{"title":"z阶顶点代数的Nakayama引理及其应用","authors":"Hao Wang , Wei Wang","doi":"10.1016/j.aim.2025.110503","DOIUrl":null,"url":null,"abstract":"<div><div>This is the first paper in a series of studies on <span><math><mi>Z</mi></math></span>-graded vertex algebras arising from Lie algebras with triangular decomposition (referred to as triangulated Lie algebras for simplicity). In this paper, we establish the Jacobson radical theory for <span><math><mi>Z</mi></math></span>-graded vertex algebras and prove Nakayama's lemma. As an application, we investigate a specific triangulated Lie algebra: <span><math><mi>g</mi><mo>=</mo><mi>C</mi><mi>f</mi><mo>⊕</mo><mi>C</mi><mi>h</mi><mo>⊕</mo><mi>C</mi><mi>e</mi></math></span> with Lie brackets <span><math><mo>[</mo><mi>h</mi><mo>,</mo><mi>e</mi><mo>]</mo><mo>=</mo><mn>2</mn><mi>e</mi><mo>,</mo><mspace></mspace><mo>[</mo><mi>h</mi><mo>,</mo><mi>f</mi><mo>]</mo><mo>=</mo><mo>−</mo><mn>2</mn><mi>f</mi><mo>,</mo><mspace></mspace><mo>[</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>]</mo><mo>=</mo><mn>0</mn></math></span>. Denote the <span><math><mi>Z</mi></math></span>-graded vertex algebra constructed from <span><math><mi>g</mi></math></span> by <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>l</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. Using Nakayama's lemma, we classify all the irreducible admissible <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>l</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-modules. Finally, we construct a class of indecomposable admissible <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>l</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-modules.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110503"},"PeriodicalIF":1.5000,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Nakayama's lemma for Z-graded vertex algebras and its applications\",\"authors\":\"Hao Wang , Wei Wang\",\"doi\":\"10.1016/j.aim.2025.110503\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>This is the first paper in a series of studies on <span><math><mi>Z</mi></math></span>-graded vertex algebras arising from Lie algebras with triangular decomposition (referred to as triangulated Lie algebras for simplicity). In this paper, we establish the Jacobson radical theory for <span><math><mi>Z</mi></math></span>-graded vertex algebras and prove Nakayama's lemma. As an application, we investigate a specific triangulated Lie algebra: <span><math><mi>g</mi><mo>=</mo><mi>C</mi><mi>f</mi><mo>⊕</mo><mi>C</mi><mi>h</mi><mo>⊕</mo><mi>C</mi><mi>e</mi></math></span> with Lie brackets <span><math><mo>[</mo><mi>h</mi><mo>,</mo><mi>e</mi><mo>]</mo><mo>=</mo><mn>2</mn><mi>e</mi><mo>,</mo><mspace></mspace><mo>[</mo><mi>h</mi><mo>,</mo><mi>f</mi><mo>]</mo><mo>=</mo><mo>−</mo><mn>2</mn><mi>f</mi><mo>,</mo><mspace></mspace><mo>[</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo>]</mo><mo>=</mo><mn>0</mn></math></span>. Denote the <span><math><mi>Z</mi></math></span>-graded vertex algebra constructed from <span><math><mi>g</mi></math></span> by <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>l</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>. Using Nakayama's lemma, we classify all the irreducible admissible <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>l</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-modules. Finally, we construct a class of indecomposable admissible <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>(</mo><mi>l</mi><mo>,</mo><mn>0</mn><mo>)</mo></math></span>-modules.</div></div>\",\"PeriodicalId\":50860,\"journal\":{\"name\":\"Advances in Mathematics\",\"volume\":\"480 \",\"pages\":\"Article 110503\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2025-09-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0001870825004013\",\"RegionNum\":1,\"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":"Advances in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0001870825004013","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Nakayama's lemma for Z-graded vertex algebras and its applications
This is the first paper in a series of studies on -graded vertex algebras arising from Lie algebras with triangular decomposition (referred to as triangulated Lie algebras for simplicity). In this paper, we establish the Jacobson radical theory for -graded vertex algebras and prove Nakayama's lemma. As an application, we investigate a specific triangulated Lie algebra: with Lie brackets . Denote the -graded vertex algebra constructed from by . Using Nakayama's lemma, we classify all the irreducible admissible -modules. Finally, we construct a class of indecomposable admissible -modules.
期刊介绍:
Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.