2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63489An alternating sum of compositions of N vector fields is called N-commutator. In general N-commutator of N vector fields is no longer a vector field. Question: is it possible to find N=N(n), such that N-commutator of any vector fields in $Vect(n)$ is once again a vector field. Theorem: $n^2+2n-2$-commutator is a such one. A theory of N-commutators with emphasis on 5-and 6-commutators on two dimensional manifold is developed.44 pagesRings and AlgebrasMathematical PhysicsQuantum Algebra17B66, 17C50, 22E65text