Schauenburg bialgebroid, also called Ehresmann-Schauenburg bialgebroid, is a noncommutative generalization of the algebra of functions on an Atiyah groupoid (also called gauge or Ehresmann groupoid) of a principal bundle where the principal bundle is replaced by a Hopf-Galois extension (on the level of algebra of functions).

Definition

It is an associative bialgebroid whose structure, given a Hopf-Galois extension, is described in Brzeziński-Wisbauer2003, 34.14. The description there is by first constructing an associated coring, the Ehresmann coring (which is a more general construction, defined for any coalgebra-Galois extension which is faithfully flat as a left module over the base of the extension), and then making the bialgebroid from it.

Literature

T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.

Peter Schauenburg, Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules, Appl. Cat. Str.6:193–222 (1998)

Piotr M. Hajac, Tomasz Maszczyk, Pullbacks and nontriviality of associated noncommutative vector bundles, arxiv/1601.00021

Tomasz Brzeziński, Joost Vercruysse, Bimodule herds, J. Algebra 321:9, (2009) 2670-2704 arXiv:0805.2510doi

Xiao Han, Giovanni Landi, On the gauge group of Galois objects, arxiv/2002.06097

Xiao Han, Twisted Ehresmann Schauenburg bialgebroids, arxiv/2009.02764; Quantum principal bundles, gauge groupoids and coherent Hopf 2-algebras, PhD thesis, SISSA 2019/20 pdf

Ludwik Dabrowski, Giovanni Landi?, Jacopo Zanchettin, Hopf algebroids and twists for quantum projective spaces, arXiv:2302.12073