so(2, 1;R)

Consider the one-dimensional harmonic oscillator with degrees of freedom a, a† which satisfythe Heisenberg algebra [a, a†] = 1 upon setting ~ = 1, and have the following Hamiltonianand additional generators,

H = a†a+1

2J+ = (a†)2 J− = (a)2 (4.25)

They satisfy the structure relations of so(2, 1;R) given by,

[H, J±] = ±2J± [J+, J−] = −4H (4.26)

57

Clearly, J+ is a raising operator for H while J− is a lowering operator. The quadraticCasimir take the form,

C = H2 − 1

2J+, J− = H2 − 2H − J+J− (4.27)

and on the harmonic oscillator representation evaluates to C = −34. Clearly H is positive,

and there must thus be a lowest energy state, which we shall denote |0〉 such that,

H|0〉 = E0|0〉 J−|0〉 = 0 (4.28)

for an as yet undetermined ground state energy E0. Applying C to |0〉 gives,

−3

4= E2

0 − 2E0 or E±0 = 1± 1

2(4.29)

Applying raising operators gives the excited states |n〉 = Jn+|0〉 with energy,

E±n = 1± 1

2+ 2n n ≥ n (4.30)

The interpretation is as follows: while the operator a† raises the energy by one unit, theraising operator J+ which is in the algebra so(2, 1;R) raises the energy by two units sinceit is quadratic in a†. The two different “ground states” correspond to the even and oddwave functions of the harmonic oscillator, and are the lowest weight states of differentrepresentations of so(2, 1;R). Thus the entire spectrum of the harmonic oscillator is thedirect sum of two irreducible representations.

4.5 Unitary representations of so(2, 1;R)

We consider a quantum system in one dimension which generalizes the harmonic oscillator,governed by the Hamiltonian H, and the following additional generators,

H =p2

2+

g

2x2+x2

2

K =p2

2+

g

2x2− x2

2

D =i

2(xp+ px) (4.31)

where x, p satisfy the Heisenberg algebra, [x, p] = i. The operator D generates dilations,and has been normalized so that [D, x] = x and [D, p] = −p. From a quantum point ofview, the operators H,K and iD are self-adjoint for g ∈ R, assuming some restrictions

58

on the values of g to be spelled out later. Together with the operators K and D, theHamiltonian forms the so(2, 1;R) algebra, with the following structure relations,

[D,H] = −2K [D,K] = −2H [H,K] = 2D (4.32)

For g ≥ 0, the spectrum of H is discrete, real, and positive, and we choose to work in abasis where H is diagonal. Next, we organize the operators K and D in combinations thatraise and lower the eigenvalues of H, i.e. that diagonalize AdH ,

J± = K ±D [H, J±] = ±2J± [J+, J−] = −4H (4.33)

Since K is self-adjoint and D is anti-self-ajoint, we see that (J±)†−J∓, as in the case of theharmonic oscillator. One verifies using the structure relations that the quadratic Casimiroperator is given by

C = H2 −K2 +D2 = H2 − 2H − J+J− (4.34)

The calculation of C for the particular quantum system considered here may be simplifiedby using the canonical commutation relations [x, p] = i, and we find that C is proportionalto the identity operator, and given by C = g − 3

4. Clearly, the spectrum of H is real and

bounded from below, so there must be a state of lowest eigenvalue of E0,

H|0〉 = E0|0〉 J−|0〉 = 0 (4.35)

The value of E0 is determined by using the value of the Casimir, and we get the entirespectrum from applying the raising operator,

E±0 = 1± 1

2

√1 + 4g E±n = 1± 1

2

√1 + 4g + 2n n ∈ Z+ (4.36)

The spectrum makes sense even when g is negative as long as g ≥ −14. The entire spectrum

again would appear to be the direct sum of two irreducible representations, as was the casefor the harmonic oscillator. However, it is clear from the Hamiltonian that just its potentialpart is bounded from below by

√g, so that the energy E±0 must be bounded from below

by√g. This is always the case for the E+

0 branch, but holds for the E−0 branch only when−1

4≤ g ≤ 9

64. The energy also needs to be bounded from below by the harmonic oscillator

value, and the combined bound gives that the E−0 branch can exist only for −14≤ g ≤ 0.

Theorem 9 The irreducible unitary representations of SO(2, 1;R) are as follows.

1. discrete series: C ≥ −1 and there is a lowest (or highest) energy state with energyE0 ≥ 0 (or ≥ −1

4in the above example);

2. continuous series: C < −1 and 0 ≤ E0 < 2;

3. complementary series: C = −1 + λ2 and λ ∈ R.

59

4.6 An example of the continuous series

Let me now give an example of a physically relevant representation that produces thecontinuous series. We note that SO(2, 1) acts transitively on the hyperboloids,

H± = (x, y, z) ∈ R2 x2 + y2 − z2 = ±1 (4.37)

While H+ is connected by not simply connected, H− has two connected components, eachof which is simply connected. Note that the compact subgroup SO(2) of SO(2, 1) rotates(x, y). Both surfaces H± have constant negative curvature. To find the representations inthe continuous series, we diagonalize the Laplace-Beltrami operator acting on functions oneither one of these hyperboloids,

∆g = − 1√g∂m√g gmn∂n (4.38)

and the Laplace operator represents the Casimir operator. We consider the connectedcomponent of H− which is characterized by z ≥ 1, and introduce the following coordinates,

H− :

x = shϕ cos θy = shϕ sin θz = chϕ

(4.39)

where 0 ≤ θ〈2π and 0 ≤ ϕ <∞. The metric is readily obtained,

ds2 = dϕ2 + sh2ϕdθ2 (4.40)

and so is the Laplace operator,

∆ = − ∂2

∂ϕ2− chϕ

shϕ

∂

∂ϕ− 1

sh2ϕ

∂2

∂θ2(4.41)

The differential operators of SO(2, 1) in this representation are given by,

T3 = −2i∂

∂θ

T± = 2e±iθ(i∂

∂ϕ∓ chϕ

shϕ

∂

∂θ

)(4.42)

Extra factors of 2 have been inserted so that the algebra is the same as in the precedingexamples, and the structure relations are as follows,

[T3, T±] = ±2T± [T+, T−] = −4T3 (4.43)

The Casimir operator is,

C = T 23 − 2T3 − T+T− = −4∆ (4.44)

60

We now solve for the eigenvalue equation:

∆ψ = −1

4Cψ = −∂

2ψ

∂ϕ2− chϕ

shϕ

∂ψ

∂ϕ− 1

sh2ϕ

∂2ψ

∂θ2(4.45)

We shall use the following parametrization for λ2 ∈ R,

C = −1− λ2 (4.46)

and diagonalize T3 simultaneously with ∆, by introducing the eigenfunctions of ∂/∂θ,

ψλ,m(ϕ, θ) = eimθψλ,m(ϕ) (4.47)

so that the remaining eigenvalue equation reduces to,

−ψ′′λ,m −chϕ

shϕψ′λ,m +

m2

sh2ϕψλ,m =

(1

4+ λ2

)ψλ,m (4.48)

This equation is seen to be the Legendre equation by changing variables x = chϕ, andrelabeling the parameter λ in the equation as follows,

ν(ν + 1) = −1

4− λ2 ν = −1

2+ iλ (4.49)

which converts the eigenvalue equation into the standard form for the Legendre functions,

(1− x2)ψ′′ − 2xψ′ +

(ν(ν + 1)− m2

1− x2

)ψ = 0 (4.50)

The solutions are conical functions Pm− 1

2+iλ

(chϕ) and Qm− 1

2+iλ

(chϕ) for which we have usual

completeness relations.

61

5 Tensor representations

Vector and tensor algebra is a topic in physics that has a somewhat separate life from grouptheory but is in fact equivalent to the representation theory for the classical groups as longas no spinors are involved. Tensor methods are organized by the pictorial Young tableauxand are often more intuitive and practical when only representations of low dimension areinvolved, and are

5.1 Tensor product representations

We consider one of the classical matrix Lie groups G, such as SL(n), SU(n), SO(n), orSp(n) (for n even), and start out with their respective defining representations, which areall of dimension n. The associated vector space Vn is Rn or Cn depending on the group.The defining representation matrix will be denoted R(g) for g ∈ G and U(g) and n × nmatrix. The group G acts on Vn by linear transformations on its vectors,

G : Vn → Vn v → R(g)v ∈ Vn v ∈ Vn (5.1)

or in component notation,

vi →n∑

i′=1

R(g)ii′ vi′ i = 1, · · · , n (5.2)

The transformation of Vn under the defining representation R of G induces a transformationon the tensor product space Vn ⊗ Vn under the tensor product representation R⊗R,

G : Vn ⊗ Vn → Vn ⊗ Vn v1 ⊗ v2 →(R(g)⊗R(g)

)(v1 ⊗ v2) (5.3)

Note that the group element g is the same in both factors. Equivalently, in components,

vi1 vj2 →

n∑i′,j′=1

R(g)ii′R(g)jj′ vi′

1 vj′

2 (5.4)

By taking linear combinations in Vn ⊗ Vn of tensor products of vectors we obtain generalelement w ∈ Vn ⊗ Vn whose components may be denoted by wij for i, j = 1, · · · , n and isreferred to as a tensor of rank 2 since it has two indices of the defining representation, anddimension n2. It transforms under R⊗R by,

G : Vn ⊗ Vn → Vn ⊗ Vn w → (R(g)⊗R(g))w (5.5)

or in components,

wij →n∑

i′,j′=1

R(g)ii′R(g)jj′ wi′j′ (5.6)

62

The process may be repeated by taking the tensor product of r copies of Vn to obtain tensorsof rank r which transform under the r-fold tensor product of the defining representation R.

G : Vn ⊗ · · · ⊗ Vn︸ ︷︷ ︸r

→ Vn ⊗ · · · ⊗ Vn︸ ︷︷ ︸r

w → R(g)⊗ · · · ⊗R(g)︸ ︷︷ ︸r

w (5.7)

or in components,

wi1,··· ,ir →n∑

i1,··· ,ir=1

R(g)i1j1 · · ·R(g)ir jr wj1,··· ,jr (5.8)

When no confusion is expected to arise, we shall henceforth omit exhibiting the g-dependence.The representations thus constructed are tensor product representations of the definingrepresentation. For the groups SL(n), SU(n) and Sp(n) we will show later that all finite-dimensional representations may be constructed from tensor product representations. Forthe groups SO(n), however, tensor product representations will give only a subset of allrepresentations, as these groups have also spinor representations which cannot be obtainedfrom tensor product representations.

