Integrable hierarchies and the modular class. (English) Zbl 1147.53065

Damianou, Pantelis A.; Fernandes, Rui Loja

Ann. Inst. Fourier 58, No. 1, 107-137 (2008).

Authors’ 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 this in detail with the harmonic oscillator, the Calogero-Moser system, the classical Toda lattice and various Bogoyavlensky-Toda lattices.

Reviewer: Nicolai K. Smolentsev (Kemerovo)

MSC:

53D17	Poisson manifolds; Poisson groupoids and algebroids
37J35	Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests

Keywords:

Poisson-Nijhenhuis manifolds; modular class; integrable hierarchies

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