A non-abelian Hom-Leibniz tensor product and applications

Abstract

The notion of non-abelian Hom-Leibniz tensor product is introduced and some properties are established. This tensor product is used in the description of the universal (α-)central extensions of Hom-Leibniz algebras. We also give its application to the Hochschild homology of Hom-associative algebras.

Universal α-central extensions of Hom-Leibniz n-algebras

“…From this fact, our aim in this article is to introduce and characterize universal α-central extensions of Hom-Leibniz n-algebras. In case n = 2 we recover the corresponding results on universal α-central extensions of Hom-Leibniz algebras in [14,15]. Moreover, in case α = id we recover results on universal central extensions of Leibniz n-algebras in [12].…”

“…Remark 4.2 If α L = (id L ), then (M 1 * · · · * M n , α M 1 * ··· * Mn ) coincides with the non-abelian tensor product of Leibniz n-algebras introduced in [12]. In case n = 2, we recover a particular case of the non-abelian tensor product of Hom-Leibniz algebras given in [15].…”

“…In case n = 2, the universal central extension in Theorem 4.3 provides the universal central extension of a Hom-Leibniz algebra given in [15]. where ψ(l 1 * · · · * l n ) = [l 1 , .…”

“…In section 4 we introduce the concept of non-abelian tensor product of Hom-Leibniz n-algebras that generalizes the non-abelian tensor product of Leibniz (n)algebras in [12,15], and we establish its relationship with the universal central extension.…”

Universal central extensions and non-abelian tensor product of hom-Lie–Rinehart algebras

“…A non-abelian tensor product of Lie algebras was introduced in [El91] and its relationship with universal central extensions of Lie algebras was described. Later on, this study was extended to Leibniz algebras, Lie-Rinehart algebras, hom-Lie algebras and hom-Leibniz algebras in [Gn99], [CGM], [CKP1], and [CKP2], respectively.…”