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


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 ).


n-Lie algebra Jacobian complete integrability differential system Frobenius theorem 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

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

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