  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.
