On classification of discrete,scalar-valued Poisson brackets |
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Authors: | E. Parodi |
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Affiliation: | SISSA, Via Bonomea 265, 34136, Trieste, Italy |
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Abstract: | We address the problem of classifying discrete differential-geometric Poisson brackets (dDGPBs) of any fixed order on a target space of dimension 1. We prove 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 a cubic PB of the standard Volterra lattice by discrete Miura-type transformations. Finally, by improving a lattice consolidation procedure, we obtain new families of non-degenerate, vector-valued and first-order dDGPBs that can be considered in the framework of admissible Lie–Poisson group theory. |
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Keywords: | Discrete Poisson brackets Discrete Miura transformations Lie&ndash Poisson groups |
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