The $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems

TL;DR

This paper classifies variational discrete-time systems with $rak{sl}_2(R)$ coalgebra symmetry, revealing new superintegrable systems and conjecturing completeness of these cases within this algebraic framework.

Contribution

It provides a comprehensive classification of superintegrable discrete-time systems with $rak{sl}_2(R)$ coalgebra symmetry, including new systems and a conjecture on their completeness.

Findings

Identification of several superintegrable discrete-time systems

Introduction of new superintegrable models

Conjecture on the exhaustiveness of the classification

Abstract

In this paper, we classify all the variational discrete-time systems in quasi-standard form in $N$ degrees of freedom admitting coalgebra symmetry with respect to the generic realisation of the Lie-Poisson algebra $\mathfrak{sl}_{2}(\mathbb{R})$. This approach naturally yields several quasi-maximally and maximally superintegrable discrete-time systems, both known and new. We conjecture that this exhausts the (super)integrable cases associated with this algebraic construction.