nLab Schauenburg bialgebroid

Motivation

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 faithfully flat $H$-Hopf-Galois extension $U\hookrightarrow E$, is described in Brzeziński-Wisbauer2003, 34.14. The description there is by first constructing an associated coring, the Ehresmann coring $B= (E\otimes_k E)^{co H}$ (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 noticing that in this case $B$ is a subalgebra of $E\otimes_k E^{op}$ and equipping it with source $s:U\to E$, $a\mapsto a\otimes 1$ and target $t:U\to E^{op}$, $a\mapsto 1\otimes a$.

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 (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