Quantum isometry group of the $n$-tori

Abstract

We show that the quantum isometry group (introduced by Goswami) of the $n$-tori $\mathbb {T}^{n}$ coincides with its classical isometry group; i.e. there does not exist any faithful isometric action on $\mathbb {T}^{n}$ by a genuine (noncommutative as a $C^{\ast }$-algebra) compact quantum group. Moreover, using an earlier result, 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.