Skip to main content

Quantum G-spaces and Heisenberg algebra

  • I. Quantum Groups, Deformation Theory And Representation Theory
  • Conference paper
  • First Online:
Quantum Groups

Part of the book series: Lecture Notes in Mathematics ((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.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Drinfeld V.G., Quantum groups, Proc. ICM-86 (Berkeley) 1 (1987), 798–820.

    MathSciNet  Google Scholar 

  2. Jimbo N., Quantum R-matrix related to the generalized Toda system: an algebraic approach, Lect. Notes in Phys. 246 (1985), 335–360, Springer.

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  4. Berezin F.A., A general concept of quantization, Commun. Math. Phys. 40 (1975), 153–174.

    Article  MathSciNet  MATH  Google Scholar 

  5. Podleś P., Quantum spheres, Lett. Math. Phys. 14 (1987), 193–202.

    Article  MathSciNet  MATH  Google Scholar 

  6. Mimachi K., Nuomi M., Quantum 2-spheres and big q-Jacobi polynomials, Commun. Math. Phys. 128 (1990), 521–531.

    Article  MathSciNet  MATH  Google Scholar 

  7. Mimachi K., Nuomi M., Spherical functions on a family of quantum 3-spheres, Preprint (1990).

    Google Scholar 

  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.

    Article  MATH  Google Scholar 

  9. Pusz W., Woronowicz S.L., Twisted second quantization, Rep.Math.Phys. 27 no. 2 (1989), 231–257.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  14. Vilenkin N.Ya., Special functions and group representation theory., Transl. of Math. Monographs Amer.Math.Soc. 22 (1968).

    Google Scholar 

  15. Titchmarsch E.C., Eigenfunction expansions associated with second-order differential equitions.-Vol. 1, Oxford University Press, 1946.

    Google Scholar 

  16. Majid Sh., Quasi-triangular Hopf algebras and Yang-Baxter equations, Int.J.Mod.Phys.A 5 (1990), 1–91.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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. Korogodsky L.I., Quantum projective spaces, spheres and hyperboloids, Preprint of Inst. for Theor. Phys. ITP-90-27P (1991), Kiev.

    Google Scholar 

  23. Jurco B., On coherent states for the simplest quantum groups, Lett.Math.Phys. 21 (1991), 51–58.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Petr P. Kulish

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag

About this paper

Cite this paper

Korogodsky, L.I., Vaksman, L.L. (1992). Quantum G-spaces and Heisenberg algebra. In: Kulish, P.P. (eds) Quantum Groups. Lecture Notes in Mathematics, vol 1510. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0101178

Download citation

  • DOI: https://doi.org/10.1007/BFb0101178

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55305-2

  • Online ISBN: 978-3-540-47020-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics