On classification of discrete, scalar-valued Poisson Brackets

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.

💡 Research Summary

The paper addresses the classification of discrete differential‑geometric Poisson brackets (dDGPBs) defined on an infinite lattice with scalar‑valued fields (target space dimension N = 1). Starting from the general local Poisson bracket formula
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