On classification of discrete, scalar-valued Poisson Brackets

TL;DR

This paper classifies discrete differential-geometric Poisson brackets of fixed order on one-dimensional target spaces, linking them to projective hypersurface intersections and reducing them to standard forms, while introducing new families via lattice procedures.

Contribution

It provides a complete classification of these Poisson brackets, establishes their correspondence with geometric intersections, and introduces new families through an improved lattice method.

Findings

Poisson brackets correspond to intersection points of projective hypersurfaces.

All brackets can be reduced to cubic PB of the Volterra lattice.

New families of non-degenerate, vector-valued, first-order brackets are constructed.

Abstract

We address the problem of classifying discrete differential-geometric Poisson brackets (dDGPBs) of any fixed order on target space of dimension 1. It is proved that these Poisson brackets (PBs) are in one-to-one correspondence with the intersection points of certain projective hypersurfaces. In addition, they can be reduced to cubic PB of standard Volterra lattice by discrete Miura-type transformations. Finally, improving a consolidation lattice procedure, we obtain new families of non-degenerate, vector-valued and first order dDGPBs, which can be considered in the framework of admissible Lie-Poisson group theory.