5.2 Symmetrization and anti-symmetrization

The tensor product representations constructed above are in general reducible. Sym-metrization and anti-symmetrization always provides operations which will reduce therepresentations into direct sums of representations of smaller dimension, though the rep-resentations obtained this way may or may not be irreducible. We begin by symmetrizingand anti-symmetrizing the tensor product of the representation spaces Vn⊗Vn by definingthe operations of symmetrization S and anti-symmetrization A on Vn ⊗ Vn,

S(v1 ⊗ v2) =1

2(v1 ⊗ v2 + v2 ⊗ v1) S(vi1 v

j2) =

1

2(vi1v

j2 + vj1v

i2)

A(v1 ⊗ v2) =1

2(v1 ⊗ v2 − v2 ⊗ v1) A(vi1 v

j2) =

1

2(vi1v

j2 − v

j1vi2) (5.9)

The symmetrized and anti-symmetrized tensor product representations then act as follows,

RS : S(vi1vj2) →

n∑k,`=1

1

2(Ri

kRj` +Rj

kRi`)S(vk1v

`2)

RA : A(vi1vj2) →

n∑k,`=1

1

2(Ri

kRj` −Rj

kRi`)A(vk1v

`2) (5.10)

The dimensions of the representations RS and RA are respectively given by 12n(n+ 1) and

12n(n− 1) whose sum is indeed the dimension of R⊗R, which is n2.

63

5.3 Representations of SU(3)

Denote the defining representation of SU(3) by R : SU(3)→ GL(3;C). Actually, since Ris a representation of SU(3), then so is the complex conjugate representation R∗. We shallnow show that the complex conjugate representation R∗ is equivalent to the antisymmetrictensor product of R with itself. To see this concretely, define the totally anti-symmetrictensor in three indices,

εijk =

(−)σ if (ijk) = σ(123) σ ∈ S3

0 otherwise(5.11)

Next, we consider the relation detR = 1, expressed in terms of the ε-symbol,

3∑α,β,γ=1

εαβγ RiαRj

βRkγ = εijk detR = εijk (5.12)

Multiplying by (R†)δk on both sides, summing over k and using the unitarity relation∑

k(R†)δ

kRkγ = δδ

γ, we have,

3∑α,β=1

εαβδRiαRj

β =3∑

k=1

εijk(R†)δ

k (5.13)

Multiply by εij` on both sides and sum over i, j using the relation,

3∑i,j=1

εijkεij` = 2δk` (5.14)

gives,

3∑i,j=1

3∑α,β=1

εijkεαβδRiαRj

β = 2(R†)δγ = 2R∗γδ (5.15)

Therefore, the representation R∗ is equivalent to the anti-symmetrized tensor product oftwo defining representations.

We have learned something interesting: each time we anti-symmetrize the tensor prod-uct of two defining representations, we may replace that representation by its equivalent R∗,while the totally anti-symmetrized tensor product of three R gives a singlet representation.We shall use the convention that the vectors in the vector space under which the definingrepresentation transforms are denoted with a lower index such as vi, while a vector in thedual vector space which transforms under R∗ will be denoted with a barred upper index,vi. Thus, a general tensor of SU(3) is of the form,

Ti1···ipi1···ip (5.16)

64

To reduce the tensor, we separately symmetrize it in the upper and lower indices. To fullyreduce the tensor, we remove all the traces between upper and lower indices.

To count the dimension of the irreducible representation with p defining and p anti-defining representations, we first ignore the trace condition and count the dimension ofjust the tensor ti1···ip . Since we have total symmetrization, it suffices to count the numberof ways we can put the three possible values i = 1, 2, 3 in three boxes with the sum ofthe lengths of the boxes being n. Denote by p1, p2, p3 the number of values 1, 2, and 3respectively, so that p1 + p2 + p3 = n. Each pi satisfies 0 ≤ pi ≤ n. thus the dimension is,

dim(t) =n∑

p1=0

n−p1∑p2=0

1 =n∑

p1=0

(n+ 1− p1) =(p+ 2)!

2! p!(5.17)

Thus, the dimension of the tensor T fully symmetrized in upper and lower indices, butignoring the trace is given by,

(p+ 2)!

2! p!

(p+ 2)!

2! p!(5.18)

Removing the traces is equivalent to removing the tensor with any pair of upper and lowerindices contracted in view of the symmetry of the tensor. Thus, the dimension of theirreducible tensor T is given by,

D(p, p) =(p+ 2)!

2! p!

(p+ 2)!

2! p!− (p+ 1)!

2! (p− 1)!

(p+ 1)!

2! (p− 1)!

=1

2(p+ 1)(p+ 1)(p+ p+ 2) (5.19)

We can check this formula for example on the adjoint representation, which is of the formTij so that p = p = 1, and we find dim(T ) = 8, which is the famous meson multiplet of

π0, π±, K±, K0, K0, η, when only the u, d, s quarks are included. Another is the multipletin which the proton and neutron live, which is the symmetric tensor product of 3 definingrepresentations, so that p = 3 and p = 0, and we get dim(T ) = 10. This is the multipletthat contains the Ω− consisting of three s quarks, which was predicted on the basis ofSU(3) group theory by Murray Gell-Mann for which he got the Nobel Prize.

Young tableaux provide a pictorial image of irreducible representations. One begins bydenoting the defining representation by a single square 2, and the trivial representation bya center dot •. Taking the tensor product with another copy of the defining representationis indicated by adjoining another box, either to the right or underneath the first box. In thevertical alignment of boxes, the tensor product is anti-symmetrized, while in the horizontalalignment, they are symmetrized. For SU(3), the anti-symmetric tensor product of threedefining representations is the trivial representation, so non-trivial representations can have

65

· · ·· · ·

· · ·p p

either one or two vertically aligned boxes, and the most general irreducible representationlabelled by (p, p) constructed above has the following Young tableaux,

On may now also give a pictorial representation for tensor multiplication of an arbitraryrepresentation by the defining representation: the result is the sum of three irreduciblerepresentations, each one obtained by adding a square in each row to the utmost left.Adding the square in the third row makes the corresponding column a singlet and producesthe representation (p, p− 1). Adding the square in the second row diminishes p by one andincreases p by one, so we have the representation (p− 1, p+ 1), and finally in the first rowwe get (p+ 1, p). One verifies that the dimensions add up:

3D(p, p− 1) = D(p− 1, p+ 1) +D(p, p) +D(p+ 1, p) (5.20)

There are rules for taking the tensor product not just with the defining representation, butwith a general representation. We shall see an easier route using weights.

5.4 Representations of SU(n)

For the case of SU(n), we introduce the totally anti-symmetric symbol in n indices,

εi1···in =

(−)σ if ik = σ(k) σ ∈ Sn

0 otherwise(5.21)

The determinant relation is now,∑α1,··· ,αn

εα1···αnRi1α1 · · ·Rin

αn = εi1···in detR = εi1···in (5.22)

Multiplying by (R†)kin and summing over in gives,∑

α1,··· ,αn

εα1···αn−1kRi1α1 · · ·Rin−1

αn−1 =∑in

(R†)kinεi1···in (5.23)

The remaining step is straightforward and now proves that R∗ is equivalent to the (n− 1)-fold anti-symmetric tensor product of the defining representation.

Are the symmetric/antisymmetric reductions now irreducible representations? Thisdepends upon what the group is. In general, for representations of

G`(n,C) S,A

SU(n) S,A, δjiSO(2n+ 1), SO(2n) δijSp(2n) Jij

66

By isolating the parts proportional to these tensors, they are further reduced.

Some illustrative examples.

SO(n) n

Ti1...ip (5.24)

Now you can take the trace as well on symmetric indices

T i1...ip−2

ip−1ip= Ti1...ipδ

ip−1ip (5.25)

There is also the fully antisymmetric tensor, but it does not lead to new reductions.

Sp(2n) One has the defining representation 2n.Now there is a trace with respect to Jij: on antisymmetric indices

Ti1...ip−2 =Ti1...ip Jip−1ip [ip−1ip] (5.26)

For exceptional groups (they are all subgroups of classical groups) there is usually yetanother invariant tensor, which further reduces the representations.

67

6 Spinor representations

In this section, we discuss spinor representations of the orthogonal Lie algebras so(d;R)and so(d − 1, 1;R), starting from the respective Clifford algebras. We begin with a briefreview of the spinor representation of the group of rotations SO(3;R).

6.1 Spinor representations of SO(3;R)

The fundamental representation of SO(3;R) has dimension 3. All the higher dimensionalrepresentations may be obtained by tensor product decomposition, and give us the irre-ducible representations of integer spin j. So where do the spinor representations enter ?There are two ways of looking at it. First, look at the generators of the Lie algebra so(3;R),

L1 =

0 0 00 0 −10 1 0

L2 =

0 0 10 0 0−1 0 0

L3 =

0 −1 01 0 00 0 0

(6.1)

which satisfy the following structure relations,

[L1, L2] = L3 [L2, L3] = L1 [L3, L1] = L2 (6.2)

The group elements g ∈ SO(3;R) may then be parametrized by Euler angles, α, β, γ,

g(α, β, γ) = eαL3eβL1eγL3 0 ≤ α, β < π, 0 ≤ γ < 2π (6.3)

It may be verified that the above range of parameters covers SO(3;R) exactly once. Fromthe point of view of the Lie algebra, the spinor representations enter because the structurerelations are also satisfied by setting La = −iσa

2where σa are the Pauli matrices. This

gives a 2-dimensional representation R which cannot be obtained from the tensor productof the defining representation of SO(3;R). But actually, it is not a genuine representationof SO(3;R), because the Euler angle parametrization now gives,

R(g) = e−iασ3/2e−iβσ1/2e−iγσ3/2 0 ≤ α, β < π, 0 ≤ γ < 2π (6.4)

While g was periodic with period 2π in γ, this is not the case for R(g) since we have,

R(g(α, β, γ + 2π)) = −R(g(α, β, γ + 2π)) (6.5)

so that R(g) is actually double-valued function of g, and the representation R is said tobe double-valued or to be a projective representation of SO(3;R). But R(g) is a single-valued representation of SU(2), and in fact coincides with its defining representation. Thesign ambiguity is absent in SO(3;R) because it is isomorphic to SU(2)/Z2, the Z2 exactlyaccounting for the sign indeterminacy.

68

6.2 The Clifford Algebra

The generalization to spinor representations for higher rank orthogonal groups makes use ofthe Clifford algebra whose representations induce spinor representation in the orthogonalgroups. We begin by recalling the parametrization and the structure relations of so(d).The generators Mµν are labelled by a composite index (µν) with 1 ≤ µ, ν ≤ d and they areanti-symmetric, so that Mνµ = −Mµν . The independent generators may thus be chosen tobe Mµν with 1 ≤ µ < ν ≤ d. Their matrix elements are given by,

(Mµν)αβ = δµαδνβ − δµβδνα (6.6)

The structure relations are given by,

[Mµν ,Mρσ] = δνρMµσ − δνσMµρ − δµρMνσ + δµσMνρ (6.7)

Note that one should think of the Kronecker symbols δµν etc as representing the identitymatrix in so(d;R). It is possible to generalize the construction of spinors to so(p, q;R)with pq 6= 0 and δµν will then be replaced by the matrix elements of Ip,q, and indeed weshall do so for the Minkowski metric later. In the special case of d = 3, we may recast thegenerators Mµν in terms of the notation La used earlier for so(3) as follows,

L1 = M32 L2 = M13 L3 = M21 (6.8)

We shall now introduce the Clifford algebra to produce representations of this Lie algebra.

