Quantum symmetries of classical spaces

TL;DR

This paper introduces a method for constructing faithful actions of genuine compact quantum groups on classical spaces, revealing new symmetries and isometry groups in classical topology.

Contribution

It provides a general scheme for quantum symmetries of classical spaces and demonstrates the existence of genuine quantum group actions and isometry groups on such spaces.

Findings

Classical connected spaces can admit faithful actions by genuine quantum groups.

A spectral triple on a classical space can have a quantum isometry group that is a genuine quantum group.

New quantum symmetries of classical spaces are established.

Abstract

We give a general scheme for constructing faithful actions of genuine (noncommutative as $C^*$ algebra) compact quantum groups on classical topological spaces. Using this, we show that: (i) a compact connected classical space can have a faithful action by a genuine compact quantum group, and (ii) there exists a spectral triple on a classical connected compact space for which the quantum group of orientation and volume preserving isometries (in the sense of \cite{qorient}) is a genuine quantum group.