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 | References | Similar Articles | Additional Information

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 and the Hecke algebra of double cosets of a parabolic subgroup of 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