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 TMF, 2002, Volume 133, Number 1, Pages 3–23 (Mi tmf377)

Polynomial Poisson Algebras with Regular Structure of Symplectic Leaves

A. V. Odesskiiab, V. N. Rubtsovcb

a L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
b Université d'Angers
c Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: We study polynomial Poisson algebras with some regularity conditions. Linear (Lie–Berezin–Kirillov) structures on dual spaces of semisimple Lie algebras, quadratic Sklyanin elliptic algebras, and the polynomial algebras recently described by Bondal, Dubrovin, and Ugaglia belong to this class. We establish some simple determinant relations between the brackets and Casimir functions of these algebras. In particular, these relations imply that the sum of degrees of the Casimir functions coincides with the dimension of the algebra in the Sklyanin elliptic algebras. We present some interesting examples of these algebras and show that some of them arise naturally in the Hamiltonian integrable systems. A new class of two-body integrable systems admitting an elliptic dependence on both coordinates and momenta is among these examples.

Keywords: polynomial Poisson structures, elliptic algebras, integrable systems

DOI: https://doi.org/10.4213/tmf377

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English version:
Theoretical and Mathematical Physics, 2002, 133:1, 1321–1337

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Citation: A. V. Odesskii, V. N. Rubtsov, “Polynomial Poisson Algebras with Regular Structure of Symplectic Leaves”, TMF, 133:1 (2002), 3–23; Theoret. and Math. Phys., 133:1 (2002), 1321–1337

Citation in format AMSBIB
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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. V. Odesskii, “Elliptic algebras”, Russian Math. Surveys, 57:6 (2002), 1127–1162
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