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

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 $N$.

54 pages

References in corpus (8)

• Superintegrable Systems in Darboux spaces
• Quantum mechanics on spaces of nonconstant curvature: the oscillator problem and superintegrability
• Superintegrability of the Caged Anisotropic Oscillator
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• A new exactly solvable quantum model in N dimensions
• On two superintegrable nonlinear oscillators in N dimensions
• Demkov-Fradkin tensor for curved harmonic oscillators
• Discrete-time systems in quasi-standard form and the ${\mathfrak{h}}_6$ coalgebra symmetry

Cited by in corpus (1)

• Discrete-time maximally superintegrable systems and deformed symmetry algebras: the Calogero-Moser case