Noncommutative principal bundles through twist deformation

TL;DR

This paper develops a framework for constructing noncommutative principal bundles via twist deformations, enabling noncommutative fibers and base spaces, with applications to Hopf-Galois extensions and sheaf theory.

Contribution

It introduces a method to deform principal bundles using Drinfeld twists, creating noncommutative geometries with both noncommutative fibers and bases, expanding the toolkit for noncommutative geometry.

Findings

Constructed noncommutative principal bundles with twist deformations.

Established equivalences of module categories via natural isomorphisms.

Presented examples and a sheaf-theoretic approach.

Abstract

We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf-Galois extensions as twists of Hopf-Galois extensions. A sheaf approach is also considered, and examples presented.