{"title":"On the Dimension of Non-Abelian Tensor Squares of $n$-Lie Algebras","authors":"F. Saeedi, Nafiseh Akbarossadat","doi":"10.5556/J.TKJM.52.2021.3373","DOIUrl":null,"url":null,"abstract":"Let $L$ be an $n$-Lie algebra over a field $\\F$. In this paper, we introduce the notion of non-abelian tensor square $L\\otimes L$ of $L$ and define the central ideal $L\\square L$ of it. Using techniques from group theory and Lie algebras, we show that that $L\\square L\\cong L^{ab}\\square L^{ab}$. Also, we establish the short exact sequence\\[0\\lra\\M(L)\\lra\\frac{L\\otimes L}{L\\square L}\\lra L^2\\lra0\\]and apply it to compute an upper bound for the dimension of non-abelian tensor square of $L$.","PeriodicalId":45776,"journal":{"name":"Tamkang Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tamkang Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5556/J.TKJM.52.2021.3373","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
Abstract
Let $L$ be an $n$-Lie algebra over a field $\F$. In this paper, we introduce the notion of non-abelian tensor square $L\otimes L$ of $L$ and define the central ideal $L\square L$ of it. Using techniques from group theory and Lie algebras, we show that that $L\square L\cong L^{ab}\square L^{ab}$. Also, we establish the short exact sequence\[0\lra\M(L)\lra\frac{L\otimes L}{L\square L}\lra L^2\lra0\]and apply it to compute an upper bound for the dimension of non-abelian tensor square of $L$.
期刊介绍:
To promote research interactions between local and overseas researchers, the Department has been publishing an international mathematics journal, the Tamkang Journal of Mathematics. The journal started as a biannual journal in 1970 and is devoted to high-quality original research papers in pure and applied mathematics. In 1985 it has become a quarterly journal. The four issues are out for distribution at the end of March, June, September and December. The articles published in Tamkang Journal of Mathematics cover diverse mathematical disciplines. Submission of papers comes from all over the world. All articles are subjected to peer review from an international pool of referees.