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

TL;DR

This paper explores how discretizing the Calogero-Moser system results in deformed symmetry algebras, revealing new algebraic structures and connections to Bell polynomials, advancing understanding of superintegrable systems.

Contribution

It characterizes the symmetry algebra of the discretized Calogero-Moser system and shows it as a deformation of the continuous algebra, linking discrete superintegrability to polynomial algebraic structures.

Findings

Discretization induces a nontrivial deformation of symmetry algebra.

Deformation parameter corresponds to the discretization parameter.

Connection established between symmetry algebras and Bell polynomials.

Abstract

We determine the complete structure of the symmetry algebras associated with the N-body Calogero-Moser system and its maximally superintegrable discretization. We prove that the discretization naturally leads to a nontrivial deformation of the continuous symmetry algebra, with the discretization parameter playing the r\^ole of a deformation parameter. This phenomenon illustrates how discrete superintegrable systems can be viewed as natural sources of deformed polynomial algebraic structures. As a byproduct of these results, we also reveal a connection between the above-mentioned symmetry algebras and the Bell polynomials, as a consequence of the trace properties.