On crossed modules of pre-Lie algebras

TL;DR

This paper extends Gerstenhaber's isomorphism relating third cohomology and crossed modules from Lie algebras to the broader context of pre-Lie algebras and their modules.

Contribution

It generalizes the known correspondence between third cohomology and crossed modules from Lie algebras to pre-Lie algebras.

Findings

Established an isomorphism between $H^3$ of pre-Lie algebras and crossed modules.

Extended Gerstenhaber's result to a new algebraic setting.

Provides a framework for understanding cohomology and modules in pre-Lie algebra theory.

Abstract

Given a Lie algebra $\mathfrak g$ and a $\mathfrak{g}$-module $V$, it is due to Gerstenhaber that there is an isomorphism between $H^3(\mathfrak{g}, V )$ and the group of equivalence classes of crossed modules with kernel $V$ and cokernel $\mathfrak {g}$. The goal of this article is to obtain this result for pre-Lie algebras and their modules.