Quantum isometry groups of symmetric groups

TL;DR

This paper determines the quantum isometry groups of spectral triples on symmetric groups, revealing they are related to a doubling of group algebras, and explores how generator choices affect these groups.

Contribution

It identifies the quantum isometry groups for symmetric groups with specific length functions and analyzes their dependence on generator choices, linking to non-commutative geometry.

Findings

Quantum isometry groups are isomorphic to a doubling of symmetric group algebras.

Different generator choices yield non-isomorphic quantum isometry groups for S_3.

The quantum isometry groups correspond to certain non-commutative Hopf algebras of dimension 12.

Abstract

We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of S_n on the choice of generators in the case n=3. We show that two different choices of generators lead to non-isomorphic quantum isometry groups which exhaust the list of non-commutative non-cocommutative semisimple Hopf algebras of dimension 12. This provides non-commutative geometric interpretation of these Hopf algebras.