AI 中文总结
研究无散多项式向量场李代数导出代数的结构，确定其精确场分量，并刻画特征正情形下的交换化与Frobenius扭转。
AI 中文摘要
我们研究 $k^n$（$n\ge3$）上次数至少为2的系数、按系数次数分级的无散多项式向量场的李代数 $L_{\ge2}$。在任意域上，其次数 $d\ge3$ 的导出代数是精确场空间，即与体积形式的缩并是精确形式的那些场，并且它已由与二次场的括号积张成。在特征零情形，这恰为整个次数 $d$ 分量。在特征 $p>0$ 情形，次数高于2的交换化非零恰好出现在满足 $d\equiv1-n\pmod p$ 且 $d\ge(p-1)(n-1)$ 的次数 $d$ 上，Cartier 下降将其等同于一个有理 $\mathrm{GL}_n$-模的 Frobenius 扭转，再张量上行列式的一个幂。对于 $p\ge5$，每个精确分量由前一个分量与二次场取括号积得到。
英文摘要
We study the Lie algebra $L_{\ge2}$ of divergence-free polynomial vector fields on $k^n$, $n\ge3$, with coefficients of degree at least two, graded by coefficient degree. Over every field its derived algebra in degree $d\ge3$ 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 $d\equiv1-n\pmod p$, and Cartier descent identifies it with a Frobenius twist of a rational $\mathrm{GL}_n$-module, tensored with a power of the determinant. For $p\ge5$ each exact component is obtained from the previous one by bracketing with quadratic fields.
Comments19 pages
