Representations and formal deformations of Hom-Leibniz algebras

Abstract

In this paper, some results on representations of Hom-Leibniz algebras are found. Specifically the adjoint representation and trivial representation of Hom-Leibniz algebras are studied in detail. Deformations and central extensions of Hom-Leibniz algebras are also studied as applications.

On Hom-Leibniz superalgebras

“…The notion of Hom-Leibniz algebras was introduced by Makhlouf and Silvestrov [8]. Hom-Leibniz algebras were studied extensively in [3,6,12,21]. Furthermore, Hom-Leibniz algebras were generalized to Hom-Leibniz superalgebras by literature [9,10,14,15] as a noncommutative generalization of Hom-Lie superalgebras.…”

Representations and relative Rota-Baxter operators of Hom-Leibniz Poisson algebras

“…Since then, other types of Hom-algebras have emerged; in particular, Hom-Leibniz algebras [15] were introduced as a non-commutative version of Hom-Lie algebras as well as Hom-Poisson algebras [18] which have simultaneousely Hom-associative algebra and Hom-Lie algebra structures and satisfying a certain compatibility condition. Hom-Leibniz algebras have been widely studied from the point of view of representation and cohomology theory [5], deformation theory [2] [16] in recent years. The same is true for Hom-associative algebras since, the definition of Hochschildtype cohomology and the study of the one parameter formal deformation theory for these type of Hom-algebras are given [1], [14].…”

“…(2) Similarly, a Hom-Leibniz Poisson algebra (A, •, [, ], α) is called a Rota-Baxter Hom-Leibniz Poisson algebra if ((A, •, α) is a Rota-Baxter Hom-associative algebra and (A, [, ], α) is a Rota-Baxter Hom-Leibniz algebra.…”