Fibre product approach to index pairings for the generic Hopf fibration of SU_q(2)

TL;DR

This paper introduces a fibre product method to describe quantum line bundles over Podles spheres and the Hopf fibration of SU_q(2), simplifying index pairing calculations in noncommutative geometry.

Contribution

It presents a novel fibre product construction for quantum line bundles and provides explicit $K_0$-class representatives, streamlining index computations compared to previous methods.

Findings

Explicit $K_0$-class representatives for quantum line bundles

Simplified index pairing calculations using fibre product projections

Connection between quantum bundles and classical Bott projections

Abstract

A fibre product construction is used to give a description of quantum line bundles over the generic Podles spheres by gluing two quantum discs along their boundaries. Representatives of the corresponding $K_0$-classes are given in terms of 1-dimensional projections belonging to the C*-algebra, and in terms of analogues of the classical Bott projections. The $K_0$-classes of quantum line bundles derived from the generic Hopf fibration of quantum SU(2) are determined and the index pairing is computed. It is argued that taking the projections obtained from the fibre product construction yields a significant simplification of earlier index computations.