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1 These notes are incomplete they will be updated regularly. LIE GROUPS, LIE ALGEBRAS, AND REPRESENTATIONS SPRING SEMESTER 2008 RICHARD A. WENTWORTH Contents 1. Lie groups and Lie algebras Definition and examples of Lie groups Left invariant vector fields and Lie algebras The exponential map 6 2. Representations of Lie groups and Lie algebras Definitions and examples The adjoint representation Semisimplicity and Schur s lemma Haar measure and Weyl s trick Characters Examples Peter-Weyl theorem 9 3. Maximal tori Existence Weyl group Cartan subalgebras 9 4. Classification of complex semisimple Lie algebras Roots Dynkin diagrams Stiefel diagrams Classification 9 5. Weyl character formula Weights and multiplicities Examples 9... without fantasy one would never become a mathematician, and what gave me a place among the mathematicians of our day, despite my lack of knowledge and form, was the audacity of my thinking. - Sophus Lie Date: February 4, 2008 (last revision).

2 2 RICHARD A. WENTWORTH 1. Lie groups and Lie algebras 1.1. Definition and examples of Lie groups. Definition 1.1. A (real, complex) Lie group G is a group that has the structure of a (real, complex) differentiable manifold so that the multiplication and inverse maps are smooth. Some examples: The simplest Lie group is perhaps R with its additive structure. The simplest compact example is the circle U(1) = R/Z. More generally, take finite dimensional vector spaces such as R n, and those vector spaces divided by lattices R n /Γ, Γ Z n. Let V be an n-dimensional vector space over K = R, C, or H. Then GL(V ) = {invertible endomorphisms A : V V } Note that invertibility is equivalent to the non-vanishing of det A, which is smooth (in fact, polynomial) in the entries of A. Hence, GL(V ) is an open submanifold of the space of all endomorphisms End(V ). The manifold structure on the latter comes from the identification with the vector space of matrix entries. We will mostly denote these groups GL(n, R), GL(n, C), and GL(n, H). Let SL(V ) GL(V ) denote the subgroup with det A = 1. To see the manifold structure, by the implicit function theorem we need to show that 1 is a regular value for the determinant map. By an important formula, at a point A with det A = 1, (1.1) D det(a)x = D log det(a)x = Tr(A 1 X) and this is clearly surjective, since it is not identically zero. Let O(n) GL(n, R) be the group of length preserving matrices. This is the orthogonal group. We have O(n) = {A GL(n, R) : AA T = I} To see the manifold structure, we need to show that zero is a regular value for the map F : GL(n, R) Sym(n, R) defined by F (A) = AA T I, where Sym(n, R) are the symmetric matrices. We have (1.2) DF (A)X = XA T + AX T and this is surjective, for if Y is symmetric, A O(n), and X = Y A/2, then DF (A)X = Y. In a similar way, one defines the unitary and symplectic groups U(n) and Sp(n). More precisely, these are subgroups of GL(n, C) and GL(n, H) preserving hermitian and quaternionic norms. E.g. U(n) = {A GL(n, C) : AA = I} where A = A T. We can also define the groups SO(n) and SU(n) by the requirement det A = 1. Notice the following:

