Definition and Existence of Quantum Isometry Groups

Part of the Infosys Science Foundation Series book series (ISFS)


Under some reasonable assumptions on a spectral triple \( (\mathcal{A}^{\infty }, \mathcal{H}, D) \) (which we call admissibility), we prove the existence of a universal object in the category of compact quantum groups admitting coaction on the closure of \( \mathcal{A}^{\infty } \) which commutes with the (noncommutative) Laplacian. This universal object is called the quantum isometry group w.r.t. the Laplacian. Moreover, we discuss analogous formulations of the quantum group of orientation (and volume or a given real structure) preserving isometries. Sufficient conditions under which the action of the quantum isometry group keeps the \( C^* \) algebra invariant and is a \( C^* \) action are given. We also mention some sufficient conditions for the existence of the quantum group of orientation preserving isometries without fixing a choice of the ‘volume-form’.


Dirac Operator Quantum Group Isometric Action Compact Type Compact Quantum Group 
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.
    Goswami, D.: Quantum group of isometries in classical and non commutative geometry. Comm. Math. Phys. 285(1), 141–160 (2009)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Wang, S.: Quantum symmetry groups of finite spaces. Comm. Math. Phys. 195, 195–211 (1998)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Connes, A.; Moscovici, H.: Type III and spectral triples, traces in number theory, geometry and quantum fields. Aspects Mathematics, vol. E38, pp. 57–71. Friedr. Vieweg, Wiesbaden (2008)Google Scholar
  4. 4.
    Woronowicz, S.L.: Pseudogroups, pseudospaces and Pontryagin duality. In: Proceedings of the International Conference on Mathematical Physics, Lecture Notes in Physics, vol. 116, pp. 407–412. Lausane (1979)Google Scholar
  5. 5.
    Soltan, P.M.: Quantum families of maps and quantum semigroups on finite quantum spaces, preprint. J. Geom. Phys. 59(3), 354–368 (2009)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chakraborty, P.S., Pal, A.: Equivariant spectral triples on the quantum \(SU(2)\) group. K. Theory 28, 107–126 (2003)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Dabrowski, L., D’Andrea, F., Landi, G., Wagner, E.: Dirac operators on all Podles’ quantum spheres. J. Noncomm. Geom. 1, 213–239 (2007)MATHMathSciNetGoogle Scholar
  8. 8.
    Bhowmick, J., Goswami, D.: Quantum group of orientation preserving Riemannian isometries. J. Funct. Anal. 257(8), 2530–2572 (2009)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Fröhlich, J., Grandjean, O., Recknagel, A.: Supersymmetric quantum theory and non-commutative geometry. Comm. Math. Phys. 203(1), 119–184 (1999)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Dabrowski, L.: Geometry of quantum spheres. J. Geom. Phys. 56(1), 86–107 (2006)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer (India) Pvt. Ltd 2016

Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteKolkataIndia

Personalised recommendations