18.747: Problem set 5; due Thursday, March 16

1. Show that W is a simple Lie algebra (i.e. it has no two-sided ideals other than 0, W). Deduce from this that W has no nontrivial finite dimensional modules (i.e. the action of W in any such module is zero).

2. For 𝔤 = sl(3) compute the index of \frac{1}{2}Qˇ in Λ. Compute the diagonal matrix exp(2πi · 2ρ_{0}). Describe explicitly which dominant weights in Λ have the form s(ρ+ν), ν ∈ \frac{1}{2}Qˇ .

3. Check that every central extension of a finite dimensional simple Lie algebra is trivial.

4. Show that if an irreducible module L(λ) over a symmetrizable Kac-Moody Lie algebra is unitarizable, then λ is a positive integral weight.