Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators

TL;DR

This paper explores a parametric Lie algebra that unifies the symmetry structures of continuous and discrete isotropic oscillators, revealing explicit isomorphisms and novel algebraic properties relevant to superintegrable systems.

Contribution

It introduces a comprehensive Lie algebra framework for isotropic oscillators, including nonlinear deformations and discretizations, with explicit isomorphisms to classical matrix Lie algebras.

Findings

Explicit isomorphisms to $ _N$, $ ext{gl}_N( ext{R})$, and semidirect sums involving $ ext{so}_N( ext{R})$.

Identification of the Lie algebra in discretized and continuous models, including superintegrable discretizations.

Formulas valid for arbitrary $N$, applicable to $ ext{su}_N$ and $ ext{sl}_N( ext{R})$.

Abstract

We present a detailed study of a parametric Lie algebra encompassing the symmetry algebras of various models, both continuous and discrete. This algebraic structure characterizes the isotropic oscillator (with positive, purely imaginary, and zero frequency) and one of its possible nonlinear deformations. We demonstrate a novel occurrence of this Lie algebra in the framework of maximally superintegrable discretizations of the isotropic harmonic oscillator. In particular, we also show that the continuous model and one of its discretizations admit a Nambu-Hamiltonian structure. Through an in-depth analysis of the properties characterizing the Lie algebra in the abstract setting, for different values of the parameter, we find explicit expressions of the Killing forms and construct explicit isomorphism maps to $\mathfrak{u}_N$, $\mathfrak{gl}_N(\mathbb{R})$, and a semidirect sum of $\mathfrak{so}_N(\mathbb{R})$ with $\mathbb{R}^{N(N+1)/2}$. Notably, due to the above isomorphisms, our formulas hold true for $\mathfrak{su}_N$ and $\mathfrak{sl}_N(\mathbb{R})$ and are valid for arbitrary $N$.