{"title":"An exact sequence for the graded Picent","authors":"Andrei Marcus, Virgilius-Aurelian Minuță","doi":"10.1515/jgth-2023-0040","DOIUrl":null,"url":null,"abstract":"To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>Picent</m:mi> <m:mi>gr</m:mi> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0001.png\" /> <jats:tex-math>\\mathrm{Picent}^{\\mathrm{gr}}(A)</jats:tex-math> </jats:alternatives> </jats:inline-formula> of isomorphism classes of invertible 𝐺-graded <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0002.png\" /> <jats:tex-math>(A,A)</jats:tex-math> </jats:alternatives> </jats:inline-formula>-bimodules over the centralizer of 𝐵 in 𝐴. Our main result is a <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>Picent</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0003.png\" /> <jats:tex-math>\\mathrm{Picent}</jats:tex-math> </jats:alternatives> </jats:inline-formula> version of the Beattie–del Río exact sequence, involving Dade’s group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0004.png\" /> <jats:tex-math>G[B]</jats:tex-math> </jats:alternatives> </jats:inline-formula>, which relates <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>Picent</m:mi> <m:mi>gr</m:mi> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0001.png\" /> <jats:tex-math>\\mathrm{Picent}^{\\mathrm{gr}}(A)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Picent</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_jgth-2023-0040_ineq_0006.png\" /> <jats:tex-math>\\mathrm{Picent}(B)</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and group cohomology.","PeriodicalId":50188,"journal":{"name":"Journal of Group Theory","volume":"15 38","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Group Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/jgth-2023-0040","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group Picentgr(A)\mathrm{Picent}^{\mathrm{gr}}(A) of isomorphism classes of invertible 𝐺-graded (A,A)(A,A)-bimodules over the centralizer of 𝐵 in 𝐴. Our main result is a Picent\mathrm{Picent} version of the Beattie–del Río exact sequence, involving Dade’s group G[B]G[B], which relates Picentgr(A)\mathrm{Picent}^{\mathrm{gr}}(A), Picent(B)\mathrm{Picent}(B), and group cohomology.
期刊介绍：
The Journal of Group Theory is devoted to the publication of original research articles in all aspects of group theory. Articles concerning applications of group theory and articles from research areas which have a significant impact on group theory will also be considered.
Topics:
Group Theory-
Representation Theory of Groups-
Computational Aspects of Group Theory-
Combinatorics and Graph Theory-
Algebra and Number Theory
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