Damianou, Pantelis A.; Fernandes, Rui Loja Integrable hierarchies and the modular class. (English) Zbl 1147.53065 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) Cited in 10 Documents 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 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Numdam EuDML References: [1] Agrotis, M.; Damianou, P. A., The modular hierarchy of the Toda lattice · Zbl 1146.53053 [2] Bolsinov, A. V.; Borisov, A. V., Compatible Poisson brackets on Lie algebras, Matem notes, 72, 1, 10-30 (2002) · Zbl 1042.37041 · doi:10.1023/A:1019856702638 [3] Bolsinov, A. V.; Matveev, V. 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