Kupershmidt operators on Hom-Malcev algebras and their deformation

Abstract

The main feature of Hom-algebras is that the identities defining the structures are twisted by linear maps. The purpose of this paper is to introduce and study a Hom-type generalization of pre-Malcev algebras, called Hom-pre-Malcev algebras. We also introduce the notion of Kupershmidt operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev and Hom-pre-Malcev algebras using Kupershmidt operators. Hom-pre-Malcev algebras generalize Hom-pre-Lie algebras to the Hom-alternative setting and fit into a bigger framework with a close relationship with Hom-pre-alternative algebras. Finally, we establish a deformation theory of Kupershmidt operators on a Hom-Malcev algebra in consistence with the general principles of deformation theories and introduce the notion of Nijenhuis elements.

Dendrification of Hom-Malcev algebras

“…We study in the second section the representation of Hom-Malcev algebras and recall the definitions and some properties of Hom-pre-Malcev algebras. The notion of a representation of a Hom-pre-Malcev algebra was introduced in [23]. However, one need to add some complicated conditions to obtain the dual representation, which is not convenient in applications.…”

“…Now we recall the definition of Hom-pre-Malcev algebra and exhibit construction results in terms of O-operators on Hom-Malcev algebras (for more details see [23]) Definition 2.2. Let A be a vector space with a bilinear product "…”