On universal central extensions of Hom_Leibniz algebras

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.