# 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 $\mathfrak{g}$ 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 (http://planetmath.org/RepresentationLieAlgebra) 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 : $H^{1}(\mathfrak{g},M)=0$.

• Second Lemma : $H^{2}(\mathfrak{g},M)=0$.

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 $H^{1}(\mathfrak{g},\mathrm{Hom}(M,N){})$. 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 $H^{2}(\mathfrak{g},M)$ classifies extensions of $\mathfrak{g}$ by $M$ with a specified action of $\mathfrak{g}$ on $M$.

Title cohomology of semi-simple Lie algebras CohomologyOfSemisimpleLieAlgebras 2013-03-22 13:51:13 2013-03-22 13:51:13 asteroid (17536) asteroid (17536) 8 asteroid (17536) Feature msc 17B20 msc 17B56 Whitehead’s lemmas Whitehead’s first lemma Whitehead’s second lemma