Homework 9. (Due April 24)

1) Let L' be a Lie subalgebra of a finite dimensional Lie algebra L over k. 

(i) Using the PBW theorem show that the enveloping algebra U(L') is naturally a subalgebra 
of the enveloping algebra U(L). 

(ii) Let c:U(L')\to k be a homomorphism of (associative) k-algebras such that c(1)=1. Using 
the PBW theorem show that the vector space J spanned by
{uu'-c(u')u; u\in U(L),u'\in U(L')} does not contain 1.
Hence U(L)/J is nonzero. Show that U(L)/J has a natural structure of L-module. 

2) Let L be a finite dimensional Lie algebra over k. Let U(L) be its enveloping algebra.
Let x,y be nonzero elements of U(L). Using the PBW theorem show that xy is nonzero in U(L).