Algebras and Representation Theory

, Volume 6, Issue 2, pp 169–192 | Cite as

Quantum Real Projective Space, Disc and Spheres

  • Piotr M. Hajac
  • Rainer Matthes
  • Wojciech Szymański
Article

Abstract

We define the C*-algebra of quantum real projective space RP q 2, classify its irreducible representations, and compute its K-theory. We also show that the q-disc of Klimek and Lesniewski can be obtained as a non-Galois Z2-quotient of the equator Podleś quantum sphere. On the way, we provide the Cartesian coordinates for all Podleś quantum spheres and determine an explicit form of isomorphisms between the C*-algebras of the equilateral spheres and the C*-algebra of the equator one.

C*-representations K-theory 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [B-B86]
    Blackadar, B.: K-Theory for Operator AlgeÄras, Springer-Verlag, New York, 1986.Google Scholar
  2. [BM00]
    Brzeziński, T. and Majid, S.: Quantum geometry of algeÄra factorisations and coalgeÄra Äundles, Comm. Math. Phys. 213 (2000), 491–521.Google Scholar
  3. [CM00]
    Calow, D. and Matthes, R.: Covering and gluing of algeÄras and differential algeÄras, J. Geom. Phys. 32 (2000), 364–396.Google Scholar
  4. [DGH01]
    Dabrowski, 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.Google Scholar
  5. [DL90]
    DuÄois-Violette, M. and Launer, G.: The quantum group of a non-degenerate Äilinear form, Phys. Lett. B 245 (1990), 175–177.Google Scholar
  6. [DS88]
    Dunford, N. and Schwartz, J. T.: Linear Operators Part I: General Theory, Wiley, New York, 1988.Google Scholar
  7. [H-PM96]
    Hajac, P. M.: Strong connections on quantum principal Äundles, Comm. Math. Phys. 182 (1996), 579–617.Google Scholar
  8. [KR97]
    Kadison, R. V. and Ringrose, J. R.: Fundamentals of the Theory of Operator AlgeÄras, Volume I: Elementary Theory, Volume II: Advanced Theory, Grad. Stud. Math. 15, 16, Amer. Math. Soc., Providence, 1997.Google Scholar
  9. [K-M78]
    KarouÄi, M.: K-Theory, Springer-Verlag, New York, 1978.Google Scholar
  10. [KL92]
    Klimek, S. and Lesniewski, A.: Quantum Riemann surfaces I. The unit disc, Comm. Math. Phys. 146 (1992), 103–122.Google Scholar
  11. [KL93]
    Klimek, S. and Lesniewski, A.: A two-parameter quantum deformation of the unit disc J. Funct. Anal. 115 (1993), 1–23.Google Scholar
  12. [KS97]
    Klimyk, A. and Schmüdgen, K.: Quantum Groups and their Representations, Springer-Verlag, Berlin, 1997.Google Scholar
  13. [MNW91]
    Masuda, T., Nakagami, Y. and WatanaÄe, J.: Noncommutative differential geometry on the quantum two sphere of Podleś. I: An algeÄraic viewpoint, K-Theory 5 (1991), 151–175.Google Scholar
  14. [M-S93]
    Montgomery, S.: Hopf AlgeÄras and their Actions on Rings, Reg. Conf. Ser. Math. 82, Amer. Math. Soc., Providence, 1993.Google Scholar
  15. [NN94]
    Nagy, G. and Nica, A.: On the ‘quantum disc’ and a ‘non-commutative circle’, In: R. Curto and P. E. T. Jorgensen (eds), AlgeÄraic Methods in Operator Theory, Birkhäuser, Basel, 1994, pp. 276–290.Google Scholar
  16. [P-P87]
    Podleś, P.: Quantum spheres, Lett. Math. Phys. 14 (1987), 193–202.Google Scholar
  17. [P-P89]
    Podleś, P.: Differential calculus on quantum spheres, Lett. Math. Phys. 18 (1989), 107–119.Google Scholar
  18. [S-K90]
    Schmüdgen, K.: UnÄounded Operator AlgeÄras and Representation Theory, Birkhäuser, Basel, 1990.Google Scholar
  19. [S-A91]
    Sheu, A. J.-L.: Quantization of the Poisson SU(2) and its Poisson homogeneous space– the 2-sphere, Comm. Math. Phys. 135 (1991), 217–232.Google Scholar
  20. [W-Y90]
    Watatani, Y.: Index for C*-suÄalgeÄras, Mem. Amer. Math. Soc. 424 83 (1990).Google Scholar
  21. [W-NE93]
    Wegge-Olsen, N. E.: K-Theory and C*-AlgeÄras, Oxford Univ. Press, Oxford, 1993.Google Scholar
  22. [W-SL91]
    Woronowicz, S. L.: New quantum deformation of SL(2, ℂ). Hopf algeÄra level, Rep. Math. Phys. 30 (1991), 259–269.Google Scholar
  23. [WZ94]
    Woronowicz, S. L. and Zakrzewski, S.: Quantum deformations of the Lorentz group. The Hopf *-algeÄra level, Compositio Math. 90 (1994), 211–243.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Piotr M. Hajac
    • 1
    • 2
  • Rainer Matthes
    • 3
    • 4
  • Wojciech Szymański
    • 5
  1. 1.Polish Academy of SciencesMathematical InstituteWarsawPoland
  2. 2.Department of Mathematical Methods in PhysicsWarsaw UniversityWarsawPoland
  3. 3.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  4. 4.Institute of Theoretical PhysicsLeipzig UniversityLeipzigGermany
  5. 5.School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia

Personalised recommendations