Homework 10. (Due Nov.26). \otimes=tensor product sign \oplus =direct sum sign 1) 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.) 2) Let F_n be a family of vector spaces indexed by integers n. Let F be the direct sum of all F_n. Let F^+ be the direct sum of all F_n,n>0. Let F^- be the direct sum of all F_n,n<0. Assume that we are given a Lie algebra structure on F such that [F_n,F_m] is contained in F_{n+m} for any n,m. Assume also that F is generated as a Lie algebra by F_{-1},F_0,F_1. Note that F_0,F^+,F^- are Lie subalgebras of F. Let Z be a subspace of F^- such that [F_1,Z]=0, [F_0,Z] contained in Z. Let J be the ideal of F^- generated by Z that is the subspace spanned by [...[[z,x_1],x_2]...,x_s] for various z\in Z and x_1,x_2,...x_s in F_{n_1},...,F_{n_s} (with n_1,n_2,....,n_s<0). Show that J^- is an ideal of F.