Embedding of the Racah Algebra R($\boldsymbol{n}$) and Superintegrability

TL;DR

This paper explores the embedding of the Racah algebra R(n) within larger quadratic algebras in classical and quantum superintegrable systems, providing explicit constructions for key models and revealing complex algebraic relations.

Contribution

It demonstrates how the Racah algebra R(n) naturally appears inside larger quadratic algebras for superintegrable models, with explicit constructions for the Smorodinsky-Winternitz and Kepler-Coulomb systems.

Findings

R(n) algebra is embedded in larger quadratic algebras for superintegrable systems.

Explicit symmetry algebra constructions for the Smorodinsky-Winternitz and Kepler-Coulomb models.

High-order algebraic relations among generators are characterized in classical and quantum frameworks.

Abstract

The rank-$1$ Racah algebra $R(3)$ plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the $3$-parameter system on the $2$-sphere from which all second-order conformally flat superintegrable models in $2$D can be obtained by means of suitable limits and contractions. A higher rank generalization of $R(3)$, the so-called rank $n-2$ Racah algebra $R(n)$, has been considered recently and showed to be the symmetry algebra of the general superintegrable model on the $(n-1)$-sphere. In the present work, we show that such an algebraic structure naturally arises as embedded inside a larger quadratic algebra characterizing $n$D superintegrable models with non-central terms. This is shown both in classical and quantum mechanics through suitable (symplectic or differential) realisations of the Racah and additional generators. Among the main results, we present an explicit construction of the complete symmetry algebras for two families of $n$-dimensional maximally superintegrable models, the Smorodinsky-Winternitz system and the generalized Kepler-Coulomb system. For both families, the underlying symmetry algebras are higher-rank quadratic algebras containing the Racah algebra $R(n)$ as subalgebra. High-order algebraic relations among the generators of the full quadratic algebras are also obtained both in the classical and quantum frameworks. These results should shed new light to the further understanding of the structures of quadratic algebras in the context of superintegrable systems.