A semisimple Lie algebra {\mathfrak{g}} is one that has no nonzero abelian ideals. This is equivalent to the absence of solvable ideals. Indeed, if {\mathfrak{g}} had a solvable ideal {\mathfrak{z}}, we could consider the derived series of {\mathfrak{z}}, i.e. {D^1\mathfrak{z} = [\mathfrak{z},\mathfrak{z}], D^n\mathfrak{z} = [D^{n-1}\mathfrak{z}, D^{n-1}\mathfrak{z}]}. These are ideals because, by the Jacobi identity, the Lie product of two ideals is an ideal. These {D^n \mathfrak{z}} eventually become zero by the hypothesis of solvability, and the last nonzero one is abelian.

One justification for the epithet “semisimple” is that the category of finite-dimensional representations is in fact semisimple, i.e. that any exact sequence of {\mathfrak{g}} representations for {\mathfrak{g}} semisimple

\displaystyle 0 \rightarrow V' \rightarrow V \rightarrow V'' \rightarrow 0

splits. This is what happens for finite groups, because by Maschke’s theorem the group algebra of a finite group is semisimple. Nevertheless, the enveloping algebra {U\mathfrak{g}} is not generally semisimple; we are restricting ourselves to finite-dimensional {U \mathfrak{g}}-modules.

Cartan’s criterion

Before getting there, we will prove a basic criterion for semisimplicity.

 

Theorem 1 (Cartan) The Lie algebra {\mathfrak{g}} is semisimple if and only if its Killing form is nondegenerate.

 

The proof turns out to be a relatively easy consequence of Cartan’s criterion for solvability, which I’ve already given a two–post spiel on. (more…)