Infinitesimal (BiHom-)bialgebras of Any Weight (I): Basic Definitions and Properties

Abstract

The purpose of this paper is to introduce and study λ-infinitesimal BiHom-bialgebras (abbr. λ-infBH-bialgebra) and some related structures. They can be seen as an extension of λ-infinitesimal bialgebras considered by Ebrahimi-Fard, including Joni–Rota’s infinitesimal bialgebras as well as Loday–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 (resp. coalgebra) to be a λ-infBH-bialgebra and the notion of λ-infBH-Hopf module is introduced and discussed. It is proved, in connection with nonhomogeneous (co)associative BiHom-Yang-Baxter equation, that every (left BiHom-)module (resp., comodule) over an (anti-)quasitriangular (resp., (anti-)coquasitriangular) λ-infBH-bialgebra carries a structure of λ-infBH-Hopf module. Moreover, two approaches to construct BiHom-pre-Lie (co)algebras from λ-infBH-bialgebras are presented.

Keywords

Infinitesimal bialgebra / infinitesimal Hopf module / quasitriangular infinitesimal bialgebra / 17B61 / 17D30 / 17B38 / 17A30 / 16T10

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