The Clifford algebra in d dimensions, for Euclidean signature, is defined by,

γµ, γν = 2δµνI µ, ν = 1, · · · , d (6.9)

Now let’s assume that we can construct a representation of the Clifford algebra (we shalldo so explicitly soon). The claim is then that if γµ forms a representation of the Cliffordalgebra, then we automatically obtain a representation of the Lie algebra so(d) with thefollowing representation matrices,

Mµν =1

4[γµ, γν ] =

1

4(γµγν − γνγµ) (6.10)

This result is so simple and so fundamental that it is worth proving it explicitly by com-puting the commutator,

[Mµν ,Mρσ] =1

4[γµγν , γργσ] (6.11)

where in each entry of the commutator, we have used the Clifford algebra relations toconvert γνγµ into −γµγν plus the identity matrix which cancels out of the commutator.Next, we write out the commutator,

[Mµν ,Mρσ] =1

4γµγνγργσ −

1

4γργσγµγν (6.12)

69

By systematically permuting the γ-matrices to the same position with the help of theClifford relations, we get,

[Mµν ,Mρσ] =1

4γµγν , γργσ −

1

4γργσ, γµγν −

1

4γµ, γργνγσ +

1

4γργµγσ, γν (6.13)

Using now the Clifford algebra relation to convert all anti-commutators, we find,

[Mµν ,Mρσ] =1

2δνργµγσ −

1

2δµσγργν −

1

2δµργνγσ +

1

2δνσγργµ (6.14)

Using the Clifford algebra relations to anti-symmetrize the products of γ-matrices we pro-duce terms proportional to the identity matrix, which all cancel. We find that the matricesM indeed obey the structure relations of so(d;R). To see why the representation matriceshave anything to do with spinors, we need to construct representations of the γ-matrices.

6.3 Representations of the Clifford algebra

For d = 3 the representation of the Clifford algebra is in terms of the Pauli matrices,

σ1 =

(0 11 0

)σ2 =

(0 −ii 0

)σ3 =

(1 00 −1

)(6.15)

The Pauli matrices may also be given in the raising/lowering basis,

σ+ =1

2(σ1 + iσ2) =

(0 10 0

)σ− =

1

2(σ1 − iσ2) =

(0 01 0

)(6.16)

where they satisfy [σ3, σ±] = ±σ± and [σ+, σ−] = σ3. For d = 4, it was Dirac who, in oneof his strokes of genius, wrote down the Dirac representation in terms of 4× 4 matrices forthe case of the Minkowski metric. What he used was, effectively, taking the tensor productof the Pauli matrices, though he did not use that terminology. He used this representationto construct the Dirac equation which led to a revolution in physics.

For general d we proceed by transforming the Clifford algebra into an algebra offermions, by defining the following operators, for j = 1, · · ·n,

bj =1

2(γ2j−1 − iγ2j) b†j =

1

2(γ2j−1 + iγ2j) (6.17)

when d = 2n is even. When d = 2n + 1 is odd, we add b0 = b†0 = γ2n+1. Expressing theClifford algebra relations in terms of the bi and b†i , we find the following algebra,

bj, b†k = δjkI bj, bk = b†j, b†k = 0 j, k = 1, · · · , n (6.18)

70

when d is even, and when d is odd we supplement the above relations with the anti-commutators with b0,

(b0)2 = I bj, b0 = b†j, b0 = 0 for d odd (6.19)

The algebra for d = 2n even consists of n complex fermion operators which are independentof one another.

This means that their representation is a tensor product of the representations for eachone of the fermion oscillator pairs. We are interested in finite-dimensional representations,because we wish to construct finite-dimensional representations of so(d). To construct therepresentation for a single pair, bi, b

†i we use the fact that it is finite-dimensional so that

there must be some state |−〉i 6= 0 in the representation for which bi|−〉i = 0. Applying b†ito |−〉i, we obtain a new state |+〉i = b†i |−〉i. This state is different from |−〉i because whenwe apply bi to it, the result is |−〉i which is non-zero while applying bi to |−〉i gave zero.But now were are finished because no other states are obtained by applying the generatorsbi or b†i . Thus the irreducible representation is two-dimensional, and in fact it is just thealgebra of σ± and σ3. To obtain the representation of the full Clifford algebra for arbitraryd, we take the tensor product ground state,

|0〉 = |−〉1 ⊗ |−〉2 ⊗ · · · ⊗ |−〉n (6.20)

It is annihilated by all the bi for i = 1, · · · , n, and is an eigenstate of b0 if d is odd. Themost general state is a linear combination of the states,

(b†1)m1(b†2)m2 · · · (b†n)mn|0〉 (6.21)

where each mi can take the values 0 or 1, giving a total of 2n states. Thus the representationhas dimension 2n = 2[n

2], and it is irreducible for all d.

We may equivalently return to the representation in terms of the original γ-matrices.The states map as follows,

|+〉i → u+ =

(10

)|−〉i → u− =

(01

)(6.22)

and the ground state is given by,

|0〉 → u− ⊗ u− ⊗ · · · ⊗ u−︸ ︷︷ ︸n

(6.23)

The individual generators map as follows bi → σ− and b†i → σ+, from which we can nowconstruct the representations of the b and b† operators,

b1 = σ− ⊗ I ⊗ I ⊗ · · · ⊗ I ⊗ I b†1 = σ+ ⊗ I ⊗ I ⊗ · · · ⊗ I ⊗ I (6.24)

b2 = σ3 ⊗ σ− ⊗ I ⊗ · · · ⊗ I ⊗ I b†2 = σ3 ⊗ σ+ ⊗ I ⊗ · · · ⊗ I ⊗ I· · · · · ·bn = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ− b†n = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ+

71

Finally, converting σ± into σ1 and σ2 and bi and β†i into γµ, we obtain, the expressions forthe γ matrices,

γ1 = σ1 ⊗ I ⊗ I ⊗ · · · ⊗ I ⊗ I γ2 = σ2 ⊗ I ⊗ I ⊗ · · · ⊗ I ⊗ I (6.25)

γ3 = σ3 ⊗ σ1 ⊗ I ⊗ · · · ⊗ I ⊗ I γ4 = σ3 ⊗ σ2 ⊗ I ⊗ · · · ⊗ I ⊗ Iγ5 = σ3 ⊗ σ3 ⊗ σ1 ⊗ · · · ⊗ I ⊗ I γ6 = σ3 ⊗ σ3 ⊗ σ2 ⊗ · · · ⊗ I ⊗ I· · · · · ·

γ2n−1 = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ1 γ2n = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ2

We also define the matrix

γ = σ3 ⊗ σ3 ⊗ σ3 ⊗ · · · ⊗ σ3 ⊗ σ3 (6.26)

which satisfies

γ2 = I γ, γj = 0 (6.27)

If d is even, so that d = 2n, then γµ, µ = 1, · · · , 2n defined above, indeed satisfy (6.9). If dis odd, we define γd = γ, and then γµ, i = 1, · · · , 2n + 1 satisfies the Clifford algebra. Byconstruction, all γµ are Hermitian, ㆵ = γµ.

The γ-matrices form only a small subset of all 2n×2n matrices. All the other generatorsof 2n × 2n matrices are obtained by anti-symmetrized products of the basic γµ,

γµ1µ2···µp ≡ γ[µ1γµ2 · · · γµp] (6.28)

where [· · · ] denotes anti-symmetrization of the p indices. The identity I together with allmatrices γµ1µ2···µp for 1 ≤ p ≤ n forms a complete basis, with the following properties,

(γµ1µ2···µp)† = (−1)12p(p−1)γµ1µ2···µp

γ γµ1µ2···µm = (−1)pγµ1µ2···µp γ (6.29)

6.4 Spinor representations of so(d)

We have shown earlier that the generators Σµν = 14[γµ, γν ] satisfy the structure relations of

so(d). Here we shall provide more details and show why this is the spinor representation.We shall denote the representation matrices of the spinor representation of SO(d) by S(g)where g is an element of SO(d). Near the identity g ≈ I, we have,

gµν = δν

ν + ωµν +O(ω2)

S(g) = I +1

2

d∑µ,ν=1

ωµνΣµν +O(ω2) (6.30)

72

The spinor representation S(g) leaves the γ-matrices invariant,

d∑ν=1

gµνS(g)γνS(g)−1 = γµ (6.31)

a relation which follows from integrating the infinitesimal version,

[Σµν , γρ] = δµργν − δνργµ (6.32)

Notice that the generators have been normalized here to be anti-hermitian.

The reason this representation is a spinor representation may be seen by performing arotation in just a single direction, say by,

Σ12 =i

2σ3 ⊗ I ⊗ I ⊗ · · · ⊗ I︸ ︷︷ ︸

n−1

(6.33)

But this matrix is proportional to a Pauli matrix with coefficient 1/2, so that a rotation by2π will again produce a minus sign. This double-valuedness is the characteristic propertyof a spinor representation. For low rank, it may be understood as it was for the SO(3;R)case in terms of SU(2), as we have,

SO(4;R) = SU(2)× SU(2)/Z2

SO(5;R) = USp(4)/Z2

SO(6;R) = SU(4)/Z2 (6.34)

In each case the Z2 identification takes care of the double-valuedness. But the specialidentifications end there. For d ≥ 7, one sometimes refers to the simply-connected doublecover of SO(d) as Spin(d), so that SO(d) = Spin(d)/Z2.

6.5 Reducibility and Weyl spinor representations

For d = 2n + 1 odd, the Dirac spinor representation is irreducible. However, for d = 2neven, the chirality matrix γ commutes with the representation Σµν ,

[γ,Σµν ] = 0 (6.35)

Since γ2 = I but trγ = 0, we see that γ is not proportional to the identity matrix.Putting these two observations together, we see that the Dirac spinor representation Σmust reducible into two representations of half the dimension of 2n. The components arereferred to as Weyl spinor representations Σ+ and Σ−, and are defined by projection ontothe ±1 eigenspaces of γ by,

Σ± ≡ 1

2(I ± γ)Σµν (6.36)

each of which is irreducible.

73

6.6 Charge and complex conjugation

If the matrices γµ satisfy the Clifford algebra relations, then so must their complex con-jugate matrices γ∗µ. The matrices γµ are not all real. Since we have found only a singleirreducible representation of the Clifford algebra, and hence γ∗µ must actually correspondto an equivalent representation. Therefore, there exists an invertible matrix C such that,3

γ∗µ = −CγµC−1 (6.37)

for all 1 ≤ µ ≤ d. The matrix C is referred to as the charge conjugation matrix. We maycompute C explicitly, in the basis given above for example for d = 2n even, by observingthat half of the matrices γµ are real symmetric while the other half are imaginary anti-symmetric. To find C, we choose a basis, such as the one above, and we have

CγµC−1 = −γµ µ odd

CγµC−1 = +γµ µ even (6.38)

It is easy to find C, up to an overall multiplicative factor, by considering the product ofthe n symmetric Dirac matrices, together with a power of Γ. Choosing the multiplicativefactor as follows, we find,

C = (γ)n γ1 γ3 γ5 · · · γ2n−1 (6.39)

The matrix C then satisfies,

C†C = CtC = I

Ct = (−1)12n(n+1)C

C2 = (−1)12n(n+1)I (6.40)

The complex conjugation properties of the representation Σ of so(d) are then give as follows,

(Σ±ij)∗

=

C Σ±ij C

−1 n even

C Σ∓ij C−1 n odd

(6.41)

For n even, the Weyl spinor representations Σ± are self-conjugates, while for n odd, theyare complex conjugates of one another. An analogous result may be derived for odd d.

