The Podles spheres converge to the sphere

TL;DR

This paper proves that Podles spheres converge to classical spheres in quantum Gromov-Hausdorff distance as the deformation parameter approaches 1, and constructs q-deformed fuzzy spheres that converge to Podles spheres.

Contribution

It establishes convergence of Podles spheres to classical spheres and introduces q-deformed fuzzy spheres, extending classical results into the quantum setting.

Findings

Podles spheres converge to classical 2-sphere as q approaches 1

Q-deformed fuzzy spheres converge to Podles spheres as their dimension increases

Provides a quantum analogue of Rieffel's classical convergence result

Abstract

We prove that the Podles spheres $S_q^2$ converge in quantum Gromov-Hausdorff distance to the classical 2-sphere as the deformation parameter $q$ tends to 1. Moreover, we construct a $q$-deformed analogue of the fuzzy spheres, and prove that they converge to $S_q^2$ as their linear dimension tends to infinity, thus providing a quantum counterpart to a classical result of Rieffel.