A note on Lie algebra cohomology

Abstract

Given a finite dimensional Lie algebra $L$, let $I$ be the augmentation ideal in the universal enveloping algebra $U\left(L\right)$. We study the conditions on $L$ under which the $Ext$-groups $Ext\left(k,k\right)$ for the trivial $L$-module $k$ are the same when computed in the category of all $U\left(L\right)$-modules or in the category of $I$-torsion $U\left(L\right)$-modules. An application to cohomology of equivariant sheaves is given.