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Surfaces and the Sklyanin bracket. (English) Zbl 1041.37034

Summary: We discuss the Lie-Poisson group structure associated to splittings of the loop group \(LGL(N,\mathbb{C})\), due to Sklyanin. Concentrating on the finite-dimensional leaves of the associated Poisson structure, we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a \(\mathbb{C}^*\)-invariant Poisson structure. In particular, Sklyanin’s Lie-Poisson structure admits a suitable abelianisation once one passes to an appropriate spectral curve. The Sklyanin structure is then equivalent to one considered by Mukai, Tyurin and Bottacin on a moduli space of sheaves on the Poisson surface. The abelianization procedure gives rise to natural Darboux coordinates for these leaves, as well as separation of variables for the integrable Hamiltonian systems associated to invariant functions on the group.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14C05 Parametrization (Chow and Hilbert schemes)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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