Noncompact quantum algebra $u_q(2,1)$

Publication date: 17 November 2003

Abstract: The structure positive of unitary irreducible representations of the noncompact

u_{q} (2, 1)

quantum algebra that are related to a positive discrete series is examined. With the aid of projection operators for the

s u_{q} (2)

subalgebra, a

q

-analog of the Gel'fand--Graev formulas is derived in the basis corresponding to the reduction

u_{q} (2, 1) o s u_{q} (2) i m e s u (1)

. Projection operators for the

s u_{q} (1, 1)

subalgebra are employed to study the same representations for the reduction

u_{q} (2, 1) o u (1) i m e s s u_{q} (1, 1)

. The matrix elements of the generators of the

u_{q} (2, 1)

algebra are computed in this new basis. A general analytic expression for an element of the transformation bracket

< U | T >_{q}

between the bases associated with above two reductions (the elements of this matrix are referred to as

q

-Weyl coefficients) is obtained for a general case where the deformation parameter

q

is not equal to a root of unity. It is shown explicitly that, apart from a phase,

q

-Weyl coefficients coincide with the

q

-Racah coefficients for the

s u_{q} (2)

quantum algebra.

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