Existence and examples of quantum isometry group for a class of compact metric spaces

TL;DR

This paper defines quantum isometry groups for certain compact metric spaces, proves their existence under specific conditions, and provides concrete examples including quantum permutation groups and free wreath products.

Contribution

It generalizes the concept of quantum isometry groups to a broader class of metric spaces and establishes their existence with explicit examples.

Findings

Existence of universal quantum isometry groups for specific metric spaces.

Construction of examples including Wang's quantum permutation group.

Identification of quantum isometry groups for spaces formed by topological joins of intervals.

Abstract

We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces $(X,d)$ which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on $(X,d)$. In fact, our existence theorem applies to a larger class, namely for any compact metric space $(X,d)$ which admits a one-to-one continuous map $f : X \raro \IR^n$ for some $n$ such that $d_0(f(x),f(y))=\phi(d(x,y))$ (where $d_0$ is the Euclidean metric) for some homeomorphism $\phi$ of $\IR^+$. As concrete examples, we obtain Wang's quantum permutation group $\cls_n^+$ and also the free wreath product of $\IZ_2$ by $\cls_n^+$ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in \cite{huang1}.