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

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