N-commutators

Abstract

The N-commutator $s_N(X_1, \ldots, X_N) = \mathop{\sum}\limits_{\sigma\in {\mathfrak{S}}_N}\text{\rm sign}\, \sigma\, X_{\sigma(1)}\cdots X_{\sigma(N)}$ is conjecturally a well-defined nontrivial operation on $W(n)=Der \ {\mathbb K}[x]$ for x = (x ₁, ... , x _n ) if and only if N = n ² + 2n - 2. This is proved for n = 2 and confirmed by computer experiments for n < 5.

Under 2- and 5-commutators the algebra of divergence-free vector fields in two dimensions is an sh-Lie (strong homotopic Lie) algebra in the sense of Stasheff. Similarly, W(2) is an sh-Lie algebra with respect to 2- and 6-commutators.