Homework 10. (Due May 1). 

\otimes=tensor product sign
\oplus =direct sum sign

Let L be the Lie algebra over complex numbers with basis e,f,h and bracket
such that [e,f]=h, [h,e]=2e, [h,f]=-2f. For n in {0,1,2,...} let V^n be a 
simple L-module of dimension n+1. For n,m in {0,1,2,...} we regard  
V^n\otimes V^m as an L-module in the usual way.

(a) Show that the vector space {w\in V^n\otimes V^m| ew=0} has a basis
{w_a|a\in A} where A={n+m,n+m-2,n+m-4,...,n+m-2\min(n,m)} and hw_a=aw_a for all a. 

(b) Deduce that the L-module V^n\otimes V^m is isomorphic to the L-module 
\oplus_{a\in A}V^a. (Hint: apply Weyl's theorem to the semisimple Lie algebra
L and to its module V^n\otimes V^m.)