Cohomology and the controlling algebra of crossed homomorphisms on 3-Lie algebras

Pub Date : 2024-04-17 DOI:10.1142/s0219498825502317
Shuai Hou, Meiyan Hu, Lina Song, Yanqiu Zhou
{"title":"Cohomology and the controlling algebra of crossed homomorphisms on 3-Lie algebras","authors":"Shuai Hou, Meiyan Hu, Lina Song, Yanqiu Zhou","doi":"10.1142/s0219498825502317","DOIUrl":null,"url":null,"abstract":"<p>In this paper, first we give the notion of a crossed homomorphism on a <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>-Lie algebra with respect to an action on another <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>-Lie algebra, and characterize it using a homomorphism from a 3-Lie algebra to the semidirect product 3-Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota–Baxter operators of weight <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn></math></span><span></span> on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span><span></span>-algebra whose Maurer–Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span><span></span>-algebra that controls deformations of a given crossed homomorphism on <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn></math></span><span></span>-Lie algebras.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825502317","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

In this paper, first we give the notion of a crossed homomorphism on a 3-Lie algebra with respect to an action on another 3-Lie algebra, and characterize it using a homomorphism from a 3-Lie algebra to the semidirect product 3-Lie algebra. We also establish the relationship between crossed homomorphisms and relative Rota–Baxter operators of weight 1 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on 3-Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an L-algebra whose Maurer–Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted L-algebra that controls deformations of a given crossed homomorphism on 3-Lie algebras.

分享
查看原文
同调与 3-李代数上交叉同态的控制代数
在本文中,我们首先给出了3-Lie代数上相对于另一个3-Lie代数上的作用的交叉同态的概念,并用从3-Lie代数到半直接积3-Lie代数的同态来描述它的特征。我们还建立了交叉同态与 3-Lie 代数上权重为 1 的相对 Rota-Baxter 算子之间的关系。接下来,我们构建了 3-Lie 代数上交叉同态的同调理论,并利用第二同调群对交叉同态的无限小变形进行了分类。最后,我们利用高阶导出括号,构造了一个 L∞-algebra ,其毛勒-卡尔坦元素是交叉同态。因此,我们得到了控制给定交叉同态在 3-Lie 代数上变形的扭曲 L∞-algebra 。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
×
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