cohomology of semi-simple Lie algebras

There are some important facts that make the cohomology of semi-simple Lie algebras easier to deal with than general Lie algebra cohomology. In particular, there are a number of vanishing theorems.

First of all, let $𝔤$ be a finite-dimensional semi-simple Lie algebra over a field $𝕂$ of characteristic $0$.

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Theorem [Whitehead] - Let $M$ be an irreducible $𝔤$-module (http://planetmath.org/RepresentationLieAlgebra) of dimension greater than $1$. Then all the cohomology groups with coefficients in $M$ are trivial, i.e. $H^n(𝔤,M)=0$ for all $n\in \mathbb{N}$.

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Thus, the only interesting cohomology groups with coefficients in an irreducible $𝔤$-module are $H^n(𝔤,𝕂)$. For arbitrary $𝔤$-modules there are still two vanishing results, which are usually called Whitehead’s lemmas.

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Whitehead’s Lemmas - Let $M$ be a finite-dimensional $𝔤$-module. Then

• First Lemma : $H^1(𝔤,M)=0$.

• Second Lemma : $H^2(𝔤,M)=0$.

Whitehead’s lemmas lead to two very important results. From the vanishing of $H^1$, we can derive Weyl’s theorem, the fact that representations of semi-simple Lie algebras are completely reducible, since extensions of $M$ by $N$ are classified by $H^1(𝔤,\mathrm{Hom}(M,N))$. And from the vanishing of $H^2$, we obtain Levi’s theorem, which that every Lie algebra is a split extension of a semi-simple algebra by a solvable algebra since $H^2(𝔤,M)$ classifies extensions of $𝔤$ by $M$ with a specified action of $𝔤$ on $M$.