Integrable hierarchies and the modular class
[Hiérarchies intégrables et la classe modulaire]
Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 107-137.

Il est bien connu que les variétés de Poisson-Nijenhuis, introduites par Kosmannn-Schwarzbach et Magri, constituent le contexte le plus approprié pour étudier de nombreuses hiérarchies intégrables. Afin de définir une telle hiérarchie, on explicite d’habitude, en plus de la variété de Poisson-Nijenhuis, un champ de vecteurs bi-hamiltonien. Dans cet article, nous montrons qu’à toute variété de Poisson-Nijenhuis nous pouvons associer un et un seul champ de vecteurs canonique qui, sous des hypothèses convenables, définit une hiérarchie intégrable de flots. Ce champ de vecteurs est la classe modulaire de la variété de Poisson-Nijenhuis. Cette classe possède un représentant défini de manière canonique qui, sous une hypothèse cohomologique, est un champ de vecteurs bi-hamiltonien. Dans de nombreux exemples, la hiérarchie associée de flots reproduit les hiérarchies intégrables classiques. Nous illustrons en détails ce fait à l’aide de l’oscillateur harmonique, du système de Calogero-Moser, du réseau de Toda classique et des divers réseaux de Bogoyavlensky-Toda.

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.

DOI : https://doi.org/10.5802/aif.2346
Classification : 53D17,  37J35
Mots clés : Variétés de Poisson-Nijhenhuis, classe modulaire, hiérarchies intégrables
@article{AIF_2008__58_1_107_0,
     author = {Damianou, Pantelis A. and Fernandes, Rui Loja},
     title = {Integrable hierarchies  and the modular class},
     journal = {Annales de l'Institut Fourier},
     pages = {107--137},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {1},
     year = {2008},
     doi = {10.5802/aif.2346},
     mrnumber = {2401218},
     zbl = {1147.53065},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2346/}
}
Damianou, Pantelis A.; Fernandes, Rui Loja. Integrable hierarchies  and the modular class. Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 107-137. doi : 10.5802/aif.2346. http://www.numdam.org/articles/10.5802/aif.2346/

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