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 α-central extension, universal α-central extension and α-perfect Hom-Leibniz algebra. We prove that an α-perfect Hom-Lie algebra admits a universal α-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both. In case α = Id we recover the corresponding results on universal central extensions of Leibniz 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].…”

“…We denote by n HomLeib the category of Hom-Leibniz n-algebras. In case n = 2, identity (1) is the Hom-Leibniz identity (2.1) in [14], so Hom-Leibniz 2-algebras are exactly Hom-Leibniz algebras and we use the notation HomLeib instead of 2 HomLeib. Example 2.4 (a) When the maps (α i ) 1≤i≤n−1 in Definition 2.1 are all of them the identity maps, then one recovers the definition of Leibniz n-algebra [16].…”

“…Now we construct a chain complex for Hom-Leibniz n-algebras in order to compute its homology with trivial coefficients. Firstly we recall this complex for Hom-Leibniz homology [14].…”

“…The homology of the chain complex (CL α * (L, M), d * ) is called homology of the Hom-Leibniz algebra (L, [−, −], α L ) with coefficients in the Hom-co-representation (M, α M ) [14] and is denoted by…”

“…In this way, deformations of algebras of Lie type were considered, among others, in [21,24,25,31,30]. Deformations of algebras of Leibniz type were considered, among others, in [18,20,14,22,24]. The generalizations of n-ary algebra structures, such as Hom-Leibniz n-algebras (or n-ary Hom-Nambu) and Hom-Lie n-algebras (or n-ary Hom-Nambu-Lie), have been introduced in [4] by Ataguema, Makhlouf, and Silvestrov.…”