Some characterizations of Hom-Leibniz algebras

Abstract

Some basic properties of Hom-Leibniz algebras are found. These properties are the Hom-analogue of corresponding well-known properties of Leibniz algebras. Considering the Hom-Akivis algebra associated to a given Hom-Leibniz algebra, it is observed that the Hom-Akivis identity leads to an additional property of Hom-Leibniz algebras, which in turn gives a necessary and sufficient condition for Hom-Lie admissibility of Hom-Leibniz algebras. A necessary and sufficient condition for Hom-power associativity of Hom-Leibniz algebras is also found.

Hom-Lie-Yamaguti structures on Hom-Leibniz algebras

“…This proof is based on a specific ternary operation that can be considered on a given Hom-Leibniz algebra (this product is the Hom-analogue of the ternary operation considered in [14] on a left Leibniz algebra L that produces, along with the skew-symmetrization, a LY structure on L). Also note that our proof below essentially relies on some properties characterizing Hom-Leibniz algebras, obtained in [12]. We conclude, as an illustration of our result, by some constructions of Hom-LY algebras from twisted Leibniz algebras (incidentally, this produces examples of left Hom-Leibniz algebras).…”

“…We recall some basic notions, introduced in [8], [11], [18], [23], [24], related to Homalgebras. We also recall from [12] a characterization of the Hom-Akivis algebra associated with a given Hom-Leibniz algebra.…”

On the Universal $$\alpha $$ α -Central Extension of the Semi-direct Product of Hom-Leibniz Algebras

“…From the introducing paper, the investigation of several kinds of Hom-structures is in progress (for instance, see [1,2,3,4,12,14,17,18] and references given therein). Naturally, the non-skew-symmetric version of Hom-Lie algebras, the so called Hom-Leibniz algebras, was considered as well (see [2,5,7,9,13,14,15]). A Hom-Leibniz algebra is un triple (L, [−, −], α L ) consisting of a K-vector space L, a bilinear map [−, −] : L × L → L and a homomorphism of K-vector spaces α L : L → L satisfying the Hom-Leibniz identity:…”