Skip to main content
SpringerLink
Log in

A Locally Trivial Quantum Hopf Fibration

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

The irreducible *-representations of the polynomial algebra \(\mathcal{O}(S^{3}_{pq})\) of the quantum3-sphere introduced by Calow and Matthes are classified. The K-groups of its universal C *-algebra are shown to coincide with their classical counterparts. The U(1)-action on \(\mathcal{O}(S^{3}_{pq})\) corresponding for p=1=q to the classical Hopf fibration is proven to be Galois (free). The thus obtained locally trivial Hopf–Galois extension is shown to be equivariantly projective (admitting a strong connection) and non-cleft. The latter is proven by determining an appropriate pairing of cyclic cohomology and K-theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Brzezinski, T. and Hajac, P. M.: The Chern–Galois character, C. R. Math. Acad. Sci. Paris 338(2) (2004), 113–116.

    MathSciNet  MATH  Google Scholar 

  2. Brzezinski, T., Hajac, P. M., Matthes, R. and Szymanski, W.: The Chern character and index theory for principal extensions of noncommutative algebras, see http://www.fuw.edu.pl/~pmh for a preliminary version.

  3. Brzezinski, T. and Majid, S.: Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157 (1993), 591–638; Erratum 167 (1995), 235.

    Article  MathSciNet  MATH  Google Scholar 

  4. Brzezinski, T. and Majid, S.: Quantum geometry of algebra factorisations and coalgebra bundles, Comm. Math. Phys. 213 (2000), 491–521.

    Article  MathSciNet  MATH  Google Scholar 

  5. Budzynski, R. J. and Kondracki, W.: Quantum principal fibre bundles: Topological aspects, Rep. Math. Phys. 37 (1996), 365–385.

    Article  MathSciNet  Google Scholar 

  6. Calow, D. and Matthes, R.: Covering and gluing of algebras and differential algebras, J. Geom. Phys. 32 (2000), 364–396.

    Article  MathSciNet  MATH  Google Scholar 

  7. Calow, D. and Matthes, R.: Connections on locally trivial quantum principal fibre bundles, J. Geom. Phys. 41 (2002), 114–165.

    Article  MathSciNet  MATH  Google Scholar 

  8. Cartan, H. and Eilenberg, S.: Homological Algebra, Princeton University Press, Princeton, 1956.

    MATH  Google Scholar 

  9. D \(\mbox{\c{a}}\) browski, L.: The garden of quantum spheres, In: Noncommutative Geometry and Quantum Groups (Warsaw, 2001), Banach Center Publ. 61, Polish Acad. Sci., Warsaw, 2003, pp. 37–48.

  10. D \(\mbox{\c{a}}\) browski, L., Grosse, H. and Hajac, P.M.: Strong connections and Chern–Connes pairing in the Hopf–Galois theory, Comm. Math. Phys. 220 (2001), 301–331.

  11. D \(\mbox{\c{a}}\) browski, L., Hajac, P.M. and Siniscalco, P.: Explicit Hopf–Galois description of \(\mathrm{SL}_{\mathrm{e}_{\frac{2\pi i}{3}}}(2)\) -induced Frobenius homomorphisms, In: D. Kastler, M. Rosso and T. Schucker (eds), Enlarged Proceedings of the ISI GUCCIA Workshop on Quantum Groups, Noncommutative Geometry and Fundamental Physical Interactions, Nova Science Pub, Inc., Commack–New York, 1999, pp. 279–298.

  12. Durdevic, M.: Geometry of quantum principal bundles. I, Comm. Math. Phys. 175 (1996), 457–520.

    Article  MathSciNet  Google Scholar 

  13. Ellwood, D. A.: A new characterisation of principal actions, J. Funct. Anal. 173 (2000), 49–60.

    Article  MATH  MathSciNet  Google Scholar 

  14. Hajac, P. M. and Majid, S.: Projective module description of the q-monopole, Comm. Math. Phys. 206 (1999), 247–264.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hajac, P. M., Matthes, R. and Szymanski, W.: Quantum real projective space, disc and sphere, Algebr. Represent. Theory 6 (2003), 169–192.

    Article  MathSciNet  MATH  Google Scholar 

  16. Hajac, P. M., Matthes, R. and Szymanski, W.: Chern numbers for two families of noncommutative Hopf fibrations, C. R. Math. Acad. Sci. Paris 336 (2003), 925–930.

    MathSciNet  MATH  Google Scholar 

  17. Hong, J. H. and Szymanski, W.: Quantum spheres and projective spaces as graph algebras, Comm. Math. Phys. 232 (2002), 157–188.

    Article  MathSciNet  MATH  Google Scholar 

  18. Hong, J. H. and Szymanski, W.: The primitive ideal space of the C *-algebras of infinite graphs, J. Math. Soc. Japan 56 (2004), 45–64.

