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 ).
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Original Russian Text Copyright © 2006 Dzhumadil’daev A.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 4, pp. 780–790, July–August, 2006.
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Dzhumadil’daev, A. The n-lie property of the Jacobian as a condition for complete integrability. Sib Math J 47, 643–652 (2006). https://doi.org/10.1007/s11202-006-0075-9
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DOI: https://doi.org/10.1007/s11202-006-0075-9