The Pullbacks of Principal Coactions

TL;DR

This paper demonstrates that principal coactions are stable under certain pullback operations, enabling new analyses of noncommutative geometries and quantum principal bundles.

Contribution

It establishes the closure of principal coactions under one-surjective pullbacks and applies this to noncommutative line bundles and quantum deformations.

Findings

Index computation for noncommutative line bundles over Podles sphere

Construction of coalgebraic noncommutative deformations of U(1)-bundles

Extension of principal coaction theory beyond comodule algebras

Abstract

We prove that the class of principal coactions is closed under one-surjective pullbacks in an appropriate category of algebras equipped with left and right coactions. This allows us to handle cases of C*-algebras lacking two different non-trivial ideals. It also allows us to go beyond the category of comodule algebras. As an example of the former, we carry out an index computation for noncommutative line bundles over the standard Podles sphere using the Mayer-Vietoris type arguments afforded by a one-surjective pullback presentation of the C*-algebra of this quantum sphere. To instantiate the latter, we define a family of coalgebraic noncommutative deformations of the U(1)-principal bundle S^7 --> CP^3.