3 LIE GROUPS 3 The real groups SU(n) and SL(n, R) are both subgroups of the complex group SL(n, C). SU(n) is compact whereas SL(n, R) is non-compact. SU(n) SL(n, R) = SO(n). This structure is an important feature that holds in great generality. For example, suppose A SU(2). Then a b A = c d Since A = A 1, we have SU(2) = A 1 d b = c a { } a b GL(2, C) : a b ā 2 + b 2 = 1 In this way, we see that diffeomorphically SU(2) S 3. Notice that U(1) SU(2) by the requirement b = 0. The quotient space SU(2)/ U(1) S 2, and this is known as the Hopf fibration. We can see this as follows. SU(2) acts on S 2 = P 1 by a b [z b ā 1, z 2 ] = [az 1 + bz 2, bz 1 + āz 2 ] This action is clearly transitive, and the stabilizer of [1, 0] is precisely the U(1) above. This realizes P 1 as a homogeneous space. The requirement that a, b R implies { a b SU(2) SL(2, R) = b a } GL(2, R) : a 2 + b 2 = 1 U(1) On the other hand, SL(2, R) acts on the upper half space H 2 (not the quaternions!) by linear fractional transformations: τ = (x, y) Aτ = aτ + b cτ + d It is easy to see that this is a transitive action, and that the stabilizer of τ = i, say, is the U(1) above. In this way we see that SL(2, R)/ U(1) is diffeomorphic to the upper half plane, which is an open set in S 2 = SU(2)/ U(1). This is an example of the Borel embedding Left invariant vector fields and Lie algebras. Definition 1.2. Let G be a connected Lie group, g = T e G. Then g is called the Lie algebra of G. An important fact is that the tangent bundle of a Lie group is trivial T G G g. A trivialization is given by the following. Any g G gives rise to two diffeomorphisms of G by multiplication on the left or on the right. l g : G G : l g (h) = gh r g : G G : r g (h) = hg Now the derivative Dl g : T h G T gh G. We make the following Definition 1.3. A vector field X g on G is called left invariant if Dl g (X h ) = X gh for all g, h G.

4 4 RICHARD A. WENTWORTH The space of left invariant vector fields is isomorphic to g by setting X g = Dl g (X), for X g. This gives the desired trivialization. We will use the abbreviation l.i.v.f. for left invariant vector field. Remark 1.4. For a matrix group such as the ones we have considered above, Dl g is simply multiplication by g. Hence, X g = gx. For any smooth manifold M, there is a product on the space of vector fields X(M) on M defined as follows. For X, Y X(M) and a function f on M, Let (1.3) [X, Y ](f) = X(Y (f)) Y (X(f)) To be explicit, in local coordinates e i, write X = X i e i Y = Y i e i Then in these coordinates The bracket satisfies (1) [X, Y ] = [Y, X] [X, Y ] = i=1 {X i Y j i,j=1 e i i=1 } Y i Xj e i e j (2) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 (Jacobi identity). Given a Lie group, g = T e G has been identified with the space of l.i.v.f. s, and so g inherits the bracket operation. Namely, define [, ] on g by (1.4) [X, Y ] g = [X g, Y g ] We now abstract this. Definition 1.5. A finite dimensional real (complex) vector space g with a bilinear multiplication [, ] g satisfying properties (1) and (2) above is called a real (complex) Lie algebra. Hence, if G is a Lie group then g = T e G with bracket (1.4) is a Lie algebra. Conversely, every (finite dimensional) Lie algebra arises in this way. Remark 1.6. If g is a real Lie algebra, the complexification g C = g R C is a complex Lie algebra. This is an important construction. Proposition 1.7. For matrix groups, the Lie bracket is given by the commutator of matrix multiplication. Proof. It suffices to prove this for GL(n). For coordinates we choose the elementary matrices: x ij (A) = A ij. If X = n i,j=1 Xij, then by Remark 1.4, X x ij g = n i,j=1 gik X kj. It follows that x ij Xg pq x ij = δ pi X jq g=e g=e

