Homework Set #2

1. Show that every term in the derived series of L is an ideal of L.
2. Show the equivalence of the two definitions of Lie algebra representation that I discussed in class: a Lie algebra homomorphism L→End(V) and an action L×V→V satisfying [x y].v = x.(y.v)- y.(x.v).
3. Find all ideals of L₁⊕L₂ where L₁ and L₂ are simple Lie algebras.
4. Prove that every non-abelian 2-dimensional Lie algebra is solvable but not nilpotent.
5. An ideal is said to be nilpotent if it is nilpotent as a Lie algebra. Prove that the sum of two nilpotent ideals is again a nilpotent ideal.

Optional Bonus Problem: Let A be a finite-dimensional complex associative algebra. Prove that there is a neighborhood U of 0 in A such that if x and y in U both commute with z=[x y], then exp(x)exp(y)exp(-x)exp(-y) = exp(z).

Revised September 5, 2007.