{"title":"Rigidity results for Lie algebras admitting a post-Lie algebra structure","authors":"D. Burde, K. Dekimpe, Mina Monadjem","doi":"10.1142/s0218196722500679","DOIUrl":null,"url":null,"abstract":"We study rigidity questions for pairs of Lie algebras $(\\mathfrak{g},\\mathfrak{n})$ admitting a post-Lie algebra structure. We show that if $\\mathfrak{g}$ is semisimple and $\\mathfrak{n}$ is arbitrary, then we have rigidity in the sense that $\\mathfrak{g}$ and $\\mathfrak{n}$ must be isomorphic. The proof uses a result on the decomposition of a Lie algebra $\\mathfrak{g}=\\mathfrak{s}_1\\dotplus \\mathfrak{s}_2$ as the direct vector space sum of two semisimple subalgebras. We show that $\\mathfrak{g}$ must be semisimple and hence isomorphic to the direct Lie algebra sum $\\mathfrak{g}\\cong \\mathfrak{s}_1\\oplus \\mathfrak{s}_2$. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras $(\\mathfrak{g},\\mathfrak{n})$. We prove additional existence results for pairs $(\\mathfrak{g},\\mathfrak{n})$, where $\\mathfrak{g}$ is complete, and for pairs, where $\\mathfrak{g}$ is reductive with $1$-dimensional center and $\\mathfrak{n}$ is solvable or nilpotent.","PeriodicalId":13615,"journal":{"name":"Int. J. Algebra Comput.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Int. J. Algebra Comput.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1142/s0218196722500679","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
We study rigidity questions for pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$ admitting a post-Lie algebra structure. We show that if $\mathfrak{g}$ is semisimple and $\mathfrak{n}$ is arbitrary, then we have rigidity in the sense that $\mathfrak{g}$ and $\mathfrak{n}$ must be isomorphic. The proof uses a result on the decomposition of a Lie algebra $\mathfrak{g}=\mathfrak{s}_1\dotplus \mathfrak{s}_2$ as the direct vector space sum of two semisimple subalgebras. We show that $\mathfrak{g}$ must be semisimple and hence isomorphic to the direct Lie algebra sum $\mathfrak{g}\cong \mathfrak{s}_1\oplus \mathfrak{s}_2$. This solves some open existence questions for post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$. We prove additional existence results for pairs $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{g}$ is complete, and for pairs, where $\mathfrak{g}$ is reductive with $1$-dimensional center and $\mathfrak{n}$ is solvable or nilpotent.