Abstract:
We discuss the Lie Poisson group structures associated to splittings of the loop group LGL(N,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 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.
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Received: 8 August 2001/Accepted: 29 April 2002 Published online: 14 October 2002
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ID="★" The first author of this article would like to thank NSERC and FCAR for their support
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ID="★★" The second author was partially supported by NSF grant number DMS-9802532
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Hurtubise, J., Markman, E. Surfaces and the Sklyanin Bracket. Commun. Math. Phys. 230, 485–502 (2002). https://doi.org/10.1007/s00220-002-0700-9
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DOI: https://doi.org/10.1007/s00220-002-0700-9