Superintegrability in two dimensions and the Racah-Wilson algebra

TL;DR

This paper explores the connection between superintegrable systems on the 2-sphere and the Racah-Wilson algebra, revealing algebraic structures underlying special functions and orthogonal polynomials.

Contribution

It establishes an explicit isomorphism between the Racah-Wilson algebra and the invariance algebra of a 2D superintegrable system, linking algebraic and analytical frameworks.

Findings

Identifies the Hamiltonian and symmetry generators with Casimir operators of su(1,1) algebras.

Shows the Racah-Wilson algebra's role in the structure of superintegrable systems.

Explains the appearance of Racah polynomials as overlap coefficients.

Abstract

The analysis of the most general second-order superintegrable system in two dimensions: the generic 3-parameter model on the 2-sphere, is cast in the framework of the Racah problem for the su(1,1) algebra. The Hamiltonian of the 3-parameter system and the generators of its quadratic symmetry algebra are seen to correspond to the total and intermediate Casimir operators of the combination of three su(1,1) algebras, respectively. The construction makes explicit the isomorphism between the Racah-Wilson algebra, which is the fundamental algebraic structure behind the Racah problem for su(1,1), and the invariance algebra of the generic 3-parameter system. It also provides an explanation for the occurrence of the Racah polynomials as overlap coefficients in this context. The irreducible representations of the Racah-Wilson algebra are reviewed as well as their connection with the Askey scheme of classical orthogonal polynomials.