3The sign prefactor is a matter of convention since, if C satisfies the above equation, then Cγ satisfiesthe equation with the opposite sign, γ∗µ = γtµ = (Cγ) γµ(Cγ)−1.

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6.7 Spinor representations of so(d− 1, 1;R)

When the metric is changed from Euclidean signature (+ + · · ·+) to Minkowski signature(+ · · ·+−) with metric ηµν , the structure equation becomes,

[Mµν ,Mρσ] = ηνρMµσ − ηνσMµρ − ηµρMνσ + ηµσMνρ (6.42)

The Clifford algebra is defined with respect to the metric η as well and we have,

Γµ,Γν = 2ηµνI (6.43)

and the spinor representation matrices take the form,

Σµν =1

4[Γµ,Γν ] (6.44)

We use capital letters here to represent the γ-matrices to clearly distinguish them from thecase of Euclidean signature. It is straightforward to obtain a representation of the Cliffordalgebra matrices Γµ in terms of γµ by setting,

Γd = i γd

Γµ = γµ for µ = 1, · · · , d− 1 (6.45)

For even d, we define the matrix Γ such that Γ2 = I. The representation Σ obtained thisway is a spinor representation for the same reasons as with Euclidean signature. For evend, the Dirac spinor representation is reducible using the matrix γ in terms of Weyl spinorrepresentations Σ± with half the dimension of 2n.

What changes when the signature is changed is the reality conditions that one can im-pose on the representations. Complex conjugation may still be defined in an SO(d−1, 1;R)-invariant way, but there is now a further distinction one can make since transposition isnot related to complex conjugation by Hermiticity. Thus, if Γµ satisfy the Clifford algebrarelations, then so must Γ∗µ and Γtµ, but these representations must be equivalent to Γµ, sothat there must exist invertible matrices C and B such that,4

Γtµ = −C ΓµC−1

Γ∗µ = +B ΓµB−1 (6.46)

To find C, we must pick a basis, such as the one above, and we have

C ΓµC−1 = −Γµ µ odd

C ΓµC−1 = +Γµ µ even (6.47)

4The sign prefactors are a matter of convention since, if B and C satisfy the above equations, then ΓBand ΓC satisfy the equation with the opposite sign, Γtµ = +(ΓC)Γµ(ΓC)−1, and Γ∗

µ = −(ΓB)Γµ(ΓB)−1.

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For even d = 2n it is easy to find C by considering the product,

C = (Γ)nΓ1Γ3Γ5 · · ·Γ2n−1 (6.48)

The matrix C is then the same matrix as in the Euclidean case and satisfies,

C†C = CtC = I

Ct = (−1)12n(n+1)C

C2 = (−1)12n(n+1)I (6.49)

Complex conjugation on the other hand behaves differently, and is most easily gotten asthe composition of Hermitian conjugation and charge conjugation,

Γ∗µ = (Γ†µ)t = −(Γt0)−1ΓtµΓ0 = Γ0CΓµ(Γ0C)−1 (6.50)

Using the definition of the matrix B and Shur’s lemma, we find that B must be proportionalto CΓ0. We shall define B as follows,

B = CΓ0 (6.51)

For even d = 2n, the complex conjugation properties of the representation are,

(Σ±µν)∗

=

B Σ±µν B

−1 n odd

B Σ∓µν B−1 n even

(6.52)

Thus, for n odd, the spinor representations ψ+ and ψ− are self-conjugates, while for n even,they are complex conjugates of one another.

The spinors ψ∗ and Bψ and ΓBψ transform under the same representation. Thus, wemay impose the following reality conditions,

ψ∗ = Bψ (6.53)

which requires the consistency condition B∗B = I. In view of B∗ = B we have,

B∗B = −(−1)12n(n+1)I = I (6.54)

has the solutions n ≡ 1, 2 (mod 4). Alternatively, one may impose the reality condition,

ψ∗ = ΓBψ (6.55)

which requires that the consistency condition (ΓB)∗(ΓB) = I be obeyed, or

(ΓB)∗(ΓB) = +(−1)12n(n−1)I = I (6.56)

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has the solutions n ≡ 0, 1 (mod 4).

The Majorana and Weyl conditions may be enforced when we have Majorana spinors,and [B,Γ] = 0. So, this can happen only when n ≡ 0, 1, 2 (mod 4). To fulfill the remainingcondition, we require,

B = ΓB Γ = Γ Γ0C Γ = (−1)n+1B (6.57)

so that only n ≡ 1 (mod 4) i.e. d ≡ 2 (mod 8) survives.

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7 Roots, weights, and representations

The construction given in the earlier section for the representations of sl(2;C) basically con-tains all the ingredients for the construction of the representations of all finite-dimensionalrepresentations of an arbitrary semi-simple Lie algebra G over C. The construction for Liealgebras of rank higher than 1 is considerably more complicated, but may be handled withgreat ease using the method of roots and weights due to Cartan.

The adjoint representation of G is such a representation, whose matrices are the struc-ture constants of G and whose dimension is that of G. The classification of all represen-tations thus contains the classification of all semi-simple Lie algebras. The power of themethod of roots and weights is illustrated by Cartan’s classification theorem (1920) of allsimple finite-dimensional Lie algebras, and by Dynkin’s complete theory of representations.

7.1 The Cartan-Killing form

We consider a Lie algebra G over C of finite dimension n = dimC G and rank r. In a basisX1, · · · , Xn of G the structure relations are given by the structure constants fab

c,

[Xa, Xb] =n∑c=1

fabcXc (7.1)

The Cartan-Killing form is the bilinear symmetric form, or metric, γab on G defined by,

γab = tr (AdXaAdXb) =

n∑c,d=1

facdfbd

c (7.2)

Cartan’s definition of a semi-simple Lie algebra is that its Cartan-Killing form is invertible,in which case we denote the inverse by γab,

n∑b=1

γabγbc = δb

c (7.3)

We shall denote the inner product of two vectors va, wb by,

(v, w) =n∑

a,b=1

γabvawb (7.4)

One could, of course, take the convention to choose γ proportional to the identity ma-trix, as is done in Georgi’s book. However, this choice leads to unnecessarily complicatednormalizations of the Cartan generators.

An alternative and equivalent definition of a complex semi-simple Lie algebra is that itis the complexification of the Lie algebra K of a compact matrix Lie group. This definition,advocated for example in Brian Hall’s book, is especially convenient to use.

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Theorem 10 Let G be a semi-simple Lie algebra with Cartan-Killing form γab.

1. The Cartan-Killing form is invariant under the action of G;

2. The combination fabc =∑

d fabdγdc is totally anti-symmetric in its indices;

3. For an arbitrary representation ρ of G, we define a symmetric bilinear form by,

tr(ρ(Xa)ρ(Xb)

)= γ(ρ)ab (7.5)

If G is a simple Lie algebra (i.e. its adjoint representation is irreducible), then for anyrepresentation ρ of G the form γ(ρ) is proportional to the Cartan-Killing form,

γ(ρ)ab = K(ρ)γab (7.6)

where K(ρ) depends only on ρ and equals 1 for the adjoint representation.

To prove these results, we start from the following identity, valid in any representation,

0 = tr(

[ρ(Xa), ρ(Xb)ρ(Xc)])

=n∑d=1

fabdγ(ρ)dc +

n∑d=1

facdγ(ρ)db (7.7)

This relation proves invariance of γ(ρ)ab in any representation ρ. Taking ρ to be theadjoint representation shows that fabc, defined above, is anti-symmetric in its last twoindices and is thus anti-symmetric in all indices. To prove 3, we write (7.7) in matrix formFaγ(ρ) + (Faγ(ρ))t = 0 using (Fa)b

c = fabc. Eliminating F t

a between the relations for anarbitrary representation ρ and the adjoint representation we find,

Faγ(ρ)γ−1 = γ(ρ)γ−1Fa (7.8)

When G is a simple Lie algebra, its adjoint representation is irreducible, and by Shur’slemma γ(ρ)γ−1 must be proportional to the identity matrix, thereby proving 3.

7.2 Weights

The Cartan subalgebra H of G is defined to be the maximal Abelian subalgebra of G forwhich the adjoint map is diagonalizable. The dimension r of the vector space H over C isdefined to be the rank of G. We denote by H1, · · · , Hr a choice of basis generators of Hwhich, by definition, mutually commute,

[Hi, Hj] = 0 for all i, j = 1, · · · , r (7.9)

Since G is the complexification of the Lie algebra K of a compact Lie group, by the abovedefinition of a semi-simple Lie algebra, we may choose the Cartan generators to be the Her-mitian generators of K. Furthermore, since every finite-dimensional representation of K is

79

equivalent to a unitary representation (by theorem 3.6), we may choose the representationsmatrices ρ(Hi) in any finite-dimensional representation of the complexified Lie algebra Gto be Hermitian as well. For sl(2;C) the Cartan generator may be chosen to be J3. Forsl(3;C) there will be two such generators, which we denoted λ3, λ8 in Problem set 2.

Consider an arbitrary representation ρ of G, whose dimension N is finite. The repre-sentation matrices ρ(Hi) mutually commute,

[ρ(Hi), ρ(Hj)] = 0 for all i, j = 1, · · · , n (7.10)

and may be chosen to be Hermitian, as argued above. Since they mutually commute andare Hermitian, they may be diagonalized in the same basis and have real eigenvalues. Weshall label the states of the representation ρ, in the basis where all ρ(Hi) are diagonal, bythe real eigenvalues µi of the matrices ρ(Hi),

ρ(Hi)|ρ;µµµ〉 = mi|ρ;µµµ〉 µµµ = (m1, · · · ,mn) (7.11)

The array µµµ of eigenvalues is referred to as the weight vector of the state |ρ;µµµ〉. We shallshow later that, when ρ is an irreducible representation, all the states are uniquely labelledby the set of all weight vectors. This result is familiar from quantum mechanics, where oneuses a maximal set of commuting observables to generate all the quantum numbers.

7.3 Roots

Since the generators Hi mutually commute and are Hermitian, we may choose the remaininggenerators of G to diagonalize simultaneously the adjoint maps,

[Hi, Eααα] = aiEααα ααα = (a1, · · · , an) (7.12)

where each ai is real, and ααα is referred to as the root vector. Diagonalizing the adjoint mapgenerally requires complexifying the Lie algebra. For the case of sl(2;C) there were twosuch generators which we denoted J± = J1 ± iJ2 which clearly required introducing i ∈ C.The generators satisfied [J3, J±] = ±J± so in this case the root vectors would be (±1).

The root vectors ααα are identical to the weight vectors of the adjoint representation.Denoting the subspace of G generated by Eααα by Gααα, and the space of all root vectors ααα byR, we have the Cartain decomposition of the Lie algebra G as follows,

G = H⊕⊕ααα∈R

Gααα (7.13)

The result follows from the Hermiticity of the generators Hi of H. In addition, we provethe following results.

