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

Abstract

We study representations of $U_q(su(1,1))$ 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 $U_q(su(1,1))$ 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.

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• Intertwiners of $U_q^{\prime }(\widehat{sl}(2))$-representations and the vector-valued big $q$-Jacobi transform