Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras

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 Schrödinger algebras $\hat{S}(n)$ 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.

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• Algebraic (super-)integrability from commutants of subalgebras in universal enveloping 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
• On the construction of polynomial Poisson algebras: a novel grading approach
• Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras
• The Lie algebra of the lowest transitively differential group of degree three