
SIGMA 3 (2007), 064, 12 pages arXiv:0705.0276
http://dx.doi.org/10.3842/SIGMA.2007.064
Degenerate Series Representations of the qDeformed 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 qdeformed algebra so'_{q}(r,s) is a real
form of the qdeformed algebra U_{q}'(so(n,C)),
n = r + s, which differs from the quantum algebra
U_{q}(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:
qdeformed algebras; irreducible representations; reducible representations.
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References
 Gavrilik A.M., Klimyk A.U., qdeformed orthogonal and
pseudoorthogonal algebras and their representations, Lett. Math. Phys.
21 (1991), 215220.
 Drinfeld V.G., Hopf algebras and quantum YangBaxter equation,
Sov. Math. Dokl. 32 (1985), 254258.
 Jimbo M., A qdifference analogue of U_{q}(gl(N + 1)) and
the YangBaxter equations, Lett. Math. Phys. 10 (1985), 6369.
 Klimyk A.U., Schmüdgen K., Quantum groups and their
representations, Springer, Berlin, 1997.
 Klimyk A.U., Kachurik I.I., Spectra,
eigenvectors and overlap functions for representation operators of
qdeformed algebras, Comm. Math. Phys. 175 (1996), 89111.
 Nelson J., Regge T., 2 + 1 gravity for genus s > 1,
Comm. Math. Phys. 141 (1991), 211223.
 Noumi M., Macdonald's symmetric polynomials as zonal
spherical functions on quantum homogeneous spaces, Adv. Math. 123
(1996), 1677.
 Noumi M., Umeda T., Wakayama M.,
Dual pairs, spherical harmonics and a Capelli identity
in quantum group theory, Compos. Math. 104 (1996), 227277.
 Iorgov N.Z., Klimyk A.U., The qLaplace
operator and qharmonic polynomials on the quantum vector space,
J. Math. Phys. 42 (2001), 13261345.
 Bullock D., Przytycki J.H., Multiplicative
structure of Kauffman bracket skein module quantization,
math.QA/9902117.
 Twietmeyer E., Real forms of U_{q}(g),
Lett. Math. Phys. 49 (1992), 4958.
 Dobrev V.K., Canonical qdeformation of noncompact
Lie (super)algebras, J. Phys. A: Math. Gen. 26 (1993),
13171329.
 Celegini E., Giachetti R., Reyman A., Sorace E., Tarlini M.,
SO_{q}(n + 1,n1) as a real form of SO_{q}(2n,C),
Lett. Math. Phys. 23 (1991), 4544.
 Raczka R., Limic N., Niederle J., Discrete degenerate representations
of the noncompact rotation groups,
J. Math. Phys. 7 (1966), 18611876.
 Molchanov V.F., Representations of pseudoorthogonal groups
associated with a cone, Math. USSR Sbornik 10 (1970), 353347.
 Klimyk A.U., Matrix elements and ClebschGordan
coefficients of group representations, Naukova Dumka, Kiev, 1979.
 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), 174.
 Gavrilik A.M., Klimyk A.U., Representations of
qdeformed algebras U_{q}(so_{2,1}) and U_{q}(so_{3,1}),
J. Math. Phys. 35 (1994), 36703686.
 Kachurik I.I., Klimyk A.U., Representations of the
qdeformed algebra U_{q}(so_{r,2}), Dokl. Akad. Nauk
Ukrainy, Ser. A (1995), no. 9, 1820.
 Schmüdgen K., Unbounded operator algebras and
representation theory, Birkhäuser, Basel, 1990.
 Ostrovskyi V., Samoilenko Yu., Introduction to the theory
of representations of finitely presented *algebras,
Reviers in Math. and Math. Phys. 11 (1999), 1261.

