Some properties of the Schur multiplier and stem covers of Leibniz crossed modules

TL;DR

This paper explores the properties of the Schur multiplier and stem covers in Leibniz crossed modules, establishing exact sequences, existence, structure, and connections with Lie crossed modules to deepen understanding of their algebraic properties.

Contribution

It introduces a six-term exact sequence for Leibniz crossed modules and characterizes the structure and existence of stem covers, linking Leibniz and Lie crossed modules.

Findings

Established a six-term exact sequence for Leibniz crossed modules.

Proved the existence and characterized the structure of stem covers.

Connected stem covers of Leibniz and Lie crossed modules.

Abstract

In this article we investigate the interplay between stem covers, the Schur multiplier of Leibniz crossed modules and the non-abelian exterior product of Leibniz algebras. In concrete, we obtain a six-term exact sequence associated to a central extension of Leibniz crossed modules, which is useful to characterize stem covers. We show the existence of stem covers and determine the structure of all stem covers of Leibniz crossed modules. Also, we give the connection between the stem cover of a Lie crossed module in the categories of Lie and Leibniz crossed modules respectively.