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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. I. Man'ko, G. Marmo, S. Solimeno, and F. Zaccaria, Int. J. Mod. Phys. A, 8, 3577 (1993).
V. I. Man'ko, G. Marmo, S. Solimeno, and F. Zaccaria, Phys. Lett. A, 176, 173 (1993).
V. I. Man'ko, G. Marmo, and F. Zaccaria, Phys. Lett. A, 191, 13 (1994).
V. I. Man'ko and G. M. Tino, Phys. Lett. A, 202, 24 (1995).
J. Lukierski, A. Nowicki, and H. Ruegg, Phys. Lett. B, 293, 344 (1992).
M. Pillin, J. Math. Phys., 35, 2804 (1994).
U. H. Niederer and L. O'Raifeartaigh, Fortschr. Phys., 22, 111; 131 (1974).
C. Fronsdal, Phys. Rev. D, 12, 3819 (1975).
M. Flato and C. Fronsdal, Phys. Lett. B, 97, 236 (1980).
P. A. M. Dirac, J. Math. Phys., 4, 901 (1963).
D. P. Zhelobenko, Dokl. Akad. Nauk SSSR, 126, 635 (1959).
I. M. Gel'fand and V. A. Ponomarev, Usp. Mat. Nauk, 23, 3 (1968).
D. N. Verma, Bull. Am. Math. Soc., 74, 160 (1968).
I. N. Bernstein, I. M. Gel'fand, and S. I. Gel'fand, Funct. Anal. Appl., 5, 1 (1971).
B. Gruber and A. U. Klimyk, J. Math. Phys., 16, 1816 (1975).
B. Gruber and A. U. Klimyk, J. Math. Phys., 19, 2009 (1978).
B. Gruber, A. U. Klimyk, and Yu. F. Smirnov, Nuovo Cim. A, 69, 97 (1982).
B. Gruber and Yu. F. Smirnov, Lett. Math. Phys., 4, 367 (1980).
A. V. Turbiner, Commun. Math. Phys., 118, 467 (1988).
V. K. Dobrev, Lett. Math. Phys., 22, 251 (1991).
Yu. F. Smirnov, V. N. Tolstoy, and Yu. I. Kharitonov, Yad. Fiz., 54, 721 (1991).
N. Jacobson, Lie Algebras, Wiley Interscience, New York (1962).
Yu. F. Smirnov and Yu. I. Kharitonov, Yad. Fiz., 64, 2256 (2001).
V. A. Knyr, Yu. F. Smirnov, and V. N. Tolstoy, Repts. Math. Phys., 8, 343 (1975).
Author information
Authors and Affiliations
Rights 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
Issue Date:
DOI: https://doi.org/10.1023/A:1022569409239