# Clebsch-Gordan Coefficients for $U_{q}(su(1,1))$ and $U_{q}(sl(2))$, and Linearization Formula of Matrix Elements

### Youichi Shibukawa

Hokkaido University, Sapporo, Japan

## Abstract

The tensor product of two representations of the discrete series and the limit of the discrete series of $U_{q}(su(1,1))$ is decomposed into the direct sum of irreducible components of $U_{q}(sl(1,1))$, 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 $U_{q}(sl(2))$ and a representation of the discrete series of $U_{q}(su(1,1))$ is decomposed into the direct sum of irreducible components of $U_{q}(sl(2))$, 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.

## Cite this article

Youichi Shibukawa, Clebsch-Gordan Coefficients for $U_{q}(su(1,1))$ and $U_{q}(sl(2))$, and Linearization Formula of Matrix Elements. Publ. Res. Inst. Math. Sci. 28 (1992), no. 5, pp. 775–807

DOI 10.2977/PRIMS/1195167936