The purpose of this post is to show that the category of finite-dimensional representations of a semismple Lie algebra is  a semisimple category; there is thus an analogy with Maschke’s theorem, except in this case the proofs are more complicated.  They can be simplified somewhat if one uses the cohomology of Lie algebras (i.e., appropriate Ext groups), which I may talk more about, but most likely only later.  Here we will give the proofs based on linear algebra.

The first step is to construct certain central elements in the enveloping algebra.

Casimir elements

Let ${B}$ be a nondegenerate invariant bilinear form on the Lie algebra ${\mathfrak{g}}$. (E.g. ${\mathfrak{g}}$ could be semisimple and ${B}$ the Killing form.) Given a basis ${e_i \in \mathfrak{g}}$, we can consider the dual basis ${f_j}$ with respect to it, i.e. such that ${B(e_i, f_j) = \delta_{ij}}$. Consider the Casimir element

$\displaystyle C := \sum e_i f_i \in U \mathfrak{g}.$

I claim that ${C}$ is independent of the choice of ${e_i}$ and is in the center of the enveloping algebra. First off, consider the isomorphisms of ${\mathfrak{g}}$-modules,

$\displaystyle \hom_k( \mathfrak{g}, \mathfrak{g}) \simeq \mathfrak{g} \otimes \mathfrak{g}^{\vee} \simeq \mathfrak{g} \otimes \mathfrak{g} .$

The last one is given by the form ${B}$. (more…)