Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems

Abstract

The notion of hidden symmetry algebra used in the context of exactly solvable systems is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By construction, these algebras do not depend on the choice of realizations by vector fields of the underlying Lie algebra, allowing to propose a procedure to analyze polynomial algebras as those subspaces in an enveloping algebra that commute with a given algebraic Hamiltonian.

12 pages

References in corpus (1)

• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials

Cited by in corpus (6)

• Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
• Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras
• Polynomial algebras from Lie algebra reduction chains $\mathfrak{g}\supset {\mathfrak{g}}^{\prime }$
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Polynomial algebra from the Lie algebra reduction chain $\mathfrak{su}(4)\supset \mathfrak{su}(2)\times \mathfrak{su}(2)$: The supermultiplet model
• Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras