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

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  • Clebsch-Gordan Coefficients for \mathscr{U}<SUB><I>q</I></SUB>(<I>su</I>(1, 1)) and \mathscr{U}<SUB><I>q</I></SUB>(<I>sl</I>(2)), and Linearization Formula of Matrix Elements

説明

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

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