Infinitesimal (BiHom-)bialgebras of any weight (II): Representations

Abstract: The aim of this paper is to investigate representation theory of infinitesimal (BiHom-)bialgebras of any weight $\l$ (abbr. $\l$-inf(BH)-bialgebras). Firstly, inspired by the well-known Majid-Radford's bosonization theory in Hopf algebra theory, we present a class of $\l$-inf(BH)-bialgebras, named $\l$-inf(BH)-biproduct bialgebras, consisting of an inf(BH)-product algebra structure and an inf(BH)-coproduct coalgebra structure, which induces a structure of a $\l$-inf(BH)-Hopf bimodule over a $\l$-inf(BH)-bialgebra. Secondly, we explore relationships among $\l$-inf(BH)-Hopf bimodules, $\l$-Rota-Baxter (BiHom-)bimodules, (BiHom-)dendriform bimodules and (BiHom-)pre-Lie bimodules. Finally, we provide two kinds of general Gelfand-Dorfman theorems related to BiHom-Novikov algebras.