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 $0$ .

$\\,$

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 $n\\in\\mathbb{N}$ .

$\\,$

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.

$\\,$

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 $M$ .

单词	CohomologyOfSemisimpleLieAlgebras
释义	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 $0$ . $\\,$ 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 $n\\in\\mathbb{N}$ . $\\,$ 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. $\\,$ 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 $M$ .
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