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

Full-text PDF

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.

**[A]**Robert D.M. Accola,*Vanishing Properties of Theta Functions for Abelian Covers of Riemann Surfaces*, Advances in the Theory of Riemann Surfaces, Proceedings of the 1969 Stony Brook Conference, Princeton University Press, Princeton, N.J., 1971, pp. 7-18. MR**44:5448****[AvM1]**M. Adler and P. van Moerbeke,*Completely Integrable Systems, Euclidean Lie Algebras, and Curves*, Advances in Math.**38**(1980), 267-317. MR**83m:58041****[AvM2]**-,*Linearization of Hamiltonian Systems, Jacobi Varieties, and Representation Theory*, Advances in Math.**38**(1980), 318-379. MR**83m:58042****[BE]**Robert J. Baston and Michael G. Eastwood,*The Penrose Transform: Its Interaction with Representation Theory*, Oxford University Press, New York, N.Y., 1989. MR**92j:32112****[BL]**Ch. Birkenhake and H. Lange,*Norm-endomorphisms of abelian subvarieties*, Classification of Irregular Varieties, Lecture Notes in Mathematics, vol. 1515, Springer-Verlag, Berlin, 1992, pp. 21-32. MR**93g:14051****[B]**Armand Borel, et al.,*Seminar on Transformation Groups*, Annals of Mathematics Studies, Number 46, Princeton University Press, Princeton, N.J., 1960. MR**22:7129****[BMP]**M.R. Bremner, R.V. Moody, and J. Patera,*Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras*, Pure and Applied Mathematics, vol. 90, Marcel Dekker, Inc., New York, N.Y., 1985. MR**86f:17002****[BD]**Theodor Bröcker and Tammo tom Dieck,*Representations of Compact Lie Groups*, GTM, vol. 98, Springer-Verlag, New York, N.Y., 1985. MR**86i:22023****[CR]**Charles W. Curtis and Irving Reiner,*Methods of Representation Theory*, vol. I, John Wiley & Sons, New York, N.Y., 1981. MR**82i:20001****[G]**Phillip A. Griffiths,*Linearizing Flows and a Cohomological Interpretation of Lax Equations*, American Journal of Mathematics**107**(1985), 1445-1483. MR**87e:58048****[GH]**Phillip A. Griffiths and Joseph Harris,*Principles of Algebraic Geometry*, John Wiley & Sons, New York, N.Y., 1978. MR**80b:14001****[Ha]**Morton Hamermesh,*Group Theory and its Application to Physical Problems*, Dover Publications, Inc., New York, N.Y., 1962. MR**25:132****[H1]**James E. Humphreys,*Reflection Groups and Coxeter Groups*, Cambridge University Press, Cambridge, Great Britain, 1990.**[H2]**-,*Introduction to Lie algebras and representation theory*, GTM, vol. 9, Springer-Verlag, New York, N.Y., 1972. MR**48:2197****[K]**Vassil I. Kanev,*Spectral Curves, Simple Lie Algebras, and Prym-Tjurin Varieties*, Proceedings of Symposia in Pure Mathematics**49**(1989), 627-645. MR**91b:14028****[Kr]**Aloys Krieg,*Hecke Algebras*, Memoirs of the American Mathematical Society**87**(435) (1990). MR**90m:16024****[M]**William Massey,*Algebraic Topology: An Introduction*, Harcourt-Brace, New York, N.Y., 1967. MR**35:2271****[Mc1]**Andrew McDaniel,*Lie Algebra Representations and the Toda lattice*, thesis, Brandeis University (1985).**[Mc2]**A. McDaniel,*Representations of and the Toda lattice*, Duke Math. Journal**56**(1988), 47-99. MR**89e:58061****[MS]**A. McDaniel and L. Smolinsky,*A Lie Theoretic Galois Theory for the Spectral Curves of an Integrable System: I*, Communications in Mathematical Physics**149**(1992), 127-148. MR**94f:58065****[MS1]**A. McDaniel and L. Smolinsky,*The Flow of the Periodic Toda Lattice*, preprint.**[OP]**M.A. Olshanetsky and A.M. Perelomov,*Classical Integrable Systems Related to Lie Algebras*, Physics Reports**71**(1981), 313-400. MR**83d:58032****[P]**Stefanos Pantazis,*Prym Varieties and the Geodesic Flow on SO(n)*, Mathematische Annalen**273**(1986), 297-315. MR**87g:58054****[R]**John F. X. Ries,*Splittable Jacobi Varieties*, Curves, Jacobians, and Abelian Varieties, Contemporary Mathematics, vol. 136, American Mathematical Society, Providence, R.I., 1992, pp. 305-326. MR**93j:14031****[vMM]**P. van Moerbeke and D. Mumford,*The Spectrum of Difference Operators and Algebraic Curves*, Acta Math.**143**(1979), 93-154. MR**80e:58028**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
58F05,
58F07

Retrieve articles in all journals with MSC (1991): 58F05, 58F07

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