Topics in Noncommutative Geometry

TL;DR

This review explores noncommutative principal U(1)-bundles, their construction via Hopf-Galois theory, and extends into nonassociative geometry with applications to quantum spaces and cochain quantization.

Contribution

It provides a comprehensive overview of noncommutative principal bundles, including new constructions and applications to quantum and nonassociative geometries.

Findings

Construction of noncommutative U(1)-bundles via Pimsner's method

Examples involving quantum lens spaces and quantum weighted projective spaces

Discussion of cochain quantization and applications to octonions and noncommutative tori

Abstract

The leitmotiv of this review is noncommutative principal U(1)-bundles and associated line bundles. In the first part I give a brief introduction to Hopf-Galois theory and its applications, from field extensions to principal group actions. I then recall Woronowicz' definition of compact quantum group and the notion of noncommutative principal bundle. When the structure group is U(1), there is a construction due to Pimsner that allows to get the total space of a "bundle" (more precisely, a strongly graded C*-algebra) from the base space and a noncommutative "line bundle" (a self-Morita equivalence bimodule). As an example of this construction, I will discuss the U(1)-principal bundles of quantum lens spaces over quantum weighted projective space. The second part is a peek into the realm of nonassociative geometry: after a review of some properties of Hopf cochains and cocycles, I will discuss the theory of cochain quantization and its applications, from Albuquerque-Majid example of octonions, to "line bundles" on the noncommutative torus.