Commuting variety of r-tuples over the Witt algebra
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摘要: 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.Abstract: 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,$\mathfrak{sl}_2$ .-
Key words:
- Witt algebra /
- irreducible component /
- dimension /
- commuting variety of r-tuples /
- normal variety
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