Homework Set #4

1. Show that the quotient of T(L) by the two-sided ideal generated by {x⊗y − y⊗x − [x y] | x,y∈L} is indeed the universal enveloping algebra of L.
2. Show that V→ Sym(V) is the universal commutative algebra admitting a linear map from V.
3. Let L denote the Lie algebra of upper triangular 3×3 matrices with zeroes along the diagonal. Let A denote the Weyl algebra of differential operators with polynomial coefficients, i.e., the subalgebra of all C-linear transformations C[x]→C[x] generated by multiplication by x and differentiation with respect to x. Prove that the Weyl algebra is a quotient algebra of U(L).
4. Find the dimension of the degree 4 graded piece of the free Lie algebra on a two-dimensional vector space V = Span(x,y).