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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A Lie theoretic Galois theory
for the spectral curves
of an integrable system. II


Authors: Andrew McDaniel and Lawrence Smolinsky
Journal: Trans. Amer. Math. Soc. 349 (1997), 713-746
MSC (1991): Primary 58F05, 58F07
DOI: https://doi.org/10.1090/S0002-9947-97-01853-9
MathSciNet review: 1401779
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Abstract: In the study of integrable systems of ODE's arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group $W$ and the Hecke algebra of double cosets of a parabolic subgroup of $W.$ For each algebra a subtorus is identified that agrees with Kanev's Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.


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Additional Information

Andrew McDaniel
Affiliation: Department of Mathematics, Georgetown University, Washington, D.C. 20057
Email: Mcdaniea@guvax.georgetown.edu

Lawrence Smolinsky
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
Email: smolinsk@math.lsu.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01853-9
Received by editor(s): February 22, 1995
Additional Notes: The second author was partially supported by a Louisiana Education Quality Support Fund grant LEQSF(90-93)-RD-A-10.
Article copyright: © Copyright 1997 American Mathematical Society