Abstract
For generic rational coadjoint orbits in the dual \(\tilde gl(r)^{ + *}\) of the positive half of the loop algebra \(\tilde gl(r)^{ + *}\), the natural divisor coordinates associated to the eigenvector line bundles over the spectral curves project to Darboux coordinates on the Gl(r)-reduced space. The geometry of the embedding of these curves in an ambient ruled surface suggests an intrinsic definition of symplectic structure on the space of pairs (spectral curves, duals of eigenvector line bundles) based on Serre duality. It is shown that this coincides with the reduced Kostant-Kirillov structure. For all Hamiltonians generating isospectral flows, these Darboux coordinates allow one to deduce a completely separated Liouville generating function, with the corresponding canonical transformation to linearizing variables identified as the Abel map.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Adams, M., Harnad, J. and Hurtubise, J.,Isospectral H amiltonian Flows in Finite and Infinite Dimensions II. Integration of Flows, Commun. Math. Phys. (1990, in press).
———, Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras, CRM preprint (1990).
———, Liouville Generating Function for Isospectral Flow in Loop Algebras, CRM preprint 1652 (1990). to appear in “Integrable and Superin- tegrable Systems” (ed. B. Kupershmidt), World Scientific, Singapore (1990).
Adams, M., Hamad, J. and Previato, E., Isospectral Hamiltonian Flows in Finite and Infinite Dimensions I. Generalised Moser Systems and Moment Maps into Loop Algebras, Commun. Math. Phys. 117 (1988), 451–500.
Adler, M. and van Moerbeke, P., Completely Integrable Systems, Euclidean Lie Algebras, and Curves, Adv. Math. 38 (1980), 267–317.
Beauville, A., Jacobiennes des courbes spectrales et systèmes Hamiltoniens complètement integrables, Preprint (1989).
Dickey, L.A., Integrable Nonlinear Equations and Liouville’s Theorem I, II, Commun. Math. Phys. 82, 345–360; ibid. 82 (1981), 360–375.
Flaschka, H., Newell, A.C. and Ratiu, T., Kac-Moody Algebras and Soliton Equations II. Lax Equations Associated to Physica 9D (1983), p. 300.
Griffiths, P. and Harris, J., “Principles of Algebraic Geometry,” Wiley, New York, 1978.
Krichever, I.M., Methods of Algebraic Geometry in the Theory of Nonlinear Equations, Russ. Math. Surveys 32 (1980), 53–79.
van Moerbeke, P. and Mumford, D., The Spectrum of Difference Operators and Algebraic Curves, Acta Math. 143 (1979), 93–154.
Reiman, A.G., and Semenov-Tian-Shansky, M.A., Reduction of Hamiltonian systems, Affine Lie algebras and Lax Equations I, II, Invent. Math. 54 (1979), 81–100; ibid. 63 (1981), 423–432.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Adams, M.R., Harnad, J., Hurtubise, J. (1991). Coadjoint Orbits, Spectral Curves and Darboux Coordinates. In: Ratiu, T. (eds) The Geometry of Hamiltonian Systems. Mathematical Sciences Research Institute Publications, vol 22. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9725-0_2
Download citation
DOI: https://doi.org/10.1007/978-1-4613-9725-0_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4613-9727-4
Online ISBN: 978-1-4613-9725-0
eBook Packages: Springer Book Archive