Cohomology of perfect Lie algebras

TL;DR

This paper investigates the adjoint cohomology of perfect Lie algebras over complex numbers, providing explicit results for a family involving $ ext{sl}_2( ext{C})$ and classifying small-dimensional cases.

Contribution

It offers explicit cohomology computations for a family of perfect Lie algebras and classifies all perfect Lie algebras up to dimension 9.

Findings

Explicit cohomology formulas for $ ext{sl}_2( ext{C}) times V_m$

Classification of perfect Lie algebras of dimension ≤ 9

Cohomology computations for small-dimensional cases

Abstract

We study the adjoint cohomology of perfect Lie algebras over the complex numbers. For the family of perfect Lie algebras $\mathfrak{g}=\mathfrak{sl}_2(\Bbb C)\ltimes V_m$ we obtain some explicit results for $H^k(\mathfrak{g},\mathfrak{g})$ with $k\ge 0$. Here $V_m$ is the irreducible representation of $\mathfrak{sl}_2(\Bbb C)$ 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 $\mathfrak{sl}_2(\Bbb C)\ltimes V_m$, 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.