5 Hence, plugging into the expression above, [X g, Y g ] = g=e Examples: = = i,j,p,q=1 i,j,p,q=1 i,j,p,q=1 = XY Y X LIE GROUPS 5 ij Y pq {X x ij } Y ij Xpq x ij x pq { X ij δ pi Y jq Y ij δ pi X jq} x pq { X pj Y jq Y pj X jq} x pq If G is abelian, g is a trivial Lie algebra, i.e. all brackets vanish. Indeed, suppose G is a matrix group. If g(t) and h(s) are curves through the identity with ġ(0) = X, ḣ(0) = Y, then 0 = ( t s ghg 1 h 1) = ( s=0 t gy g 1 Y ) = XY Y X For the general argument, see Proposition 1.13 below. Since GL(V ) is an open subset of the vector space of endomorphisms, gl(v ) End(V ). By Proposition 1.7, the Lie algebra structure is simply the commutator of matrix multipication. Suppose dim V = n. A basis is given by the elementary matrices {e(i, j)} n i,j=1, i.e. (e(i, j)) pq = δ ip δ qj. Since (e(i, j)e(p, q)) kl = (e(i, j)) kr (e(p, q)) rl = we have r=1 (1.5) [e(i, j), e(p, q)] = δ jp e(i, q) δ qi e(p, j) From (1.1) we see that δ ik δ rj δ rp δ ql = δ ik δ jp δ ql = δ jp e(i, q) l=1 sl(v ) = {X End(V ) : Tr(X) = 0} Notice that any endomorphism can be decomposed into traceless and trace pieces: X = (X 1n ) Tr X I + 1 n Tr X I Hence, if V is a vector space over K we have a canonical decomposition of Lie algebras: From (1.2) we see that gl(v ) = K sl(v ) o(n) = {X End(R n ) : X T = X}

6 6 RICHARD A. WENTWORTH i.e. the skew-symmetric matrices. Also, so(n) consists of the traceless, skew-symmetric matrices. Similarly, u(n) = {X End(C n ) : X = X} or skew-hermitian matrices, and su(n) are traceless, skew-hermtian. A basis for su(2) is given by σ 1 = 1 0 i 2 i 0 with commutation relations σ 2 = 1 2 ( 0 ) σ 3 = 1 2 [σ 1, σ 2 ] = σ 3 [σ 2, σ 3 ] = σ 1 [σ 3, σ 1 ] = σ 2 These (up to scale) are the relations for the cross product on R 3. For sl(2, R), take X = Y = and one checks the same commutation relations Z = i 0 0 i [X, Y ] = 2Y [X, Z] = 2Z [Y, Z] = X Notice the 2-dimensional subalgebras. It is clear that su(2) does not have 2-dimensional subalgebras, so sl(2, R) and su(2) are not isomorphic. On the other hand, their complexifications are: 1.3. The exponential map. su(2) C sl(2, R) C sl(2, C) Definition 1.8. A one parameter subgroup (1-PS) of a Lie group G is a homomorphism R G. A 1-PS is determined by an element of g in the following way. Given X g, let γ(t) be a solution to the differential equation γ(t) = X γ(t) γ(0) = e i.e. γ(t) is the flow through the identity of the l.i.v.f. of X. Lemma 1.9. The map t γ(t) defines a homomorphism R G. In particular, γ( t) = γ(t) 1. Proof. We need to show γ(s + t) = γ(s)γ(t) for all t, s. Fix s and let Then σ(t) = γ(s)γ(t) ρ(t) = γ X (s + t) σ(t) = Dl γ(s) γ(t) = Dl γ(s) X γ(t) = Dl γ(s) Dl γ(t) X = Dl γ(s)γ(t) X = X σ(t)

