Some Aspects of $q$- and $qp$-Boson Calculus

Abstract

A set of compatible formulas for the Clebsch-Gordan coefficients of the quantum algebra $U_q({\mathrm{su}}_2)$ is given in this paper. These formulas are $q$-deformations of known formulas, as for instance: Wigner, van der Waerden, and Racah formulas. They serve as starting points for deriving various realizations of the unit tensor of $U_q({\mathrm{su}}_2)$ in terms of $q$-boson operators. The passage from the one-parameter quantum algebra $U_q({\mathrm{su}}_2)$ to the two-parameter quantum algebra $U_{qp}({\mathrm{u}}_2)$ is discussed at the level of Clebsch-Gordan coefficients.

15 pages, Tex File