Construction of Hom-Pre-Jordan algebras and Hom-J-dendriform algebras

TL;DR

This paper introduces Hom-pre-Jordan and Hom-J-dendriform algebras, generalizing Hom-Jordan algebras, and explores their structures, representations, and relations to Rota-Baxter operators and $\\\mathcal{O}$-operators.

Contribution

It defines new algebraic structures, establishes their properties, and connects them to existing concepts like Rota-Baxter operators and Hom-pre-Lie algebras.

Findings

Hom-pre-Jordan algebras' anti-commutator forms a Hom-Jordan algebra

Hom-Pre-Jordan algebras serve as underlying structures for Hom-Jordan algebras

Hom-J-dendriform algebras are analogues of Hom-dendriform algebras with specific properties

Abstract

The aim of this work is to introduce and study the notions of Hom-pre-Jordan algebra and Hom-J-dendriform algebra which generalize Hom-Jordan algebras. Hom-Pre-Jordan algebras are regarded as the underlying algebraic structures of the Hom-Jordan algebras behind the Rota-Baxter operators and $\mathcal{O}$-operators introduced in this paper. Hom-Pre-Jordan algebras are also analogues of Hom-pre-Lie algebras for Hom-Jordan algebras. The anti commutator of a Hom-pre-Jordan algebra is a Hom-Jordan algebra and the left multiplication operator gives a representation of a Hom-Jordan algebra. On the other hand, a Hom-J-dendriform algebra is a Hom-Jordan algebraic analogue of a Hom-dendriform algebra such that the anti-commutator of the sum of the two operations is a Hom-pre-Jordan algebra.