Exercise 4, § 1

Let $\mathfrak{g}$ be a Lie algebra. $A\mathfrak{g}$-module E is said to be locally finite if it is a union of finite dimensional $\mathfrak{g}$-submodules; equivalently, every $\mathfrak{g}$-submodule of E of finite type (as a $U(\mathfrak{g}$)-module) is finite dimensional.

a) Let $0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$ be an exact sequence of $\mathfrak{g}$-modules. Show that, if $E'$ and $E''$ are locally finite, so is E (reduce to the case in which E is of finite type, and use the fact that $U(\mathfrak{g})$ is noetherian, cf. Chap. I, §2, no. 6).

b) Assume that $\mathfrak{g}$ is semi-simple. Show that E is locally finite if and only if E is a direct sum of finite dimensional simple $\mathfrak{g}$-modules.

c) Assume that $\mathfrak{g}=\mathfrak{s}$. Show that E is locally finite if and only if the following conditions are satisfied:

$c_1) E$ has a basis consisting of eigenvectors of H,

$c_2)$ the endomorphisms $X_{+E}$ and $X_{-E}$ are locally nilpotent.

(Begin by showing that, if E satisfies $c_1$) and $c_2)$, and is not reduced to 0, it contains a primitive element $e$ of integer weight $m\geq 0$; prove next that $X^{m+1}e= 0$, and hence that E contains $V(m)$. Conclude by applying a) to the largest locally finite submodule $E'$ of E.)