The Derived Algebra of Nonlinear Polynomial Divergence-Free Vector Fields

Let k be a field, V = k^n with n ≥ 3, and A = Sym(V*) = k[x_1,...,x_n]. Let L_d be the space of homogeneous divergence-free polynomial vector fields of coefficient degree d, and let E_d ⊂ L_d consist of the fields whose contraction with a volume form is exact. For every d ≥ 3, we prove that [L_2,L_{d−1}] = E_d. The proof is weightwise: on each weight space, the bracket map becomes a Koszul contraction by a single covector. This argument is valid in every characteristic. Consequently, the derived algebra of L_{≥2} = ⊕_{d≥2} L_d has degree-d part E_d for d ≥ 3. In characteristic p > 0, the Cartier isomorphism identifies (L_d/E_d) ⊗ det(V*), in coefficient degree d = pq − (n − 1) ≥ 3, with the pullback along relative Frobenius of Sym^{q−n+1}((V^{(1)})*) ⊗ Λ^{n−1}((V^{(1)})*), where q ≥ n − 1. The quotient vanishes in all other degrees d ≥ 3. These components have dimension n·binom(q,n−1) and determine the Hilbert series and the minimal homogeneous generators of L_{≥2}.