关于可分裂线性李代数的说明

IF 1 3区 数学 Q1 MATHEMATICS
Zhiguang Hu, Haichuan Bai
{"title":"关于可分裂线性李代数的说明","authors":"Zhiguang Hu,&nbsp;Haichuan Bai","doi":"10.1016/j.laa.2024.07.016","DOIUrl":null,"url":null,"abstract":"<div><p>A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra <span><math><mi>g</mi></math></span> is splittable if and only if <span><math><mi>g</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>n</mi></math></span>, where <span><math><mi>a</mi></math></span> is an abelian subalgebra of <span><math><mi>g</mi></math></span> composed of semisimple elements and <span><math><mi>n</mi></math></span> is the ideal of all nilpotent matrices of <span><math><mi>g</mi></math></span>. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A note on splittable linear Lie algebras\",\"authors\":\"Zhiguang Hu,&nbsp;Haichuan Bai\",\"doi\":\"10.1016/j.laa.2024.07.016\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra <span><math><mi>g</mi></math></span> is splittable if and only if <span><math><mi>g</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>n</mi></math></span>, where <span><math><mi>a</mi></math></span> is an abelian subalgebra of <span><math><mi>g</mi></math></span> composed of semisimple elements and <span><math><mi>n</mi></math></span> is the ideal of all nilpotent matrices of <span><math><mi>g</mi></math></span>. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements.</p></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-07-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Linear Algebra and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0024379524003069\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003069","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

如果线性李代数包含每个元素的半纯部分和零纯部分,那么它就是可分裂的。我们很早就知道,当且仅当 g=a+n 时,一个可解线性李代数 g 是可分裂的,其中 a 是由半简单元素组成的 g 的无性子代数,n 是 g 的所有零势矩阵的理想数。此外,我们还确定了由半简单元素或零能元素组成的线性李代数的结构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A note on splittable linear Lie algebras

A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra g is splittable if and only if g=a+n, where a is an abelian subalgebra of g composed of semisimple elements and n is the ideal of all nilpotent matrices of g. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
2.20
自引率
9.10%
发文量
333
审稿时长
13.8 months
期刊介绍: Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信