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

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, and show that any CQG action on a compact Riemannian manifold which is isometric in the geometric sense of [12] automatically satisfies the isometry condition of the present article. We also prove for certain special class of metric measure spaces the existence of the universal object in the category of those compact quantum groups which act isometrically and in a measure-preserving way.

[1]  Adam G. Skalski,et al.  Quantum symmetry groups of C*‐algebras equipped with orthogonal filtrations , 2011, 1109.6184.

[2]  J. Quaegebeur,et al.  Isometric coactions of compact quantum groups on compact quantum metric spaces , 2010, 1007.0363.

[3]  Adam G. Skalski,et al.  Quantum Isometry Groups of 0- Dimensional Manifolds , 2008, 0807.4288.

[4]  Debashish Goswami,et al.  Quantum Group of Orientation preserving Riemannian Isometries , 2008, 0806.3687.

[5]  Jyotishman Bhowmick.  Quantum isometry group of the n-tori , 2008, 0803.4434.

[6]  J. Bichon.  Algebraic quantum permutation groups , 2007, 0710.1521.

[7]  Debashish Goswami,et al.  Quantum Isometry Groups: Examples and Computations , 2007, 0707.2648.

[8]  Debashish Goswami.  Quantum Group of Isometries in Classical and Noncommutative Geometry , 2007, 0704.0041.

[9]  P. Sołtan.  Quantum families of maps and quantum semigroups on finite quantum spaces , 2006, math/0610922.

[10]  J. Bichon,et al.  Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups , 2005, math/0502018.

[11]  T. Banica.  Quantum automorphism groups of homogeneous graphs , 2003, math/0311402.

[12]  T. Banica.  Quantum automorphism groups of small metric spaces , 2003, math/0304025.

[13]  J. Kustermans,et al.  Locally compact quantum groups in the von Neumann algebraic setting , 2000, math/0005219.

[14]  Julien Bichon,et al.  Quantum automorphism groups of finite graphs , 1999, math/9902029.

[15]  Shuzhou Wang,et al.  Quantum Symmetry Groups of Finite Spaces , 1998, math/9807091.

[16]  Ann Maes,et al.  Notes on Compact Quantum Groups , 1998, math/9803122.

[17]  J. Norris,et al.  Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds , 1997 .

[18]  Shuzhou Wang,et al.  Free products of compact quantum groups , 1995 .

[19]  P. Podlés.  Symmetries of quantum spaces. Subgroups and quotient spaces of quantumSU(2) andSO(3) groups , 1994, hep-th/9402069.

[20]  S. Woronowicz,et al.  Compact matrix pseudogroups , 1987 .