{"title":"Morita equivalence and globalization for partial Hopf actions on nonunital algebras","authors":"Marcelo Muniz Alves, Tiago Luiz Ferrazza","doi":"10.1142/s0219498825502627","DOIUrl":null,"url":null,"abstract":"<p>In this work, we investigate partial actions of a Hopf algebra <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> on nonunital algebras and the associated partial smash products, with the objective of providing a framework where one may obtain results for both <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mo>𝕜</mo></math></span><span></span>-algebras with local units and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mo>𝕜</mo></math></span><span></span>-categories. We show that our partial actions correspond to nonunital algebras in the category of partial representations of <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span>. The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and we show that if two symmetrical partial actions are Morita equivalent then their standard globalizations are also Morita equivalent. Particularizing to the case of a partial action on an algebra with local units, we obtain several strong results on equivalences of categories of modules of partial smash products of algebras and partial smash products of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mo>𝕜</mo></math></span><span></span>-categories.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825502627","RegionNum":3,"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
In this work, we investigate partial actions of a Hopf algebra on nonunital algebras and the associated partial smash products, with the objective of providing a framework where one may obtain results for both -algebras with local units and -categories. We show that our partial actions correspond to nonunital algebras in the category of partial representations of . The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and we show that if two symmetrical partial actions are Morita equivalent then their standard globalizations are also Morita equivalent. Particularizing to the case of a partial action on an algebra with local units, we obtain several strong results on equivalences of categories of modules of partial smash products of algebras and partial smash products of -categories.
在这篇论文中,我们研究了霍普夫代数 H 在非空格代数上的部分作用以及相关的部分粉碎乘积,目的是提供一个框架,在这个框架中,我们既可以得到有局部单元的𝕜代数的结果,也可以得到𝕜范畴的结果。我们详细研究了部分作用的全局化存在性这一核心问题,并为一般关联代数的最小全局化的存在性(和唯一性)提供了充分条件。我们扩展了阿巴迪、多库恰耶夫、埃塞尔和西蒙以前的成果,定义了部分霍普夫作用的莫里塔等价性,并证明如果两个对称的部分作用是莫里塔等价的,那么它们的标准全局化也是莫里塔等价的。特别是在具有局部单元的代数上的部分作用的情况下,我们得到了关于代数的部分粉碎乘积的模块类别和𝕜类别的部分粉碎乘积的等价性的几个强有力的结果。
期刊介绍:
The Journal of Algebra and Its Applications will publish papers both on theoretical and on applied aspects of Algebra. There is special interest in papers that point out innovative links between areas of Algebra and fields of application. As the field of Algebra continues to experience tremendous growth and diversification, we intend to provide the mathematical community with a central source for information on both the theoretical and the applied aspects of the discipline. While the journal will be primarily devoted to the publication of original research, extraordinary expository articles that encourage communication between algebraists and experts on areas of application as well as those presenting the state of the art on a given algebraic sub-discipline will be considered.