Hopf bimodules for bialgebroids
Sophie Chemla, Niels Kowalzig
TL;DR
The paper generalizes classical Hopf module theory from Hopf algebras to left bialgebroids by defining four Hopf module variants and introducing Hopf bimodules (tetramodules) without requiring antipodes. It proves a Fundamental Theorem via Hopf-Galois comodules and establishes that the category of Hopf bimodules is equivalent to Yetter-Drinfel'd module categories, via coinvariants and induction functors, under suitable flatness/projectivity assumptions. It further develops monoidal and braided monoidal structures, showing two independent monoidal pictures on Hopf bimodules become braided-equivalent to Yetter-Drinfel'd module categories and to the monoidal centre of left U-modules. These results extend Schauenburg-type equivalences to bialgebroids, offering a robust framework for bialgebroid (co)homology, deformation theory, and representation theory in noncommutative base settings.
Abstract
Generalising a result for Hopf algebras, we not only define the four possible types of Hopf modules in the bialgebroid setting but also yield the notion of two-sided two-cosided Hopf modules, also known as Hopf bimodules or tetramodules, in this realm. By explicitly formulating a fundamental theorem for Hopf modules via the concept of Hopf-Galois comodules, we prove that the category of Hopf bimodules can be endowed with the structure of a (pre-)braided monoidal category in two different ways, which, in turn, are shown to be both braided monoidally equivalent to the category of Yetter-Drinfel'd modules, that is, to the monoidal centre of the category of left bialgebroid modules.