Exercise 3, § 7

Let $g$ be a Lie algebra and $U$ its enveloping algebra.

(a) Show that for all $u \in U$ ($u \neq 0$) there exists a finite-dimensional representation $\pi$ of $U$ such that $\pi(u) \neq 0$. (Use Exercise 2 and Exercise 5 (b) of § 3.)

(b) Deduce from (a) that if $g$ is semi-simple there exists an infinity of inequivalent finite-dimensional simple representations of $g$. (Let $\rho_1, \ldots, \rho_n$ be finite-dimensional simple representations of $g$, $N_1, \ldots, N_n$ the annihilators of the corresponding U-modules and $N = \bigcap_{i=1}^n N_i$. Show that $N$ is of finite co-dimension in U. Let $n \in \mathbf{N}$, $n \neq 0$. Apply (a) to $n$.)