关于Witt代数基本子代数的一点注记

设g=W_1是特征p3的代数闭域k上的Witt代数.本文确定了g的极大基本子代数.进一步具体给出了最大维数的基本子代数的G共轭类,这里G是g的自同构群.从而证明了最大维数为(p-1)/2的基本子代数射影簇E((p-1)/2,g)是不可约的且是一维的.更进一步,证明了E(1,g)是不可约的,具有维数p-2,而E(2,g)是等维的,共有(p-3)/2个不可约分支,且每个不可约分支的维数是p-4.而当3≤r≤(p-3)/2时,E(r,g)是可约的.给出了E(r,g)(3≤r≤(p-3)/2)维数的一个下界.

A Note on Elementary Subalgebras of the Witt Algebra

Let $mathfrak{g}=W_1$ be the Witt algebra over an algebraically closedfield $k$ of characteristic $p>3$. Maximal elementary subalgebras of$mathfrak{g}$ are determined. Moreover, $G$ conjugacy classes of elementarysubalgebras of maximal dimension under the automorphism group of$mathfrak{g}$ are precisely given. As a consequence, the projective variety${mathbb{E}}big(frac{p-1}{2}, mathfrak{g}big)$ of elementary subalgebras ofmaximal dimension $frac{p-1}{2}$ is shown to be irreducible andone-dimensional. Moreover, we show that ${mathbb{E}}(1,mathfrak{g})$ isirreducible and has dimension $p-2$, ${mathbb{E}}(2,mathfrak{g})$ isequidimensional and has $frac{p-3}{2}$ irreducible components withthe same dimension $p-4$. While ${mathbb{E}}(r,mathfrak{g})$ is reduciblefor $3leq rleq frac{p-3}{2}$. A lower bound for the dimension of${mathbb{E}}(r,mathfrak{g})$ big($3leq rleq frac{p-3}{2}$big) is given.