Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schr\"odinger algebras

TL;DR

This paper presents a method to identify quadratic algebras as commutants of algebraic Hamiltonians within the enveloping algebra of Schrödinger algebras, applicable to non-semisimple Lie algebras and tested on various cases.

Contribution

It introduces a new analytical procedure to determine quadratic algebras as commutants in the enveloping algebra, independent of specific realizations, and applicable to non-semisimple Lie algebras.

Findings

Method successfully applied to Schrödinger algebras 0(n)

Identifies minimal quadratic algebras without explicit enveloping algebra manipulation

Validates approach for extended Cartan solvable cases

Abstract

We discuss a procedure to determine finite sets $\mathcal{M}$ within the commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie algebra $\mathfrak{g}$ such that their generators define a quadratic algebra. Although independent from any realization of Lie algebras by differential operators, the method is partially based on an analytical approach, and uses the coadjoint representation of the Lie algebra $\mathfrak{g}$. The procedure, valid for non-semisimple algebras, is tested for the centrally extended Schr\"odinger algebras $\widehat{S}(n)$ for various different choices of algebraic Hamiltonian. For the so-called extended Cartan solvable case, it is shown how the existence of minimal quadratic algebras can be inferred without explicitly manipulating the enveloping algebra.