×

zbMATH — the first resource for mathematics

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

MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
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
[3] Bolsinov, A. V.; Matveev, V. S., Geometrical interpretation of beneti systems, J. of Geom. and Phys., 44, 489-506, (2003) · Zbl 1010.37035
[4] Casati, P.; Falqui, G.; Magri, F.; Pedroni, M.; Springer-Verlag, Integrability of Nonlinear Systems, 495, Eight lectures on integrable systems, 209-250, (1997), Lecture Notes in Physics, Berlin · Zbl 0907.58031
[5] Caseiro, R., Master integrals, superintegrability and quadratic algebras, Bull. Sci. Math., 126, 8, 617-630, (2002) · Zbl 1010.37033
[6] Caseiro, R.; Françoise, J. P., Algebraically linearizable dynamical systems, Textos Mat. Ser. B, 32, 35-45, (2002), Coimbra · Zbl 1057.35069
[7] Damianou, P. A., Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices, Rev. Math. Phys., 16, 2, 175-241, (2004) · Zbl 1053.37061
[8] Damianou, P. A., On the bi-Hamiltonian structure of Bogoyavlensky-Toda lattices, Nonlinearity, 17, 2, 397-413, (2004) · Zbl 1052.37045
[9] Dufour, J. P.; Haraki, A., Rotationnels et structures de Poisson quadratiques, C. R. Acad. Sci. Paris Ser. I Math., 312, 1, 137-140, (1991) · Zbl 0719.58001
[10] Dufour, J. P.; Zung, N. T., Poisson structures and their normal forms, Progress in Mathematics, (2005) · Zbl 1082.53078
[11] Evens, S.; Lu, J.-H.; Weinstein, A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford, 50, 2, 417-436, (1999) · Zbl 0968.58014
[12] Faybusovich, L.; Gekhtman, M., Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices, Phys. Lett. A, 272, 4, 236-244, (2000) · Zbl 1115.37336
[13] Fernandes, R. L., On the mastersymmetries and bi-Hamiltonian structure of the Toda lattice, J. Phys. A, 26, 3797-3803, (1993) · Zbl 0811.58035
[14] Fernandes, R. L., Connections in Poisson geometry I: holonomy and invariants, J. Diff. Geom., 54, 303-366, (2000) · Zbl 1036.53060
[15] Fernandes, R. L., Lie algebroids, holonomy and characteristic classes, Adv. Math., 170, 1, 119-179, (2002) · Zbl 1007.22007
[16] Flaschka, H., The Toda lattice. I. existence of integrals, Phys. Rev. B, 9, 3, 1924-1925, (1974) · Zbl 0942.37504
[17] Grabowski, J.; Marmo, G.; Michor, P., Homology and modular classes of Lie algebroids, Ann. Inst. Fourier, 56, 69-83, (2006) · Zbl 1141.17018
[18] Grabowski, J.; Marmo, G.; Perelomov, A. M., Poisson structures towards a classification, Modern Phys. Lett. A, 8, 1719-1733, (1993) · Zbl 1020.37529
[19] Kosmann-Schwarzbach, Y.; Winternitz, P., Systèmes dynamiques non linéaires: intégrabilité et comportement qualitatif, 102, Géométrie des systèmes bihamiltoniens, 185-216, (1986), Presses de l’Université de Montréal · Zbl 0624.58009
[20] Kosmann-Schwarzbach, Y.; Magri, F., Poisson-Nijenhuis structures, Ann. Inst. Henri Poincaré, 53, 35-81, (1990) · Zbl 0707.58048
[21] Kosmann-Schwarzbach, Y.; Weinstein, A., Relative modular classes of Lie algebroids, C. R. Math. Acad. Sci. Paris, 341, 8, 509-514, (2005) · Zbl 1080.22001
[22] Koszul, J. L., Crochet de Schouten-Nijenhuis et cohomologie, Astérisque Numéro Hors Série, 257-271, (1985) · Zbl 0615.58029
[23] Lichnerowicz, A., LES variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geom., 12, 253-300, (1977) · Zbl 0405.53024
[24] Liu, Z.; Xu, P., On quadratic Poisson structures, Lett. Math. Phys., 26, 33-42, (1992) · Zbl 0773.58007
[25] Nunes da Costa, J.; Damianou, P. A., Toda systems and exponents of simple Lie groups, Bull. Sci. Math., 125, 1, 49-69, (2001) · Zbl 1017.37028
[26] Oevel, W, Topics in Soliton Theory and Exactly Solvable non-linear Equations, A geometrical approach to integrable systems admitting time dependent invariants, 108-124, (1987), World Scientific Publ., Singapore · Zbl 0736.35119
[27] Ranada, M. F., Superintegrability of the Calogero-Moser system: constants of motion, master symmetries, and time-dependent symmetries, J. Math. Phys., 40, 1, 236-247, (1999) · Zbl 0956.37041
[28] Vaisman, I., Lectures on the geometry of Poisson manifolds, 118, (1994), Progress in Mathematics, Basel · Zbl 0810.53019
[29] Weinstein, A., The modular automorphism group of a Poisson manifold, J. Geom. Phys., 23, 3-4, 379-394, (1997) · Zbl 0902.58013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.