Homework Set #9

1. Prove that if L is a simple Lie algebra, then dim Hom_L(L,L) = 1.
2. Assuming Problem 1, prove that the Casimir elements associated to different representations φ_i of a simple Lie algebra L are all proportional to one another.
3. Let L be a Lie algebra and V a finite-dimensional L-representation. Prove that V is a semisimple representation if and only if every L-subrepresentation of V has a complement.
4. Let L be a semisimple Lie algebra and K a semisimple Lie subalgebra of L. Prove that the Jordan decomposition of every x∈K in K is the same as its decomposition in L.