On the quantum algebra $su_q(1,1)$ from a Special Function standpoint

TL;DR

This paper explores the quantum algebra su_q(1,1) through special functions, deriving explicit Clebsch-Gordan coefficients using q-hypergeometric series and unveiling new properties of these coefficients.

Contribution

It provides a new special function approach to compute and analyze Clebsch-Gordan coefficients for su_q(1,1), including explicit formulas and novel properties.

Findings

Explicit representation of Clebsch-Gordan coefficients as symmetric q-hypergeometric series

Derivation of new properties of Clebsch-Gordan coefficients

Unified approach connecting quantum algebra and special functions

Abstract

In this paper, we study the tensor product of two unitary irreducible representations, as well as the tensor product of a unitary irreducible representation with a finite-dimensional one, and determine the corresponding Clebsch-Gordan coefficients. Using von Neumann's projection operator method, we obtain an explicit representation of these coefficients, which allows us to express them as symmetric q-hypergeometric series. Finally, by leveraging the properties of the q-hypergeometric function, we derive several properties of the Clebsch-Gordan coefficients, including a number of new results, in a unified and straightforward manner.