A natural extension of the universal enveloping algebra functor to crossed modules of Leibniz algebras

TL;DR

This paper extends the universal enveloping algebra functor to crossed modules of Leibniz algebras, establishing a natural generalization and an isomorphism with representation categories, addressing challenges due to the non-existence of actors.

Contribution

It introduces a universal enveloping crossed module construction for Leibniz algebras and relates it to representation categories, overcoming the lack of actors in Leibniz crossed modules.

Findings

Constructed the universal enveloping crossed module of Leibniz algebras.

Proved an isomorphism between Leibniz crossed module representations and modules over its enveloping algebra.

Connected the framework to Lie crossed modules via the Loday-Pirashvili category.

Abstract

The universal enveloping algebra functor between Leibniz and associative algebras defined by Loday and Pirashvili is extended to crossed modules. We prove that the universal enveloping crossed module of algebras of a crossed module of Leibniz algebras is its natural generalization. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Our approach is particularly interesting since the actor in the category of Leibniz crossed modules does not exist in general, so the technique used in the proof for the Lie case cannot be applied. Finally we move on to the framework of the Loday-Pirashvili category $\mathcal{LM}$ in order to comprehend this universal enveloping crossed module in terms of the Lie crossed modules case.