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 $\mathfrak{g}$ be a finite-dimensional semi-simple Lie algebra over a field $\mathbb{K}$ of characteristic $0$ .

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

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Thus, the only interesting cohomology groups with coefficients in an irreducible $\mathfrak{g}$ -module are $H^n(\mathfrak{g}, \mathbb{K})$ . For arbitrary $\mathfrak{g}$ -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 $\mathfrak{g}$ -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 $H^1(\g,\Hom MN{})$ . And from the vanishing of $H^2$ , we obtain Levi's theorem, which states 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 $M$ .