18.747: Problem set 1; due Thursday, February 16

1. a) Let g be a Lie algebra, and g = h ⊕ ⨁_{α∈h^{∗}} g_{α} be a decomposition, such that

h is an abelian subalgebra, the commutator action of h on g_{α} is given by α and g_{0} = h. Show that every derivation of g is a sum of an inner derivation and a derivation preserving the decomposition. [See Kac, exercise 1.9 for a hint].

b) Let A be a square matrix with no zero rows. Let c ⊂ h ⊂ g(A) be the center (a.k.a. the kernel of A). Deduce from (a) that every derivation of g_{ext}(A)/c is inner.

2. Let Lg = g[t, t^{−1}] be the loop algebra of a simple Lie algebra g. Let z be a nonzero complex number and V be a finite dimensional representation of g. Define a representation V (z) of Lg by a(n)|_{V} = z^{n}a|_{V}, where a(n) = at^{n}. This is called an evaluation representation.

(a) Consider the representation V_{1}(z_{1}) ⊗ ... ⊗ V_{n}(z_{n}), where V_{i} are irreducible nontrivial representations of g. Show that this representation is irreducible if and only if z_{i}. are distinct.

(b) When are two such irreducible representations isomorphic? Hint. Consider the eigenvalues of h(n).

3. (harder) Show that any irreducible finite dimensional representation of Lg has the form as in problem 2.

4. Can you give an example of a reducible but not decomposable finite dimensional representation of Lg? Hint. Consider Lg/(t − 1)^{2}Lg.