Infinitesimal (BiHom-)bialgebras of any weight (I): Basic definitions and properties

TL;DR

This paper introduces and studies $mbda$-infinitesimal BiHom-bialgebras, extending existing algebraic structures, and explores their properties, constructions, and related modules, with applications to BiHom-pre-Lie algebras.

Contribution

It defines $mbda$-infinitesimal BiHom-bialgebras, provides construction methods, and links them to BiHom-Hopf modules and BiHom-pre-Lie algebras, extending prior algebraic frameworks.

Findings

Introduced $mbda$-infBH-bialgebras and their properties.

Established connections with BiHom-Hopf modules and (co)quasitriangular structures.

Presented methods to construct BiHom-pre-Lie (co)algebras from these bialgebras.

Abstract

The purpose of this paper is to introduce and study $\lambda$-infinitesimal BiHom-bialgebras (abbr. $\l$-infBH-bialgebra) and some related structures. They can be seen as an extension of $\l$-infinitesimal bialgebras considered by Ebrahimi-Fard, including Joni and Rota's infinitesimal bialgebras as well as Loday and Ronco's infinitesimal bialgebras, and including also infinitesimal BiHom-bialgebras introduced by Liu, Makhlouf, Menini, Panaite. In this paper, we provide various relevant constructions and new concepts. Two ways are provided for a unitary (resp. counitary) algebra (coalgebra) to be a $\l$-infBH-bialgebra and the notion of $\l$-infBH-Hopf module is introduced and discussed. It is proved, in connexion with nonhomogeneous (co)associative BiHom-Yang-Baxter equation, that every (left BiHom-)module (resp. comodule) over a (anti-)quasitriangular (resp. (anti-)coquasitriangular) $\l$-infBH-bialgebra carries a structure of $\l$-infBH-Hopf module. Moreover, two approaches to construct BiHom-pre-Lie (co)algebras from $\l$-infBH-bialgebras are presented.