Quantum principal bundles over quantum real projective spaces

TL;DR

This paper constructs two hierarchies of quantum principal bundles over quantum real projective spaces with different structure groups, analyzing their triviality using Hopf-Galois extension criteria.

Contribution

It introduces new hierarchies of quantum principal bundles over quantum real projective spaces with U(1) and SU_q(2) as structure groups, derived via prolongation methods.

Findings

Bundles with U(1) and SU_q(2) structure groups are constructed.

The triviality of these bundles is characterized using cleft Hopf-Galois extension criteria.

Hierarchies are obtained from bundles with cyclic group of order 2 as fiber.

Abstract

Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with U(1) as a structure group, the other has the quantum group $SU_q(2)$ as a fibre. Both hierarchies are obtained by the process of prolongation from bundles with the cyclic group of order 2 as a fibre. The triviality or otherwise of these bundles is determined by using a general criterion for a prolongation of a comodule algebra to be a cleft Hopf-Galois extension.