Explicit isomorphisms for the symmetry algebras of continuous and discrete isotropic oscillators
Pavel Drozdov, Giorgio Gubbiotti, Danilo Latini
TL;DR
The paper identifies a parametric Lie algebra $\mathfrak{A}_{N}(\alpha)$ that encompasses the symmetry algebras of both continuous and maximally superintegrable discretizations of the isotropic harmonic oscillator. It establishes explicit isomorphisms for the three regimes: $\alpha>0$ yields $\mathfrak{u}_{N}$, $\alpha<0$ yields $\mathfrak{gl}_{N}(\mathbb{R})$, and $\alpha=0$ yields $\mathfrak{so}_{N}(\mathbb{R})\niplus\mathbb{R}^{N(N+1)/2}$, with explicit basis changes and matrix realizations. The work also demonstrates a Nambu–Hamiltonian formulation for both the continuous IHO and its MS discretizations, and provides detailed Levi decompositions and Killing-form analyses to classify the algebras. Furthermore, it connects the discrete MS discretization to existing discretization schemes (e.g., KHK/RK) and discusses the preservation of invariants under discretization. The results offer a unified algebraic framework that can be extended to other maximally superintegrable systems and discretizations, with potential applications in symmetry analysis and numerical integrator design.
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 $N$.