Topics in Noncommutative Geometry
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.
Cite
@article{arxiv.1510.07271,
title = {Topics in Noncommutative Geometry},
author = {Francesco D'Andrea},
journal= {arXiv preprint arXiv:1510.07271},
year = {2015}
}
Comments
28 pages. Lecture notes for the autumn school "From Poisson Geometry to Quantum Fields on Noncommutative Spaces", University of Wurzburg, 5-10 October 2015