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Polynomial Poisson algebras with regular structure of symplectic leaves. (English. Russian original) Zbl 1138.53314

Theor. Math. Phys. 133, No. 1, 1321-1337 (2002); translation from Teor. Mat. Fiz. 133, No. 1, 3-23 (2002).
Summary: 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.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
17B63 Poisson algebras
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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