Math 128: Lecture 4. March 31, 2014

Transcription

1 Math 128: Lecture 4 March 31, 2014

2 Recall if g is a Lie algebra with modules M and N, then x g acts on m n M N by x(m n) = xm n + m xn. So (x) = x x Ug Ug.

3 Recall if g is a Lie algebra with modules M and N, then x g acts on m n M N by x(m n) = xm n + m xn. So (x) = x x Ug Ug. We also calculated for x, y g, So xy(m n) = x(ym n + m yn) = (xy)m n + xm yn + ym xn + m (xy)n. (xy) = (xy) 1 + x y + y x + 1 (xy)

4 Recall if g is a Lie algebra with modules M and N, then x g acts on m n M N by x(m n) = xm n + m xn. So (x) = x x Ug Ug. We also calculated for x, y g, So xy(m n) = x(ym n + m yn) = (xy)m n + xm yn + ym xn + m (xy)n. (xy) = (xy) 1 + x y + y x + 1 (xy) = (x x)(y y) = (x) (y).

5 Recall if g is a Lie algebra with modules M and N, then x g acts on m n M N by x(m n) = xm n + m xn. So (x) = x x Ug Ug. We also calculated for x, y g, So xy(m n) = x(ym n + m yn) = (xy)m n + xm yn + ym xn + m (xy)n. (xy) = (xy) 1 + x y + y x + 1 (xy) = (x x)(y y) = (x) (y). In general, if w = x 1 x 2... x l Ug with x i g, then (w) = (x 1 ) (x 2 ) (x l ).

6 Representations of sl 2 (C) Last time: Let M be a finite-dimensional simple sl 2 (C)-module. (1) h has at least one eigenvector v M, with hv = λv for some λ C.

7 Representations of sl 2 (C) Last time: Let M be a finite-dimensional simple sl 2 (C)-module. (1) h has at least one eigenvector v M, with hv = λv for some λ C. (2) Use hx = xh + [h, x] to show that x l v is also an eigenvector for h with weight λ + 2l for each l Z 0.

8 Representations of sl 2 (C) Last time: Let M be a finite-dimensional simple sl 2 (C)-module. (1) h has at least one eigenvector v M, with hv = λv for some λ C. (2) Use hx = xh + [h, x] to show that x l v is also an eigenvector for h with weight λ + 2l for each l Z 0. (3) Since the eigenvalues of h on the x l v s are distinct, the x l v s are distinct. Since M is finite dimensional, there must be a non-zero v + M with xv + = 0 and hv + = µv + for some µ C.

9 Representations of sl 2 (C) Last time: Let M be a finite-dimensional simple sl 2 (C)-module. (1) h has at least one eigenvector v M, with hv = λv for some λ C. (2) Use hx = xh + [h, x] to show that x l v is also an eigenvector for h with weight λ + 2l for each l Z 0. (3) Since the eigenvalues of h on the x l v s are distinct, the x l v s are distinct. Since M is finite dimensional, there must be a non-zero v + M with xv + = 0 and hv + = µv + for some µ C. (4) Similarly as with x, use hy = yh + [h, y] to show that y l v + is also an eigenvector for h with weight µ 2l for each l Z 0.

10 Representations of sl 2 (C) Last time: Let M be a finite-dimensional simple sl 2 (C)-module. (1) h has at least one eigenvector v M, with hv = λv for some λ C. (2) Use hx = xh + [h, x] to show that x l v is also an eigenvector for h with weight λ + 2l for each l Z 0. (3) Since the eigenvalues of h on the x l v s are distinct, the x l v s are distinct. Since M is finite dimensional, there must be a non-zero v + M with xv + = 0 and hv + = µv + for some µ C. (4) Similarly as with x, use hy = yh + [h, y] to show that y l v + is also an eigenvector for h with weight µ 2l for each l Z 0. (5) Again, since M is finite-dimensional, there must be some d Z 0 with y d v + 0 and y d+1 v + = 0.

11 Representations of sl 2 (C) Last time: Let M be a finite-dimensional simple sl 2 (C)-module. (1) h has at least one eigenvector v M, with hv = λv for some λ C. (2) Use hx = xh + [h, x] to show that x l v is also an eigenvector for h with weight λ + 2l for each l Z 0. (3) Since the eigenvalues of h on the x l v s are distinct, the x l v s are distinct. Since M is finite dimensional, there must be a non-zero v + M with xv + = 0 and hv + = µv + for some µ C. (4) Similarly as with x, use hy = yh + [h, y] to show that y l v + is also an eigenvector for h with weight µ 2l for each l Z 0. (5) Again, since M is finite-dimensional, there must be some d Z 0 with y d v + 0 and y d+1 v + = 0. So far: 0 x h h y y y h h y y v + yv + y 2 v + y d 1 v + y d v + y 0

12 In summary, the sl 2 -action is given by: h is a diagonal matrix with µ, µ 2, µ 4,..., µ 2d on the diagonal, y has 1 s on the sub-diagonal and zeros elsewhere, and x has the weights µ, 2µ 2, 3µ 6,..., d(µ (d 1)) on the super-diagonal. µ h = µ 2 µ 4... µ 2d y = µ x = 0 2µ 2 0 3µ 6... d(µ (d 1)) 0

13 Theorem The simple finite dimensional sl 2 modules L(d) are indexed by d Z 0 with basis {v +, yv +, y 2 v +,..., y d v + } and action h(y l v + ) = (d 2l)(y l v + ), x(y l v + ) = l(d + 1 l)(y l 1 v + ), y(y l v + ) = y l+1 v +, with y d+1 v + = 0. with xv + = 0 and

14 Identifying finite dimensional sl 2 -modules Fact: all f.d. sl 2 -modules are finite sums of simple modules (i.e. sums of L(d) s).

15 Identifying finite dimensional sl 2 -modules Fact: all f.d. sl 2 -modules are finite sums of simple modules (i.e. sums of L(d) s). Let M be a (not nec. simple) sl 2 -module.

16 Identifying finite dimensional sl 2 -modules Fact: all f.d. sl 2 -modules are finite sums of simple modules (i.e. sums of L(d) s). Let M be a (not nec. simple) sl 2 -module. As a Ch-module, M = µ C M µ where M µ = {m M hm = µm}, is the µ weight space. (Remember, weight = eigen... )