华东师范大学学报（自然科学版） ›› 2022, Vol. 2022 ›› Issue (2): 1-8.doi: 10.3969/j.issn.1000-5641.2022.02.001

• 数学 •    下一篇

无限维3-Pre-李代数

白瑞蒲1,2,*(), 刘山1,2   

  1. 1. 河北大学 数学与信息科学学院, 河北 保定　071002
    2. 河北大学 河北省机器学习与智能计算重点实验室, 河北 保定　071002
  • 收稿日期:2020-10-21 出版日期:2022-03-25 发布日期:2022-03-28
  • 通讯作者: 白瑞蒲 E-mail:bairuipu@hbu.edu.cn
  • 基金资助:
    河北省自然科学基金(20182011126)

Infinite dimensional 3-Pre-Lie algebras

Ruipu BAI1,2,*(), Shan LIU1,2   

  1. 1. College of Mathematics and Information Science, Hebei University, Baoding Hebei　071002, China
    2. Key Laboratory of Machine Learning and Computational Intelligence of Hebei Province, Hebei University, Baoding Hebei　071002, China
  • Received:2020-10-21 Online:2022-03-25 Published:2022-03-28
  • Contact: Ruipu BAI E-mail:bairuipu@hbu.edu.cn

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摘要：

构造3-Pre-李代数一直是一个很困难的问题, 目前关于3-Pre-李代数的例子很少. 利用单无限维3-李代数 $A_{\omega}=\langle L_m~\vert~m\in {\mathbb{Z}}\rangle$ 上所有权为0 的齐性Rota-Baxter 算子, 构造了5类不同构的3-Pre-李代数 $B_k, 0\leqslant k\leqslant4$ , 且对所构造的3-Pre-李代数的结构进行了研究, 证明了 $B_2$ 和 $B_4$ 是2类单3-Pre-李代数, $B_1$ 是具有无限多个1维理想的不可分解3-Pre-李代数, $B_3$ 是具有有限多个理想的不可分解3-Pre-李代数.

关键词: 3-Pre-李代数, 3-李代数, 齐性Rota-Baxter 算子

Abstract:

Constructing 3-Pre-Lie algebras has always been a difficult problem; until now, there have been very few examples of 3-Pre-Lie algebras. In this paper, we use homogenous Rota-Baxter operators of weight zero on the infinite dimensional 3-Lie algebra $A_{\omega}=\langle L_m | m\in {\mathbb{Z}}\rangle$ to construct 3-Pre-Lie algebras $B_k,~0\leqslant k\leqslant 4$ , and we subsequently discuss the structure. It is shown that $B_2$ and $B_4$ are non-isomorphic simple 3-Pre-Lie algebras, $B_1$ is an indecomposable 3-Pre-Lie algebra with infinitely many one-dimensional ideals, and $B_3$ is an indecomposable 3-Pre-Lie algebra with finitely many ideals.

Key words: 3-Pre-Lie algebras, 3-Lie algebras, homogenous Rota-Baxter operator

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