Quantum isometry group of dual of finitely generated discrete groups - II

Arnab Mandal

doi:10.5802/ambp.361

Abstract

As a continuation of the programme of [13], we carry out explicit computations of ℚ(Γ,S), the quantum isometry group of the canonical spectral triple on C r * (Γ) coming from the word length function corresponding to a finite generating set S, for several interesting examples of Γ not covered by the previous work [13]. These include the braid group of 3 generators, ℤ 4 *n etc. Moreover, we give an alternative description of the quantum groups H s + (n,0) and K n + (studied in [3], [4]) in terms of free wreath product. In the last section we give several new examples of groups for which ℚ(Γ) turns out to be a doubling of C * (Γ).

Highlights

It is a very important and interesting problem in the theory of quantum groups and noncommutative geometry to study ‘quantum symmetries’ of various classical and quantum structures
S.Wang pioneered this by defining quantum permutation groups of finite sets and quantum automorphism groups of finite dimensional matrix algebras
In [13] together with Goswami we studied the quantum isometry groups of such spectral triples in a systematic and unified way

Summary

Introduction

It is a very important and interesting problem in the theory of quantum groups and noncommutative geometry to study ‘quantum symmetries’ of various classical and quantum structures. In [11] Goswami extended such constructions to the set-up of possibly infinite dimensional C∗-algebras, and more interestingly, that of spectral triples a la Connes [10], by defining and studying quantum isometry groups of spectral triples. This led to the study of such quantum isometry groups by many authors including Goswami, Bhowmick, Skalski, Banica, Bichon, Soltan, Das, Joardar and others. In the last section we present more examples of groups as in [14], [18], Section 5 of [13] where Q(Γ, S) turns out to be a doubling of C∗(Γ)

Preliminaries

Compact quantum groups and free wreath product

Some facts about quantum isometry groups

QISO computation of the braid group

A B 0 0 E F

Lamplighter Group