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Faithful actions of locally compact quantum groups on classical spaces

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Abstract

We construct examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on connected classical (noncompact) smooth manifolds. However, none of these actions can be isometric in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009), leading to the conjecture that the result obtained by Goswami and Joardar (Rigidity of action of compact quantum groups on compact, connected manifolds, 2013. arXiv:1309.1294) about nonexistence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the noncompact case as well.

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References

  1. Baaj, S., Skandalis, G., Vaes, S.: Non-semi-regular quantum groups coming from number theory. Commun. Math. Phys. 235(1), 139–167 (2003). doi:10.1007/s00220-002-0780-6

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Baaj, S., Vaes, S.: Double crossed products of locally compact quantum groups. J. Inst. Math. Jussieu 4(1), 135–173 (2005). doi:10.1017/S1474748005000034

    Article  MathSciNet  MATH  Google Scholar 

  3. Drinfel’d, V.G.: Quantum groups. Proc. Int. Cong. Math. 1(2), 798–820 (1987). (Berkeley, Calif., 1986)

    MATH  Google Scholar 

  4. Etingof, P., Walton, C.: Semisimple Hopf actions on commutative domains. Adv. Math. 251, 47–61 (2014). doi:10.1016/j.aim.2013.10.008

    Article  MathSciNet  MATH  Google Scholar 

  5. Goswami, D.: Quantum group of isometries in classical and noncommutative geometry. Commun. Math. Phys. 285(1), 141–160 (2009). doi:10.1007/s00220-008-0461-1

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Goswami, D., Joardar, S.: Rigidity of action of compact quantum groups on compact, connected manifolds (2013). Eprint. arXiv:1309.1294

  7. Huang, H.: Faithful compact quantum group actions on connected compact metrizable spaces. J. Geom. Phys. 70, 232–236 (2013). doi:10.1016/j.geomphys.2013.03.027

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Huang, H.: Invariant subsets under compact quantum group actions (2013). Eprint arXiv:1210.5782v2

  9. Jimbo, M.: A \(q\)-difference analogue of \(U(\mathfrak{g})\) and the Yang-Baxter equation. Lett. Math. Phys. 10(1), 63–69 (1985). doi:10.1007/BF00704588

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Kustermans, J., Vaes, S.: Locally compact quantum groups. Ann. Sci. École Norm. Sup. (4) 33(6), 837–934 (2000). doi:10.1016/0012-9593(00)01055-7

    Article  MathSciNet  MATH  Google Scholar 

  11. Kustermans, J., Vaes, S.: Locally compact quantum groups in the von Neumann algebraic setting. Math. Scand. 92(1), 68–92 (2003) http://www.mscand.dk/article.php?id=198

  12. Manin, Y.I.: Quantum Groups and Noncommutative Geometry. Université de Montréal, Centre de Recherches Mathématiques, Montreal (1988)

    MATH  Google Scholar 

  13. Masuda, T., Nakagami, Y., Woronowicz, S.L.: A \(C^*\)-algebraic framework for quantum groups. Int. J. Math 14(9), 903–1001 (2003). doi:10.1142/S0129167X03002071

    Article  MathSciNet  MATH  Google Scholar 

  14. Meyer, R., Roy, S., Woronowicz, S.L.: Homomorphisms of quantum groups. Münster J. Math. 5:1–24 (2012) http://nbn-resolving.de/urn:nbn:de:hbz:6-88399662599

  15. Sołtan, P.M.: Examples of non-compact quantum group actions. J. Math. Anal. Appl. 372(1), 224–236 (2010). doi:10.1016/j.jmaa.2010.06.045

    Article  MathSciNet  MATH  Google Scholar 

  16. Sołtan, P.M.: New quantum “\(az+b\)” groups. Rev. Math. Phys. 17(3), 313–364 (2005). doi:10.1142/S0129055X05002339

    Article  MathSciNet  MATH  Google Scholar 

  17. Vaes, S., Vainerman, L.: Extensions of locally compact quantum groups and the bicrossed product construction. Adv. Math. 175(1), 1–101 (2003). doi:10.1016/S0001-8708(02)00040-3

    Article  MathSciNet  MATH  Google Scholar 

  18. Vaes, S., Van Daele, A.: Hopf \(C^*\)-algebras. Proc. London Math. Soc. (3). 82(2), 337–384 (2001). doi:10.1112/S002461150101276X

    Article  MATH  Google Scholar 

  19. Woronowicz, S.L.: Compact quantum groups, Symétries quantiques (Les Houches, 1995), North-Holland. Amsterdam, pp. 845–884 (1998)

  20. Woronowicz, S.L.: From multiplicative unitaries to quantum groups. Int. J. Math. 7(1), 127–149 (1996). doi:10.1142/S0129167X96000086

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Statistics and Mathematics Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India

    Debashish Goswami

  2. School of Mathematical Sciences, National Institute of Science Education and Research Bhubaneswar, HBNI, Jatni, 752050, India

    Sutanu Roy

Authors
  1. Debashish Goswami
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  2. Sutanu Roy
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Correspondence to Debashish Goswami.

Additional information

Debashish Goswamir was partially supported by Swarnajayanti Grant and J.C. Bose fellowship given by D.S.T., Government of India. Sutanu Roy was partially supported by the visiting scientist program at the Indian Statistical Institute, Kolkata and Inspire faculty award given by D.S.T., Government of India.

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Goswami, D., Roy, S. Faithful actions of locally compact quantum groups on classical spaces. Lett Math Phys 107, 1375–1390 (2017). https://doi.org/10.1007/s11005-017-0951-1

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