Skip to main content
Book cover

Integrable Systems, Quantum Groups, and Quantum Field Theories pp 83–94Cite as

Quantum Algebras and Quantum Groups in q-Special Function Theory

  • Chapter

  • 540 Accesses

Part of the NATO ASI Series book series (ASIC,volume 409)

Abstract

The quantum algebra and quantum group interpretation of q-special functions is reviewed. Taking the algebra U q (sl(2)) as example, we shall see how its representation theory can be used to make advances in the study of the q-hypergeometric series 2ø1(a, b; c; q, z).

Keywords

  • Matrix Element
  • Quantum Group
  • Addition Formula
  • Quantum Algebra
  • Basic Hypergeometric Series

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.

Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada and the Fonds FCAR of Québec.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-94-011-1980-1_4
  • Chapter length: 12 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   39.99
Price excludes VAT (USA)
  • ISBN: 978-94-011-1980-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   54.99
Price excludes VAT (USA)
Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Miller, W., Lie Theory and Special Functions, (Academic Press, New York, 1968)

    MATH  Google Scholar 

  2. Agarwal, A.K., Kalnins, E.G. and Miller, W., Canonical equations and symmetry techniques for q-series, SIAM J. Math. Anal. 18, 1519–1538 (1987)

    MathSciNet  MATH  Google Scholar 

  3. Floreanini, R. and Vinet, L., q-Orthogonal polynomials and the oscillator quantum group, Lett. Math. Phys. 22, 45–54 (1991)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  4. Floreanini, R. and Vinet, L., The metaplectic representation of su q (1,1) and the q-Gegenbauer polynomials, J. Math. Phys. 33, 1358–1363 (1992)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  5. Floreanini, R. and Vinet, L., q-Conformal quantum mechanics and q-special functions, Phys. Lett. B 277, 442–446 (1992)

    CrossRef  MathSciNet  ADS  Google Scholar 

  6. Floreanini, R. and Vinet, L., Quantum algebras and q-special functions, Ann. of Phys., to appear

    Google Scholar 

  7. Floreanini, R. and Vinet, L., Addition formulas for q-Bessel functions, University of Montreal-preprint, J. Math. Phys., to appear

    Google Scholar 

  8. Floreanini, R. and Vinet, L., Representations of quantum algebras and q-special functions, Proceedings of the II International Wigner Symposium, Dobrev, V. and Scherer, W., eds., (Springer-Verlag, Berlin, 1992), to appear

    Google Scholar 

  9. Floreanini, R. and Vinet, L., On the quantum group and quantum algebra approach to q-special functions, University of Montreal-preprint, UdeM-LPNTH86, 1992

    Google Scholar 

  10. Floreanini, R. and Vinet, L., Generalized q-Bessel functions, University of Montreal-preprint, UdeM-LPN-TH87, 1992

    Google Scholar 

  11. Floreanini, R. and Vinet, L., Using quantum algebras in q-special function theory, University of Montreal-preprint, UdeM-LPN-TH90, 1992

    Google Scholar 

  12. Kalnins, E.G., Manocha, H.L. and Miller, W., Models of q-algebra representations: I. Tensor products of special unitary and oscillator algebras, J. Math. Phys. 33, 2365–2383 (1992)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  13. Kalnins, E.G, Miller, W., and Mukherjee, S., Models of q-algebra representations: the group of plane motions, University of Minnesota preprint, 1992

    Google Scholar 

  14. Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Saburi, Y. and Ueno, K., Unitary representations of the quantum group SU q (1, 1): structure of the dual space of U q (sl(2)), Lett. Math. Phys. 19, 187–194 (1990)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  15. Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M., Saburi, Y. and Ueno, K., Unitary representations of the quantum group SU q (1,1): II-matrix elements of unitary representations and the basic hypergeometric functions, ibid. 19, 195–204 (1990)

    MathSciNet  MATH  Google Scholar 

  16. Gasper, G. and Rahman, M., Basic Hypergeometric Series, (Cambridge University Press, Cambridge, 1990)

    MATH  Google Scholar 

  17. Drinfel’d, V.G., Quantum groups, in: Proceedings of the International Congress of Mathematicians, Berkeley (1986), vol. 1, pp. 798–820, (The American Mathematical Society, Providence, 1987)

    Google Scholar 

  18. Jimbo, M., A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10, 63–69 (1985)

    CrossRef  MathSciNet  ADS  MATH  Google Scholar 

  19. Jimbo, M., A q-analogue of U(gl(N + 1)), Hecke algebra and the Yang-Baxter equation, ibid. 11, 247–252 (1986)

    MathSciNet  MATH  Google Scholar 

  20. Vaksman, L.L. and Soibelman, Ya.S., Algebra of functions of the quantum group SU(2), Funct. Anal. Appl. 22, 1–14 (1988)

    MathSciNet  MATH  Google Scholar 

  21. Masuda, T., Mimachi, K., Nakagami, Y., Noumi, M. and Ueno, K., Representations of the quantum group SU q (2) and the little q-Jacobi polynomials, J. Funct. Anal. 99, 357–386 (1991)

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. Koornwinder, T.H., Representations of the twisted SU(2) quantum group and some q-hypergeometric orthogonal polynomials, Nederl. Akad. Wetensch. Proc. Ser. A92, 97–117 (1989)

    MathSciNet  Google Scholar 

  23. Carlitz, L., Some polynomials related to theta functions, Annali di Matematica Pura ed Applicata (4) 41, 359–373 (1955)

    Google Scholar 

  24. Carlitz, L., Some polynomials related to theta functions, Duke Math. J. 24, 521–527 (1957)

    CrossRef  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto Nazionale di Fisica Nucleare, Sezione di Trieste Dipartimento di Fisica Teorica, Università di Trieste, Strada Costiera 11, 34014, Trieste, Italy

    Roberto Floreanini

  2. Laboratoire de Physique Nucléaire and Centre de Recherches Mathématiques, Université de Montréal, Montréal, Canada, H3C 3J7

    Luc Vinet

Authors
  1. Roberto Floreanini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Luc Vinet
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Física Teórica, Universidad Complutense de Madrid, Madrid, Spain

    L. A. Ibort & M. A. Rodríguez & 

Rights and permissions

Reprints and Permissions

Copyright information

© 1993 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Floreanini, R., Vinet, L. (1993). Quantum Algebras and Quantum Groups in q-Special Function Theory. In: Ibort, L.A., Rodríguez, M.A. (eds) Integrable Systems, Quantum Groups, and Quantum Field Theories. NATO ASI Series, vol 409. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1980-1_4

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-1980-1_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4874-3

  • Online ISBN: 978-94-011-1980-1

  • eBook Packages: Springer Book Archive