Quantum Analogs of Tensor Product Representations of $\mathfrak{su}(1,1)$

We study representations of $\mathcal U_q(\mathfrak{su}(1,1))$ that can be considered as quantum analogs of tensor products of irreducible $*$-representations of the Lie algebra $\mathfrak{su}(1,1)$. We determine the decomposition of these representations into irreducible $*$-representations of $\mathcal U_q(\mathfrak{su}(1,1))$ 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.