{"title":"A characterisation of Lie algebras using ideals and subalgebras","authors":"Vladimir Dotsenko, Xabier García-Martínez","doi":"10.1112/blms.13062","DOIUrl":null,"url":null,"abstract":"<p>We prove that if, for a non-trivial variety of non-associative algebras, every subalgebra of every free algebra is free and <span></span><math>\n <semantics>\n <msup>\n <mi>I</mi>\n <mn>2</mn>\n </msup>\n <annotation>$I^2$</annotation>\n </semantics></math> is an ideal whenever <span></span><math>\n <semantics>\n <mi>I</mi>\n <annotation>$I$</annotation>\n </semantics></math> is an ideal, then this variety coincides with the variety of all Lie algebras.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"56 7","pages":"2408-2423"},"PeriodicalIF":0.8000,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13062","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove that if, for a non-trivial variety of non-associative algebras, every subalgebra of every free algebra is free and is an ideal whenever is an ideal, then this variety coincides with the variety of all Lie algebras.
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