Homework Set #7

1. Explicitly compute the restriction of the Killing form on sl(n) to the diagonal.
2. Show that if L is semisimple and H is a maximal toral subalgebra, then H is self-normalizing.
3. Show that if V₁ and V₂ are finite dimensional representations of sl(2) which are isomorphic as representations of H, then they are isomorphic as representations of sl(2). You may assume that every finite dimensional representation of sl(2) is semisimple as sl(2)-representation and as H-representation.
4. Show that if V is a finite-dimensional representation of sl(2) and h acts nilpotently on V, then all of sl(2) acts trivially on V. You may not assume that h acts semisimply on V.