{"title":"Invariant Projective Properties Under the Action of the Lie Group \\(\\textrm{SL}(3;\\mathbb {R})\\) on \\(\\mathbb{R}\\mathbb{P}^2\\)","authors":"Debapriya Biswas, Sandipan Dutta","doi":"10.1007/s40010-023-00813-3","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we define the projective action of the Lie group <span>\\(\\textrm{SL}(3;\\mathbb {R})\\)</span> on <span>\\(\\mathbb{R}\\mathbb{P}^2\\)</span>. We have considered all the one-parameter subgroups (up to conjugacy) of <span>\\(\\textrm{SL}(3;\\mathbb {R})\\)</span> and constructed their orbits in two-dimensional homogeneous space by defining the projective action. We obtain the underlying geometry under this action of <span>\\(\\textrm{SL}(3;\\mathbb {R})\\)</span> by finding the corresponding invariant projective properties. We also discuss whether the action of <span>\\(\\textrm{SL}(3;\\mathbb {R})\\)</span> is triply transitive and to find the possible fixed points under the action.</p></div>","PeriodicalId":744,"journal":{"name":"Proceedings of the National Academy of Sciences, India Section A: Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40010-023-00813-3.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the National Academy of Sciences, India Section A: Physical Sciences","FirstCategoryId":"103","ListUrlMain":"https://link.springer.com/article/10.1007/s40010-023-00813-3","RegionNum":4,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we define the projective action of the Lie group \(\textrm{SL}(3;\mathbb {R})\) on \(\mathbb{R}\mathbb{P}^2\). We have considered all the one-parameter subgroups (up to conjugacy) of \(\textrm{SL}(3;\mathbb {R})\) and constructed their orbits in two-dimensional homogeneous space by defining the projective action. We obtain the underlying geometry under this action of \(\textrm{SL}(3;\mathbb {R})\) by finding the corresponding invariant projective properties. We also discuss whether the action of \(\textrm{SL}(3;\mathbb {R})\) is triply transitive and to find the possible fixed points under the action.