On the gauge group of Galois objects

TL;DR

This paper explores the structure of gauge groups in noncommutative principal bundles, linking them to bisections of bialgebroids and providing new insights into Galois objects and their automorphisms.

Contribution

It introduces a framework connecting the gauge group of noncommutative bundles to bisections of bialgebroids, especially for Galois objects with Hopf algebra structures.

Findings

Gauge group is isomorphic to bisections when the base algebra is central.

Provides a crossed module structure for bisections and automorphisms.

Includes examples with group Hopf algebras and Taft algebras.

Abstract

We study the Ehresmann--Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the classical gauge groupoid of a principal bundle. When the base algebra is in the centre of the total space algebra, the gauge group of the noncommutative principal bundle is isomorphic to the group of bisections of the bialgebroid. In particular we consider Galois objects (non-trivial noncommutative bundles over a point in a sense) for which the bialgebroid is a Hopf algebra. For these we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include Galois objects of group Hopf algebras and of Taft algebras.