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Journal of Mathematical Sciences

, Volume 144, Issue 2, pp 3926–3937 | Cite as

Factorization of loop algebras over so(4) and integrable nonlinear differential equations

  • O. V. Efimovskaya
Article
  • 23 Downloads

Abstract

We consider factoring subalgebras for loop algebras over so(4). Given a factoring subalgebra, we find (in terms of coefficients of commutator relations) an explicit form of (1) the corresponding system of the chiral-field-equation type, (2) the corresponding two-spin model of the Landau-Lifshitz equation, and (3) the corresponding Hamiltonian system of ordinary differential equations with homogeneous quadratic Hamiltonian and linear so(4)-Poisson brackets.

Keywords

Hamiltonian System Commutator Relation Poisson Bracket Laurent Series Loop Algebra 
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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References

  1. 1.
    I. V. Cherednik, “Functional realizations of basis representations of factoring Lie groups and algebras,” Funkts. Anal. Prilozhen., 19, No. 3, 36–52 (1985).MathSciNetGoogle Scholar
  2. 2.
    B. A. Dubrovin, I. M. Krichever, and S. P. Novikov, “Integrable systems,” in: Itogi Nauki i Tekh., Sovr. Probl. Mat., Fundam. Napr., 4, All-Union Institute for Scientific and Technical Information, Moscow (1985), pp. 179–284.Google Scholar
  3. 3.
    B. A. Dubrovin, S. P. Novikov, and V. B. Matveev, “Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties,” Usp. Mat. Nauk, 31, No. 1, 107–136 (1976).MathSciNetGoogle Scholar
  4. 4.
    O. V. Efimovskaya and V. V. Sokolov, “Decompositions of the loop algebra over so(4) and integrable models of the chiral equation type,” Fund. Prikl. Mat., 10, No. 1, 39–47 (2004).zbMATHGoogle Scholar
  5. 5.
    L. D. Faddeev and L. A. Takhtajan, Hamiltonian Approach in the Theory of Solitons [in Russian], Nauka, Moscow (1986).Google Scholar
  6. 6.
    P. I. Golod, “Hamiltonian systems on orbits of affine Lie groups and nonlinear integrable equations,” in: Physics of Many-Particle Systems, Vol. 7, Naukova Dumka, Kiev (1985), pp. 30–39.Google Scholar
  7. 7.
    I. Z. Golubchik and V. V. Sokolov, “Generalized Heisenberg equations on ℤ-graded Lie algebras,” Teor. Mat. Fiz., 120, No. 2, 248–255 (1999).MathSciNetGoogle Scholar
  8. 8.
    I. Z. Golubchik and V. V. Sokolov, “Compatible Lie brackets and integrable equations of the type of the principal chiral field model,” Funkts. Anal. Prilozhen., 36, No. 3, 9–19 (2002).MathSciNetGoogle Scholar
  9. 9.
    I. Z. Golubchik and V. V. Sokolov, “Factorization of the loop algebra and integrable top-like systems,” Teor. Mat. Fiz., 141, No. 1, 3–23 (2004).MathSciNetGoogle Scholar
  10. 10.
    I. Z. Golubchik and V. V. Sokolov, “Factorization of the loop algebras and compatible Lie brackets,” J. Nonlinear Math. Phys., 12, No. 1, 343–350 (2005).CrossRefMathSciNetGoogle Scholar
  11. 11.
    P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Comm. Pure Appl. Math., 21, No. 5, 467–490 (1968).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Problem Method [in Russian], Nauka, Moscow (1980).zbMATHGoogle Scholar
  13. 13.
    A. G. Reyman and M. A. Semenov-Tian-Shansky Integrable System. Group-Theoretical Approach, R&C Dynamics, Izhevsk (2003).Google Scholar
  14. 14.
    M. A. Semenov-Tian-Shansky, “What is the classical r-matrix?” Funkts. Anal. Prilozhen., 17, No. 4, 17–33 (1983).Google Scholar
  15. 15.
    V. V. Sokolov, “On decompositions of the loop algebra over so(3) into a sum of two subalgebras,” Dokl. Ross. Akad. Nauk, 397, No. 3, 321–324 (2004).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • O. V. Efimovskaya
    • 1
  1. 1.Moscow State UniversityVorobievy Gory, MoscowRussia

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