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:

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 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 Cr(g) 是可约的, 共有 p12 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, Cr(g) 既不是正规的也不是Cohen-Macaulay. 这些结果不同于典型李代数 sl2 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 g be the Witt algebra over an algebraically closed field of characteristic p>3 , and rZ2 . The commuting variety Cr(g) of r -tuples over g is defined as the collection of all r -tuples of pairwise commuting elements in g . In contrast with Ngo’s work in 2014, for the case of classical Lie algebras, we show that the variety Cr(g) is reducible, and there are a total of p12 irreducible components. Moreover, the variety Cr(g) is not equidimensional. All irreducible components and their dimensions are precisely determined. In particular, the variety Cr(g) is neither normal nor Cohen-Macaulay. These results are different from those for the case of classical Lie algebra, sl2 .

Key words: Witt algebra, irreducible component, dimension, commuting variety of r-tuples, normal variety

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