Goswami, Debashish
(2015)
*Existence and examples of quantum isometry groups for a class of compact metric spaces*
Advances in Mathematics, 280
.
pp. 340-359.
ISSN 0001-8708

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1016/j.aim.2015.03.024

## 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→R^{n} for some n such that d_{0}(f(x),f(y))= φ(d(x,y)) (where d_{0}( is the Euclidean metric) for some homeomorphism φ of R^{+}. As concrete examples, we obtain Wang's quantum permutation group S_{n}^{+} and also the free wreath product of Z_{2} by S_{n}^{+} as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in [13].

Item Type: | Article |
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Source: | Copyright of this article belongs to Elsevier Science. |

Keywords: | Quantum Isometry; Compact Quantum Group; Metric Space |

ID Code: | 102092 |

Deposited On: | 01 Feb 2018 04:02 |

Last Modified: | 01 Feb 2018 04:02 |

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