On crossed modules of pre-Lie algebras

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.