The Podles sphere as a spectral metric space

TL;DR

This paper explores the spectral metric structure of the Podles sphere, a quantum space related to SU(2), demonstrating it as a compact quantum metric space using a specific Dirac operator.

Contribution

It shows that the Podles sphere can be endowed with a quantum metric structure via a seminorm derived from a particular Dirac operator, extending spectral triple analysis.

Findings

Podles sphere admits a compact quantum metric space structure

The Dirac operator interprets the sphere as a 0-dimensional space

Seminorm from commutators defines the quantum metric structure

Abstract

We study the spectral metric aspects of the standard Podles sphere, which is a homogeneous space for quantum SU(2). The point of departure is the real equivariant spectral triple investigated by Dabrowski and Sitarz. The Dirac operator of this spectral triple interprets the standard Podles sphere as a 0-dimensional space and is therefore not isospectral to the Dirac operator on the 2-sphere. We show that the seminorm coming from commutators with this Dirac operator provides the Podles sphere with the structure of a compact quantum metric space in the sense of Rieffel.