{"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":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 2","pages":"1477 - 1511"},"PeriodicalIF":0.5000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebras and Representation Theory","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-024-10266-w","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","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.
期刊介绍：
Algebras and Representation Theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including Lie algebras and superalgebras, rings of differential operators, group rings and algebras, C*-algebras and Hopf algebras, with particular emphasis on quantum groups.
The journal contains high level, significant and original research papers, as well as expository survey papers written by specialists who present the state-of-the-art of well-defined subjects or subdomains. Occasionally, special issues on specific subjects are published as well, the latter allowing specialists and non-specialists to quickly get acquainted with new developments and topics within the field of rings, algebras and their applications.
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