Noncompact u q(2, 1) quantum algebra: Discrete series of highest weight irreducible representations

Abstract

The structure of unitary irreducible representations of the noncompact uq(2, 1) quantum algebra that are related to a negative discrete series is examined. With the aid of projection operators for the suq(2) subalgebra, a q analog of the Gelfand-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 brackets q between the bases associated with the 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, the q Weyl coefficients coincide with the q Racah coefficients for the suq(2) quantum algebra.