Quantum Isometry Group of the n tori

TL;DR

This paper proves that the quantum isometry group of classical n tori is the same as its classical isometry group, and extends this to noncommutative tori via Rieffel deformation.

Contribution

It establishes the quantum isometry group of classical n tori as the classical isometry group and describes the quantum isometry group of noncommutative tori as a Rieffel deformation.

Findings

Quantum isometry group of classical n tori equals the classical isometry group.

Quantum isometry group of noncommutative n tori is a Rieffel deformation of the classical case.

The results connect quantum symmetries with classical geometric symmetries.

Abstract

We show that the Quantum Isometry Group(as introduced in \cite{goswami}) of the n tori is the classical isometry group. Moreover, using a result in \cite{bhowmick goswami}, we conclude that the Quantum Isometry group of the noncommutative n tori is a Rieffel deformation of the Quantum Isometry Group of the commutative n tori.