Lie groups and Lie algebras: Questions 3

1. sl(2,C)

(Depends on the video where we worked out a similar example).

Let C 2 be the standard representation of little s l 2 C and let X be the matrix 0, 1; 0, 0 and Y be the matrix 0, 0; 1, 0 be the usual matrices. Calculate the action of Sym 3 X and Sym 3 Y on Sym 3 C 3 explicitly.

2. Exterior powers

(Depends on the video about symmetric powers.)

Given a representation R from G to G L V and an integer n, define the alternating map Alt from (V to the nth tensor power) to (V to the nth tensor power) by Alt of v_1 tensor dot dot dot tensor v_n equals one over n factorial times the sum over permutations sigma of the sign of sigma times v_(sigma of 1) tensor dot dot dot tensor v_(sigma of n) where the sign of sigma is the sign of the permutation (1 for even permutations, minus 1 for odd permutations).

• Show that Alt is a morphism of representations. Its image is called the nth exterior power of V, written Lambda n of V. This is a subrepresentation of V to the nth tensor power.

• Show that if v_i = v_j for some i not equal to j then Alt of v_1 tensor dot dot dot tensor v_n equals 0.

3. More on exterior powers

(Depends on previous question).

• Define v wedge w to be v tensor w minus w tensor v, which equals 2 times Alt v tensor w and more generally v_1 wedge dot dot dot wedge v_n equals n factorial times Alt of v_1 tensor dot dot dot tensor v_n. With this notation:

  1. Let V equals C 2. Write down a basis for Lambda 2 of V and show that V tensor square equals Sym 2 V direct sum Lambda 2 V.

  2. Let W be C 3. Write down a basis for Lambda 2 W. Is it true that W tensor squared equals Sym 2 W direct sum Lambda 2 W? (Hint: what are the dimensions of these vector spaces?)

  3. Let U be C n. Write down a basis of Lambda n U.

• Given integers k and n, what is the dimension of Lambda k C n?

4. Even more on exterior powers

(Depends on previous question).

Let e_1 up to e_n be a basis for C n. If v_1 equals sum of c_{1,i}e_i, v_2 equals sum of c_{2,i}e_i, dot dot dot, v_n equals sum of c_{n,i}e_i, show that v_1 wedge dot dot dot wedge v_n equals det of C times e_1 wedge dot dot dot wedge e_n, where C is the matrix with entries c_{i, j}. (Hint: Look at the formula for the determinant in index notation.)

5. You're probably sick of them by now

(Depends on previous question).

Let C 2 be the standard representation of SU(2). Find the weight space decomposition of Lambda 2 of Sym 2 of C 2 and hence show that Lambda 2 of Sym 2 of C 2 is isomorphic to Sym 2 of C 2.

6. Pure tensors

(Depends on the video where we introduced tensor powers).

Let V and W be vector spaces. By tensor, I mean an element of V tensor W. A tensor of the form little v tensor little w is called a pure tensor.

Let V and W be C 2 and let e_1, e_2 be a basis.

• Show that e_1 tensor e_1 + e_2 tensor e_1 minus e_1 tensor e_2 minus e_2 tensor e_2 is a pure tensor.

• If v = ae_1 + be_2 and w = ce_1+de_2, write out v tensor w.

• Using the previous part of the question, show that if P e_1 tensor e_1 + Q e_1 tensor e_2 + R e_2 tensor e_1 + S e_2 tensor e_2 is a pure tensor then P S = Q R.

• Write down a tensor which is not pure and justify your answer.

7. Morphisms, part 1

(Depends on the video in which we defined morphisms of representations.)

Suppose that R from G to G L V and S from G to G L W are representations of a group G and that L from V to W is a morphisms of representations from R to S.

• Show that the kernel of L is a subrepresentation of V.

• Assuming there exists an invariant Hermitian inner product on V, and assuming that L is surjective, show that V is isomorphic to ker L direct sum W.

• The adjoint representation of a matrix group on its Lie algebra is the representation big Ad from G to G L of little g defined by Ad of g applied to X equals g X g inverse. Show that the trace map trace from little g to R is a morphism from the adjoint representation to the trivial 1-dimensional representation. Give an example of a group G for which this morphism is zero and an example of a group G for which this morphism is surjective.

8. Morphisms, part 2

(Depends on previous question and the video where we defined symmetric powers.)

