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

Abstract

We prove that an associative commutative algebra U with derivations D 1 , . . . , D n ∈ Der U 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-ary algebras: a review with applications

“…For n = 3, equation ( 172) is the expression of the Nambu bracket {f 1 , f 2 , f 3 } studied in [38] although, of course, Nambu also mentioned the general n case. The fact that the Jacobian defines an n-Lie algebra structure was already noticed by Filippov in his original paper [16] (see also [42] and [186] for further analysis).…”