nLab Schauenburg bialgebroid


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).


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.


  • 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 (1998) 193–222 [doi:10.1023/A:1008608028634]

  • 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:0805.2510 doi

  • 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

category: algebra

Last revised on September 1, 2023 at 21:11:51. See the history of this page for a list of all contributions to it.