Clebsch-Gordan Coefficients for ${\mathcal{U}}_q(su(1,1))$ and ${\mathcal{U}}_q(sl(2))$, and Linearization Formula of Matrix Elements

Abstract

The tensor product of two representations of the discrete series and the limit of the discrete series of ${\mathcal{U}}_q(su(1,1))$ is decomposed into the direct sum of irreducible components of ${\mathcal{U}}_q(sl(1,1))$, and the Clebsch–Gordan coefficients with respect to this decomposition are computed in two ways. In some cases, the tensor product of an irreducible unitary representation of ${\mathcal{U}}_q(sl(2))$ and a representation of the discrete series of ${\mathcal{U}}_q(su(1,1))$ is decomposed into the direct sum of irreducible components of ${\mathcal{U}}_q(sl(2))$, and the Clebsch–Gordan coefficients with respect to this decomposition are calculated, too. Making use of these coefficients, the linearization formula of the matrix elements is obtained.