The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres

Abstract

We characterize the symmetry algebra of the generic superintegrable system on a pseudo-sphere corresponding to the homogeneous space $SO(p,q+1)/SO(p,q)$ where $p+q=\mathcal{N}$, $\mathcal{N}\in \mathbb{N}$. We show that this algebra is independent of the signature $(p,q+1)$ of the metric and that it is the same as the Racah algebra $\mathcal{R}(\mathcal{N}+1)$. The spectrum obtained from $\mathcal{R}(\mathcal{N}+1)$ via the Daskaloyannis method depends on undetermined signs that can be associated to the signatures. Two examples are worked out explicitly for the cases $SO(2,1)/SO(2)$ and $SO(3)/SO(2)$ where it is shown that their spectrum obtained by means of separation of variables coincide with particular choices of the signs corresponding to the specific signatures of the spectrum for the symmetry algebra $\mathcal{R}(3)$.

arXiv comment11 pages

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Cited by in corpus (9)

• Embedding of the Racah Algebra R($\mathbf{n}$) and Superintegrability
• Racah algebras, the centralizer $Z_n({\mathfrak{sl}}_2)$ and its Hilbert-Poincaré series
• Racah Algebra $R(n)$ from Coalgebraic Structures and Chains of $R(3)$ Substructures
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• Demkov-Fradkin tensor for curved harmonic oscillators
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Generalized quadratic commutator algebras of PBW-type
• Non-Hermitian superintegrable systems