On some algebraic formulations within universal enveloping algebras related to superintegrability

Abstract

We report on some recent purely algebraic approaches to superintegrable systems from the perspective of subspaces of commuting polynomials in the enveloping algebras of Lie algebras that generate quadratic (and eventually higher-order) algebras. In this context, two algebraic formulations are possible; a first one strongly dependent on representation theory, as well as a second formal approach that focuses on the explicit construction within commutants of algebraic integrals for appropriate algebraic Hamiltonians defined in terms of suitable subalgebras. The potential use in this context of the notion of virtual copies of Lie algebras is briefly commented.