Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 007, 16 pages      math-ph/0610048
Contribution to the Vadim Kuznetsov Memorial Issue

Multi-Hamiltonian Structures on Beauville's Integrable System and Its Variant

Rei Inoue a and Yukiko Konishi b
a) Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
b) Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8914, Japan

Received October 24, 2006, in final form December 29, 2006; Published online January 08, 2007

We study Beauville's completely integrable system and its variant from a viewpoint of multi-Hamiltonian structures. We also relate our result to the previously known Poisson structures on the Mumford system and the even Mumford system.

Key words: completely integrable system; Mumford system; multi-Hamiltonian structure.

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  1. Beauville A., Jacobiennes des courbes spectrales et systèmes hamiltoniens complètement intégrables, Acta Math. 164 (1990), 211-235.
  2. Donagi R., Markman E., Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 1-119.
  3. Fernandes R.L., Vanhaecke P., Hyperelliptic Prym varieties and integrable systems, Comm. Math. Phys. 221 (2001), 169-196, math-ph/0011051.
  4. Fu B., Champs de vecteurs invariants par translation sur les jacobiennes affines des courbes spectrales, C. R. Math. Acad. Sci. Paris 337 (2003), no. 2, 105-110.
  5. Inoue R., Konishi Y., Yamazaki T., Jacobian variety and integrable system - after Mumford, Baeuville and Vanhaecke, J. Phys. Geom., to appear, math-ph/0512033.
  6. Mumford D., Tata lectures on theta II, Birkhäuser, 1984.
  7. Nakayashiki A., Smirnov F.A., Cohomologies of affine Jacobi varieties and integrable systems, Comm. Math. Phys. 217 (2001), 623-652, math-ph/0001017.
  8. Pendroni M., Vanhaecke P., A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure, Regul. Chaotic Dyn. 3 (1998), 132-160.
  9. Reyman A.G., Semenov-Tian-Shansky M.A., Group-theoretical methods in the theory of finite-dimensional integrable systems, Encyclopedia of Mathematical Sciences, Vol. 16, Springer-Verlag, Berlin Heidelgerg, 1994, 116-225.
  10. Smirnov F.A., Zeitlin V., Affine Jacobi varieties of spectral curves and integrable models, math-ph/0203037.
  11. Vanhaecke P., Linearising two-dimensional integrable systems and the construction of action-angle variables, Math. Z. 211 (1992), 265-313.
  12. Vanhaecke P., Integrable systems in the realm of algebraic geometry, Lecture Notes in Math., Vol. 1638, Springer, Berlin, 2001.

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