A Note on Elementary Subalgebras of the Witt Algebra

DOI：

 作者 单位 王晓明 上海海洋大学信息学院, 上海 201306.

设$\mathfrak{g}=W_1$是特征$p>3$的代数闭域$k$上的Witt代数. 本文确定了$\mathfrak{g}$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$\mathfrak{g}$的自同构群. 从而证明了最大维数为$\frac{p-1}{2}$的基本子代数射影簇 ${\mathbb{E}}\big(\frac{p-1}{2}, \mathfrak{g}\big)$是不可约的且是一维的. 更进一步, 证明了${\mathbb{E}}(1,\mathfrak{g})$是不可约的, 具有维数$p-2$, 而${\mathbb{E}}(2,\mathfrak{g})$是等维的, 共有 $\frac{p-3}{2}$个不可约分支, 且每个不可约分支的维数是$p-4$. 而当 $3\leq r\leq \frac{p-3}{2}$时, ${\mathbb{E}}(r,\mathfrak{g})$是可约的. 给出了 ${\mathbb{E}}(r,\mathfrak{g})$ \big($3\leq r\leq \frac{p-3}{2}$\big) 维数的一个下界.

Let $\mathfrak{g}=W_1$ be the Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Maximal elementary subalgebras of $\mathfrak{g}$ are determined. Moreover, $G$ conjugacy classes of elementary subalgebras 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 of maximal dimension $\frac{p-1}{2}$ is shown to be irreducible and one-dimensional. Moreover, we show that ${\mathbb{E}}(1,\mathfrak{g})$ is irreducible and has dimension $p-2$, ${\mathbb{E}}(2,\mathfrak{g})$ is equidimensional and has $\frac{p-3}{2}$ irreducible components with the same dimension $p-4$. While ${\mathbb{E}}(r,\mathfrak{g})$ is reducible for $3\leq r\leq \frac{p-3}{2}$. A lower bound for the dimension of ${\mathbb{E}}(r,\mathfrak{g})$ \big($3\leq r\leq \frac{p-3}{2}$\big) is given.