The Derived Algebra of Nonlinear Polynomial Divergence-Free Vector Fields

We study the Lie algebra $L_{\ge 2}$ of divergence-free polynomial vector fields on $k^n$, $n\ge 3$, with coefficients of degree at least two, graded by coefficient degree. Over every field its derived algebra in degree $d\ge 3$ is the space of exact fields, those whose contraction with the volume form is an exact form, and it is already spanned by brackets with quadratic fields. In characteristic zero this is the whole degree-$d$ component. In characteristic $p>0$ the abelianization is nonzero above degree two exactly in the degrees $d\ge (p-1)(n-1)$ with $d\equiv 1-n(\mathrm{m}\mathrm{o}\mathrm{d}p)$, and Cartier descent identifies it with a Frobenius twist of a rational ${\mathrm{G}\mathrm{L}}_n$-module, tensored with a power of the determinant. For $p\ge 5$ each exact component is obtained from the previous one by bracketing with quadratic fields.