Hom-pre-Malcev and Hom-M-Dendriform algebras

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 and M-dendriform algebras, called Hom-pre-Malcev algebras and Hom-M-dendriform algebras. We also introduce the notion of $\mathcal{O}$-operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev, Hom-pre-Malcev and Hom-M-dendriform algebras using $\mathcal{O}$-operators. Hom-pre-Malcev algebras and Hom-M-dendriform algebras generalize Hom-pre-Lie algebras and Hom-L-dendriform algebras respectively to the alternative setting and fit into a bigger framework with a close relationship with Hom-pre-alternative algebras and Hom-alternative quadri-algebras respectively.