Surfaces and the Sklyanin bracket

Abstract

We discuss the Lie Poisson groups structures associated to splittings of the loop group LGL(N), 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 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.