Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 064, 12 pages      arXiv:0705.0276      https://doi.org/10.3842/SIGMA.2007.064

Degenerate Series Representations of the q-Deformed Algebra so'q(r,s)

Valentyna A. Groza
National Aviation University, 1 Komarov Ave., 03058 Kyiv, Ukraine

Received January 26, 2007, in final form April 18, 2007; Published online May 02, 2007

Abstract
The q-deformed algebra so'q(r,s) is a real form of the q-deformed algebra Uq'(so(n,C)), n = r + s, which differs from the quantum algebra Uq(so(n,C)) of Drinfeld and Jimbo. We study representations of the most degenerate series of the algebra so'q(r,s). The formulas of action of operators of these representations upon the basis corresponding to restriction of representations onto the subalgebra so'q(r) × so'q(s) are given. Most of these representations are irreducible. Reducible representations appear under some conditions for the parameters determining the representations. All irreducible constituents which appear in reducible representations of the degenerate series are found. All *-representations of so'q(r,s) are separated in the set of irreducible representations obtained in the paper.

Key words: q-deformed algebras; irreducible representations; reducible representations.

pdf (235 kb)   ps (163 kb)   tex (13 kb)

References

  1. Gavrilik A.M., Klimyk A.U., q-deformed orthogonal and pseudo-orthogonal algebras and their representations, Lett. Math. Phys. 21 (1991), 215-220.
  2. Drinfeld V.G., Hopf algebras and quantum Yang-Baxter equation, Sov. Math. Dokl. 32 (1985), 254-258.
  3. Jimbo M., A q-difference analogue of Uq(gl(N + 1)) and the Yang-Baxter equations, Lett. Math. Phys. 10 (1985), 63-69.
  4. Klimyk A.U., Schmüdgen K., Quantum groups and their representations, Springer, Berlin, 1997.
  5. Klimyk A.U., Kachurik I.I., Spectra, eigenvectors and overlap functions for representation operators of q-deformed algebras, Comm. Math. Phys. 175 (1996), 89-111.
  6. Nelson J., Regge T., 2 + 1 gravity for genus s > 1, Comm. Math. Phys. 141 (1991), 211-223.
  7. Noumi M., Macdonald's symmetric polynomials as zonal spherical functions on quantum homogeneous spaces, Adv. Math. 123 (1996), 16-77.
  8. Noumi M., Umeda T., Wakayama M., Dual pairs, spherical harmonics and a Capelli identity in quantum group theory, Compos. Math. 104 (1996), 227-277.
  9. Iorgov N.Z., Klimyk A.U., The q-Laplace operator and q-harmonic polynomials on the quantum vector space, J. Math. Phys. 42 (2001), 1326-1345.
  10. Bullock D., Przytycki J.H., Multiplicative structure of Kauffman bracket skein module quantization, math.QA/9902117.
  11. Twietmeyer E., Real forms of Uq(g), Lett. Math. Phys. 49 (1992), 49-58.
  12. Dobrev V.K., Canonical q-deformation of noncompact Lie (super)algebras, J. Phys. A: Math. Gen. 26 (1993), 1317-1329.
  13. Celegini E., Giachetti R., Reyman A., Sorace E., Tarlini M., SOq(n + 1,n-1) as a real form of SOq(2n,C), Lett. Math. Phys. 23 (1991), 45-44.
  14. Raczka R., Limic N., Niederle J., Discrete degenerate representations of the noncompact rotation groups, J. Math. Phys. 7 (1966), 1861-1876.
  15. Molchanov V.F., Representations of pseudo-orthogonal groups associated with a cone, Math. USSR Sbornik 10 (1970), 353-347.
  16. Klimyk A.U., Matrix elements and Clebsch-Gordan coefficients of group representations, Naukova Dumka, Kiev, 1979.
  17. Howe R.E., Tan E.C., Homogeneous functions on light cone: the infinitesimal structure of some degenerate principal series representations, Bull. Amer. Math. Soc. 28 (1993), 1-74.
  18. Gavrilik A.M., Klimyk A.U., Representations of q-deformed algebras Uq(so2,1) and Uq(so3,1), J. Math. Phys. 35 (1994), 3670-3686.
  19. Kachurik I.I., Klimyk A.U., Representations of the q-deformed algebra Uq(sor,2), Dokl. Akad. Nauk Ukrainy, Ser. A (1995), no. 9, 18-20.
  20. Schmüdgen K., Unbounded operator algebras and representation theory, Birkhäuser, Basel, 1990.
  21. Ostrovskyi V., Samoilenko Yu., Introduction to the theory of representations of finitely presented *-algebras, Reviers in Math. and Math. Phys. 11 (1999), 1-261.

Previous article   Next article   Contents of Volume 3 (2007)