Advertisement

Siberian Mathematical Journal

, Volume 47, Issue 4, pp 643–652 | Cite as

The n-lie property of the Jacobian as a condition for complete integrability

  • A. Dzhumadil’daev
Article

Abstract

We prove that an associative commutative algebra U with derivations D 1, ..., D n ε DerU is an n-Lie algebra with respect to the n-multiplication D 1 ^ ⋯ ^ D n if the system {D 1, ..., D n } is in involution. In the case of pairwise commuting derivations this fact was established by V. T. Filippov. One more formulation of the Frobenius condition for complete integrability is obtained in terms of n-Lie multiplications. A differential system {D 1, ..., D n } of rank n on a manifold M m is in involution if and only if the space of smooth functions on M is an n-Lie algebra with respect to the Jacobian Det(D i u j ).

Keywords

n-Lie algebra Jacobian complete integrability differential system Frobenius theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Nambu Y., “Generalized Hamiltonian mechanics,” Phys. Rev., 7, No. 8, 2405–2412 (1973).zbMATHMathSciNetGoogle Scholar
  2. 2.
    Filippov V. T., “n-Lie algebras,” Siberian Math. J., 26, No. 6, 879–891 (1985).zbMATHCrossRefGoogle Scholar
  3. 3.
    Filippov V. T., “On an n-Lie algebra of Jacobians,” Siberian Math. J., 39, No. 3, 573–581 (1998).zbMATHMathSciNetGoogle Scholar
  4. 4.
    Pozhidaev A. P., “Monomial n-Lie algebras,” Algebra and Logic, 37, No. 5, 307–322 (1998).zbMATHMathSciNetGoogle Scholar
  5. 5.
    Dzhumadil’daev A. S., “Identities and derivations for jacobian algebras,” Contemp. Math., 315, 245–278 (2002).MathSciNetGoogle Scholar
  6. 6.
    Flanders H., Differential Forms with Applications to the Physical Sciences, Acad. Press, New York; London (1963).Google Scholar
  7. 7.
    Treves F., Introduction to Pseudodifferential and Fourier Integral Operators, Plenum Press, New York; London (1982).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. Dzhumadil’daev
    • 1
  1. 1.Institute of MathematicsKazakh-British University AlmatyKazakhstan

Personalised recommendations