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Theorem 11 Let G be a semi-simple Lie algebra with Cartan subalgebraH = H1, · · ·Hr.

1. If ααα is a root, then so is −ααα and we have E−ααα = (Eααα)†.

2. Opposite root generators close onto a sl(2;C) subalgebra as follows,

[Eααα, E−ααα] = Hααα Hααα = (ααα,H) =r∑

i,j=1

γijaiHj

[Hααα, E±ααα] = ±(ααα,ααα)E±ααα (7.14)

3. The commutator of two generators associated with roots ααα,βββ 6= −ααα vanishes if ααα+βββis not a root, and gives a generator Eααα+βββ if ααα + βββ is a root,

[Eααα, Eβββ] ∼ Eααα+βββ (7.15)

4. The root spaces Gααα are one-dimensional and the only root proportional to ααα is −ααα.

To prove 1. we take the adjoint of [Hi, Eααα] = aiEααα and use the fact that Hi is Hermitian,and ai is real. It follows that (Eααα)† has eigenvalue −ai. To prove 2, we use the Jacobiidentity to evaluate the following commutator,

[Hi, [Eααα, E−ααα]] = [Eααα, [Hi, E−ααα]]− [E−ααα, [Hi, Eααα]]

= −ai[Eααα, E−ααα]− ai[E−ααα, Eααα] = 0 (7.16)

Hence, [Eααα, E−ααα] must be in the Cartan subalgebra of G, and there exist bi such that,

[Eααα, E−ααα] =n∑i=1

biHi (7.17)

Taking the inner product with Hi and using the Cartan-Killing form,

tr(Hi[Eααα, E−ααα]

)=

r∑j=1

γijbj (7.18)

But the left hand side may be rearranged as tr([Hi, Eααα]E−ααα) which evaluates to aitr(EαααE−ααα).It will be convenient to normalize the generators Eααα so that,

tr(EαααE−ααα) = 1 (7.19)

so that bi =∑

j γijaj for all i = 1, · · · , r. To prove 3. we consider the commutator [Eααα, Eβββ]

for ααα + βββ 6= 0, and evaluate its commutator with Hi using the Jacobi identity,

[Hi, [Eααα, Eβββ]] = [[Hi, Eααα], Eβββ] + [Eααα, [Hi, Eβββ]] = (ai + bi)[Eααα, Eβββ] (7.20)

If ααα + βββ is not a root, then [Eααα, Eβββ] has no components along Eγ for any root γ, and itcan have no components along Hj since the left side vanishes, but the right side would not.Thus if ααα + βββ is not a root, then [Eααα, Eβββ] = 0. Finally, if ααα + βββ is a root, then we have[Eααα, Eβββ] ∼ Eααα+βββ, as stated above. Point 4. will be proven later.

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7.4 Raising and Lowering operators

A generator Eααα of G corresponding to a root ααα modifies the weights of a representation ρ.To see this, we use the structure relations satisfied by the representation matrices,

[ρ(Hi), ρ(Eααα)] = aiρ(Eααα) (7.21)

and evaluate the weight vector µµµ = (m1, · · · ,mr) of a state ρ(Eααα)|ρ;µµµ〉 as follows,

ρ(Hi)ρ(Eααα)|ρ;µµµ〉 =(

[ρ(Hi), ρ(Eααα)] + ρ(Eααα)ρ(Hi))|ρ;µµµ〉

= (ai +mi)Eααα|ρ;µµµ〉 (7.22)

Since all the states in an irreducible representation ρ were uniquely labelled by their weightvector, we conclude that we must have,

ρ(Eααα)|ρ;µµµ〉 = Nα(ρ;µµµ)|ρ;µµµ+ααα〉 (7.23)

where Nααα(ρ;µµµ) are normalization constants which may vanish. Since we have,

[ρ(Eααα), ρ(E−ααα)] = ρ(Hααα) =n∑

i,j=1

γijaiρ(Hj) (7.24)

for any representation, we obtain relations between the normalizations Nααα(ρ;µµµ) by applyingthis relation to an arbitrary state |ρ;µµµ〉 6= 0,

ρ(Eααα)ρ(E−ααα)|ρ;µµµ〉 − ρ(E−ααα)ρ(Eααα)|ρ;µµµ〉 =n∑

i,j=1

γijaiρ(Hj)|ρ;µµµ〉 (7.25)

we obtain the following fundamental relation,

Nα(ρ;µµµ−ααα)N−ααα(ρ;µµµ)−Nααα(ρ;µµµ)N−ααα(ρ;µµµ+ααα) = (ααα,µµµ) (7.26)

7.5 Finite-dimensional representations

For a finite-dimensional representation ρ, the vector space generated by applying ρ(Eααα)to any given state transforming under the representation ρ must have finite dimension.Consider the state |ρ;µµµ〉 associated with weight vector µµµ, and the vector space generatedby applying a single root generator multiple times,

ρ(Eααα)κ|ρ;µµµ〉 =κ∏k=1

Nααα(ρ;µµµ+ (k − 1)ααα) |ρ;µµµ+ κααα〉

ρ(E−ααα)κ|ρ;µµµ〉 =κ∏`=1

N−ααα(ρ;µµµ− (`− 1)ααα) |ρ;µµµ− κααα〉 (7.27)

82

with κ ∈ N. We shall denote by p and q the largest values of κ such that the states satisfy|ρ;µµµ+ pααα〉 6= 0 and |ρ;µ− qααα〉 6= 0 and,

Eααα|ρ;µµµ+ pααα〉 = 0 E−ααα|ρ;µµµ− qααα〉 = 0 (7.28)

Since the representation ρ has finite dimension, we must have 1 ≤ p, q < ∞. Now we usethe fundamental formula (7.26) for all weights µµµ+ kααα with −q ≤ k ≤ p ,

Nα(ρ;µµµ+ (k − 1)ααα)N−ααα(µµµ+ kααα)−Nααα(ρ;µµµ+ kααα)N−ααα(µµµ+ (k + 1)ααα) = (α,µµµ+ kααα) (7.29)

Summing the contributions over this range,

p∑k=−q

Nααα(ρ;µµµ+ (k − 1)ααα)N−α(µµµ+ kααα)

−p∑

k=−q

Nααα(ρ;µµµ+ kααα)N−ααα(µµµ+ (k + 1)ααα) =

p∑k=−q

(ααα,µµµ+ kααα) (7.30)

Changing summation variables k → k − 1 in the second sum, we see that all the termsexcept at the end points of the summation on the left side cancel one another, and we areleft with,

Nααα(ρ;µµµ− (q + 1)ααα)N−ααα(µµµ− qααα)

−Nααα(ρ;µµµ+ pααα)N−ααα(µµµ+ (p+ 1)ααα) =

p∑k=−q

(ααα,µµµ+ kααα) (7.31)

In view of the definition of p and q we have Nααα(ρ;µµµ+ pααα) = 0 and N−ααα(ρ;µµµ− qααα) = 0, sothat the left side cancels. The vanishing of the right side then gives the equation,

(p+ q + 1)(ααα,µµµ) +

(1

2p(p+ 1)− 1

2q(q + 1)

)(ααα,ααα) = 0 (7.32)

Dividing through by (p+ q + 1) gives the following fundamental formula,

2(ααα,µµµ)

(ααα,ααα)= q − p (7.33)

Since q, p ∈ N, we see that the weights µµµ lie in a lattice that is dual to the root lattice.Note that in this ratio of inner products the overall normalization of γab cancels out, andone may use the Cartan Killing form defined in an arbitrary representation.

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7.6 The example of SU(3)

The choice Gell-Mann made for the generators of SU(3) is as follows.

T1 =1

2

0 1 01 0 00 0 0

T2 =1

2

0 −i 0i 0 00 0 0

T3 =1

2

1 0 00 −1 00 0 0

T4 =

1

2

0 0 10 0 01 0 0

T5 =1

2

0 0 −i0 0 0i 0 0

T6 =

1

2

0 0 00 0 10 1 0

T7 =1

2

0 0 00 0 −i0 i 0

T8 =1

2√

3

1 0 00 1 00 0 −2

The matrices Ta form an orthogonal basis under the inner product provided by the

trace, and their normalization has been chosen so that they satisfy,

tr(TaTb) =1

2δab (7.34)

Thus the inner product (u, v) may be simply replaced by the vector inner product u · v.Ultimately, we shall prefer the basis in which T8 is replaced by the generator,

T ′8 =1

2

0 0 00 1 00 0 −1

=

√3

2T8 −

1

2T3 (7.35)

at the cost of having tr(T ′8T3) = −14, and thus a non-diagonal metric. We observe the

following subalgebras isomorphic to su(2) or so(3),

S1 = T1, T2, T3S2 = T4, T5, T3 + T ′8S3 = T6, T7, T

′8

S4 = T2, T5, T7 (7.36)

The sub-algebras S1,S2,S3 are in the two-dimensional representation of su(2), and areequivalent to one another, but S4 is in the three-dimensional representation of su(2) or inthe defining representation of so(3) and is inequivalent to the first three.

Now we proceed to choosing a Cartan subalgebra, and constructing the roots of su(3)and the weights of arbitrary representations of su(3). The rank is 2, and clearly we willchoose the Cartan subalgebra generators to be H1 = T3 and H2 = T8. The states in anyrepresentation ρ will be labelled by the weight vector µ = (m1,m2),

Hi|ρ;µ〉 = mi|ρ;µ〉 i = 1, 2 (7.37)

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The states of the defining representation 3 in the standard basis of C3 correspond to thefollowing basis in terms of the weight vectors λ of the defining representation,1

00

= |3;λ1〉

010

= |3;λ2〉

001

= |3;λ3〉 (7.38)

where the weight vectors are given by,

λ1 =

(1

2,

1

2√

3

)λ2 =

(−1

2,

1

2√

3

)λ3 =

(0, − 1√

3

)(7.39)

For the adjoint representation, we choose the following linear combinations which diago-nalize AdH1 and AdH2 ,

T1 ± i T2 = E±α1

T4 ± i T5 = E±α3

T6 ± i T7 = E±α2 (7.40)

where the root vectors αi are given by,

α1 = (1, 0) α2 =

(−1

2,

√3

2

)α3 =

(1

2,

√3

2

)(7.41)

Notice that each root satisfies αi · αi = 1, so that the master formula becomes,

2α · β ∈ Z (7.42)

for any pair of roots. We readily verify that 2α1 · α3 = 2α2 · α3 = −2α1 · α2 = 1. But themaster formula tells us more than that. If µ is a weight, and if µ+ pα and µ− qα are thelargest addition and subtraction of µ by the root α, then p and q satisfy,

2α · µ = q − p (7.43)

Taking µ = α2 and α = α1 we have 2α ·µ = q− p = −1. This means that p ≥ 1, so α1 +α2

must be a root, and indeed α1 + α2 = α3. We can choose as a basis of all roots the vectorsα1 and α2. If 2α1 · µ ∈ Z and 2α2 · µ ∈ Z, then it follows that 2α3 · µ ∈ Z.

