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Journal of the London Mathematical Society 1997 56(3):504-518; doi:10.1112/S0024610797005590
© 1997 by London Mathematical Society
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© The London Mathematical Society

The Algebraic Geometry of the Kostant–Kirillov Form

J. C. Hurtubise

Department of Mathematics, McGill University 805 Sherbrooke Street West, Montreal, Canada H3A 2K6. E-mail: hurtubis{at}math.mcgill.ca

Received 12 January 1995. Revision received 27 May 1995.

Large classes of integrable Hamiltonian systems can be expressed as systems over coadjoint orbits in a loop algebra defined over a semi-simple Lie algebra g. These systems can then be integrated via the classical, symplectic Liouville–Arnold method. On the other hand, the existence of spectral curves as constants of motion allows one to integrate these systems in terms of flows of line bundles on the curves. This note links the symplectic geometry of the coadjoint orbits with the algebraic geometry of these curves for arbitrary semi-simple g, which then allows us to reconcile the two integration methods.


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