Lecture 1: Lie Groups and Lie Algebras - Stanford...

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Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020

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Page 1: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Lecture 1: Lie Groups and Lie Algebras

Daniel Bump

April 7, 2020

Page 2: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Lie groups and Lie algebras

A Lie group is a manifold G that is also a group, in which themultiplication map m : G× G −→ G and the inverse mapG −→ G are smooth. We will also encounter complex analyticgroups in which G is a complex manfold, and the multiplicationand inverse maps are holomorphic maps.

A Lie algebra over a field F (which will always be R or C in thiscourse) is a vector space g together with a bilinear Lie bracketoperation [X,Y] defined for X,Y ∈ g, that is skew-symmetric:

[X,Y] = −[Y,X]

and satisfies the Jacobi relation

[X, [Y,Z]] + [Y, [Z,X]] + [Z, [X,Y]] = 0.

Page 3: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Representations of Lie groups

A main topic will be representations, which in this class will bemainly finite-dimensional.

If G is a Lie group (or more general topological group), arepresentation is a complex vector space V with a homorphismπ : G −→ GL(V). If G is a complex analytic Lie group, werequire π to be holomorphic. (If G is an ordinary Lie group, wemake no such requirement.)

Page 4: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Representations of Lie algebras

If g is a Lie algebra, then a representation is a complex vectorspace g together with a homorphism π : g −→ End(V) such that

π([X,Y]) = π(X)π(Y)− π(Y)π(X), X,Y ∈ g.

If g is a complex Lie algebra, we further requireπ : g −→ End(V) to be complex linear. If g is a real Lie algebra,we make no such requirement.

Page 5: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Complexification of a real Lie algebra

If g is a real Lie algebra, then gC = C⊗ g is naturally a complexvector space containing a copy of g embedded by X 7→ 1⊗ X.We may extend the bracket operation to gC by linearity, and gCbecomes a complex Lie algebra, called the complexificationof g.

[a⊗ X, b⊗ Y] = ab⊗ [X,Y], , a, b ∈ C,X,Y ∈ g.

For example, the complexification of g = gl(n,R) is gl(n,C).

Page 6: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Complexification of representations

Let π : g −→ End(V) be a representation. Then we may extendπ to a representation of gC by linearity. Conversely, restricting arepresentation of gC to g gives a representation of g.

Explanation: If W is a complex Vector space,

HomR(g,W) ∼= HomC(gC,W).

We apply this with W = End(V) and note that

π ∈ HomC(gC,W)

is a Lie algebra homomorphism (representation) if and only ifits restriction to g is.

Page 7: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Complexification of representations (continued)

With W = End(V):

HomR(g,End(V)) ∼= HomC(gC,End(V).

Representations are just Lie algebra homomorphisms on eitherside of this isomorphism, so Lie algebra homomorphismscorrespond to Lie algebra homomorphisms.

Thus the representations of g and gC are really the same,making use of our convention that a representation of gC iscomplex linear, while a representation of g is only required to bereal linear.

Page 8: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

The Lie algebras gl(n,F), F = R,C

If A is an associative algebra (such as Matn(R)) then we maydefine a bracket on A by

[X,Y] = XY − YX.

The Lie algebra axioms may be easily verified. We will denotethis Lie algebra as Lie(A). For example if A = EndF(V) where Vis and F-vector space, we will denote Lie(A) as gln(F) orgl(n,F). The reason for this notation will become clear later.

The complexification of gl(n,R) is gl(n,C). So with ourconventions, the representations of gl(n,R) are the same asrepresentations of gl(n,C).

Page 9: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

The Unitary Lie algebra

Another Lie algebra u(n) consists of n× n skew-Hermitianmatrices, that is, complex n× n matrices X that satisfy X = −tX.Although these are complex matrices, this real vector space isnot closed under complex multiplication. It is thus a real Liealgebra but not a complex one.

Let g0 = u(n). We find that gl(n,C) = g0 ⊕ ig0. This means thatgC = gl(n,C) ∼= C⊗ g0 is the complexification of g0 as well as ofg = gl(n,C). So the representations of gl(n,C) are also thesame as the representations of g0. We reiterate our caveat, if gis a complex Lie algebra, we further require a representationπ : g −→ End(V) to be complex linear. With this caveat, thethree Lie algebras (two real and one complex) g, gC and g0 allhave the same representation theories.

Page 10: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Forms

If g1 and g2 are real Lie algebras with the samecomplexification, we say that gi are real forms of one another.From the above discussion, there is a natural bijection ofrepresentations of g1 and g2; it is easy to see that irreduciblerepresentations correspond.

The classification of forms is a topic in Galois cohomology. Thisconcept applies to other things, such as algebraic group.

For example the real Lie algebras gl(n,R) and u(n) plexificationare called real forms of one another.

Page 11: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Examples of Lie groups

Our first Lie group is GL(n,R). This is a noncompact Lie groupof dimension n2.

A second example is GL(n,C). This group (unlike GL(n,R)) is acomplex analytic group: it is a complex manifold, and themultiplication and inverse maps are holomorphic. Thedimension of GL(n,C) as a complex manifold is n2, but itsdimension as a real Lie group is 2n2.

The third example is the unitary group

U(n) ={

g ∈ GL(n,C)|g · tg−1 = I}

where g is the complex conjugate. The group U(n) is compact.ven though complex numbers figure in its definition, it is not acomplex analytic Lie group.

Page 12: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Weyl’s unitarian trick

Weyl made use of this situation to reduce representation theoryof noncompact groups to compact groups. The followingcategories are essentially the same:

Finite-dimensional representations of GL(n,R);Finite-dimensional analytic representations of GL(n,C);Finite-dimensional representations of U(n).

This is analogous to the similar relationship between the Liealgebras gl(n,R), gl(n,C) and u(n).

