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Quantum G-spaces and Heisenberg algebra

  • L. I. Korogodsky
  • L. L. Vaksman
I. Quantum Groups, Deformation Theory And Representation Theory
Part of the Lecture Notes in Mathematics book series (LNM, volume 1510)

Abstract

In this paper we construct an isomorphism between the quantum Heisenberg algebra and a quantum function algebra. Some applications to the representation theory of quantum groups SU (n, 1) and SU (n + 1) are given.

Keywords

Quantum Group Function Algebra Geometric Progression Weyl Algebra Geometric Realization 
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.

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References

  1. [1]
    Drinfeld V.G., Quantum groups, Proc. ICM-86 (Berkeley) 1 (1987), 798–820.MathSciNetGoogle Scholar
  2. [2]
    Jimbo N., Quantum R-matrix related to the generalized Toda system: an algebraic approach, Lect. Notes in Phys. 246 (1985), 335–360, Springer.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Reshetikhin N.Y., Takhtajan L.A., Faddeev L.D., Quantization of Lie groups and Lie algebras, Algebra Anal. 1 (1989), 178–206. (in Russian)MathSciNetGoogle Scholar
  4. [4]
    Berezin F.A., A general concept of quantization, Commun. Math. Phys. 40 (1975), 153–174.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Podleś P., Quantum spheres, Lett. Math. Phys. 14 (1987), 193–202.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Mimachi K., Nuomi M., Quantum 2-spheres and big q-Jacobi polynomials, Commun. Math. Phys. 128 (1990), 521–531.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Mimachi K., Nuomi M., Spherical functions on a family of quantum 3-spheres, Preprint (1990).Google Scholar
  8. [8]
    Hayashi T., Q-analogues of Clifford and Weyl algebras. Spinor and oscillator representation of quantum enveloping algebras, Commun. Math. Phys. 127 (1990), 129–144.CrossRefzbMATHGoogle Scholar
  9. [9]
    Pusz W., Woronowicz S.L., Twisted second quantization, Rep.Math.Phys. 27 no. 2 (1989), 231–257.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Biedenharn L.C., The quantum group SU q(2) and a q-analogue of the boson operator, J.Phys.A.:-Math.Gen. 22 (1989), L873–878.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Masuda T., Mimachi K., Nakagami Y., Nuomi M., Saburi Y., Ueno K., Unitary representations of the quantum groups SU q (1, 1), I.II.-Lett.Math.Phys. 19 (1990), 187–204.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Ueno K., Spectral analysis for the Casimir operator on the quantum group SU q(1,1), Proc. Japan Acad.,ser. A. 66 (1990), 42–44.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Birkhoff G.D., The generalized Riemann problem for linear differential and q-difference equations, Proc. Amer. Acad. Arts and Sci. 49 (1913), 521–568.CrossRefGoogle Scholar
  14. [14]
    Vilenkin N.Ya., Special functions and group representation theory., Transl. of Math. Monographs Amer.Math.Soc. 22 (1968).Google Scholar
  15. [15]
    Titchmarsch E.C., Eigenfunction expansions associated with second-order differential equitions.-Vol. 1, Oxford University Press, 1946.Google Scholar
  16. [16]
    Majid Sh., Quasi-triangular Hopf algebras and Yang-Baxter equations, Int.J.Mod.Phys.A 5 (1990), 1–91.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Koelink H.T., Koornwinder T.H., The Clebsch-Gordan coefficients for the quantum group SU q(2) and q-Hahn polynomials, Proc.Kon.Ned.Akad.van Wetensch A 92, no. 4 (1989), 443–456.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Kachurik I.I., Klimyk A.U., On Clebsch-Gordan coefficients of quantum algebra U q(SU2), Preprint of Inst. for Theor. Phys. ITP-89-51E (1989), Kiev.Google Scholar
  19. [19]
    Vaksman L.L., Q-analogues of Clebsch-Gordan coefficients and a function algebra of the quantum group SU (2), Soviet Math. Dokl. 306 (1989), 269–271. (in Russian)MathSciNetzbMATHGoogle Scholar
  20. [20]
    Vaksman L.L., Soibelman Ya.S., On algebras of functions on quantum group SU(N) and odd dimensional quantum spheres., Algebra Anal. 5 (1990), 101–120. (in Russian)MathSciNetGoogle Scholar
  21. [21]
    Vaksman L.L., Korogodsky L.I., Harmonic analysis on quantum hyperboloids, Preprint of Inst. for Theor. Phys. ITP-90-27P (1990). (in Russian)Google Scholar
  22. [22]
    Korogodsky L.I., Quantum projective spaces, spheres and hyperboloids, Preprint of Inst. for Theor. Phys. ITP-90-27P (1991), Kiev.Google Scholar
  23. [23]
    Jurco B., On coherent states for the simplest quantum groups, Lett.Math.Phys. 21 (1991), 51–58.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • L. I. Korogodsky
    • 1
  • L. L. Vaksman
    • 1
  1. 1.Department of MathematicsRostov State UniversityRostov-na-DounUSSR

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