Skip to main content
SpringerLink
Log in

Quantum Algebra U q(gl(3)) and Nonlinear Optics

  • Published:
Journal of Russian Laser Research Aims and scope

Abstract

Indecomposable representations are investigated for the U q(gl(3)) quantum algebra. The matrix elements are explicitly determined for the elementary representations, and the extremal vectors which characterize invariant subspaces are given in explicit form. Quotient spaces are used to derive other representations from the elementary representations, including the finite-dimensional irreducible representations and infinite-dimensional representations which are bounded above. Applications to nonlinear-optical phenomena are discussed.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. I. Man'ko, G. Marmo, S. Solimeno, and F. Zaccaria, Int. J. Mod. Phys. A, 8, 3577 (1993).

    Google Scholar 

  2. V. I. Man'ko, G. Marmo, S. Solimeno, and F. Zaccaria, Phys. Lett. A, 176, 173 (1993).

    Google Scholar 

  3. V. I. Man'ko, G. Marmo, and F. Zaccaria, Phys. Lett. A, 191, 13 (1994).

    Google Scholar 

  4. V. I. Man'ko and G. M. Tino, Phys. Lett. A, 202, 24 (1995).

    Google Scholar 

  5. J. Lukierski, A. Nowicki, and H. Ruegg, Phys. Lett. B, 293, 344 (1992).

    Google Scholar 

  6. M. Pillin, J. Math. Phys., 35, 2804 (1994).

    Google Scholar 

  7. U. H. Niederer and L. O'Raifeartaigh, Fortschr. Phys., 22, 111; 131 (1974).

    Google Scholar 

  8. C. Fronsdal, Phys. Rev. D, 12, 3819 (1975).

    Google Scholar 

  9. M. Flato and C. Fronsdal, Phys. Lett. B, 97, 236 (1980).

    Google Scholar 

  10. P. A. M. Dirac, J. Math. Phys., 4, 901 (1963).

    Google Scholar 

  11. D. P. Zhelobenko, Dokl. Akad. Nauk SSSR, 126, 635 (1959).

    Google Scholar 

  12. I. M. Gel'fand and V. A. Ponomarev, Usp. Mat. Nauk, 23, 3 (1968).

    Google Scholar 

  13. D. N. Verma, Bull. Am. Math. Soc., 74, 160 (1968).

    Google Scholar 

  14. I. N. Bernstein, I. M. Gel'fand, and S. I. Gel'fand, Funct. Anal. Appl., 5, 1 (1971).

    Google Scholar 

  15. B. Gruber and A. U. Klimyk, J. Math. Phys., 16, 1816 (1975).

    Google Scholar 

  16. B. Gruber and A. U. Klimyk, J. Math. Phys., 19, 2009 (1978).

    Google Scholar 

  17. B. Gruber, A. U. Klimyk, and Yu. F. Smirnov, Nuovo Cim. A, 69, 97 (1982).

    Google Scholar 

  18. B. Gruber and Yu. F. Smirnov, Lett. Math. Phys., 4, 367 (1980).

    Google Scholar 

  19. A. V. Turbiner, Commun. Math. Phys., 118, 467 (1988).

    Google Scholar 

  20. V. K. Dobrev, Lett. Math. Phys., 22, 251 (1991).

    Google Scholar 

  21. Yu. F. Smirnov, V. N. Tolstoy, and Yu. I. Kharitonov, Yad. Fiz., 54, 721 (1991).

    Google Scholar 

  22. N. Jacobson, Lie Algebras, Wiley Interscience, New York (1962).

    Google Scholar 

  23. Yu. F. Smirnov and Yu. I. Kharitonov, Yad. Fiz., 64, 2256 (2001).

    Google Scholar 

  24. V. A. Knyr, Yu. F. Smirnov, and V. N. Tolstoy, Repts. Math. Phys., 8, 343 (1975).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Southern Illinois University, Carbondale, Illinois, 62901

    B. Gruber

  2. D. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State University, Vorob'evy Gory, 119899, Moscow, Russia

    Yu. F. Smirnov

  3. St. Petersburg Institute of Nuclear Physics of the Russian Academy of Sciences, 188350, Gatchina, Russia

    Yu. I. Kharitonov

Authors
  1. B. Gruber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. F. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu. I. Kharitonov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gruber, B., Smirnov, Y.F. & Kharitonov, Y.I. Quantum Algebra U q(gl(3)) and Nonlinear Optics. Journal of Russian Laser Research 24, 56–68 (2003). https://doi.org/10.1023/A:1022569409239

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022569409239

Search

Navigation