Caveat: the group GL(n,R) has infinite-dimensionalrepresentations that are irreducible (even unitary) but U(n)does not, so this device is restricted to finite-dimensionalrepresentations.

Page 13: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

The Lie algebra of a Lie group

In the rest of today’s lecture, we will explain how one associatesa Lie algebra Lie(G) with a Lie group G.

We recall that if G is a manifold, to every point g ∈ G there is aTangent space Tg(G).

TheoremIf G is a Lie group the tangent space T1(G) of G at the identityhas a natural Lie algebra structure.

We will explain this.

Page 14: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Vector fields

Let G be a manifold. A vector field is a rule g→ Xg thatassociates to every g ∈ G a tangent vector Xg ∈ Tg(G). We askthat the map g→ Xg be smooth in a suitable sense. The disjointunion

T(G) =⋃g∈G

Tg(G)

is the tangent bundle and a vector field is just a smooth sectionof the tangent bundle.

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Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Vector fields as derivations

If Xg is a vector field, then we may associate with Xg aderivation of the ring C∞(G) as follows. If g ∈ G and f ∈ C∞(G)then we may differentiate f along any path tangent to Xg andobtain a new value X(f ). By the Leibnitz rule for derivatives

X(f1f2) = X(f1)f2 + f1X(f2).

Thus X is a derivation of C∞(G).

Conversely, it may be shown that any derivation of C∞(G)arises from a vector field.

Page 16: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Lie bracket of vector fields

Let R be an algebra over a field and let Der(R) be the set ofderivations of R, that is, linear maps such that

D(f1f2) = D(f1)f2 + f1D(f2).

It is an easy algebraic calculation that if D1 and D2 are in Der(R)then so is [D1,D2] and Der(R) is thus a Lie algebra.

Interpreting vector fields as derivations of C∞(G) (with G amanifold) this gives a bracket operation on vector fields, and sothe space of vector fields become a Lie algebra. It is of courseinfinite-dimensional.

Page 17: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Left invariant vector fields

Now let G be a group. If g ∈ G then left translation λg : G→ G isthe map λg(h) = gh. It induces automorphisms of the tangentbundle, also denoted λg, and hence λg acts on vector fields.

PropositionEvery X ∈ T1(G) may be extended uniquely to a vector field thatis invariant under left translation.

(This is actually obvious.) So:

T1(G) is isomorphic to the space of left invariant vector fields,which are closed under the Lie bracket, and hence form a Liealgebra.

This is Lie(G).

Page 18: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

The Lie algebra of GL(n,R)

The above discussion doesn’t give guidance as to how tocompute with the Lie algebra, but that is easily overcome. Forexample here are some specific Lie groups, and realizations oftheir Lie algebras.

If G = GL(n,R), we will show that the Lie algebra gl(n,R) of Gmay be identified with the Lie algebra of the associative algebraMatn(R), which we have previously denoted gl(n,R).

Page 19: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Matrices as tangent vectors

It is easy to see how X ∈ Matn(R) can be identified with atangent vector to G at the identity, namely X is the vectortangent to the path t 7→ etX = I + tX + 1

2 t2X2 + . . . at t = 0. Orequivalently X is the vector tangent to the path t 7→ I + tX att = 0, which is tangent to etX.

We must interpret X ∈ Matn(R) as a derivation of C∞(G) alongthe left-invariant vector field that is tangent to the path t 7→ etX atthe identity. The correct formula for this is

(Xf )(g) =ddt

f (getX)|t=0

Since the curves t 7→ getX and g(I + tX) are tangent at t = 0,

(Xf )(g) =ddt

f (g(I + tX))|t=0.

Page 20: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Computing the Lie bracket

Now let us compute [X,Y]. We will prove

[X,Y] = X · Y − Y · X

where the dot denotes matrix multiplication. This should not beconfused with the defining formula [X,Y] = XY − YX where nowthe “multiplication” is composition of mapsX,Y : C∞(G) −→ C∞(G).

Let us compute X(Y(f )). We want a Taylor expansion for thefunction f (geX) near X = 0 and (with g fixed) we will write this inthe form

f (g(I + X)) = c0 + c1(X) + B(X,X) + R(X),

where c1(X) is linear, B(X,Y) is a symmetric bilinear form, andthe remainder R(X) vanishes to the third order.

Page 21: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Computing X(f )(g)

First we proveXf (g) = c1(X) .

Indeed(Xf )(g) =

ddt

f (g(I + tX))|t=0

=ddt

(c0 + c1(tX) + B(tX, tX) + R(tX))|t=0 = c1(X),

picking off the linear term.

Page 22: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

Computing X(Y(f ))(g)

NextX(Yf )(g) =

ddt

((Yf )(g(I + tX)))|t=0 =

ddt

ddu

f (g(I + tX)(I + uY))|t=u=0.

Now the value here of f is

c0 + c1(tX + uY + tuXY)

+B(tX + uY + tuXY, tX + uY + tuXY) + R(. . .)

and we are picking off the coefficient of tu. Thus

X(Yf (g)) = c1(X · Y) + B(X,Y) + B(Y,X).

Page 23: Lecture 1: Lie Groups and Lie Algebras - Stanford Universitysporadic.stanford.edu/Math210C/lecture1.pdf · Lecture 1: Lie Groups and Lie Algebras Daniel Bump April 7, 2020. Basic

Basic definitions Lie algebra representations The Lie algebra of a Lie group Making the bracket explicit

The end of the computation

Therefore

X(Yf (g))− Y(Xf (g)) = c1(X · Y − Y · X).

Remembering our computation of Xf , we have proved

[X,Y] = X · Y − Y · X

where · is matrix multiplication.

This shows that the Lie algebra of GL(n,R) is gl(n,R).