Kupershmidt operators on Hom-Malcev algebras and their deformation

TL;DR

This paper introduces Hom-pre-Malcev algebras as a Hom-type generalization of pre-Malcev algebras, explores Kupershmidt operators within this framework, and develops a deformation theory including Nijenhuis elements.

Contribution

It defines Hom-pre-Malcev algebras, introduces Kupershmidt operators for Hom-Malcev and Hom-pre-Malcev algebras, and establishes a deformation theory for these operators.

Findings

Hom-pre-Malcev algebras generalize Hom-pre-Lie algebras.

Kupershmidt operators connect Hom-Malcev and Hom-pre-Malcev algebras.

A deformation theory for Kupershmidt operators is developed.

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.