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.