Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems

TL;DR

This paper re-examines hidden symmetry algebras in exactly solvable systems, proposing an algebraic approach to analyze quadratic algebras generated by commuting polynomials, independent of specific realizations.

Contribution

It introduces a purely algebraic method to construct and analyze quadratic symmetry algebras in superintegrable systems, independent of vector field realizations.

Findings

Identifies subspaces of commuting polynomials forming finite-dimensional quadratic algebras.

Provides a realization-independent procedure for analyzing polynomial symmetry algebras.

Enhances understanding of algebraic structures underlying 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.