\v{C}ech cocycles for quantum principal bundles

TL;DR

This paper develops a framework using noncommutative localization and cech cocycles to describe quantum principal bundles and associated vector bundles on noncommutative schemes, with applications to quantum groups.

Contribution

It introduces a method to describe cocycle data for quantum principal bundles using noncommutative localization theory and extends this to associated vector bundles.

Findings

Framework for quantum principal bundles using cech cocycles

Application to noncommutative schemes with Hopf algebra structure

Examples related to quantum groups

Abstract

In other to study connections and gauge theories on noncommutative spaces it is useful to use the local trivializations of principal bundles. In this note we show how to use noncommutative localization theory to describe a simple version of cocycle data for the bundles on noncommutative schemes with Hopf algebras in the role of the structure group which locally look like Hopf algebraic smash products. We also show how to use these \v{C}ech cocycles for associated vector bundles. We sketch briefly some examples related to quantum groups.