Integrable hierarchies and the modular class

Pantelis A. Damianou[1]; Rui Loja Fernandes[2]

  • [1] University of Cyprus Department of Mathematics and Statistics P.O. Box 20537, 1678 Nicosia (Cyprus)
  • [2] Instituto Superior Técnico Departamento de Matemática Lisboa (Portugal)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 1, page 107-137
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Damianou, Pantelis A., and Fernandes, Rui Loja. "Integrable hierarchies and the modular class." Annales de l’institut Fourier 58.1 (2008): 107-137. <http://eudml.org/doc/10306>.

@article{Damianou2008,
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.},
affiliation = {University of Cyprus Department of Mathematics and Statistics P.O. Box 20537, 1678 Nicosia (Cyprus); Instituto Superior Técnico Departamento de Matemática Lisboa (Portugal)},
author = {Damianou, Pantelis A., Fernandes, Rui Loja},
journal = {Annales de l’institut Fourier},
keywords = {Poisson-Nijhenhuis manifolds; modular class; integrable hierarchies},
language = {eng},
number = {1},
pages = {107-137},
publisher = {Association des Annales de l’institut Fourier},
title = {Integrable hierarchies and the modular class},
url = {http://eudml.org/doc/10306},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Damianou, Pantelis A.
AU - Fernandes, Rui Loja
TI - Integrable hierarchies and the modular class
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 1
SP - 107
EP - 137
AB - 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.
LA - eng
KW - Poisson-Nijhenhuis manifolds; modular class; integrable hierarchies
UR - http://eudml.org/doc/10306
ER -

References

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