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

TL;DR

This paper presents compatible formulas for the Clebsch-Gordan coefficients of quantum algebras $U_q(su_2)$ and $U_{qp}(u_2)$, extending classical formulas through $q$-deformations and exploring realizations via $q$-boson operators.

Contribution

It introduces new compatible formulas for quantum algebra Clebsch-Gordan coefficients and discusses their $q$-deformations and realizations in terms of $q$-boson operators.

Findings

Derived $q$-deformed Clebsch-Gordan formulas for $U_q(su_2)$.

Established relations between $U_q(su_2)$ and $U_{qp}(u_2)$ coefficients.

Provided frameworks for realizing tensor operators with $q$-bosons.

Abstract

A set of compatible formulas for the Clebsch-Gordan coefficients of the quantum algebra $U_{q}({\rm su}_2)$ 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 $U_{q}({\rm su}_2)$ in terms of $q$-boson operators. The passage from the one-parameter quantum algebra $U_{q }({\rm su}_2)$ to the two-parameter quantum algebra $U_{qp}({\rm u}_2)$ is discussed at the level of Clebsch-Gordan coefficients.