华东师范大学学报(自然科学版) ›› 2021, Vol. 2021 ›› Issue (3): 1-7.doi: 10.3969/j.issn.1000-5641.2021.03.001

• 数学 • 上一篇    下一篇

Witt代数的r元组交换簇

姚裕丰*(), 张雅静   

  1. 上海海事大学 数学系, 上海 201306
  • 收稿日期:2020-01-12 出版日期:2021-05-25 发布日期:2021-05-26
  • 通讯作者: 姚裕丰 E-mail:yfyao@shmtu.edu.cn
  • 基金资助:
    国家自然科学基金(11771279, 11671138, 12071136)

Commuting variety of r-tuples over the Witt algebra

Yufeng YAO*(), Yajing ZHANG   

  1. Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China
  • Received:2020-01-12 Online:2021-05-25 Published:2021-05-26
  • Contact: Yufeng YAO E-mail:yfyao@shmtu.edu.cn

摘要:

${\mathfrak{g}}$ 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.

关键词: Witt代数, 不可约分支, 维数, r元组交换簇, 正规簇

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

中图分类号: