Quantum Analogs of Tensor Product Representations of su(1,1)

TL;DR

This paper explores quantum analogs of tensor product representations of su(1,1), decomposing them into irreducibles and linking them to big q-Jacobi polynomials as quantum Clebsch-Gordan coefficients.

Contribution

It introduces a method to decompose quantum tensor product representations of U_q(su(1,1)) into irreducibles using Casimir operator diagonalization, connecting to special functions.

Findings

Decomposition of quantum tensor product representations into irreducibles.

Identification of big q-Jacobi polynomials as quantum Clebsch-Gordan coefficients.

Explicit spectral analysis of the Casimir operator.

Abstract

We study representations of $U_q(su(1,1))$ that can be considered as quantum analogs of tensor products of irreducible *-representations of the Lie algebra $su(1,1)$. We determine the decomposition of these representations into irreducible *-representations of $U_q(su(1,1))$ by diagonalizing the action of the Casimir operator on suitable subspaces of the representation spaces. This leads to an interpretation of the big $q$-Jacobi polynomials and big $q$-Jacobi functions as quantum analogs of Clebsch-Gordan coefficients.