Math 319 Problem Set 12: Lie algebras Lie Groups 1. De¯ne the Lie bracket of two matrices A and B in M(n; R) by [A; B] = AB ¡ BA: In this problem you will verify that this bracket operation has the properties required to make M(n; R) a Lie algebra. a. The bracket is anti-symmetric: [A; B] = ¡[B; A] and [A; A] = 0n for all A; B 2 M(n; R). b. The bracket is bilinear: [A; B + C] = [A; B] + [A; C] [A + B; C] = [A; C] + [B; C] [rA; B] = r[A; B] = [A; rB] for all A; B; C 2 M(n; R) and all r 2 R. c. The bracket satis¯es the Jacobi identity: [A; [B; C]] + [B; [C; A]] + [C; [A; B]] = 0n for all A; B; C 2 M(n; R). d. Give a speci¯c example for n = 2 to show that the Lie bracket is not associative. 2. In this problem you will show that the Lie bracket de¯ned in (1) makes the tangent space of G = L(2; R) a Lie algebra. Recall from Problem Set 10 that ½· ¸ ¾ 0 b L(G) = : b 2 R b 0 You already know that L(G) is a vector space and, from (1), that the bracket has the necessary properties. So all you need to show is that L(G) is closed under the bracket operation. 3. In this problem you will show that the Lie bracket de¯ned in (1) makes the tangent space of G = SL(2; R) a Lie algebra. Recall from Problem Set 10 that L(G) consists of the 2 by 2 trace zero matrices. ½· ¸ ¾ a b L(G) = : a; b; c 2 R c ¡a Show that L(G) is closed under the bracket operation. 4. Generalize problem (3) to G = SL(n; R) for n ¸ 2 by showing that L(G) = fB 2 M(n; R) : tr(B) = 0g is closed under the bracket in (1). (Use the facts about the trace on the Linear Algebra Review sheet.) 1 T 5. Let G = O(n; R), the Euclidean case, so L(G) = fB 2 M(n; R): B + B = 0ng. Show L(G) is a Lie algebra by showing it is closed under the bracket in (1). 6. Let G = O(n; R) = fM 2 GL(n; R): M T CM = Cg, the general case of the orthogonal groups, so T L(G) = fB 2 M(n; R): B C + CB = 0ng: Show L(G) is a Lie algebra by showing it is closed under the bracket in (1). 7. Let L = R3. In this problem you will show that L is a Lie algebra via the bracket operation given by the vector cross product: [v; w] = v £ w for v; w 2 L = R3 by verifying properties (a)-(c) of problem (1). Assume facts about the vector cross product from linear algebra, including u £ (v £ w) = (u ¢ w)v ¡ (u ¢ v)w: 8. In this problem you will look at an example of a Lie algebra homomorphism. Let G be the Euclidean orthogonal group O(3; R), so L(G) consists of all the 3 by 3 skew-symmetric matrices. Let L = R3 be the Lie algebra of problem (7). De¯ne T : L ! L(G) by 2 3 0 ¡x ¡y T (v) = T (x; y; z) = 4 x 0 ¡z 5 y z 0 where v = (x; y; z) 2 L = R3. (You'll need to work with coordinates.) a. Show that T is a linear transformation. b. Show that T is one-to-one and onto. c. Show that T preserves the Lie bracket. That is, show [T (v);T (w)] = T ([v; w]): for any v; w 2 L = R3. Remember that the bracket [v; w] is given by the vector cross product, while the bracket [T (v);T (w)] is the usual bracket operation for matrices. 9. For a ¯xed M 2 GL(n; R), recall that Ad(M): M(n; R) !M(n; R) is de¯ned by Ad(M)(X) = MXM ¡1 for X 2 M(n; R). In parts (a) and (b) of this problem, you'll show that Ad(M) is a Lie algebra homomorphism | that is, Ad(M) is a linear transformation and preserves the bracket operation. a. Show that Ad(M) is a linear transformation. b. Show that Ad(M) preserves the bracket on M(n; R). c. Show that Ad(M) is invertible. (Hint: What would \un-do" Ad(M)?) 