Complexe canonique d'une alg\`ebre de Lie r\'eductive

Abstract

Let ${\goth g}$ be a finite dimensional complex reductive Lie algebra and $\dv ..$ an invariant non degenerated bilinear form on ${\goth g}\times {\goth g}$ which extends the Killing form of $[{\goth g},{\goth g}]$. We define the homology complex $C_{\bullet}({\goth g})$. Its space is the algebra $\tk {{\Bbb C}}{\e Sg}\tk {{\Bbb C}}{\e Sg}\ex {}{{\goth g}}$ where $\e Sg$ and $\ex {}{{\goth g}}$ are the symmetric and exterior algebras of ${\goth g}$. The differential of $C_{\bullet}({\goth g})$ is the $\tk {{\Bbb C}}{\e Sg}\e Sg$-derivation which associates to the element $v$ of ${\goth g}$ the function $(x,y)\mapsto \dv v{[x,y]}$ on ${\goth g}\times {\goth g}$. Then the complex $C_{\bullet}({\goth g})$ has no homology in degree strictly bigger than $\rk {\goth g}$.