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The \(n\)-Lie property of the Jacobian as a condition for complete integrability. (Russian, English) Zbl 1150.17021
Sib. Mat. Zh. 47, No. 4, 780-790 (2006); translation in Sib. Math. J. 47, No. 4, 643-652 (2006).
Summary: We prove that an associative commutative algebra \(U\) with derivations \(D_1,\dots,D_n\subset\text{ Der\,}U\) is an \(n\)-Lie algebra with respect to the \(n\)-multiplication \(D_1\wedge\dots\wedge D^n\) if the system \(\{D_1,\dots,D_n\}\) is in involution. In the case of pairwise commuting derivations this fact was established by V. T. Filippov. Another formulation for the Frobenius condition of complete integrability is obtained in terms of \(n\)-Lie multiplications. A differential system \(\{D_1,\dots,D_n\}\) 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 \(\text{Det} (D_iu_j)\).

17B63 Poisson algebras
17A42 Other \(n\)-ary compositions \((n \ge 3)\)
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
53D17 Poisson manifolds; Poisson groupoids and algebroids
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