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

2012/05/28 by Goswami, Debashish
#20G42 #58B34 #81R50 #81R60 #FOS: Mathematics #Metric Geometry (math.MG) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1205.6099

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 \IRⁿ 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 \IR⁺. As concrete examples, we obtain Wang's quantum permutation group \cls_n⁺ and also the free wreath product of \IZ₂ by \cls_n⁺ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in \citehuang1.

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