Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

TL;DR

This paper develops an algebraic framework for superintegrability using commutants in universal enveloping algebras, explicitly connecting Lie algebra structures with superintegrable models on spheres and their symmetry algebras.

Contribution

It introduces a purely algebraic method to construct polynomial symmetry algebras for superintegrable systems from Lie algebra commutants, with explicit examples for (n) and sphere models.

Findings

Explicit basis for commutants in (n) obtained

Connection established between algebraic commutants and superintegrable models

Symmetry algebras derived from quadratic and cubic polynomial algebras

Abstract

Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra $\mathfrak{sl}(n)$ is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of $\mathfrak{sl}(n)$, this provides an explicit connection with the generic superintegrable model on the $(n-1)$-dimensional sphere $\mathbb{S}^{n-1}$ and the related Racah algebra $R(n)$. In particular, we show explicitly how the models on the $2$-sphere and $3$-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of $\mathfrak{sl}(3)$ and $\mathfrak{sl}(4)$, respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.