The Gauge Group of a Noncommutative Principal Bundle and Twist Deformations

TL;DR

This paper explores the gauge group structure of noncommutative principal bundles, especially under twist deformations, showing that the gauge group remains invariant and classical in certain noncommutative settings.

Contribution

It introduces a framework for understanding gauge groups in noncommutative principal bundles and proves invariance of the gauge group under Drinfeld twist deformations.

Findings

Gauge group defined as vertical automorphisms in quasi-commutative cases

Gauge group is preserved under Drinfeld twist deformations

Noncommutative bundles from twists have classical gauge groups

Abstract

We study noncommutative principal bundles (Hopf-Galois extensions) in the context of coquasitriangular Hopf algebras and their monoidal category of comodule algebras. When the total space is quasi-commutative, and thus the base space subalgebra is central, we define the gauge group as the group of vertical automorphisms or equivalently as the group of equivariant algebra maps. We study Drinfeld twist (2-cocycle) deformations of Hopf-Galois extensions and show that the gauge group of the twisted extension is isomorphic to the gauge group of the initial extension. In particular, noncommutative principal bundles arising via twist deformation of commutative principal bundles have classical gauge group. We illustrate the theory with a few examples.