Functional Analysis and Its Applications

, Volume 17, Issue 4, pp 259–272 | Cite as

What is a classical r-matrix?

  • M. A. Semenov-Tyan-Shanskii


Functional Analysis 
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.

Literature Cited

  1. 1.
    E. K. Sklyanin, "On complete integrability of the Landau—Lifshitz equation," Preprint LOMI, E-3-79, Leningrad, LOMI (1980).Google Scholar
  2. 2.
    E. K. Sklyanin, "The quantum inverse scattering method," Zap. Nauchn. Sem. LOMI,95, 55–128 (1980).Google Scholar
  3. 3.
    A. A. Belavin and V. G. Drinfel'd, "On the solutions to the classical Yang—Baxter equation for simple Lie algebras," Funkts. Anal. 16, No. 3, 1–29 (1982).Google Scholar
  4. 4.
    4.A. A. Belavin and V. G. Drinfel'd, "Triangle equations and simple Lie algebras," Preprint ITF 1982-18, Chernogolovka, ITF.Google Scholar
  5. 5.
    V. G. Drinfel'd, "Hamiltonian structures on Lie groups, Lie bialgebras, and the geometrical meaning of the Yang—Baxter equations," Dokl. Akad. Nauk SSSR,268, No. 2, 285–287 (1983).Google Scholar
  6. 6.
    L. D. Faddeev, "Integrable models in 1 + 1 dimensional quantum field theory," Preprint S.Ph.T. 82/76, CEN Saclay (1982).Google Scholar
  7. 7.
    M. A. Semenov-Tyan-Shanskii, "Classical r-matrices and the orbit method," Zap. Nauchn. Sem. LOMI,123, 77–91 (1983).Google Scholar
  8. 8.
    A. G. Reyman, and M. A. Semenov-Tyan-Shanskii, "Algebras of currents and nonlinear partial differential equations," Dokl. Akad. Nauk SSSR,251, No. 6, 1310–1313 (1980).Google Scholar
  9. 9.
    A. G. Reyman, "Integrable Hamiltonian systems connected with graded Lie algebras," Zap. Nauchn. Sem. LOMI,95, 3–54 (1980).Google Scholar
  10. 10.
    M. Adler and P. van Moerbeke, "Completely integrable systems, Euclidean Lie algebras, and curves," Adv. Math.,38, No. 2, 267–317 (1980).Google Scholar
  11. 11.
    A. G. Reyman and M. A. Semenov-Tyan-Shanskii [Semenov-Trian-Shansky], "Reduction of Hamiltonian systems, affine Lie algebras, and Lax equations, I, II," Invent. Math.,54, No. 1, 81–100 (1979), and63, No. 3, 423–432 (1981).Google Scholar
  12. 12.
    P. P. Kulish and A. G. Reyman, "Hamiltonian structure of polynomial bundles," Zap. Nauchn. Sem. LOMI,123, 67–76 (1983).Google Scholar
  13. 13.
    S. Lie (Unter Mitwirkung von F. Engel), Theorie der Transformationsgruppen, Bd. 1–3, Teubner, Leiptzig (1888, 1890, 1893).Google Scholar
  14. 14.
    B. Kostant, "Quantization and representation theory," in: Proc. Research Symp. on Representations of Lie groups, Oxford 1977, London Math. Soc. Lecture Notes Series, Vol. 34 (1979).Google Scholar
  15. 15.
    M. Adler, "On a trace functional for formal pseudodifferential operators and the symplectic structure for the KdV type equations," Invent. Math.,50, No. 2, 219–248 (1979).Google Scholar
  16. 16.
    N. I. Muskhelishvili, Singular Integral Equations [in Russian], Nauka, Moscow (1968).Google Scholar
  17. 17.
    E. K. Sklyanin, "Some algebraic structures connected with the Yang—Baxter equation," Funkts. Anal.,16, No. 4, 27–34 (1982).Google Scholar
  18. 18.
    I. M. Gel'fand and I. Ya. Dorfman, "Hamiltonian operators and the classical Yang—Baxter equation," Funkts. Anal., 16, No. 4, 1–9 (1982). 1–9 (1982).Google Scholar
  19. 19.
    I. M. Gel'fand and I. Ya. Dorfman, "Schouten brackets and Hamiltonian operators," Funkts. Anal.,14, No. 3, 71–74 (1980).Google Scholar
  20. 20.
    I. M. Gel'fand and L. A. Dikii, "A family of Hamiltonian structures connected with nonlinear integrable equations," Preprint IPM Akad. Nauk SSSR, No. 136, IPM, Moscow (1978).Google Scholar
  21. 21.
    V. E. Zakharov and A. B. Shabat, "Integration of nonlinear equations of mathematical physics by the method of the inverse scattering problem II," Funkts. Anal.,13, No. 3, 13–22 (1979).Google Scholar
  22. 22.
    I. M. Glazman and Yu. I. Lyubich, Finite-Dimensional Linear Analysis in Problems [in Russian], Nauka, Moscow (1969).Google Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • M. A. Semenov-Tyan-Shanskii

There are no affiliations available

Personalised recommendations