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Annales de l’institut Fourier
Article . 2008 . Peer-reviewed
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Article . 2008
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https://dx.doi.org/10.48550/ar...
Article . 2006
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Integrable hierarchies and the modular class

Integrable hierarchies and the modular class
Authors: Damianou, Pantelis A.; Fernandes, Rui Loja;

Integrable hierarchies and the modular class

Abstract

It is well-known that the Poisson-Nijenhuis manifolds, introduced by Kosmann-Schwarzbach and Magri form the appropriate setting for studying many classical integrable hierarchies. In order to define the hierarchy, one usually specifies in addition to the Poisson-Nijenhuis manifold a bi-hamiltonian vector field. In this paper we show that to every Poisson-Nijenhuis manifold one can associate a canonical vector field (no extra choices are involved!) which under an appropriate assumption defines an integrable hierarchy of flows. This vector field is the modular class of the Poisson-Nijhenhuis manifold. This class has a canonical representative which, under a cohomological assumption, is a bi-hamiltonian vector field. In many examples the associated hierarchy of flows reproduces classical integrable hierarchies. We illustrate in detail with the Harmonic Oscillator, the Calogero-Moser system, the classical Toda lattice and various Bogoyavlensky-Toda Lattices.

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Keywords

Mathematics - Differential Geometry, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, integrable hierarchies, FOS: Physical sciences, Mathematical Physics (math-ph), 53D17, 37J35, Poisson manifolds; Poisson groupoids and algebroids, Differential Geometry (math.DG), modular class, Poisson-Nijhenhuis manifolds, FOS: Mathematics, Mathematical Physics

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
11
Average
Top 10%
Top 10%
Green
gold