Now consider a more general representation ρ of su(3). Its weight vectors must satisfy,

2α1 · µ = p1 ∈ Z 2α2 · µ = p2 ∈ Z (7.44)

Solving for µ in terms of p1, p2, we find,

µ =

(p1

2,p1 + 2p2

2√

3

)(7.45)

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Comparison with the weights of the fundamental representation, we see that µ is a linearcombination with integer coefficients of the weights of teh defining representation,

µ = (p1 + p2)λ1 + p2λ2 (7.46)

This is no accident: the weights of any representation lies in the dual lattice to the rootlattice, of which the weight lattice of the defining representation is a basis. Of course, thisis not the full story since these states must truncate somewhere.

7.7 The root lattice of An = sl(n+ 1)

Although the rank of the algebra is n, it will be convenient to consider the root vectors inRn+1, generated by an orthonormal basis of vectors ei for i = 1, · · · , n + 1. Parametrizethe Lie algebra sl(n+ 1) by the matrices Mij with components,

(Mi,j)αβ = δi,αδj,β (7.47)

The commutator of two such matrices is given by,

[Mi,j,Mk,`] = δj,kMi,` − δi,`Mk,j (7.48)

We shall chose the Cartan generators to be diagonal and traceless,

Hi = Mi,i −Mi+1,i+1 (7.49)

The commutator of Hi with an arbitrary generator Mk,` is given by,

[Hi,Mk,`] =(δi,k − δi,` − δi+1,k + δi+1,`

)Mk,` (7.50)

Therefore, the root vector associated with the generator Mk,` with k 6= ` is given by,

ααα = ek − e` Eααα = Mk,` (7.51)

The set of all roots is given by,

R = ek − e` for 1 ≤ k 6= ` ≤ n+ 1 #R = n(n+ 1) (7.52)

One defines a root ααα = ek − e` to be a positive root iff k < ` and a root ααα to be a negativeroot iff −ααα is a positive root, which is equivalent to k > `. All positive roots may beexpressed as a linear combinations of the simple roots,

αααi = ei − ei+1 i = 1, · · · , n (7.53)

86

with positive integer, or zero, coefficients. To see this, we decompose the positive rootek − e` for k < ` as follows,

ek − e` =`−1∑i=k

(ei − ei+1) (7.54)

All the coefficients are either zero or 1, and hence all positive or zero, as announced.

The inner products of the simple roots are given as follows,

(αααi,αααj) =

2 if i = j−1 if |i− j| = 10 if |i− j| ≥ 2

(7.55)

Using the Dynkin diagran representation introduced earlier, in which roots are representedby nodes, two roots with inner product −1 are linked by a single link, and two roots whichare orthogonal re not linked, we have the following Dynkin diagram for An = sl(n+ 1;C).

· · ·ααα1 ααα2 ααα3 αααn−1 αααn

The Cartan matrix is the matrix of inner products of the simple roots, normalized insuch a manner that the entries have to be integers by construction, and are independentof the overall scaling of the root system,

Cij =2(αααi,αααj)

(αααi,αααi)(7.56)

Note that the Cartan matrix is symmetric when all the simple roots have the same length,as is the case for he algebra An = sl(n + 1), but will not be symmetric when simple rootsof different lengths occur in the root space. For sl(n + 1), the Cartan matrix is simplyCij = (αααi,αααj), and takes the form,

C =

2 −1 0 0 · · · 0 0−1 2 −1 0 · · · 0 00 −1 2 −1 · · · 0 0· · · · · · · · · · · · · · · −1 00 0 0 0 · · · 2 −10 0 0 0 · · · −1 2

(7.57)

7.8 The root lattice of Dn = so(2n)

The algebra so(2n) is generated by 2n × 2n anti-symmetric matrices which we denote byMab = −Mba = −M t

ab for a, b = 1, · · · , 2n and with components,

(Mab)α,β = δaαδbβ − δaβδbα (7.58)

87

Their structure relations are given as follows,

[Mab,Mcd] = δadMbc + δbcMad − δbdMac − δacMbd (7.59)

These are actually the generators of so(2n;R) which is the Lie algebra of the compact Liegroup SO(2n;R). We now complexify the algebra and choose the following generators ofthe Cartain subalgebra,

Hi = iM2i−1,2i i = 1, 2, . . . , n (7.60)

where the prefactor i =√−1 and not the index. The remaining generators may be or-

ganized into four sets, depending on the parity of each index, and we have the followingcommutation relations,

[Hi,M2j−1,2k−1] = −iδi,jM2j,2k−1 − iδi,kM2j−1,2k

[Hi,M2j,2k−1] = +iδi,jM2j−1,2k−1 − iδi,kM2j,2k

[Hi,M2j−1,2k] = −iδi,jM2j,2k + iδi,kM2j−1,2k−1

[Hi,M2j,2k] = +iδi,jM2j−1,2k + iδi,kM2j,2k−1 (7.61)

The roots are identified as follows,

ααα = η ej + η′ ek j 6= k (7.62)

where η = ±1 and independently η′ = ±1, and the associated generators are,

Eααα =1

2

(M2j−1,2k−1 + ηM2j,2k−1 + η′M2j−1,2k − ηη′M2j,2k

)(7.63)

One verifies that Eααα is indeed a simultaneous eigenstate with the following eigenvalues,

[Hi, Eααα] = (ηδi,j + η′δi,k)Eααα (7.64)

Counting the number of generators, we have n in the Cartan subalgebra and 2n(n−1) rootgenerators, adding up to 1

22n(2n− 1) which indeed is the dimension of so(2n). The roots

are usually denoted,

R = ±ej ± ek, 1 ≤ j 6= k ≤ n (7.65)

where the ± signs are independent of one another. All roots have equal length and satisfy(ααα,ααα) = 2. By the same lexicographical rule that we used for (n), the positive roots are,

R+ = ej − ek, ej + ek, 1 ≤ j < k ≤ n (7.66)

88

The simple roots system may be chosen as follows,

αααi = ei − ei+1, i = 1, · · · , n− 1

αααn = en−1 + en (7.67)

It is easy to see that every positive root is a linear combination with positive integercoefficients of simple roots. The inner products of the simple roots are given as follows,

(αααi,αααj) =

2 if i = j for 1 ≤ i, j ≤ n−1 if |i− j| = 1 for 1 ≤ i, j ≤ n− 10 if |i− j| ≥ 2 for 1 ≤ i, j ≤ n0 if i = n− 1, j = n

(7.68)

We find the following Dynkin diagram. Since all roots and all simple roots have the same

· · ·ααα1 ααα2 αααn−3 αααn−2

αααn−1

αααn

length, the Cartan matrix is symmetric and given by Cij = (αααi,αααj) and is given by a smallmodification of the one for sl(n+ 1;C),

C =

2 −1 0 0 · · · 0 0 0−1 2 −1 0 · · · 0 0 00 −1 2 −1 · · · 0 0 0· · · · · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · 2 −1 −10 0 0 0 · · · −1 2 00 0 0 0 · · · −1 0 2

(7.69)

7.9 The root lattice of Bn = so(2n+ 1)

Since so(2n) is a subalgebra of so(2n + 1) of the same rank, we shall choose the Cartangenerators of so(2n+1) to be those chosen for so(2n). The roots ±ej±ek with 1 ≤ j, k ≤ nof so(2n) are all roots of so(2n+ 1) as well. But there are more roots, which are given by,

[Hi, Eααα] = η δi,jEααα ααα = η ej η = ±1 (7.70)

with associated generators,

Eααα = M2j−1,2n+1 + ηM2j,2n+1 (7.71)

89

Hence the system of all roots is given by,

R = ±ej ± ek, 1 ≤ j 6= k ≤ n and ± ej, 1 ≤ j ≤ n (7.72)

where the ± signs are independent of one another. By the same lexicographical rule thatwe used for sl(n), the positive roots are,

R+ = ej − ek, 1 ≤ j < k ≤ n and ej, 1 ≤ j ≤ n (7.73)

The simple roots system may be chosen as follows,

αααi = ei − ei+1, i = 1, · · · , n− 1

αααn = en (7.74)

We encounter here, for the first time, roots of different lengths, and simple roots of differentlengths. The Cartan matrix is not now symmetric. For example, in the case of n = 3,namely so(7), we have,

C =

2 −1 0−1 2 −10 −2 2

(7.75)

The Dynkin diagram is given by

· · ·ααα1 ααα2 ααα3 αααn−1 αααn

7.10 The root lattice of Cn = sp(2n+ 1)

Without derivation we shall give here the root lattice of sp(2n),

R = ±ej ± ek, 1 ≤ j 6= k ≤ n and ± 2ej, 1 ≤ j ≤ n (7.76)

where the ± signs are independent of one another. By the same lexicographical rule thatwe used for sl(n+ 1;C), the positive roots are,

R+ = ej − ek, 1 ≤ j < k ≤ n and 2ej, 1 ≤ j ≤ n (7.77)

The simple roots system may be chosen as follows,

αααi = ei − ei+1, i = 1, · · · , n− 1

αααn = 2en (7.78)

The Dynkin diagram is given by Superficially, this is looks like the same Dynkin diagramas we had for so(2n+ 1), but the key difference is that the role of the long and short rootshas been exchanged. Sometimes this is indicated by an arrow pointing from the short tothe long roots.

90

· · ·ααα1 ααα2 ααα3 αααn−1 αααn

Coincidences of Lie algebras of low rank

From looking at the Dynkin diagrams of the classical Lie algebras, it is immediately manifestthat we have the following coincidences,

so(3;C) = sl(2;C) = sp(2;C)

so(4;C) = sl(2;C)⊕ sl(2;C)

so(5;C) = sp(4;C)

so(6;C) = sl(4;C) (7.79)

and that there are no other coincidences.

91

8 Representations of the classical Lie algebras

The classical Lie algebras are An = sl(n + 1;C), Bn = so(2n + 1;C), Cn = sp(2n;C), andDn = so(2n;C). Their root systems have been constructed in the preceding section. In thepresent section, we shall discuss the construction of their finite-dimensional representations,and give details for the case of sl(n+ 1;C) and so(2n;C) only.

The key tool is the following result obtained earlier. Consider a semi-simple Lie algebraG of rank n and a finite-dimensional irreducible representation ρ of G. If µµµ is a weight of ρand ααα a root of G, and µµµ+ kααα is a weight of ρ for −q ≤ k ≤ p with p, q ≥ 1, but µµµ+ kααα isnot a weight of ρ for k = p+1 and k = −q−1 then p, q satisfy the following magic formula,

q − p =2(ααα,µµµ)

(ααα,ααα)(8.1)

As established earlier, the weights of all finite-dimensional representations lie on the weightlattice which consists of all vectors µµµ such that 2(ααα,µµµ)/(ααα,ααα) ∈ Z for all roots ααα of G.

A basis of the weight lattice is given by the fundamental weights λλλi for i = 1, · · ·n,defined as duals to the simple roots αααj by,

2(αααj,λλλi)

(αααj,αααj)= δij (8.2)

The weights of every finite-dimensional representation ρ span a finite sub-lattice of thelattice of linear combinations with integer coefficients of the fundamental weights λλλi.