    MathSciNet  MATH  Google Scholar 

  19. Kadison, R. V. and Ringrose, J. R.: Fundamentals of the Theory of Operator Algebras. Volume I: Elementary Theory, Grad. Stud. Math. 15, Amer. Math. Soc., Providence, 1997.

  20. Klimek, S. and Lesniewski, A.: A two-parameter quantum deformation of the unit disc, J. Funct. Anal. 115 (1993), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  21. Klimyk, A. and Schmüdgen, K.: Quantum Groups and their Representations, Springer-Verlag, Berlin, 1997.

    MATH  Google Scholar 

  22. Kumjian, A. and Pask, D.: Higher rank graph C *-algebras, New York J. Math. 6 (2000), 1–20.

    MathSciNet  MATH  Google Scholar 

  23. Masuda, T., Nakagami, Y. and Watanabe, J.: Noncommutative differential geometry on the quantum SU(2). I. An algebraic viewpoint, K-Theory 4 (1990), 157–180.

    Article  MathSciNet  MATH  Google Scholar 

  24. Matsumoto, K.: Non-commutative three dimensional spheres, Japan J. Math. 17 (1991), 333–356.

    MATH  MathSciNet  Google Scholar 

  25. Matsumoto, K.: Non-commutative three dimensional spheres II – non-commutative Hopf fibering, Yokohama Math. J. 38 (1991), 103–111.

    MATH  MathSciNet  Google Scholar 

  26. Matsumoto, K. and Tomiyama, J.: Non-commutative lens spaces, J. Math. Soc. Japan 44 (1992), 13–41.

    Article  MathSciNet  MATH  Google Scholar 

  27. Montgomery, S.: Hopf Algebras and Their Actions on Rings, Regional Conf. Ser. Math. 82, Amer. Math. Soc., Providence, 1993.

  28. Naber, G. L.: Topology, Geometry, and Gauge Fields Foundations, Texts Appl. Math. 25, Springer-Verlag, New York, 1997.

    MATH  Google Scholar 

  29. Podles, P.: Quantum spheres, Lett. Math. Phys. 14 (1987), 193–202.

    Article  MATH  MathSciNet  Google Scholar 

  30. Podles, P.: Differential calculus on quantum spheres, Lett. Math. Phys. 18 (1989), 107–119.

    Article  MATH  MathSciNet  Google Scholar 

  31. Pedersen, G. K.: Pullback and pushout constructions in C *-algebra theory, J. Funct. Anal. 167 (1999), 243–344.

    Article  MATH  MathSciNet  Google Scholar 

  32. Pflaum, M. J.: Quantum groups on fibre bundles, Comm. Math. Phys. 166 (1994), 279–315.

    Article  MATH  MathSciNet  Google Scholar 

  33. Raeburn, I., Sims, A. and Yeend, T.: Higher-rank graphs and their C *-algebras, Proc. Edinburgh Math. Soc. (2) 46 (2003), 99–115.

    Article  MathSciNet  MATH  Google Scholar 

  34. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972.

    Google Scholar 

  35. Rieffel, M. A.: Noncommutative tori – a case study of noncommutative differentiable manifolds, In: Geometric and Topological Invariants of Elliptic Operators (Brunswick, ME, 1988), Contemp. Math. 105, Amer. Math. Soc., Providence, RI, 1990, pp. 191–211.

    Google Scholar 

  36. Schauenburg, P.: Bigalois objects over the Taft algebras, Israel J. Math. 115 (2000), 101–123.

    MATH  MathSciNet  Google Scholar 

  37. Schneider, H.-J.: Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), 167–195.

    MATH  MathSciNet  Google Scholar 

  38. Wegge-Olsen, N. E.: K-Theory and C * -Algebras, Oxford University Press, Oxford, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität München, Theresienstr. 39, Munich, 80333, Germany and Instytut Matematyczny, Polska Akademia Nauk, ul. Sniadeckich 8, Warsaw, 00-950, Poland

    Piotr M. Hajac

  2. Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Hoza 74, Warsaw, 00-682, Poland

    Piotr M. Hajac & Rainer Matthes

  3. Fachbereich Physik der TU Clausthal, Leibnizstr. 10, D-38678, Clausthal-Zellerfeld, Germany

    Rainer Matthes

  4. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, 2308, Australia

    Wojciech Szymanski

Authors
  1. Piotr M. Hajac
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rainer Matthes
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wojciech Szymanski
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Mathematics Subject Classifications (2000)

16W30, 46L87.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hajac, P.M., Matthes, R. & Szymanski, W. A Locally Trivial Quantum Hopf Fibration. Algebr Represent Theor 9, 121–146 (2006). https://doi.org/10.1007/s10468-005-3080-y

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-005-3080-y

Keywords

Search

Navigation