设 ${\mathfrak{g}}$ 是特征大于3的代数闭域上的Witt代数, $r$ 是大于等于2的整数. Witt代数的 $r$ 元组交换簇是 ${\mathfrak{g}}$ 中互相交换的 $r$ 元组的集合. 对比Ngo在2014年关于典型李代数的工作, 证明了Witt代数的 $r$ 元组交换簇 ${{\cal{C}}_{r}}\left( \mathfrak{g} \right)$ 是可约的, 共有 $\frac{p-1}{2}$ 个不可约分支, 且不是等维的; 确定了所有不可约分支及其维数. 特别地, ${{\cal{C}}_{r}}\left( \mathfrak{g} \right)$ 既不是正规的也不是Cohen-Macaulay. 这些结果不同于典型李代数 $\mathfrak{sl}_2$ 相应的结果.

Let ${\mathfrak{g}}$ be the Witt algebra over an algebraically closed field of characteristic $p>3$ , and $r\in\mathbb{Z}_{\geqslant 2}$ . The commuting variety ${{\cal{C}}_{r}}\left( \mathfrak{g} \right)$ of $r$ -tuples over ${\mathfrak{g}}$ is defined as the collection of all $r$ -tuples of pairwise commuting elements in ${\mathfrak{g}}$ . In contrast with Ngo’s work in 2014, for the case of classical Lie algebras, we show that the variety ${{\cal{C}}_{r}}\left( \mathfrak{g} \right)$ is reducible, and there are a total of $\frac{p-1}{2}$ irreducible components. Moreover, the variety $ {{\cal{C}}_{r}}\left( \mathfrak{g} \right) $ is not equidimensional. All irreducible components and their dimensions are precisely determined. In particular, the variety ${{\cal{C}}_{r}}\left( \mathfrak{g} \right)$ is neither normal nor Cohen-Macaulay. These results are different from those for the case of classical Lie algebra, $\mathfrak{sl}_2$ .