# Clebsch-Gordan Coefficients for <em>U<sub>q</sub></em>(<em>su</em>(1,1)) and <em>U<sub>q</sub></em>(<em>sl</em>(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 *Uq*(*su*(1,1)) is decomposed into the direct sum of irreducible components of *Uq*(*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 *Uq*(*su*(2)) and a representation of the discrete series of *Uq*(*su*(1,1)) is decomposed into the direct sum of irreducible components of *Uq*(*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 <em>U<sub>q</sub></em>(<em>su</em>(1,1)) and <em>U<sub>q</sub></em>(<em>sl</em>(2)), and Linearization Formula of Matrix Elements. Publ. Res. Inst. Math. Sci. 28 (1992), no. 5, pp. 775–807

DOI 10.2977/PRIMS/1195167936