cohomology of semisimple Lie algebras
There are some important facts that make the cohomology of semisimple Lie algebras easier to deal with than general Lie algebra cohomology. In particular, there are a number of vanishing theorems.
First of all, let $\U0001d524$ be a finitedimensional semisimple Lie algebra over a field $\mathbb{K}$ of characteristic $0$.
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Theorem [Whitehead]  Let $M$ be an irreducible^{} $\U0001d524$module (http://planetmath.org/RepresentationLieAlgebra) of dimension^{} greater than $1$. Then all the cohomology groups^{} with coefficients in $M$ are trivial, i.e. ${H}^{n}(\U0001d524,M)=0$ for all $n\in \mathbb{N}$.
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Thus, the only interesting cohomology groups with coefficients in an irreducible $\U0001d524$module are ${H}^{n}(\U0001d524,\mathbb{K})$. For arbitrary $\U0001d524$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 finitedimensional $\U0001d524$module. Then

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First Lemma : ${H}^{1}(\U0001d524,M)=0$.

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Second Lemma : ${H}^{2}(\U0001d524,M)=0$.
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 semisimple Lie algebras are completely reducible, since extensions of $M$ by $N$ are classified by ${H}^{1}(\U0001d524,\mathrm{Hom}(M,N))$. And from the vanishing of ${H}^{2}$, we obtain Levi’s theorem, which that every Lie algebra^{} is a split extension of a semisimple algebra by a solvable^{} algebra^{} since ${H}^{2}(\U0001d524,M)$ classifies extensions of $\U0001d524$ by $M$ with a specified action of $\U0001d524$ on $M$.
Title  cohomology of semisimple Lie algebras 

Canonical name  CohomologyOfSemisimpleLieAlgebras 
Date of creation  20130322 13:51:13 
Last modified on  20130322 13:51:13 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  8 
Author  asteroid (17536) 
Entry type  Feature 
Classification  msc 17B20 
Classification  msc 17B56 
Defines  Whitehead’s lemmas 
Defines  Whitehead’s first lemma 
Defines  Whitehead’s second lemma 