Advertisement

Quantum Groups pp 197-209 | Cite as

Induced representations and tensor operators for quantum groups

  • L. C. Biedenharn
  • M. A. Lohe
II. Quantum Groups, Symmetries Of Dynamical Systems And Conformal Field Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1510)

Abstract

The analog of the Borel-Weil construction of irreducible representations as holomorphic sections of holomorphic line bundles is constructed for quantum groups and applied to Uq(2) and Uq(3). The concept of a tensor operator for a quantum group and the corresponding q-analog to the generalized Wigner-Eckart theorem are developed and discussed with examples.

Keywords

Quantum Group Holomorphic Section Tensor Operator Holomorphic Line Bundle 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    E. Sklyanin, L. Takhtajan, and L. Faddeev, Teor. Math. Phys. 40 (1979), 194.CrossRefGoogle Scholar
  2. [2]
    P. Kulish, N. Y. Reshetikhin, Zap. Nauch. Seminarov LOMI 101 (1981), 101; ibid., J.Soviet Math. 23 (1983), 2435.MathSciNetGoogle Scholar
  3. [3]
    V. G. Drinfeld, Quantum Groups, Proc. ICM-86 (Berkeley) 1 (1986), 798; Sov. Math. Dokl. 36 (1988), 212.MathSciNetGoogle Scholar
  4. [4]
    L. Faddeev, Les Houches Lectures (1982) (1984), Amsterdam,Elsevier.Google Scholar
  5. [5]
    M. Jimbo, Lett. Math. Phys. 10 (1985 2), 63.MathSciNetCrossRefGoogle Scholar
  6. [6]
    L. C. Biedenharn, XVIIIth International Colloqium on Group Theoretical Methods in Physics, to be published by Springer-Verlag (Moscow, 4–9 June 1990)).Google Scholar
  7. [7]
    L. C. Biedenharn and M. Tarlini, Lett.Math.Phys. 20 (1990), 271.MathSciNetCrossRefGoogle Scholar
  8. [8]
    L. C. Biedenharn, J.Phys.A.Math.Gen. 22 (1989), L873.Google Scholar
  9. [9]
    A. J. Macfarlane, J.Phys.A.Math.Gen. 22 (1989), 4581.MathSciNetCrossRefGoogle Scholar
  10. [10]
    C.-P. Sun and H.-C. Fu, J.Phys.A.Math.Gen. 22 (1989), L983.Google Scholar
  11. [11]
    P. Kulish and E. Damaskinsky, J.Phys. A23 (1990), L415.Google Scholar
  12. [12]
    M. Chaichian and P. Kulish, Phys.Lett. 234B (1990), 72.MathSciNetCrossRefGoogle Scholar
  13. [13]
    M. Chaichian, P. Kulish and J. Lukierski, Phys.Lett. 237B (1990), 401.MathSciNetCrossRefGoogle Scholar
  14. [14]
    L. C. Biedenharn and M. A. Lohe, invited paper at the Rochester Conference (honoring Prof. S. Okubo), Proceedings, World Scientific, Singapore (4–5 May 1990), (to appear).Google Scholar
  15. [15]
    F. H. Jackson, Messenger Math. 33 (1909), 57–61; ibid, 62–64; Quart. J. Pure Appl. Math. 41 (1910), 192–203.Google Scholar
  16. [16]
    R. Askey and J. A. Wilson, Memoirs Amer. Math. Soc. 319 (1985).Google Scholar
  17. [17]
    S. C. Milne, Advances in Math. 72 (1988), 59–131.MathSciNetCrossRefGoogle Scholar
  18. [18]
    G. E. Andrews, Reg. Conf. Series in Math. no. 66, (Providence, R.I., AMS).Google Scholar
  19. [19]
    G. Gasper and M. Rahman, “Basic Hypergeometric Series”, Encyclopedia of Mathematics and Its Applications 35 (1990), (Cambridge University Press).Google Scholar
  20. [20]
    G. Lusztig, Adv. In Math. 70 (1988), 237.MathSciNetCrossRefGoogle Scholar
  21. [21]
    M. Rosso, Commun. Math. Phys. 117 (1989), 581.MathSciNetCrossRefGoogle Scholar
  22. [22]
    Y. Manin, Commun. Math. Phys. 123 (1989), 163.MathSciNetCrossRefGoogle Scholar
  23. [23]
    R. LeBlanc and D.J. Rowe, J. Math. Phys. 29 (1988), 767 and references cited here.MathSciNetCrossRefGoogle Scholar
  24. [24]
    M. A. Lohe and L. C. Biedenharn, “Inductive Construction of Representations of the Quantum Unitary Groups”, in preparation.Google Scholar
  25. [25]
    I. M. Gel'fand and A. V. Zelevinsky, Societé Math. de France, Astérisque, hors serie (1985), 117.Google Scholar
  26. [26]
    L. C. Biedenharn, A. Giovannini and J. D. Louck, J. Math. Phys. 8 (1967), 691.MathSciNetCrossRefGoogle Scholar
  27. [27]
    L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Encyclopedia of Mathematics and Its Applications 8 (1981), Addison Wesley; reprinted Cambridge University Press (1989).Google Scholar
  28. [28]
    A. N. Kirillov and N. Yu. Reshetikhin, Representations of the Algebra U q(sl(2)), q-Orthogonal Polynomials and Invariants of Links, Preprint, LOMI (1988).Google Scholar
  29. [29]
    M. Nomura, J. Math. Phys. 30 (1988), 2397; J. Math. Soc. Jap. 58 (1989), 2694; ibid 59 (1990), 439.MathSciNetCrossRefGoogle Scholar
  30. [30]
    L. Vaksman, Sov. Math. Dokl. 39 (1989), 467.MathSciNetGoogle Scholar
  31. [31]
    H. Ruegg, J. Math. Phys. 31 (1990), 1085.MathSciNetCrossRefGoogle Scholar
  32. [32]
    V. A. Groza, I.I. Kachurik and A. U. Klimyk, J. Math. Phys. 31 (1990), 2769.MathSciNetCrossRefGoogle Scholar
  33. [33]
    T. H. Koornwinder, Nederl. Acad. Wetensch. Proc. Ser. A92 (1989), 97.MathSciNetGoogle Scholar
  34. [34]
    A. N. Kirillov and N. Yu. Reshetikhin, Commun.Math.Phys. 134 (1991), 421–431.MathSciNetCrossRefGoogle Scholar
  35. [35]
    E. Witten, Nucl. Phys. B330 (1990), 285.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • L. C. Biedenharn
    • 1
  • M. A. Lohe
    • 1
  1. 1.Department of PhysicsDuke UniversityDurhamUSA

Personalised recommendations