Nonexistence of Genuine Smooth CQG Coactions on Classical Connected Manifolds

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


The question of existence of coactions of Hopf algebras and compact quantum groups on connected Riemannian manifolds is investigated. We give a very brief outline of the proof of some important cases of the conjecture as well as a number of examples supporting the conjecture.


Hopf Algebra Quantum Group Isometric Action Compact Quantum Group Smooth Vector Field 


  1. 1.
    Goswami, D., Joardar, S.: Rigidity of action of compact quantum groups on compact connected manifolds. arXiv:1309.1294v4
  2. 2.
    Goswami, D., Joardar, S.: An averaging trick for smooth actions of compact quantum groups on manifolds. Indian J. Pure Appl. Math. 46(4), 477–488 (2015)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Krahmer, U., Tabiri, A.: The nodal cubic is a quantum homogeneous space. arXiv:1604.00199
  4. 4.
    Podles’, P.: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum \( SU(2)\) and \( SO(3)\) groups. Commun. Math. Phys. 170, 1–20 (1995)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Podles’, P.: Quantum spheres. Lett. Math. Phys. 14, 193–202 (1987)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Etingof, P., Walton, C.: Semisimple Hopf actions on commutative domains. Adv. Math. 251, 47–61 (2014)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Huang, H.: Faithful compact quantum group actions on connected compact metrizable spaces. J. Geom. Phys. 70, 232–236 (2013)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Woronowicz, S.L., Zakrzewski, S.: Quantum \( ax + b \) group. Rev. Math. Phys. 14, Nos 7 & 8, 797–828 (2002)Google Scholar
  9. 9.
    Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. Sci. Ecole Norm. Sup. (4) 33(6), 837–934 (2000)Google Scholar
  10. 10.
    Baaj, S., Skandalis, G.: Duality for locally compact quantum groups, Mathematisches Forschungsinstitut Oberwolfach, Tagungsbericht 46/1991, \(C^*\)-algebren (1991)Google Scholar
  11. 11.
    Vaes, S., Vainerman, L.: Extensions of locally compact quantum groups and the bicrossed product construction. Adv. Math. 175(1), 1–101 (2003)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Walton, C., Wang, X.: On quantum groups associated to non-Noetherian regular algebras of dimension 2. arXiv:1503.0918
  13. 13.
    Kolsinsky, A.A.: Differential Manifolds. Academic Press (1993)Google Scholar
  14. 14.
    Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, Graduate Texts in Mathematics, vol. 137, 2nd edn. Springer, New York (2001)Google Scholar
  15. 15.
    Goswami, D.: Existence and examples of quantum isometry groups for a class of compact metric spaces. Adv. Math. 280, 340–359 (2015)CrossRefMATHMathSciNetGoogle Scholar

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© Springer (India) Pvt. Ltd 2016

Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteKolkataIndia

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