{"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 -Lie algebra with respect to an action on another -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 on 3-Lie algebras. Next we construct a cohomology theory for a crossed homomorphism on -Lie algebras and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Finally, using the higher derived brackets, we construct an -algebra whose Maurer–Cartan elements are crossed homomorphisms. Consequently, we obtain the twisted -algebra that controls deformations of a given crossed homomorphism on -Lie algebras.