Gauge groups and bialgebroids

We study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples illustrating these constructions include: Galois objects of Taft algebras, a monopole bundle over a quantum spheres and a not faithfully flat Hopf–Galois extension of commutative algebras. The latter two examples have in fact a structure of Hopf algebroid for a suitable invertible antipode.