On classification of discrete, scalarvalued Poisson brackets
Abstract
We address the problem of classifying discrete differentialgeometric Poisson brackets (dDGPBs) of any fixed order on a target space of dimension 1. We prove that these Poisson brackets (PBs) are in onetoone 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 Miuratype transformations. Finally, by improving a lattice consolidation procedure, we obtain new families of nondegenerate, vectorvalued and firstorder dDGPBs that can be considered in the framework of admissible LiePoisson group theory.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 October 2012
 DOI:
 10.1016/j.geomphys.2012.05.004
 arXiv:
 arXiv:1109.4327
 Bibcode:
 2012JGP....62.2059P
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematical Physics;
 Mathematics  Differential Geometry
 EPrint:
 24 pages