Open AccessOn some algebraic formulations within universal enveloping algebras related to superintegrability
Author(s) -
Rutwig Campoamor-Stursberg
Publication year - 2022
Publication title -
acta polytechnica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.207
H-Index - 15
eISSN - 1805-2363
pISSN - 1210-2709
DOI - 10.14311/ap.2022.62.0016
Subject(s) - mathematics , representation theory , context (archaeology) , linear subspace , algebraic number , algebra over a field , pure mathematics , lie algebra , quadratic equation , quadratic algebra , representation (politics) , algebraic structure , algebra representation , jordan algebra , mathematical analysis , paleontology , geometry , politics , political science , law , biology
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.