On realizations of polynomial algebras with three generators via deformed oscillator algebras

Abstract

We present the most general polynomial Lie algebra generated by a second order integral of motion and one of order M, construct the Casimir operator, and show how the Jacobi identity provides the existence of a realization in terms of deformed oscillator algebra. We also present the classical analog of this construction for the most general Polynomial Poisson algebra. Two specific classes of such polynomial algebras are discussed that include the symmetry algebras observed for various 2D superintegrable systems.

28 pages

References in corpus (6)

• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• A Recurrence Relation Approach to Higher Order Quantum Superintegrability
• Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
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• Models for Quadratic Algebras Associated with Second Order Superintegrable Systems in 2D
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• Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum
• The general Racah algebra as the symmetry algebra of generic systems on pseudo--spheres
• Algebraic calculations for spectrum of superintegrable system from exceptional orthogonal polynomials
• New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies
• Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces
• Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
• Superintegrable systems, polynomial algebra structures and exact derivations of spectra
• Superintegrability and Deformed Oscillator Realizations of Quantum TTW Hamiltonians on Constant-Curvature Manifolds and with Reflections in a Plane
• Infinite dimensional representations of cubic and quintic algebras and special functions
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Family of $N$-dimensional superintegrable systems and quadratic algebra structures
• Generalized quadratic commutator algebras of PBW-type
• Deformed oscillator algebra approach of some quantum superintegrable Lissajous systems on the sphere and of their rational extensions
• Quadratic algebra structure in the 5D Kepler system with non-central potentials and Yang-Coulomb monopole interaction
• Non-Hermitian superintegrable systems