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

TL;DR

This paper introduces Hom-pre-Malcev and Hom-M-dendriform algebras, generalizing existing Hom-algebra structures with twisted identities, and explores their connections via $\\mathcal{O}$-operators, expanding the framework of Hom-algebras.

Contribution

It defines Hom-pre-Malcev and Hom-M-dendriform algebras, establishes their relationships with other Hom-algebras, and introduces $\\mathcal{O}$-operators to connect these structures.

Findings

Hom-pre-Malcev and Hom-M-dendriform algebras generalize existing Hom-algebras.

Connections between these algebras are established using $\\mathcal{O}$-operators.

The new structures extend the framework of Hom-pre-Lie and Hom-L-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.