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.