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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 $$\text{Det} (D_iu_j)$$.

##### MSC:
 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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