N-commutators of vector fields

The question is studied as to whether the N-commutator sN(X1,…,XN)=∑σ∈SymNsign σXσ(1)⋯Xσ(N) is a well-defined operation on Vect(n). It is if N=n2+2n−2. A theory of N-commutators with emphasis on 5- and 6-commutators on two-dimensional manifolds is developed. Divergenceless vector fields Vect0(2) under 2- and 5-commutators generate a sh-Lie algebraic structure in the sense of Stasheff. Similar results hold for Vect(2) with 2- and 6-commutators.