• Recall that v_1 times v_2 times dot dot dot times v_p is shorthand for the averaging map applied to v_1 tensor v_2 tensor dot dot dot tensor v_p. Let R from G to G L V be a representation. Show that the map F from Sym p V tensor Sym q V to Sym (p + q) V defined on pure tensors by F of (v_1 v_2 up to v_p tensor w_1 w_2 up to w_q) equals v_1 v_2 up to v_p times w_1 w_2 up to w_q, is a morphism of representations.

• Suppose that V equals C 2 is the standard representation of SU(2), that p = 1 and q = 2. Write out explicitly the values of F from C 2 tensor Sym 2 C 2 to Sym 3 C 2 applied to a basis (e.g. elements like e_1 tensor e_1 e_2).

• Hence find the kernel of F.

• Deduce that C 2 tensor Sym 2 C 2 is isomorphic to C 2 direct sum Sym 3 C 3. (Hint: (a) Find the weight space decomposition of ker F. (b) Use the previous question. You may assume the existence of an invariant Hermitian inner product.)

9. Even more about the Heisenberg group

(Builds on Questions 4 and 11 on Sheet 2, but we recap the necessary ideas below).

Recall that H_3 is the group of 3-by-3 matrices 1, a, c; 0, 1, b; 0, 0, 1 and that N_3 is the subgroup of matrices 1, 0, c; 0, 1, 0; 0, 0, 1 with c in R. We saw last time that:

1. elements of N_3 commute with everything in H_3,

2. if R from H_3 to G L V is a representation then det of R of g equals 1 for all g in N_3.

Let G_3 inside N_3 be the subgroup of matrices for which c is an integer.

• Prove that G_3 is normal in H_3 (so that the quotient H_3 over G_3 is a well-defined group).

• Why is N_3 over G_3 isomorphic to U(1)?

• If R from H_3 over G_3 to G L V is a representation, let V equals the direct sum of subspaces W_m be the decomposition of V into weight spaces for the action of N_3 over G_3. Prove that for any g in H_3 over G_3 and any v in W_m, g v is also in W_m (so that W_m is a subrepresentation of V). (Hint: Use the fact that elements of N_3 commute with everything in H_3. If you get stuck, take a look at the proof of the lemma in this future video: it uses the same idea you will need).

• Show that the only nonzero weight space is W_0. (Hint: If g is in N_3, what is the determinant of R of g restricted to W_m?)

• Deduce that H_3 over G_3 is not isomorphic to a subgroup of G L n C for any n.

Even though it's not a matrix group, H_3 over G_3 is a perfectly nice Lie group: it has local coordinates given by the three matrix entries.

10. Fourier theory again

(Depends on Question 12 on Sheet 2 and on the video about representations of U(1).)

As before, let V be the (infinite-dimensional) vector space of complex-valued 2 pi-periodic functions f of theta (such that f of theta plus 2 pi equals f of theta) and define the homomorphism R from U(1) to G L V by setting R of e to the i phi to be the linear map which takes a function f of theta to f of (theta plus phi).

Suppose that there is a function lambda of phi such that f is a lambda of phi-eigenvector of R of e to the i phi for all phi. How do we know that lambda of phi equals e to the i n phi for some n in Z?

11. Casimir, part 1

(Depends only on the video about SL(2,\CC).)

Let X, Y, H be the usual elements of little s l 2 C satisfying the commutation relations H bracket X equals 2 X, H bracket Y equals minus 2 Y Show that if f from little s l 2 C to little g l V is a (complex linear) representation then the matrix C, defined to be f of X times f of Y plus f of Y times f of X plus a half f of H squared commutes with f of M for every M in little s l 2 C. (Hint: Check it for M = X, Y, H. Is that enough?)

WARNING! Since f is a representation of Lie algebras, we know that e.g. f of H equals f of X bracket Y equals f(X) f(Y) minus f(Y) f(X). We do not know that f of X Y equals f(X) times f(Y) (and usually this latter equation is not true, so don't use it!).

There is an in-depth project which investigates this phenomenon in more detail.

12. Casimir, part 2

(Depends on the previous question and the formula from the proof of the classification of irreducible SU(2)-representations)

In the notation from the previous question, suppose that V lambda inside V is an eigenspace of C with eigenvalue lambda. Show that V lambda is a subrepresentation. If V is irreducible and has highest weight m, show that C is the diagonal matrix (m plus a half m squared) times the identity. (Hint: evaluate C on the highest weight vector).