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

SIGMA 6 (2010), 010, 13 pages      arXiv:0912.5403      https://doi.org/10.3842/SIGMA.2010.010
Contribution to the Proceedings of the XVIIIth International Colloquium on Integrable Systems and Quantum Symmetries

q-Analog of Gelfand-Graev Basis for the Noncompact Quantum Algebra Uq(u(n,1))

Raisa M. Asherova a, Čestmír Burdík b, Miloslav Havlíček b, Yuri F. Smirnov a, c and Valeriy N. Tolstoy a, b
a) Institute of Nuclear Physics, Moscow State University, 119992 Moscow, Russia
b) Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, 12000 Prague 2, Czech Republic
c) Deceased

Received November 05, 2009, in final form January 15, 2010; Published online January 26, 2010

Abstract
For the quantum algebra Uq(gl(n+1)) in its reduction on the subalgebra Uq(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction Z-algebra Zq(gl(n+1),gl(n)) is given in terms of the generators and their defining relations. Using this Z-algebra we describe Hermitian irreducible representations of a discrete series for the noncompact quantum algebra Uq(u(n,1)) which is a real form of Uq(gl(n+1)), namely, an orthonormal Gelfand-Graev basis is constructed in an explicit form.

Key words: quantum algebra; extremal projector; reduction algebra; Shapovalov form; noncompact quantum algebra; discrete series of representations; Gelfand-Graev basis.

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