2015/10/25 by D'Andrea, Francesco
#17B37 #53D55 #FOS: Mathematics #Primary: 46L87 #Quantum Algebra (math.QA) #Secondary: 20G42
paper · doi:10.48550/arxiv.1510.07271
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.