Cohomology of perfect Lie algebras

Abstract

We study the adjoint cohomology of perfect Lie algebras over the complex numbers. For the family of perfect Lie algebras $\mathfrak{g}={\mathfrak{sl}}_2(\mathbb{C})⋉V_m$ we obtain some explicit results for $H^k(\mathfrak{g},\mathfrak{g})$ with $k\ge 0$. Here $V_m$ is the irreducible representation of ${\mathfrak{sl}}_2(\mathbb{C})$ of dimension $m+1$. For the computation of the cohomology we use the Hochschild-Serre formula, a long exact sequence in the cohomology and explicit formulas for the multiplicities of $V_k$ in the exterior product ${\Lambda }^j(V_m)$ for $j\le 4$. In general we cannot determine the total adjoint cohomology for ${\mathfrak{sl}}_2(\mathbb{C})⋉V_m$, but for some small $m$ this is possible. We also give a classification of complex perfect Lie algebras $\mathfrak{g}$ of dimension $n\le 9$ and explicitly compute the cohomology spaces $H^k(\mathfrak{g},\mathfrak{g})$ with $k=0,1,2$ for all Lie algebras from the classification list.