Deformation quantization of principal bundles

TL;DR

This paper explores how Drinfeld twist techniques can deform classical principal bundles into noncommutative versions, affecting both fibers and base spaces, with potential applications in noncommutative geometry.

Contribution

It introduces a method to deform principal bundles into noncommutative structures using Drinfeld twists, combining deformations of structure groups and automorphism groups.

Findings

Deformation of structure groups into quantum groups.

Deformation of base spaces into noncommutative spaces.

Construction of noncommutative principal bundles with both fibers and base spaces deformed.

Abstract

We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles, and more in general to the deformation of Hopf-Galois extensions. First we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next we twist deform a subgroup of the group of authomorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations we obtain noncommutative principal bundles with noncommutative fiber and base space as well.