2 10. Since M(n; R) is essentially Rn2 , parts (a) and (c) of problem 9 show that Ad: GL(n; R) ! GL(n2; R). Show that the function Ad is a group homomorphism: Ad(M1M2) = Ad(M1) ± Ad(M2) for all M1;M2 2 GL(n; R). Note that we're thinking of elements of GL(n; R) as matrices, so the binary operation on the left is matrix mul- tiplication, but we're thinking of the elements of GL(n2; R) as linear transformations, so the binary operation on the right is composition of functions. (Hint: You're to verify the equality of two functions, so you need to show that both functions have the same e®ect on X 2 M(n; R). ) 11. Use the formula ad(A)(X) = AX ¡ XA for all X 2 M(n; R) to show that the function ad preserves the Lie bracket. That is, show ad([A; B]) = [ad(A); ad(B)] for any A; B 2 M(n; R). (Use the hint in (10).) 12. In this problem you will take a careful look at the Lie algebra of G = SL(2; R), using the following basis for L(G): · ¸ · ¸ · ¸ 0 1 1 0 0 0 E = H = F = : 0 0 0 ¡1 1 0 a. Find [E,H], [F,H] and [E,F] (where the bracket operation is the one de¯ned in (1)). b. Assume ad(A)(X) = AX ¡ XA, as in problem (11). Using the basis fE; H; F g (in that order), ¯nd the matrix of E = ad(E). (The matrix will be 3 by 3.) c. Find the matrix of H = ad(H) and the matrix of F = ad(F ), as in (b). d. Find exp(E), exp(H) and exp(F). e. Find ME = exp(E), MH = exp(H), and MF = exp(F ). Because tr(MAM ¡1) = tr(M), Ad(M): L(G) ! L(G). f. Find the matrix of Ad(ME) with respect to the basis E, H, F of L(G). g. Find the matrices of Ad(MH ) and Ad(MF ) with respect to the basis E, H, F of L(G). Pattern alert: Did you ¯nd Ad(exp(X)) = exp(ad(X)) in (d){ (g)? 13. This is a di®erent approach to Proposition 4.1.3(a) in Part 4. (If we de¯ne ad as the di®erential of Ad, then we already know ad is a linear transformation.) Assume G is a Lie group and L(G) is its tangent space. Assume a bracket operation has been de¯ned on L(G) (so L(G) is closed under the bracket operation) satisfying the properties (a)-(c) in (1). For a ¯xed A 2 L(G) de¯ne ad(A): L(G) ! L(G) by ad(A)(B) = [A; B]: Use this de¯nition in terms of the bracket and the properties of the bracket in (1) to show ad(A) is a linear transformation. 3 14*. In this problem you'll use the de¯nition of the di®erential of a Lie group homo- morphism to deduce the formula for ad given in problem (11). Let G1 = GL(n; R) 2 2 and G2 = GL(n ; R). Start with Ad: G1 = GL(n; R) ! G2 = GL(n ; R) as in problem 9. Choose A 2 L(G1) = M(n; R). We know that γ1(t) = exp(tA) is a 0 smooth curve in G1 = GL(n; R) passing through the identity In with γ1(0) = A. Let 0 γ2(t) = Ad(γ1(t)). By the de¯nition of the di®erential, ad(A) = γ2(0). That is, for B 2 M(n; R), γ2(t)(B) = Ad(γ1(t))(B): Di®erentiate both sides of the equation above and then evaluate at t = 0. This calculation should show that ad(A)(B) = AB ¡ BA. 15*. Let G be a subgroup of GL(n; R) and let L(G) be the Lie algebra G. For X 2 L(G), let X = (Xij) be the matrix of the linear transformation ad(X): L(G) ! L(G) with respect to some basis of L(G). Assume that X is upper triangular, that is that Xij = 0 for i · j. Show that det(exp(X )) = 1. (Hint: use a theorem from Part 3.) 4.