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

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.

💡 Research Summary

This paper presents a novel, algebraic‑geometric method for describing quantum line bundles over the generic Podleś spheres by means of a fibre‑product construction. The authors start by recalling that the C*‑algebra of a quantum disc is the Toeplitz algebra T, equipped with the symbol map σ : T → C(S¹) which records the boundary values. By taking two copies of T and imposing the equality of their symbols, they obtain a concrete realisation of the C*‑algebra of the generic Podleś sphere S²_{q,s} as a pull‑back (fibre product): \