Infinite dimensional symmetry algebras in integrable systems

  • Harald Eichenherr
Session II — Completely Integrable Systems and Group Theory
Part of the Lecture Notes in Physics book series (LNP, volume 180)


An infinite dimensional symmetry algebra of the Heisenberg spin chain is described and some of its properties are discussed.

It is shown that the principle of gauge equivalence of Lax pairs leads to the existence of such symmetry algebras even in models which do not have a global non-Abelian symmetry. This is explained for the examples of the non-linear Schrödinger equation and the complex sine-Gordon equation. In the latter case one also uses the fact that symmetry algebra and conformal transformations commute.


Heisenberg Model Symmetry Algebra Casimir Operator Chiral Model Heisenberg Spin Chain 
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. 1a).
    W. Kinnersley, J. Math. Phys. 18 (1977) 1529Google Scholar
  2. 1b).
    1b)W. Kinnersley and D.M. Chitre, J. Math. Phys. 18 (1977) 1538Google Scholar
  3. 1c).
    I. Hauser and F.J. Ernst, Phys. Rev. D2o (1979) 362Google Scholar
  4. 2a).
    L. Dolan, Phys. Rev. Lett. 47 (1981) 1371Google Scholar
  5. 2b).
    B. Julia, Paris preprint LPTENS 81/14 K. Ueno, Kyoto University preprint RIMS-374 (1981) K. Ueno and Y. Nakamura, Kyoto university preprint RIMS-376 (1981)Google Scholar
  6. 2c.
    See also the footnote in: Th. Curtright and C. Zachos, Phys Rev. D24 (1981) 2661Google Scholar
  7. 3).
    V.E. Zakharov and A.V. Mikhailov, Sov. Phys. JETP 47 (1978) 1017Google Scholar
  8. 4).
    V.E. Zakharov and L.A. Takhtajan, Theor. Math. Phys. 38 (1979) 17Google Scholar
  9. 5).
    L.A. Takhtajan, Phys. Lett. 64A (1977) 235Google Scholar
  10. 6).
    H. Eichenherr, CERN TH-3299 (1982), t0 appear in Phys. Lett. BGoogle Scholar
  11. 7).
    C. Devchand and D.B. Fairlie, Nucl. Phys. B194 (1982) 232Google Scholar
  12. 8).
    K. Pohlmeyer, Commun. Math. Phys. 46 (1976) 207Google Scholar
  13. 9).
    M. Ademollo, A. d'Adda, R. d'Auria, E. Napolitano, S. Sciuto, P. di Vecchia, F. Gliozzi, R. Musto and F. Nicodemi, Nuovo Cimento 21A (1974) 77Google Scholar
  14. 10a).
    L.L. Chau, M.L. Ge and Y.S. Wu, Phys. Rev. D25 (1982) 1086Google Scholar
  15. 10b.
    K. Ueno and Y. Nakamura, Phys. Lett. 109B (1982) 273Google Scholar
  16. 10c.
    L. Dolan, Rockefeller preprint RU82/B20 (1982)Google Scholar
  17. 11).
    S. Okubo, J. Math. Phys. 18 (1977) 2382Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Harald Eichenherr
    • 1
  1. 1.CERNGeneva 23Switzerland

Personalised recommendations