Problem set 5. Due Friday, November 13.
Mathematics 534, Term 1, 2015. Instructor: Reichstein.
All Lie algebras and all vector spaces in the problems below are assumed to be finitedimensional and defined over the field C of the complex numbers.
1. Let λ1 , . . . , λn be complex numbers. Show from first principles that if λi1 +· · ·+λin = 0
for i = 1, . . . , n, then λ1 = · · · = λn = 0.
2. Let L be a Lie algebra and R be the radical of L. Show that [L, R] = [L, L] ∩ R.
Hint: Use Levi’s Theorem.
3. Let L be a Lie algebra of dimension ≥ 2. Assume that L is not simple. Show that
there exists an ideal (0) ( I ( L such that the exact sequence
π
0 −→ I −→ L −−→ L/I −→ (0)
splits. That is, there exists a subalgebra L0 of L such that π restricts to an isomorphism
between L0 and L/I.
4. Let V be an sl2 -module. Show that V is self-dual. That is, show that V and V ∗ are
isomorphic as sl2 -modules.
5. Let V (m) be the unique irreducible sl2 -module of dimension m + 1. Decompose
V (m) ⊗ V (n) as a direct sum of irreducible sl2 -modules.