Gauge groups and bialgebroids
arXiv:2104.09258 · doi:10.1007/s11005-021-01482-2
Abstract
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.
33 pages. Sect. 5.1 expanded. Few minor changes. arXiv admin note: substantial text overlap with arXiv:2002.06097