{"title":"同调与 3-李代数上交叉同态的控制代数","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":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":54888,\"journal\":{\"name\":\"Journal of Algebra and Its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-04-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Algebra and Its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219498825502317\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825502317","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Cohomology and the controlling algebra of crossed homomorphisms on 3-Lie algebras
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.
期刊介绍:
The Journal of Algebra and Its Applications will publish papers both on theoretical and on applied aspects of Algebra. There is special interest in papers that point out innovative links between areas of Algebra and fields of application. As the field of Algebra continues to experience tremendous growth and diversification, we intend to provide the mathematical community with a central source for information on both the theoretical and the applied aspects of the discipline. While the journal will be primarily devoted to the publication of original research, extraordinary expository articles that encourage communication between algebraists and experts on areas of application as well as those presenting the state of the art on a given algebraic sub-discipline will be considered.