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Poisson brackets on rational functions and multi-Hamiltonian structure for integrable lattices. (English) Zbl 1115.37336
Summary: We introduce a family of compatible Poisson brackets on the space of rational functions with denominator of a fixed degree and use it to derive a multi-Hamiltonian structure for a family of integrable lattice equations that includes both the standard and the relativistic Toda lattices.

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K60 Lattice dynamics; integrable lattice equations
53D17 Poisson manifolds; Poisson groupoids and algebroids
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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[1] J. Moser, Finitely Many Mass Points on the Line under the Influence of an Exponential Potential, Batelles Recontres, Springer Notes in Physics, 1974, p. 417-497.
[2] Deift, P.; Li, L.; Nanda, T.; Tomei, C., Comm. pure & appl. math., 39, 183, (1986)
[3] Berezansky, Yu.M., Rep. math. phys., 24, 21, (1986)
[4] Faybusovich, L., J. math. syst. est. control, 3, 129, (1993)
[5] Kostant, B., Adv. math., 34, 195, (1979)
[6] L. Faybusovich, M.I. Gekhtman, Elementary Toda orbits and integrable lattices, J. Math. Phys., to appear. · Zbl 1052.37051
[7] Damianou, P.A., Lett. math. phys., 20, 101, (1990)
[8] Damianou, P.A., J. math. phys., 35, 5511, (1994)
[9] Suris, Y.B., Phys. lett. A, 180, 419, (1993)
[10] M.F. Atiyah, N. Hitchin, The Geometry and Dynamics of Magnetic Monopoles, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988. · Zbl 0671.53001
[11] I.M. Gelfand, I. Zakharevich, Webs, Lenard schemes, and the local geometry of bihamiltonian Toda and Lax structures, preprint math. DG/990308. · Zbl 0986.37060
[12] Brockett, R.; Faybusovich, L., Syst. control lett., 16, 79, (1991)
[13] Suris, Y.B., Rev. math. phys., 11, 727, (1999)
[14] Kharchev, S.; Mironov, A.; Zhedanov, A., Int. J. mod. phys., 12, 2675, (1997)
[15] P.A. Fuhrmann, U. Helmke, Bezoutians, Linear Algebra Appl. 122/123/124 (1989), 1039-1097.
[16] N.I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Co. Publishing, New York, 1965. · Zbl 0135.33803
[17] Ablowitz, M.J.; Ladik, J.F., J. math. phys., 16, 598, (1975)
[18] Reyman, A.G.A.G.; Semenov-Tian-Shansky, M.A., Encyclopaedia math. sci., 16, 116, (1994)
[19] Fuchssteiner, B., Progr. theor. phys., 70, 1508, (1983)
[20] Berezansky, Yu.M.; Gekhtman, M.; Shmoish, M., Ukrain. math. J, 38, 84, (1986)
[21] Berezansky, Yu.M.; Shmoish, M., J. nolin. math. phys., 1, 116, (1994)
[22] Wilson, G., Invent. math., 113, 1, (1998)
[23] A. Kasman, Nested Bethe ansatz and finite dimensional canonical commutation relationship, math-ph/0004030. · Zbl 0991.81030
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