{"title":"通过对称四元组对 B 型和 C 型簇代数进行分类","authors":"Azzurra Ciliberti","doi":"10.1016/j.jalgebra.2024.09.015","DOIUrl":null,"url":null,"abstract":"<div><div>We express cluster variables of type <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in terms of cluster variables of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Then we associate a cluster tilted bound symmetric quiver <em>Q</em> of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> to any seed of a cluster algebra of type <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Under this correspondence, cluster variables of type <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (resp. <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>) correspond to orthogonal (resp. symplectic) indecomposable representations of <em>Q</em>. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.</div></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A categorification of cluster algebras of type B and C through symmetric quivers\",\"authors\":\"Azzurra Ciliberti\",\"doi\":\"10.1016/j.jalgebra.2024.09.015\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We express cluster variables of type <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in terms of cluster variables of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Then we associate a cluster tilted bound symmetric quiver <em>Q</em> of type <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> to any seed of a cluster algebra of type <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Under this correspondence, cluster variables of type <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (resp. <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>) correspond to orthogonal (resp. symplectic) indecomposable representations of <em>Q</em>. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.</div></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-10-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021869324005246\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021869324005246","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A categorification of cluster algebras of type B and C through symmetric quivers
We express cluster variables of type and in terms of cluster variables of type . Then we associate a cluster tilted bound symmetric quiver Q of type to any seed of a cluster algebra of type and . Under this correspondence, cluster variables of type (resp. ) correspond to orthogonal (resp. symplectic) indecomposable representations of Q. We find a Caldero-Chapoton map in this setting. We also give a categorical interpretation of the cluster expansion formula in the case of acyclic quivers.
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