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.

Theorem [Whitehead] - Let M be an irreduciblePlanetmathPlanetmath 𝔤-module ( of dimensionMathworldPlanetmath greater than 1. Then all the cohomology groupsPlanetmathPlanetmath with coefficients in M are trivial, i.e. Hn(𝔤,M)=0 for all n.

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

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

  • First Lemma : H1(𝔤,M)=0.

  • Second Lemma : H2(𝔤,M)=0.

Whitehead’s lemmas lead to two very important results. From the vanishing of H1, 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 H1(𝔤,Hom(M,N)). And from the vanishing of H2, we obtain Levi’s theorem, which that every Lie algebraMathworldPlanetmath is a split extension of a semi-simple algebra by a solvablePlanetmathPlanetmath algebraPlanetmathPlanetmathPlanetmath since H2(𝔤,M) classifies extensions of 𝔤 by M with a specified action of 𝔤 on M.

Title cohomology of semi-simple Lie algebras
Canonical name CohomologyOfSemisimpleLieAlgebras
Date of creation 2013-03-22 13:51:13
Last modified on 2013-03-22 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