10-commutator and 13-commutator

TL;DR

This paper constructs specific higher-order commutators, namely 10-commutator and 13-commutator, on vector fields in three-dimensional space, expanding the algebraic tools available for analyzing vector field compositions.

Contribution

It introduces explicit constructions of 10-commutator and 13-commutator on $Vect(3)$ and a 10-commutator on divergence-free vector fields, advancing the understanding of complex vector field operations.

Findings

Constructed 10-commutator on $Vect(3)$.

Constructed 13-commutator on $Vect(3)$.

Constructed 10-commutator on divergence-free vector fields.

Abstract

Skew-symmetric sum of $N!$ compositions of $N$ vector fields in all possible order is called $N$-commutator. We construct 10-commutator and 13-commutator on a space of vector fields $Vect(3)$ and 10-commutator on a space of divergenceless vector fields $Vect_0(3).$