{"title":"Recasting the Hazrat Conjecture: Relating Shift Equivalence to Graded Morita Equivalence","authors":"Gene Abrams, Efren Ruiz, Mark Tomforde","doi":"10.1007/s10468-024-10266-w","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>E</i> and <i>F</i> be finite graphs with no sinks, and <i>k</i> any field. We show that shift equivalence of the adjacency matrices <span>\\(A_E\\)</span> and <span>\\(A_F\\)</span>, together with an additional compatibility condition, implies that the Leavitt path algebras <span>\\(L_k(E)\\)</span> and <span>\\(L_k(F)\\)</span> are graded Morita equivalent. Along the way, we build a new type of <span>\\(L_k(E)\\)</span>–<span>\\(L_k(F)\\)</span>-bimodule (a <i>bridging bimodule</i>), which we use to establish the graded equivalence.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-024-10266-w","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let E and F be finite graphs with no sinks, and k any field. We show that shift equivalence of the adjacency matrices \(A_E\) and \(A_F\), together with an additional compatibility condition, implies that the Leavitt path algebras \(L_k(E)\) and \(L_k(F)\) are graded Morita equivalent. Along the way, we build a new type of \(L_k(E)\)–\(L_k(F)\)-bimodule (a bridging bimodule), which we use to establish the graded equivalence.
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