Some counterexamples in the theory of quantum isometry groups

TL;DR

This paper presents counterexamples in quantum isometry groups showing they may lack $C^*$-actions and universal objects, challenging assumptions in the theory of quantum symmetries of spectral triples.

Contribution

It demonstrates that quantum isometry groups can fail to have $C^*$-actions and may not possess universal objects, providing new insights into their structure.

Findings

Quantum isometry groups may lack $C^*$-actions.

They can fail to be matrix quantum groups.

The category of such isometries may lack a universal object.

Abstract

By considering spectral triples on $S^{2}_{\mu, c}$ ($c>0$) constructed by Chakraborty and Pal (\cite{chak_pal}), we show that in general the quantum group of volume and orientation preserving isometries (in the sense of \cite{goswami2}) for a spectral triple of compact type may not have a $C^*$-action, and moreover, it can fail to be a matrix quantum group. It is also proved that the category with objects consisting of those volume and orientation preserving quantum isometries which induce $C^*$-action on the $C^*$ algebra underlying the given spectral triple, may not have a universal object.