8.1 Weyl reflections and the Weyl group

The system of root and weight vectors is highly symmetric, a property which is reflectedin the fact that it is invariant under the Weyl group W (G) generated by Weyl reflections.

To quantify Weyl reflections, we start from a weight µµµ of the Lie algebra G in a finite-dimensional irreducible representation ρ and use the magic formula in the following way,

If2(ααα,µµµ)

(ααα,ααα)≥ 0 then q ≥ 2(ααα,µµµ)

(ααα,ααα)

If2(ααα,µµµ)

(ααα,ααα)≤ 0 then p ≥ −2(ααα,µµµ)

(ααα,ααα)(8.3)

When 2(ααα,µµµ)/(ααα,ααα) is negative, we can always add −2(ααα,µµµ)/(ααα,ααα) times ααα to the weightµµµ to obtain a new weight µµµ′,

µµµ′ = µµµ− 2(ααα,µµµ)

(ααα,ααα)ααα (8.4)

92

When 2(ααα,µµµ)/(ααα,ααα) is positive, we can always subtract 2(ααα,µµµ)/(ααα,ααα) times ααα to the weightµµµ to obtain a new weight which turns out to be given by the same formula as µµµ′. Definingthe following linear operation Πααα(µµµ) on the weight space,

Πααα(µµµ) = µµµ− 2(ααα,µµµ)

(ααα,ααα)ααα (8.5)

we have just shown that it will take a weight to another weight of the representation ρ.The operation Πααα reverses the sign of the root vector ααα,

Πααα(ααα) = −ααα (8.6)

and leaves invariant any vector which is orthogonal to ααα. Thus, geometrically, Πααα reflectsthe vector µµµ in the plane orthogonal to ααα, and is referred to as a Weyl reflection. Thereflection property is confirmed by showing that the square of Πααα is the identity,

Πααα

(Πααα(µµµ)

)= Πααα(µµµ)− 2(ααα,µµµ)

(ααα,ααα)Πααα(ααα) = µµµ (8.7)

The set of Weyl reflections for all roots generates a group, referred to as the Weyl groupW (G) which is a subgroup of the group of all permutations of all the roots. The Weyl groupprovides an invaluable tool for constructing the full weight diagram of a representation,given its highest weight.

8.2 Finite-dimensional irreducible representations of sl(n+ 1;C)

Using the simple roots of sl(n+ 1;C),

αααi = ei − ei+1 i = 1, · · · , n (8.8)

which all have length (αααi,ααα)i) = 2, the fundamental weights λλλj are defined to satisfy,

(ei − ei+1) · λλλj = δij 1 ≤ i, j ≤ n (8.9)

The solution is unique and given as follows,

λλλj =

j∑`=1

e` (8.10)

An arbitrary representation is specified by its highest weight vector,

λλλ =n∑i=1

qiλλλi (8.11)

The numbers qi correspond to the lengths of the horizontal arrays of boxes in the Youngtableau. For example, taking the case of sl(5;C) with n = 4, the values of qi parametrizethe Young tableau as follows.

93

q1 q2 q3 q4

8.3 Spinor representation of so(2n+ 1;C)

The simple roots of so(2n+ 1;C) are given by,

αααi = ei − ei+1 i = 1, · · · , n− 1

αααn = en (8.12)

The fundamental weights λλλj for j = 1, · · · , n − 1 obey identically the same equations asthe first n− 1 fundamental weights of sl(n+ 1;C) and are thus given by,

λλλj =

j∑`=1

e` j = 1, · · · , n− 1 (8.13)

But the fundamental weight λλλn satisfies a different equation than for sl(n + 1;C) in viewof the fact that αααn is a sorter rot, and we have, 2(αααn,λλλn) = 1 while (αi,λλλn) = 0 fori = 1, · · · , n− 1. The solution is uniquely given by,

λλλn =1

2

n∑`=1

e` (8.14)

All the weights of the representation with highest weight λλλn may be constructed by Weylreflections in the roots ei, and we get,

1

2(±e1 ± e2 ± · · · ± en) (8.15)

where the ± assignments are independent of one another. Therefore, the representation hasdimension 2n, and is referred to as the fundamental spinor representation of so(2n+ 1;C).

8.4 Spinor representations of so(2n;C)

The simple roots of so(2n;C) are given by,

αααi = ei − ei+1 i = 1, · · · , n− 1

αααn = en−1 + en (8.16)

94

The fundamental weights λλλj for j = 1, · · · , n − 2 obey identically the same equations asthe first n− 2 fundamental weights of sl(n+ 1;C) and are thus given by,

λλλj =

j∑`=1

e` j = 1, · · · , n− 2 (8.17)

But the fundamental weights λλλn−1,λλλn satisfy,

(λλλn−1,αααi) = (λλλn,αααi) = 0 i = 1, · · · , n− 2

(λλλn−1,αααn−1) = (λλλn,αααn) = 1

(λλλn−1,αααn) = (λλλn,αααn−1) = 0 (8.18)

Solving these equations, we find,

λλλn−1 =1

2(e1 + · · ·+ en−2 + en−1 − en)

λλλn =1

2(e1 + · · ·+ en−2 + en−1 + en) (8.19)

2(αααn,λλλn) = 1 while (αi,λλλn) = 0 for i = 1, · · · , n− 1. The solution is uniquely given by,

λλλn =1

2

n∑`=1

e` (8.20)

All the weights of the representation with highest weight λλλn may be constructed by Weylreflections in the roots, and we get,

λλλn−1 :1

2(±e1 ± e2 ± · · · ± en) odd number of minus signs

λλλn :1

2(±e1 ± e2 ± · · · ± en) even number of minus signs (8.21)

where the ± assignments are independent of one another, except for the overall number ofminus signs as indicated above. Therefore, we find two independent fundamental spinorrepresentations, each one having dimension dimension 2n−1.

A general finite-dimensional irreducible representation ρ of so(2n;C) is given by itshighest weight vector λλλ, which is a linear combination of the n fundamental weight vectorsλλλi with integer coefficients,

λλλ =n∑i=1

qiλλλi qi ∈ Z, qi ≥ 0 (8.22)

When qn−1 + qn is even, then the representation is single-valued and referred to as a tensorrepresentation, while when qn−1 + qn is odd, the representation is double-valued and isreferred to as a spinor representation.

95

9 The structure of semi-simple Lie algebras

A semi-simple Lie algebra is specified by its adjoint representation. Thus, in the magicformula, we take the weight vectors µµµ to be the roots, and find that the root lattice mustobey the following conditions,

M =2(ααα,βββ)

(ααα,ααα)∈ Z M ′ =

2(ααα,βββ)

(βββ,βββ)∈ Z (9.1)

which implies that we also must have, for all roots α, β,

MM ′ =4(ααα,βββ)2

(ααα,ααα) (βββ,βββ)∈ Z (9.2)

This combination is invariant under overall rescaling of the root lattice, and in fact is givenby the angle θ between the root vectors ααα,βββ,

MM ′ = 4 cos2 θ ∈ Z (9.3)

This condition is extremely restrictive, and allows for

MM ′

0

1

2

3

4

θ

90o

60o, 120o

45o, 135o

30o, 150o

0o, 180o

Dynkin

same root

Clearly these conditions provide very strong restrictions on the structure of semi-simpleLie algebras of finite dimension, and in fact lead to their complete Cartan classification,which we shall now carry out.

9.1 Some properties of simple roots

One defines a root vector to be positive as one defines in general a weight vector to bepositive. Let the root and weight vectors be given in an ordered orthonormal basis ei fori = 1, · · · , n. A root or weight µµµ is a positive root iff its first non-zero component is positive.A root or weight µµµ is a negative root if −µµµ is a positive root. The only null root or weightis µµµ = 0. With this lexicographical partial ordering, every root or weight vector is eitherpositive, negative or null.

96

One defines a simple root as a positive root which cannot be written as the sum oftwo positive roots. The entire root system is determined by giving the simple roots. Someimportant properties of simple roots are as follows.

1. A semi-simple Lie algebra of rank n has n linearly independent simple roots, whichwe shall denote by αααi for i = 1, · · · , n.

2. If ααα,βββ are simple roots, then ααα−βββ is not a root. Obviously, this is true when ααα = βββ,so consider ααα 6= βββ. Assume that ααα−βββ 6= 0 is a root. Then ether ααα−βββ is positive orβββ −ααα is positive. Consider the first and write ααα = (ααα−βββ) +βββ, but then ααα would bethe sum of two positive roots, which contradicts the fact that ααα is simple.

3. As a result, the magic formula tells us that if ααα,βββ are two simple roots, we have,

2(ααα,βββ)

(ααα,ααα)= q − p (9.4)

but we must have q = 0 since otherwise βββ − ααα would be a root, which contradictsthe item above. Hence we must have (α,βββ) ≤ 0. Thus the angle θ between differentsimple roots must obey π

2≤ θ < π. The case θ = π is excluded since it would

correspond to opposite roots which cannot both be positive.

4. The n simple roots determine all the roots of the algebra, either by using the magicformula, or by using the Weyl group.

5. Every root ααα can be uniquely decomposed as a sum of simple roots:

ααα =n∑i=1

niαααi (9.5)

where either all ni ≥ 0 for a positive root ααα, or all ni ≤ 0 for a negative root ααα.

6. All the information on the simple roots is provided by the Cartan matrix or equiva-lently by the Dynkin diagram.

9.2 Classification of finite-dimensional simple Lie algebras

Dynkin defines a Π-system to be a set of simple roots which satisfy three conditions.

(A) There are n linearly independent simple root vectors αααi for i = 1, · · · , n.

(B) Any two distinct simple roots αααi,αααj are such that,

2(αααi,αααj)

(αααi,αααi)= −pij pij ≥ 0, pij ∈ Z (9.6)

97

(C) The simple root system is indecomposable, i.e. it cannot be split into two sets ofsimple roots that are mutually orthogonal to one another. A decomposable simpleroot system corresponds to a Lie algebra which is the direct sum of two Lie algebrasand hence is not a simple Lie algebra.

We shall now enumerate all possible Π-systems. The classification theorem is simply donewith the help of a few Lemmas.

Lemma 2 The only Π system with 3 vectors are

ααα1 ααα2 ααα3 ααα1 ααα2 ααα3

The only angles allowed between two distinct simple roots are 90o, 120o, 135o, and150o. The sum of the angles between three linearly independent vectors must be < 360o.This means that one and only one angle must be 90o, since otherwise the system would bedecomposable. Suppose θ13 = 90o as in the diagrams above. The remaining angles mustsatisfy θ12 + θ23 < 270o. This requires one angle to be θ12 = 120o and the remaining angleθ23 is allowed to be either 120o or 135o, with the two cases depicted in the above figures.

Lemma 3 An indecomposable subsystem of a Π-system is a Π-system.This is obviously true.

Corollary 2 The only Π-system with a triple link is given by,

This is an immediate consequence of Lemma 3.

Lemma 4 If a Π-system contains two vectors connected by a single line, the diagramobtained by shrinking the line away and merging the two vectors into a single circle isanother Π-system.

