The ${\mathfrak{sl}}_2(\mathbb{R})$ coalgebra symmetry and the superintegrable discrete-time systems

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.

37 pages, 3 figures

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