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

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.

• The paper derives Clebsch-Gordan coefficients for suq(1,1) using a projection operator method and expresses them in symmetric q-hypergeometric series.
• It investigates tensor products of unitary irreducible and finite-dimensional representations, clarifying complex symmetry properties.
• The findings simplify computations in quantum algebra representation theory and pave the way for extended applications in quantum mechanics.

On the Quantum Algebra $su_q(1,1)$ from a Special Function Standpoint

This paper examines the quantum algebra $su_q(1,1)$ through the lens of special functions, particularly focusing on the Clebsch-Gordan coefficients associated with this algebra. It leverages the projection operator method to express these coefficients as symmetric $q$-hypergeometric series. The paper derives several properties of Clebsch-Gordan coefficients and explores their special function connections, offering new insights into the algebra's representation theory. The intricate relationship between quantum algebras and $q$-special functions is well established, providing a robust theoretical foundation for further exploration.

Tensor Products and Clebsch-Gordan Coefficients

The primary objective is to study the tensor product of two unitary irreducible representations and the tensor product involving one unitary irreducible and one finite-dimensional representation of $su_q(1,1)$. The Clebsch-Gordan coefficients for these tensor products are determined using von Neumann's projection operator method, expressed as $q$-hypergeometric series. This approach simplifies the derivation of several properties and new results concerning these coefficients. The work also corrects typographical errors in previous derivations and elucidates the symmetric nature of these coefficients.

Projection Operator Method

The paper employs the projection operator method to explicitly determine the Clebsch-Gordan coefficients. This involves expanding the projection operator in terms of the generators of $su_q(1,1)$ and resolving them using recursive methods. The symmetric $q$-hypergeometric series representation emerges naturally, allowing the authors to uncover symmetry properties and explicit expressions in significant cases.

Properties and Symmetries

By expressing the Clebsch-Gordan coefficients as $q$-hypergeometric functions, the authors elucidate the symmetry properties of these coefficients. The symmetry relation is established, offering a reciprocal perspective on these coefficients under parameter transformation. This symmetry plays a crucial role in simplifying the computations associated with these coefficients and in understanding the intrinsic nature of the algebra’s representations.

Applications and Future Directions

This work not only deepens the understanding of $su_q(1,1)$ but also paves the way for exploring other quantum algebras through special function theory. The methods and results presented are expected to facilitate applications in quantum mechanics, particularly in systems described by non-compact algebras. Future research could extend these methods to other algebras and further probe the relationships with different families of $q$-special functions.

Conclusion

The exploration of $su_q(1,1)$ from a special function standpoint delivers a comprehensive framework for understanding its representation theory. The derivation of Clebsch-Gordan coefficients via $q$-hypergeometric series represents a significant advancement, offering simplicity and elegance in expressing complex quantum algebraic relationships. This approach not only clarifies existing mathematical structures but also anticipates further developments in both theoretical and applied quantum physics.