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


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Gervais, J.-L., Matsuo, Y.: Phys. Lett.B274, 309 (1992); Commun. Math. Phys.152, 317 (1993)Google Scholar
  2. 2.
    Gervais, J.-L.: Introduction to DifferentialW-Geometry in New Developments in String Theory, Conformal Models and Topological Field Theory. Cargese Meeting 1993, Plenum ed; and Going from Conformal to light Cone inW gravity, “Strings '93” Meeting Berkeley, World Scientific ed.Google Scholar
  3. 3.
    Saveliev, M.V.: Sov. J. Theor. Math. Phys.60, 9 (1984)Google Scholar
  4. 4.
    Leznov, A.N., Saveliev, M.V.: Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems. Progress in Physics Series, V.15, Basel, Birkhaüser-Verlag, 1992Google Scholar
  5. 5.
    Bilal, A., Gervais, J.-L.: Nucl. Phys.B314, 646 (1989);B318, 579 (1989)Google Scholar
  6. 6.
    Balog, J., Fehér, L., O'Raifeartaigh, L., Forgács, P., Wipf, A.: Ann. Phys. (N.Y.)203, 76 (1990)Google Scholar
  7. 7.
    Griffiths, Ph.A., Harris, J.: Principles of Algebraic Geometry. New York: Wiley-Interscience Publ., 1978Google Scholar
  8. 8.
    Givental', A.B.: Russ. Math. Surv.44:3, 193 (1989)Google Scholar
  9. 9.
    Positsel'skii, L.E.: Sov. J. Funct. Anal. and Appl.25, No. 4, 291 (1991)Google Scholar
  10. 10.
    Yang, K.: Almost Complex Homogeneous Spaces and their Submanifolds. Singapore: World Scientific, 1987Google Scholar
  11. 11.
    Razumov, A.V., Saveliev, M.V.: Commun. Anal. & Geom.2, 461 (1994)Google Scholar
  12. 12.
    Bourbaki, N.: Elements de Mathématiques, Groupes et Algèbres de Lie. Paris: Hermann, 1968; ch. IV–VI, Paris: Hermann, 1968; and ch. VII–VIII, Paris: Hermann, 1975Google Scholar
  13. 13.
    Gilmore, R.: Lie Groups, Lie Algebras, and some of their applications. New York: Wieley-Interscience, 1974Google Scholar
  14. 14.
    Gorbatsevich, V.V., Onischik, A.L., Vinberg, E.B.: In: Lie Groups and Lie Algebras, 3. Encyclopedia of Math. Sciences, V.41, New York, 1994Google Scholar

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

Personalised recommendations