To prove this lemma, let ααα and βββ be the two vectors joined by a single line, and let Γ bethe set of all other vectors. It follows from Lemma 2 that Γ contains no vector connectedto both ααα and βββ. The vector ααα + βββ has the same length as ααα and βββ. This follows fromapplying the magic formula,

2(ααα,βββ)

(ααα,ααα)=

2(ααα,βββ)

(βββ,βββ)= −1 (9.7)

Now if γγγ ∈ Γ and connected to ααα, then it cannot be connected to βββ and hence we have(γγγ,ααα + βββ) = (γγγ,ααα), while if γγγ′ ∈ Γ and connected to βββ, then it cannot be connected to αααand hence we have (γγγ,ααα + βββ) = (γγγ,βββ). Hence the set ααα + βββ ∪ Γ is a Π-system.

98

Corollary 3 No Π-system contains more than one double link.

To prove this, consider a diagram with two double links and shrink all the single linesin between the two double links, to end up with a system with two consecutive double linkswhich contradicts Lemma 2 with the help of Lemma 3.

Corollary 4 No Π-system contains a closed loop.

Again, shrink all single lines until you get a closed loop diagram with three vectors thatcontradicts Lemma 2.

Lemma 5 If the configuration on the left is a Π-system, then the configuration on theright must also be a Π-system.

γγγ

ααα

βββααα + βββ

γγγ

To prove this, we use the fact that (ααα,βββ) = 0 and the fact that γγγ, ααα, βββ have the samelength. It then follows that also (ααα + βββ,ααα + βββ) = 2(ααα,ααα), and that its inner product withγγγ is given by,

2(γγγ,ααα + βββ)

(γγγ,γγγ)= −2

2(γγγ,ααα + βββ)

(ααα + βββ,αααβββ)= −1 (9.8)

which indeed gives the Π-systen in the right panel of the figure.

Corollary 5 The only branches in a π-system can be tri-valent with single links emanatingfrom the vertex.

Lemma 6 The following four diagrams do not correspond to Π-systems.

1 2 3 4 3 2 1

2

1 2 3 2 1

99

1 2 3 4 5 6 4 2

3

1 2 3 2 1

2 1

The numbers inside the root circles are the coefficients needed to construct a linearcombination that vanishes. For the last diagram, for example, we have,

(ααα1 + 2ααα2 + 3ααα3 + 2ααα4 + 2ααα5 +ααα6 +ααα7)2 = 0 (9.9)

hence the root vectors are linearly-dependent.

As a result of all the lemmas we have,

1. A triple link can occur only by itself, as is reflected in the algebra G2 of Theorem 9.2.

2. A double link can occur either at the end of a chain of simple links as in the algebrasBn and Cn of Theorem 9.2, or with just a single length on each side of the doublelink as in the algebra F4 of Theorem 9.2.

3. The remaining diagrams which have only single links can have at most one trivalentvertex since otherwise the single links that connected the two vertices could be shrunkto a quadrivalent vertex. This leaves chains without vertices as in the algebras An ofTheorem 9.2, or if there is one vertex then one of the links emanating from the vertexmust connected to a single root. With the diagrams that are eliminated by Lemma9.2, this leaves the algebras Dn, E8, E7, and E6 of Theorem 9.2.

100

Theorem 12 All finite-dimensional complex simple Lie algebras are given by

· · ·An dimCAn = n2 − 1

· · ·Bn dimCBn = n(2n+ 1)

· · ·Cn dimCCn = n(2n+ 1)

· · ·Dn dimCDn = n(2n− 1)

E8 dimCE8 = 248

E7 dimCE7 = 133

E6 dimCE6 = 78

F4 dimC F4 = 52

G2 dimCG2 = 14

The short roots have been indicated in blue. The root systems for the exceptionalgroups are as follows.

RE8 =± ei ± ej, 1 ≤ i < j ≤ 8;

1

2(±e1 ± e2 ± e3 ± e4 ± e5 ± e6 ± e7 ± e8)

RE7 =

± ei ± ej, 1 ≤ i < j ≤ 6;±(e7 − e8); ± 1

2(±e1 ± e2 ± e3 ± e4 ± e5 ± e6 − e7 + e8)

RE6 =

± ei ± ej, 1 ≤ i < j ≤ 5; ± 1

2(±e1 ± e2 ± e3 ± e4 ± e5 − e6 − e7 + e8)

RF4 =

± ei ± ej, 1 ≤ i < j ≤ 4; ± ei; ±

1

2(±e1 ± e2 ± e3 ± e4)

RG2 =

± (ei − ej, 1 ≤ i < j ≤ 3; (2ei − ej − ek), i, j, k = 1, 2, 3

(9.10)

101

10 Weyl’s character formulas

Consider a compact matrix group G of rank n with Lie algebra G. Two of the mostimportant problems in group theory are as follows.

1. Decomposition of the tensor product of two irreducible representations R1, R2 of G,

R1 ⊗R2 =⊕R

mR(R1, R2)R (10.1)

where the sum is over all possible irreducible representations R of G. The positiveinteger mR(R1, R2) is the multiplicity with which the representation R occurs in thetensor product R1 ⊗R2.

2. If R is an irreducible representation of the group G, and we have a subgroup G0

of G, then we want to know how R decomposes into a direct sum of irreduciblerepresentations r of the subgroup G0,

R =⊕r

mr(R)r (10.2)

where the sum is over all irreducible representations r of G0 and the positive integersmr(R) is the multiplicity with which the representation r occurs in the decompositionof R with respect to G0. This decomposition is sometimes referred to as the branchingrules for the subgroup G0.

Both questions could already have been answered by use of the method of weights or tensors,but the most natural approach is the method of group characters. Recall the definition ofthe character χR(g) of a representation R of G,

χR(g) = trR(g) χR(e) = dimR (10.3)

Recall that characters behave well under direct sum and tensor product of representations,

χR1⊕R2(g) = χR1(g) + χR2(g)

χR1⊗R2(g) = χR1(g)× χR2(g) (10.4)

for all g ∈ G. Furthermore, characters are functions only of the conjugacy class of g,

χR(h g h−1) = χR(g) for all g, h ∈ G (10.5)

The method of characters provides a very convenient solution to both the decompositionand the branching rule problem.

102

10.1 Characters on the maximal torus

By conjugation, we may diagonalize g to a diagonal matrix of phases,

g = eiH(ϕ) H(ϕ) =n∑i=1

ϕiHi (10.6)

where as earlier, we have chosen the Cartan generators Hi to be Hermitian. Since G iscompact, every finite-dimensional represnetation R of G is equivalent to a unitary repre-sentation. Thus, by the same token we may diagonalize R(g) in any representation ρ ofthe Lie algebra G of G, and choose the representation matrices ρ(Hi) again Hermitian,

R(g) = eiρ(H(ϕ)) ρ(H(ϕ)) =n∑i=1

ϕi ρ(Hi) (10.7)

Then in general a character is of the form

χR(g) =∑λλλ

nλλλei(λλλ,H(ϕ)) (10.8)

nλλλ is the multiplicity for weight λλλ in the weight diagram of representation ρ.

Now any unitary matrix may be diagonalized by a unitary transformation. For simplic-ity, we shall assume G = U(n), but the derivation can be easily generalized to SU(n) orany of the other compact Lie groups. Therefore we may decompose g as follows,

g = U+DU D =

eiϕ1 0 0 · · · 00 eiϕ2 0 · · · 00 0 eiϕ3 · · · 0· · · · · · · · · · · · · · ·0 0 0 · · · eiϕn

(10.9)

The character in a representation R is then defined by,

χR(g) = χR

eiϕ1 0. . .

0 eiϕn

(10.10)

Since characters are class functions, they only depend on the diagonal matrix D and not onU . This is a tremendous simplification. We have at our disposal the orthogonality relationsfor characters of irreducible representations (primitive characters), given for G = U(n) by,∫

G

dµ(g) χ?R1(g)χR2(g) = N δR1,R2 (10.11)

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Now we would like to express χR(g) in terms of polynomials in eiϕi . The problem is thatthe measure dµ(g) depends on D but also on the other variables U . There is a standardchange of variables that yields us the Jacobian for the change of variables

g → (U,ϕ1, · · · , ϕn) (10.12)

To do so, regard dµ(g) as coming from the invariant metric on G = U(n),

ds2 = −tr(g†dg g†dg

)(10.13)

To decompose the mesure dµ(g) it suffices to decompose the metric, using,

g†dg = −U †D?(dU)U †DU + U †D?(dD)U + U †(dU) (10.14)

In evaluating the metric, the cross terms of dD and dU cancel, and one is left with,

ds2 = −tr(D?dD)2 − 2 tr(U †dU)2 + 2 tr(D?(dUU †)D(dUU †)

)(10.15)

The metric is the sum of a metric on the variables ϕi which is independent of U , and ametric on U which depends on ϕi. Therefore, the measure factorizes and is of the form,

dµ(g) = ∆(ϕ)∆?(ϕ)n∏i=1

dϕi dµ(U) (10.16)

Each factor dU U † is accompanied by a linear function in one of the phases eiϕi and thedependence on positive and negative phases factroizes. The measure dµ(U) contains n2−nfactors, so that ∆ must be a polynomial in eiϕ1 , · · · , eiϕn of degree 1

2n(n− 1).

∆(ϕ) must vanish whenever ϕi = ϕj for i 6= j. This tells us what ∆ should be up to aϕ-independent constant,

∆(ϕi) =∏i<j

(eiϕi − eiϕj) (10.17)

which is of degree 12n(n− 1) and is the only function with those properties.

10.2 Weyl’s first formula

The character of any irreducible representation of U(n) has the following form,

χR(g) =XR(γi)

∆(γi)γi = eiϕi (10.18)

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∆(γi) =∏i<j

(γi − γj) =

∣∣∣∣∣∣∣∣∣∣∣

1 1 . . . 1γ1 γ2 γnγ2

1 γ22 . . . γ2

n...γn−1 . . . γn−1

n

∣∣∣∣∣∣∣∣∣∣∣(10.19)

and χR(γi) is an anti-symmetric polynomial in γi,

XT (γi) =

∣∣∣∣∣∣∣∣∣γ`11 γ`12 . . . γ`1n

γ`21 γ`22

...

· · · · · · ...

γ`n1 γ`n2 γ`nn

∣∣∣∣∣∣∣∣∣ (10.20)

Here

`k = mk + (n− k) k = 1, . . . , n (10.21)

with

mk =k∑i=1

qi (10.22)

and the q’s are the components as follows:

µµµ =∑i

qiλλλi (10.23)

where λλλi are the weights of the fundamental representations:

2(λλλj,αααi)

(αααi,αααi)= δij (10.24)

By letting ϕi → 0, and taking the limit, one deduces a formula for dimensions.

dim R =

∏1≤i<j≤n

(`i − `j)∏1≤i≤n−1

i !(10.25)

where we may express `i − `j directly in terms of The q-parameters by,

`i − `j = j − i+

j∑k=i+1

qk (10.26)

For SU(3), we set n = 3 and we find,

dimR =1

2(1 + q2)(1 + q3)(2 + q2 + q3) (10.27)

in agreement with the formula derived from tensor analysis and Young tableaux.

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