Quantum Isometry groups of the Podles Spheres

TL;DR

This paper characterizes the quantum isometry group of the Podles sphere, showing that it is the universal quantum group $SO_(3)$ acting as orientation and volume-preserving isometries on the spectral triple of the sphere.

Contribution

It identifies the quantum isometry group of the Podles sphere as the universal object $SO_(3)$ within a specific category of quantum groups.

Findings

Quantum isometry group of Podles sphere is $SO_(3)$.

The result applies to spectral triples on the Podles sphere.

Provides a classification of symmetries for this noncommutative space.

Abstract

For $\mu \in (0,1), c> 0,$ we identify the quantum group $SO_\mu(3)$ as the universal object in the category of compact quantum groups acting by `orientation and volume preserving isometries' in the sense of \cite{goswami2} on the natural spectral triple on the Podles sphere $S^2_{\mu, c}$ constructed by Dabrowski, D'Andrea, Landi and Wagner in \cite{{Dabrowski_et_al}}.