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

SIGMA 7 (2011), 077, 17 pages      arXiv:1104.5101      https://doi.org/10.3842/SIGMA.2011.077
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

Quantum Analogs of Tensor Product Representations of su(1,1)

Wolter Groenevelt
Delft Institute of Applied Mathematics, Technische Universiteit Delft, PO Box 5031, 2600 GA Delft, the Netherlands

Received April 28, 2011, in final form August 04, 2011; Published online August 09, 2011

Abstract
We study representations of Uq(su(1,1)) that can be considered as quantum analogs of tensor products of irreducible *-representations of the Lie algebra su(1,1). We determine the decomposition of these representations into irreducible *-representations of Uq(su(1,1)) by diagonalizing the action of the Casimir operator on suitable subspaces of the representation spaces. This leads to an interpretation of the big q-Jacobi polynomials and big q-Jacobi functions as quantum analogs of Clebsch-Gordan coefficients.

Key words: tensor product representations; Clebsch-Gordan coefficients; big q-Jacobi functions.

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