Crossed extensions of Lie algebras

TL;DR

This paper introduces crossed n-fold extensions of Lie algebras, generalizing the concept of extensions and linking them to higher Chevalley-Eilenberg cohomology groups, thus expanding the algebraic understanding of Lie algebra structures.

Contribution

It defines crossed n-fold extensions for Lie algebras and establishes their correspondence with higher Chevalley-Eilenberg cohomology groups, extending previous associative algebra results.

Findings

Crossed n-fold extensions are classified by H^{n+1}_{CE}(rak{g}, M).

The work generalizes Hochschild cohomology representations to Lie algebras.

Provides a new framework for understanding Lie algebra extensions.

Abstract

It is known that Hochschild cohomology groups are represented by crossed extensions of associative algebras. In this paper, we introduce crossed $n$-fold extensions of a Lie algebra $\mathfrak{g}$ by a module $M$, for $n \geq 2$. The equivalence classes of such extensions are represented by the $(n+1)$-th Chevalley-Eilenberg cohomology group $H^{n+1}_{CE} (\mathfrak{g}, M).$