Discrete-time maximally superintegrable systems and deformed symmetry algebras: the Calogero-Moser case
Pavel Drozdov, Giorgio Gubbiotti, Danilo Latini
TL;DR
This work addresses whether discretizing a maximally superintegrable system deforms its symmetry algebra. Using the Calogero–Moser (CM) system as a prototype, it analyzes the continuous model with Lax pair dynamics and trace-based invariants $F_k$ and $K_{m,n}$, then applies the Nijhoff–Pang discretization to obtain a discrete, symplectic counterpart with modified invariants $\widetilde K_{m,n}$ that depend on the lattice spacing $h$. The main finding is that the discrete symmetry algebra is a nontrivial deformation of the continuous polynomial algebra, with the deformation parameter $h$ and an increase in algebraic degree (to $2N$ for general $N$, and by 2 for $N=2$); this deformation vanishes in the $h\to0$ limit. Bell polynomials arise naturally in expressing higher-trace relations via Cayley–Hamilton arguments, linking trace identities to the algebraic structure. Overall, the paper demonstrates that maximally superintegrable discretizations can yield deformed polynomial Poisson algebras, suggesting new avenues for discretized integrable systems and their representation-theoretic implications.
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 rôle 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.