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Communications in Mathematical Physics

, Volume 180, Issue 2, pp 265–296 | Cite as

W-geometry of the Toda systems associated with non-exceptional simple Lie algebras

  • Jean-Loup Gervais
  • Mikhail V. Saveliev
Article

Abstract

The present paper describes theW-geometry of the Abelian finite non-periodic (conformal) Toda systems associated with theB, C andD series of the simple Lie algebras endowed with the canonical gradation. The principal tool here is a generalization of the classical Plücker embedding of theA-case to the flag manifolds associated with the fundamental representations ofB n ,C n andD n , and a direct proof that the corresponding Kähler potentials satisfy the system of two-dimensional finite non-periodic (conformal) Toda equations. It is shown that theW-geometry of the type mentioned above coincide with the differential geometry of special holomorphic (W) surfaces in target spaces which are submanifolds (quadrics) ofCP N with appropriate choices ofN. In addition, theseW-surfaces are defined to satisfy quadratic holomorphic differential conditions that ensure consistency of the generalized Plücker embedding. These conditions are automatically fulfilled when Toda equations hold.

Keywords

Neural Network Manifold Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Jean-Loup Gervais
    • 1
  • Mikhail V. Saveliev
    • 1
  1. 1.Laboratoire de Physique Théorique de l'École Normale SupérieureParis Cédex 05France

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