Discrete-time systems in quasi-standard form and the $\mathfrak{h}_6$ coalgebra symmetry

TL;DR

This paper characterizes a family of discrete-time systems with $rak{h}_6$ coalgebra symmetry, classifies those with additional quadratic invariants, and explores their integrability, relationships, and continuum limits.

Contribution

It introduces a comprehensive classification of discrete-time systems with $rak{h}_6$ symmetry and identifies cases with extra invariants, advancing understanding of their integrability.

Findings

Family of systems depending on an arbitrary potential function.

Classification of systems with additional quadratic invariants.

Discussion of integrability and continuum limits.

Abstract

In this paper, we characterize all discrete-time systems in quasi-standard form admitting coalgebra symmetry with respect to the Lie--Poisson algebra $\mathfrak{h}_{6}$. The outcome of this study is a family of systems depending on an arbitrary function of three variables, playing the r\^ole of the potential. Moreover, using a direct search approach, we classify discrete-time systems from this family that admit an additional invariant at most quadratic in the physical variables. We discuss the integrability properties of the obtained cases, their relationship with known systems, and their continuum limits.