Braided bi-Galois theory

Abstract

To keep technicalities to a minimum for the beginning of the introduction, we start by letting H be a Hopf algebra with bijective antipode over a field k. A right H-Galois object is a right H-Galois extension of the base field k, that is, a right H-comodule algebra A with A = k for which a certain canonical map β : A ⊗ A → A ⊗H is bijective. Hopf-Galois extensions are an important application and tool for the study of Hopf algebras. We refer to [9] and [6] for more background and references. Grunspan [3] has defined a quantum torsor to be an algebra T equipped with an algebra map Θ: T → T ⊗T ⊗T and an algebra map θ : T → T subject to a list of axioms; we will call Θ the torsor structure and have called θ the Grunspan map in [8]. By and large, the idea of a quantum torsor allows us to define HopfGalois objects without mentioning a Hopf algebra. The torsor structure is a sort of (triple) comultiplication, and the Grunspan map generalizes the square of the antipode of a Hopf algebra. Grunspan shows that every quantum torsor in his sense is a Hopf-Galois object over a suitably constructed Hopf algebra H. The paper [7] makes this an equivalent characterization by showing that every H-Galois extension is also a torsor. While the torsor structure is quite easy to find, the Grunspan map, written down explicitly in [7], is perhaps less obvious. In [8] it was then shown that the Grunspan map can be eliminated altogether from the axioms of a quantum torsor. Its uniqueness was already observed by Grunspan, and it can be proved to exist via constructing a HopfGalois extension from a torsor (without Grunspan map) and then a Grunspan map from the Hopf-Galois extension. In [7] the Grunspan map for an H-Galois object A was simply written down explicitly. Here now is a conceptual reason for its existence: By [5], A is an L-Hbi-Galois object (i.e. a left L-Galois object and a bicomodule) for a Hopf algebra L with bijective antipode. Also by [5], bi-Galois objects are the morphisms of a groupoid; the composition in the groupoid is the cotensor product, and