Quadratic algebra contractions and 2nd order superintegrable systems

Abstract

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. For constant curvature spaces we show that the free quadratic algebras generated by the 1st and 2nd order elements in the enveloping algebras of their Euclidean and orthogonal symmetry algebras correspond one-to-one with the possible superintegrable systems with potential defined on these spaces. We describe a contraction theory for quadratic algebras and show that for constant curvature superintegrable systems, ordinary Lie algebra contractions induce contractions of the quadratic algebras of the superintegrable systems that correspond to geometrical pointwise limits of the physical systems. One consequence is that by contracting function space realizations of representations of the generic superintegrable quantum system on the 2-sphere (which give the structure equations for Racah/Wilson polynomials) to the other superintegrable systems one obtains the full Askey scheme of orthogonal hypergeometric polynomials.

28 pages

References in corpus (2)

• Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
• Structure Theory for Second Order 2D Superintegrable Systems with 1-Parameter Potentials

Cited by in corpus (9)

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• Hidden symmetries of the Higgs oscillator and the conformal algebra
• Bôcher Contractions of Conformally Superintegrable Laplace Equations
• Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems
• Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras
• Contractions of Degenerate Quadratic Algebras, Abstract and Geometric