Noncompact quantum algebra uq(2,1)

الأصل · EN

The structure positive of unitary irreducible representations of the noncompact uq(2,1) quantum algebra that are related to a positive discrete series is examined. With the aid of projection operators for the suq(2) subalgebra, a q-analog of the Gel'fand--Graev formulas is derived in the basis corresponding to the reduction uq(2,1)→ suq(2)× u(1). Projection operators for the suq(1,1) subalgebra are employed to study the same representations for the reduction uq(2,1)→ u(1)× suq(1,1). The matrix elements of the generators of the uq(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 suq(2) quantum algebra.