Hom-Leibniz bialgebras and BiHom-Leibniz dendriform algebras

TL;DR

This paper introduces Hom-Leibniz bialgebras and BiHom-Leibniz dendriform algebras, establishing their properties, bimodules, and interrelations via $\\mathcal{O}$-operators, expanding the algebraic framework in this area.

Contribution

It defines new algebraic structures—Hom-Leibniz bialgebras and BiHom-Leibniz dendriform algebras—and explores their properties, bimodules, and connections, advancing the theory of Hom-type Leibniz algebras.

Findings

Hom-Leibniz bialgebras are equivalent to matched pairs and Manin triples of Hom-Leibniz algebras.

BiHom-Leibniz dendriform algebras and their bimodules are constructed and analyzed.

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

Abstract

The notion of a Hom-Leibniz bialgebra is introduced and it is shown that matched pairs of Hom-Leibniz algebras, Manin triples of Hom-Leibniz algebras and Hom-Leibniz bialgebras are equivalent in a certain sense. The notion of Hom-Leibniz dendriform algebra is established, their bimodules and matched pairs are defined and their properties and theorems about their interplay and construction are obtained. Furthermore, the concept of BiHom-Leibniz dendriform algebras is introduced and discussed, their bimodules and matched pairs are constructed and properties are described. Finally, the connections between all these algebraic structures using $\mathcal{O}$-operators are shown.