Gauge groups and bialgebroids

TL;DR

This paper explores the structure of gauge groups in noncommutative geometry using bialgebroids, providing new insights into their algebraic properties and examples including quantum spheres and Hopf--Galois extensions.

Contribution

It establishes an isomorphism between the gauge group of a noncommutative bundle and the bisections of its bialgebroid, introducing a crossed module structure and analyzing key examples.

Findings

Gauge group is isomorphic to bisections of the bialgebroid

Crossed module structure for bisections and automorphisms

Examples include quantum spheres and Hopf--Galois extensions

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.