Classification, centroids and derivations of two-dimensional Hom-Leibniz algebras (original) (raw)

Hom-Leibniz bialgebras and BiHom-Leibniz dendriform algebras

2021

The notion of a Hom-Leibniz bialgebra is introduced and it is shown that matched pairs of Hom-Leibniz algebras, Manin triples of Hom-Leibniz algebras and Hom-Leibniz bialgebras are equivalent in a certain sense. The notion of Hom-Leibniz dendriform algebra is established, their bimodules and matched pairs are defined and their properties and theorems about their interplay and construction are obtained. Furthermore, the concept of BiHom-Leibniz dendriform algebras is introduced and discussed, their bimodules and matched pairs are constructed and properties are described. Finally, the connections between all these algebraic structures using O-operators are shown.

On universal central extensions of Hom-Leibniz algebras

Journal of Algebra and Its Applications, 2014

In the category of Hom-Leibniz algebras we introduce the notion of Hom-co-representation as adequate coefficients to construct the chain complex from which we compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibniz 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 due to the fact that the composition of two central extensions of Hom-Leibniz algebras is not central. We also provide the recognition criteria for these kind of universal central extensions. 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 of them. In case α = Id we recover the corresponding results on universal central extensions of Leibniz algebras.

Hom Leibniz Superalgebras With Supersymmetric Invariant Nondegenerate Bilinear Forms

Journal of Mathematics Research, 2021

In this paper we study Hom Leibniz superalgebras endowed with a supersymmetric, nondegenerate and invariant bilinear form. Such Hom Leibniz superalgebras are called quadratic Hom Leibniz superalgebras. We show that every quadratic Hom Leibniz superalgebra is symmetric. After introducing representation of Hom Leibniz superalgebras, we prove that Hom Leibniz superalgebras are quadratic if and only if the adjoint representation and its dual representation are equivalent. We also extend the notion of T*-extension to Hom Leibniz superalgebras. Finally, by using double extension, we describe inductively quadratic regular Hom Leibniz superalgebras.

On n-Hom-Leibniz algebras and cohomology

Georgian Mathematical Journal, 2020

The purpose of this paper is to provide a cohomology of n-Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations.