cohomology of semi-simple Lie algebras
Theorem [Whitehead] - Let be an irreducible -module (http://planetmath.org/RepresentationLieAlgebra) of dimension greater than . Then all the cohomology groups with coefficients in are trivial, i.e. for all .
Thus, the only interesting cohomology groups with coefficients in an irreducible -module are . For arbitrary -modules there are still two vanishing results, which are usually called Whitehead’s lemmas.
Whitehead’s Lemmas - Let be a finite-dimensional -module. Then
First Lemma : .
Second Lemma : .
Whitehead’s lemmas lead to two very important results. From the vanishing of , we can derive Weyl’s theorem, the fact that representations of semi-simple Lie algebras are completely reducible, since extensions of by are classified by . And from the vanishing of , we obtain Levi’s theorem, which that every Lie algebra is a split extension of a semi-simple algebra by a solvable algebra since classifies extensions of by with a specified action of on .
|Title||cohomology of semi-simple Lie algebras|
|Date of creation||2013-03-22 13:51:13|
|Last modified on||2013-03-22 13:51:13|
|Last modified by||asteroid (17536)|
|Defines||Whitehead’s first lemma|
|Defines||Whitehead’s second lemma|