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

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→Rn for some n such that d0(f(x),f(y))=ϕ(d(x,y)) (where d0 is the Euclidean metric) for some homeomorphism ϕ of R+.As concrete examples, we obtain Wang's quantum permutation group Sn+ and also the free wreath product of Z2 by Sn+ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in [13].