Quantum Isometry Groups of Discrete Quantum Spaces

  • Debashish Goswami
  • Jyotishman Bhowmick
Part of the Infosys Science Foundation Series book series (ISFS)


We show that the definitions of quantum symmetry groups for finite metric spaces and graphs given by Banica and Bichon can be viewed as quantum isometry groups in our sense. Next, we prove that the quantum group of orientation preserving isometries of a spectral triple on some approximately finite dimensional \( C^* \) algebras arise as the inductive limit of the quantum group of orientation preserving isometries of suitable spectral triples on the constituent finite dimensional algebras.


Dirac Operator Quantum Group Inductive Limit Finite Graph Noncommutative Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Wang, S.: Quantum symmetry groups of finite spaces. Commun. Math. Phys. 195, 195–211 (1998)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bichon, J.: Quantum automorphism groups of finite graphs. Proc. Am. Math. Soc. 131(3), 665–673 (2003)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Banica, T.: Quantum automorphism groups of small metric spaces. Pac. J. Math. 219(1), 27–51 (2005)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Banica, T.: Quantum automorphism groups of homogeneous graphs. J. Funct. Anal. 224(2), 243–280 (2005)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Banica, T., Bichon, J.: Quantum automorphism groups of vertex-transitive graphs of order \(\le 11\). J. Algebraic Comb. 26, 83–105 (2007)CrossRefMATHGoogle Scholar
  6. 6.
    Banica, T., Bichon, J., Chenevier, G.: Graphs having no quantum symmetry. Ann. Inst. Fourier 57, 955–971 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Banica, T., Bichon, J.: Quantum groups acting on 4 points. J. Reine Angew. Math. 626, 75–114 (2009)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Banica, T., Bichon, J., Collins, B.: Quantum permutation groups: a survey. noncommutative harmonic analysis with applications to probability. Polish Acad. Sci. Inst. Math. 78, 13–34, Banach Center Publications, Warsaw (2007)Google Scholar
  9. 9.
    Banica, T., Moroianu, S.: On the structure of quantum permutation groups. Proc. Am. Math. Soc. 135(1), 21–29 (2007) (electronic)Google Scholar
  10. 10.
    Bichon, J.: Algebraic quantum permutation groups. Asian-Eur. J. Math. 1(1), 1–13 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Christensen, E., Ivan, C.: Spectral triples for AF \(C^*\)-algebras and metrics on the Cantor set. J. Oper. Theory 56(1), 17–46 (2006)MATHMathSciNetGoogle Scholar
  12. 12.
    Christensen, E., Ivan, C.: Sums of two dimensional spectral triples. Math. Scand. 100(1), 35–60 (2007)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Skalski, A., Soltan, P.M.: Projective limits of quantum symmetry groups and the doubling construction for Hopf algebras. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17(2), 1450012 (27 pages) (2014)Google Scholar
  14. 14.
    Rieffel, M.A.: Metrics on state spaces. Doc. Math. 4, 559–600 (1999)MATHMathSciNetGoogle Scholar
  15. 15.
    Bhowmick, J., Goswami, D., Skalski, A.: Quantum isometry groups of 0-dimensional manifolds. Trans. Am. Math. Soc. 363, 901–921 (2011)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Banica, T., Skalski, A.: Quantum symmetry groups Of \(C^*\)-algebras equipped with orthogonal filtrations. Proc. Lond. Math. Soc. 3(5), 980–1004, 106 (2013)Google Scholar
  17. 17.
    Davidson, K.R.: \( C^* \) Algebras by Examples. Hindustan Book Agency (1996)Google Scholar
  18. 18.
    Connes, A.: Compact metric spaces, Fredholm modules, and hyperfiniteness. Ergodic Theory Dynam. Syst. 9(2), 207–220 (1989)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer (India) Pvt. Ltd 2016

Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteKolkataIndia

Personalised recommendations