Quantum Isometry Groups of Classical and Quantum Spheres

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


We begin by explicitly computing the quantum isometry groups for the classical spheres and show that these coincide with the commutative \(C^*\) algebra of continuous functions on their classical isometry groups. The rest of the chapter is devoted to the computation of the quantum group of orientation preserving isometries for two different families of spectral triples on the Podles’ spheres, one constructed by Dabrowski et al and the other by Chakraborty and Pal.


Quantum Group Strong Operator Topology Compact Quantum Group Quantum Sphere Spectral Triple 
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.
    Helgason, S.: Topics in Harmonic Analysis on Homogeneous Spaces. Birkhauser, Boston, Basel, Stuttgart (1981)MATHGoogle Scholar
  2. 2.
    Etingof, P., Goswami, D., Mandal, A., Walton, C.: Hopf coactions on commutative algebras generated by a quadratically independent comodule. Commun. Algebra (To appear). arXiv:1507.08486
  3. 3.
    Schmudgen, K., Wagner, E.: Representation of cross product algebras of Podles’ quantum spheres. J. Lie Theory 17(4), 751–790 (2007)MATHGoogle Scholar
  4. 4.
    Schmudgen, K., Wagner, E.: Dirac operators and a twisted cyclic cocycle on the standard Podles’ quantum sphere. arXiv:0305051v2
  5. 5.
    Dabrowski, L., D’Andrea, F., Landi, G., Wagner, E.: Dirac operators on all Podles’ quantum spheres. J. Noncommut. Geom. 1, 213–239 (2007)MATHMathSciNetGoogle Scholar
  6. 6.
    Bhowmick, J.: Quantum isometry groups. Ph.D thesis, Indian Statistical Institute (2010). arXiv:0907.0618
  7. 7.
    Bhowmick, J., Goswami, D.: Quantum isometry groups of the Podles’ Spheres. J. Funct. Anal. 258, 2937–2960 (2010)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Soltan, P.M.: Quantum \(SO(3)\) groups and quantum group actions on \(M_2\). J. Noncommut. Geom. 4(1), 1–28 (2010)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Chakraborty, P.S., Pal, A.: Spectral triples and associated Connes-de Rham complex for the quantum SU(2) and the quantum sphere. Commun. Math. Phys. 240(3), 447–456 (2003)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Davidson, K.R.: \(C^*\) Algebras by Examples. Hindustan Book Agency (1996)Google Scholar

Copyright information

© Springer (India) Pvt. Ltd 2016

Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteKolkataIndia

Personalised recommendations