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{"title":"与维拉索罗代数有关的非无限级数列代数的表示","authors":"Chunguang Xia, Tianyu Ma, Xiao Dong, Mingjing Zhang","doi":"10.1515/forum-2023-0320","DOIUrl":null,"url":null,"abstract":"In this paper, we study representations of non-finitely graded Lie algebras <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">𝒲</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ϵ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0320_eq_0340.png\"/> <jats:tex-math>{\\mathcal{W}(\\epsilon)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> related to Virasoro algebra, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ϵ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>±</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0320_eq_0321.png\"/> <jats:tex-math>{\\epsilon=\\pm 1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Precisely speaking, we completely classify the free <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">𝒰</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>𝔥</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0320_eq_0333.png\"/> <jats:tex-math>{\\mathcal{U}(\\mathfrak{h})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules of rank one over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">𝒲</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ϵ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0320_eq_0340.png\"/> <jats:tex-math>{\\mathcal{W}(\\epsilon)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"45 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Representations of non-finitely graded Lie algebras related to Virasoro algebra\",\"authors\":\"Chunguang Xia, Tianyu Ma, Xiao Dong, Mingjing Zhang\",\"doi\":\"10.1515/forum-2023-0320\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we study representations of non-finitely graded Lie algebras <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi mathvariant=\\\"script\\\">𝒲</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>ϵ</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0320_eq_0340.png\\\"/> <jats:tex-math>{\\\\mathcal{W}(\\\\epsilon)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> related to Virasoro algebra, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>ϵ</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>±</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0320_eq_0321.png\\\"/> <jats:tex-math>{\\\\epsilon=\\\\pm 1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Precisely speaking, we completely classify the free <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi mathvariant=\\\"script\\\">𝒰</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>𝔥</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0320_eq_0333.png\\\"/> <jats:tex-math>{\\\\mathcal{U}(\\\\mathfrak{h})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules of rank one over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi mathvariant=\\\"script\\\">𝒲</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>ϵ</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_forum-2023-0320_eq_0340.png\\\"/> <jats:tex-math>{\\\\mathcal{W}(\\\\epsilon)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules.\",\"PeriodicalId\":12433,\"journal\":{\"name\":\"Forum Mathematicum\",\"volume\":\"45 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-06-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Forum Mathematicum\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/forum-2023-0320\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Forum Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/forum-2023-0320","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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