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Theoretical and Mathematical Physics

, Volume 162, Issue 2, pp 149–162 | Cite as

Symmetry algebras of Lagrangian Liouville-type systems

  • A. V. KiselevEmail author
  • J. W. van de Leur
Article

Abstract

We calculate the generators and commutation relations explicitly for higher symmetry algebras of a class of hyperbolic Lagrangian systems of Liouville type, in particular, for two-dimensional Toda chains associated with semisimple complex Lie algebras.

Keywords

symmetry two-dimensional Toda chain Liouville-type system Hamiltonian hierarchy bracket 

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References

  1. 1.
    A. N. Leznov, V. G. Smirnov, and A. B. Shabat, Theor. Math. Phys., 51, 322–330 (1982).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. B. Shabat and R. I. Yamilov, “Exponential systems of type I and Cartan matrices [in Russian],” Preprint, Bashkir Branch, Acad. Sci. USSR, Ufa (1981).Google Scholar
  3. 3.
    A. V. Zhiber and V. V. Sokolov, Russ. Math. Surveys, 56, 61–101 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. M. Guryeva and A. V. Zhiber, Theor. Math. Phys., 138, 338–355 (2004).zbMATHCrossRefGoogle Scholar
  5. 5.
    A. N. Leznov and M. V. Saveliev, Lett. Math. Phys., 3, 489–494 (1979).zbMATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    A. G. Meshkov, Theor. Math. Phys., 63, 539–545 (1985).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. B. Shabat, Phys. Lett. A, 200, 121–133 (1995).zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. 8.
    D. K. Demskoi and S. Ya. Startsev, J. Math. Sci., 136, 4378–4384 (2006).CrossRefMathSciNetGoogle Scholar
  9. 9.
    A. V. Kiselev, Theor. Math. Phys., 144, 952–960 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. V. Sokolov and S. Ya. Startsev, Theor. Math. Phys., 155, 802–811 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    V. G. Drinfel’d and V. V. Sokolov, J. Sov. Math., 30, 1975–2036 (1985).zbMATHCrossRefGoogle Scholar
  12. 12.
    A. M. Vinogradov and I. S. Krasil’shchik, eds., Symmetries and Conservation Laws for Equations of Mathematical Physics [in Russian], Faktorial, Moscow (1997); English transl.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (Trans. Math. Monogr., Vol. 182), Amer. Math. Soc., Providence, R. I. (1999).Google Scholar
  13. 13.
    P. J. Olver, Applications of Lie Groups to Differential Equations (Grad. Texts in Mat., Vol. 107), Springer, New York (1993).zbMATHGoogle Scholar
  14. 14.
    J. Krasil’shchik and A. Verbovetsky, “Homological methods in equations of mathematical physics,” arXiv: math.DG/9808130v2 (1998).Google Scholar
  15. 15.
    A. V. Kiselev, Theor. Math. Phys., 152, 963–976 (2007).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. V. Kiselev and J. W. van de Leur, “A geometric derivation of KdV-type hierarchies from root systems,” Proc. 4th Intl. Workshop “Group Analysis of Differential Equations and Integrable Systems,” Protaras, Cyprus, 26–29 October 2008 (2009) pp. 87–106; arXiv:0901.4866v1 [nlin.SI] (2009).Google Scholar
  17. 17.
    N. H. Ibragimov, A. V. Aksenov, V. A. Baikov, V. A. Chugunov, R. K. Gazizov, and A. G. Meshkov, CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 2, Applications in Engineering and Physical Sciences, CRC Press, Boca Raton, Fla. (1995).Google Scholar
  18. 18.
    S. Ya. Startsev, J. Math. Sci., 151, 3245–3253 (2008).zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    V. V. Sokolov, Russ. Math. Surveys, 43, No. 5, 165–204 (1988).zbMATHCrossRefGoogle Scholar
  20. 20.
    S. Ya. Startsev, Theor. Math. Phys., 116, 1001–1010 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    B. A. Kupershmidt and G. Wilson, Invent. Math., 62, 403–436 (1980).zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    M. Pavlov, J. Nonlinear Math. Phys., 9,Suppl. 1, 173–191 (2002).CrossRefMathSciNetADSGoogle Scholar
  23. 23.
    A. V. Kiselev and J. W. van de Leur, “Involutive distributions of operator-valued evolutionary vector fields. II,” arXiv:0904.1555v2 [math-ph] (2009).Google Scholar

Copyright information

© MAIK/Nauka 2010

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of UtrechtUtrechtThe Netherlands

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