7 LIE GROUPS 7 Hence, σ(t) is the integral curve of the l.i.v.f. of X through σ(0) = γ(s). On the other hand, ρ(t) = γ(s + t) = X γ(s+t) = X ρ(t) So ρ(t) is the integral curve of the l.i.v.f. of X through ρ(0) = γ(s). By the uniqueness of integral curves, these must coincide. Lemma Let γ(t) be the flow of X g through the identity. Then [X, Y ] g = d dt Dr γ( t) Y γ(t) Proof. Notice that Φ t (g) = gγ(t) is the flow of X g through the point g. Moreover, DΦ t Y g = Dr γ(t) Y g. Hence, by definition of the Lie derivative [X, Y ] g = L Xg Y t = d dt DΦ t Y γ(t) = d dt Dr γ( t) Y γ(t) Definition The exponential map exp : g G is defined by exp(x) = γ(1), where γ(t) is the flow of X g through the identity. Notice that d exp(0) is the identity. diffeomorphism. Here is the justification for the terminology: Claim 1. Let G be a matrix group. Then Hence, by the implicit function theorem, exp is a local exp(x) = Proof. Let E(X) be the right hand side, and σ(t) = E(tX). Then σ(0) = I and σ(t) = σ(t)x = m=0 X m m! X σ(t). Hence, σ(t) is the flow of the l.i.v.f. of X, and the claim holds. Here is one use of the exponential map. Proposition Let φ : H G be a homomorphism of Lie groups with H connected. The derivative Dφ(e) : h g is a homomorphism of Lie algebras. Moreover, any other homomorphism with the same derivative is equal to f. Proof. We first show that the derivative Dφ : T h H T φ(h) G takes l.i.v.f. s to l.i.v.f. s. Indeed, if X h, Y = Dφ(e)X, we need to show that Dφ(h)X h = Y φ(h). Let γ(t) be a curve in H, γ(0) = e, γ(0) = X. Then On the other hand, d dt φ(hγ(t)) = Dφ(h)Dl h X = Dφ(h)X h d d dt φ(h)φ(γ(t)) = Dl φ(h) dt φ(γ(t)) = Dl φ(h) Dφ(e)X = Dl φ(h) Y = Y φ(h)

8 8 RICHARD A. WENTWORTH Since φ(hγ(t)) = φ(h)φ(γ(t)), this proves the claim. It follows that Dφ(e)[X, Y ] h = Dφ(h)[X h, Y h ] h=e = [Dφ(h)X h, Dφ(h)Y h ] h=e = [(Dφ(e)X) φ(h), (Dφ(e)Y ) φ(h) ] = [Dφ(e)X, Dφ(e)Y ] g Hence, Dφ(e) is a homomorphism. Since l.i.v.f. s are sent to l.i.v.f. s, φ maps flows of these vector fields to the corresponding flows. The following diagram therefore commutes: h=e (1.6) h exp H Dφ(e) g exp G Since exp is a diffeomorphism on a neighborhood U about the origin, it follows from this diagram that if D φ(e) = Dφ(e), then φ = φ on U. Now consider { H = h H : φ(h) } = φ(h) This set is clearly closed and contains U. If h H, then since φ is a homomorphism, hu is an open neighborhood of h contained in H. Hence, H is open and closed and nonempty. It must equal H by connectivity. Proposition The Lie algebra of an abelian Lie group is trivial. Moreover, the exponential map of an abelian Lie group is a homomorphism. Proof. Fix X, Y g, and let γ(t) = exp(tx). If G is abelian, then l g = r g and Dl g = Dr g. Also, recall from Lemma 1.9 that γ( t)γ(t) = e. Now by Lemma 1.10 we have [X, Y ] g = L Xg Y g = g=e d dt Dr γ( t) Y γ(t) = d dt Dl γ( t) Y γ(t) = d dt Dl γ( t) Dl γ(t) Y = d dt Dl γ( t)γ(t) Y = d dt Dl e Y = d dt Y = 0 φ

9 LIE GROUPS 9 To prove the second statement, notice that the multiplication map µ : G G G is a homomorphism if and only if G is abelian. Also, Dµ : g g g : (X, Y ) X + Y. The result now follows from the commutativity of the diagram (1.6). 2. Representations of Lie groups and Lie algebras 2.1. Definitions and examples The adjoint representation. We may think of one vector field acting on others. Namely, for X X(M) there is a linear map (2.1) ad : X(M) End X(M) : ad X (Y ) [X, Y ] satisfying the properties for any function f on M, ad fx = f ad X ad X (fy ) = f ad X (Y ) + X(f)Y 2.3. Semisimplicity and Schur s lemma Haar measure and Weyl s trick Characters Examples Peter-Weyl theorem. 3. Maximal tori 3.1. Existence Weyl group Cartan subalgebras. 4. Classification of complex semisimple Lie algebras 4.1. Roots Dynkin diagrams Stiefel diagrams Classification. 5. Weyl character formula 5.1. Weights and multiplicities Examples.

10 10 RICHARD A. WENTWORTH address:

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