On universal central extensions of Hom_Leibniz algebras

TL;DR

This paper extends the theory of universal central extensions to Hom-Leibniz algebras, introducing new notions like $oldsymbol{ ext{ extalpha}}$-central extensions and proving existence results for $oldsymbol{ extalpha}$-perfect Hom-Lie algebras.

Contribution

It generalizes classical results on Leibniz algebras to the Hom-Leibniz setting, introducing $oldsymbol{ extalpha}$-central extensions and establishing their properties.

Findings

Existence of universal $oldsymbol{ extalpha}$-central extensions for $oldsymbol{ extalpha}$-perfect Hom-Lie algebras.

Relationships between universal extensions in Hom-Lie and Hom-Leibniz categories.

Recovery of classical Leibniz results when $oldsymbol{ extalpha} = ext{Id}$.

Abstract

In the category of Hom-Leibniz algebras we introduce the notion of representation as adequate coefficients to construct the chain complex to compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibinz algebras and generalize some classical results, nevertheless it is necessary to introduce new notions of $\alpha$-central extension, universal $\alpha$-central extension and $\alpha$-perfect Hom-Leibniz algebra. We prove that an $\alpha$-perfect Hom-Lie algebra admits a universal $\alpha$-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both. In case $\alpha = Id$ we recover the corresponding results on universal central extensions of Leibniz algebras.