Toeplitz algebras in quantum Hopf fibrations

TL;DR

This paper explores how Toeplitz algebras can be used to construct and analyze quantum Hopf fibrations, simplifying index calculations and establishing isomorphisms between different quantum bundle models.

Contribution

It introduces a novel gluing construction of quantum Hopf fibrations using Toeplitz algebras, linking different quantum bundle descriptions and simplifying K-theory index computations.

Findings

Quantum Hopf fibrations constructed via Toeplitz algebras

Isomorphism established between different quantum fibration models

Simplification of index calculations through elementary projections

Abstract

The paper presents applications of Toeplitz algebras in Noncommutative Geometry. As an example, a quantum Hopf fibration is given by gluing trivial U(1) bundles over quantum discs (or, synonymously, Toeplitz algebras) along their boundaries. The construction yields associated quantum line bundles over the generic Podles spheres which are isomorphic to those from the well-known Hopf fibration of quantum SU(2). The relation between these two versions of quantum Hopf fibrations is made precise by giving an isomorphism in the category of right U(1)-comodules and left modules over the C*-algebra of the generic Podles spheres. It is argued that the gluing construction yields a significant simplification of index computations by obtaining elementary projections as representatives of K-theory classes.