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Acta Applicandae Mathematica

, Volume 41, Issue 1–3, pp 323–339 | Cite as

Deformations of nonassociative algebras and integrable differential equations

  • V. V. Sokolov
  • S. I. Svinolupov
Article

Abstract

A new class of nonassociative algebras related to integrable PDE's and ODE's is introduced. These algebras can be regarded as a noncommutative generalization of Jordan algebras. Their deformations are investigated. Relationships between such algebras and graded Lie algebras are established.

Mathematics subject classifications (1991)

16S80 17A30 35Q58 53B05 

Key words

Jordan algebras left-symmetric algebras Lie algebras deformation of nonassociative algebras generalized chiral field equations 

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References

  1. 1.
    Svinolupov, S. L.: On the analogues of the Burgers equations,Phys. Lett. A 135(1) (1989), 32–36.Google Scholar
  2. 2.
    Svinolupov, S. I.: Jordan algebras and generalized Korteweg-de Vries equations,Teoret. Mat. Phys. 4 (1991), 46–58 (in Russian).Google Scholar
  3. 3.
    Svinolupov, S. I.: Generalized Schrodinger equations and Jordan pairs,Comm. Math. Phys. 143 (1992), 559–575.Google Scholar
  4. 4.
    Svinolupov, S. I. and Sokolov, V. V.: Jordan tops and generalization of Lie theorem,Mat. Zametki 3 (1993), 115–121 (in Russian).Google Scholar
  5. 5.
    Golubchik, I. Z., Sokolov, V. V., and Svinolupov, S. I.: New class of nonassociative algebras and a zeneralized factorization method, Preprint ESI, Vienna 53, 1993.Google Scholar
  6. 6.
    Calogero, F.: Why are certain nonlinear PDE's both widely applicable and integrable? in V. E. Zakharov (ed.),What is Integrability? Springer-Verlag, Berlin, 1991, pp. 1–62.Google Scholar
  7. 7.
    Jacobson, N.:Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ. 30, Amer. Math. Soc., Providence R.I., 1968.Google Scholar
  8. 8.
    Medina, A.: Sur quelques algèbres symétriques a gauche dont l'algèbre de Lie sous-jacente est resoluble,C. R. Acad. Sci. Paris. Ser. A. 286(3) (1978), 173–176.Google Scholar
  9. 9.
    Athorn, C. and Fordy, A. P.: Generalized KdV and MKdV equations associated with symmetric spaces,J. Phys. A 20 (1987), 1377–1386.Google Scholar
  10. 10.
    Svinolupov, S. I. and Sokolov, V. V.: Vector generalizations of classical integrable equations,Teoret. Mat. Fiz. 100(2) (1994), 214–218 (in Russian).Google Scholar
  11. 11.
    Zhevlakov, K. A., Slin'ko, A. M., Shestakov, I. P., and Shirshov, A.I.:Rings Similar to Assoiative Ones, Nauka, Moscow, 1974 (in Russian).Google Scholar
  12. 12.
    Osborn, J. M.: Commutative algebras satisfying an identity of degree four,Proc. Amer. Math. Soc. 16(4) (1965), 1114–1120.Google Scholar
  13. 13.
    Helgason, S.:Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.Google Scholar
  14. 14.
    Meyberg, K.: Jordan-Tripelsysteme und die Koecher-Konstruktion von Lie-Algebren,Math. Zeit. 115(1) (1970), 58–78.Google Scholar
  15. 15.
    Koecher, M.: Imbedding of Jordan algebras into Lie algebras, 1, 2,Amer. J. Math. 89, 90 (1967), 787–816, (1968) 476–510.Google Scholar
  16. 16.
    Drinfel'd, V. G.: Hamiltonian structure on the Lie groups and a geometric sense of classical Yang-Bxter equation,Dokl. Acad. Nauk SSSR 268(2) (1983), 285–287 (in Russian).Google Scholar
  17. 17.
    Semenov-Tian-Shansky, M. A.: What is classical r-matrix?Funkt. Anal. Prilozh. 17(4) (1983), 17–33 (in Russian).Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • V. V. Sokolov
    • 1
  • S. I. Svinolupov
    • 1
  1. 1.Mathematical Institute of Ufa Center of Russian Academy of SciencesUfaRussia

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