On the centre of crossed modules of Lie algebras

TL;DR

This paper explores the structure of crossed modules of Lie algebras, revealing their connection to cohomology and homotopy Lie algebras, and establishing an exact sequence involving their centres.

Contribution

It establishes a relationship between crossed modules of Lie algebras and their centres, including an exact sequence involving cohomology and homotopy Lie algebras.

Findings

Crossed modules of Lie algebras fit into an exact sequence.

The relationship involves cohomology of homotopy Lie algebras.

Provides new insights into the structure of Lie algebra crossed modules.

Abstract

This paper studies the relationship between crossed modules of Lie algebras and their centres. We show that any crossed module \(\partial : L_1\to L_0\) of Lie algebras fits in an exact sequence involving cohomology of the homotopy Lie algebras \(\pi_0(L_*)\) and \(\pi_1(L_*)\).