Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107841
Shuichi Harako
In 1993, Kontsevich introduced the symplectic derivation Lie algebras related to various geometric objects including moduli spaces of graphs and of Riemann surfaces, graph homologies, Hamiltonian vector fields, etc. Each of them is a graded algebra, so that its Chevalley-Eilenberg chain complex has another -grading, called weight, than the usual homological degree. We focus on one of the Lie algebras , called the “commutative case”, and its positive weight part . The symplectic invariant homology of is closely related to the commutative graph homology, hence some computational results are obtained from the viewpoint of graph homology theory. On the other hand, the details of the entire homology group are not completely known. We determine by decomposing it by weight and using the classical representation theory of the symplectic groups.
{"title":"The second homology group of the commutative case of Kontsevich's symplectic derivation Lie algebra","authors":"Shuichi Harako","doi":"10.1016/j.jpaa.2024.107841","DOIUrl":"10.1016/j.jpaa.2024.107841","url":null,"abstract":"<div><div>In 1993, Kontsevich introduced the symplectic derivation Lie algebras related to various geometric objects including moduli spaces of graphs and of Riemann surfaces, graph homologies, Hamiltonian vector fields, etc. Each of them is a graded algebra, so that its Chevalley-Eilenberg chain complex has another <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></math></span>-grading, called weight, than the usual homological degree. We focus on one of the Lie algebras <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>, called the “commutative case”, and its positive weight part <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>⊂</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span>. The symplectic invariant homology of <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> is closely related to the commutative graph homology, hence some computational results are obtained from the viewpoint of graph homology theory. On the other hand, the details of the entire homology group <span><math><msub><mrow><mi>H</mi></mrow><mrow><mo>•</mo></mrow></msub><mo>(</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>)</mo></math></span> are not completely known. We determine <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msubsup><mrow><mi>c</mi></mrow><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>)</mo></math></span> by decomposing it by weight and using the classical representation theory of the symplectic groups.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107841"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107849
Yasmeen S. Baki
Historically, the study of graded (twisted or otherwise) Calabi–Yau algebras has meant the study of such algebras under an -grading. In this paper, we propose a suitable definition for a twisted G-graded Calabi–Yau algebra, for G an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a G-graded algebra is twisted Calabi–Yau if and only if it is G-graded twisted Calabi–Yau. In the second half of the paper, we prove that localizations of twisted Calabi–Yau algebras at elements which form both left and right denominator sets remain twisted Calabi–Yau. As such, we obtain a large class of -graded twisted Calabi–Yau algebras arising as localizations of Artin–Schelter regular algebras. Throughout the paper, we survey a number of concrete examples of G-graded twisted Calabi–Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras.
{"title":"Group-graded twisted Calabi–Yau algebras","authors":"Yasmeen S. Baki","doi":"10.1016/j.jpaa.2024.107849","DOIUrl":"10.1016/j.jpaa.2024.107849","url":null,"abstract":"<div><div>Historically, the study of graded (twisted or otherwise) Calabi–Yau algebras has meant the study of such algebras under an <span><math><mi>N</mi></math></span>-grading. In this paper, we propose a suitable definition for a twisted <em>G</em>-graded Calabi–Yau algebra, for <em>G</em> an arbitrary abelian group. Building on the work of Reyes and Rogalski, we show that a <em>G</em>-graded algebra is twisted Calabi–Yau if and only if it is <em>G</em>-graded twisted Calabi–Yau. In the second half of the paper, we prove that localizations of twisted Calabi–Yau algebras at elements which form both left and right denominator sets remain twisted Calabi–Yau. As such, we obtain a large class of <span><math><mi>Z</mi></math></span>-graded twisted Calabi–Yau algebras arising as localizations of Artin–Schelter regular algebras. Throughout the paper, we survey a number of concrete examples of <em>G</em>-graded twisted Calabi–Yau algebras, including the Weyl algebras, families of generalized Weyl algebras, and universal enveloping algebras of finite dimensional Lie algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107849"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107847
Arno van den Essen , Jan Schoone
The theorem of Duistermaat and Van der Kallen from 1998 proved the first case of the Mathieu conjecture. Using the theory of Mathieu-Zhao spaces, we can reformulate this theorem as Ker L is a Mathieu-Zhao space where L is the linear map . In this paper, we generalize this result (for ) to all non-trivial linear maps such that for some .
{"title":"Kernels of linear maps","authors":"Arno van den Essen , Jan Schoone","doi":"10.1016/j.jpaa.2024.107847","DOIUrl":"10.1016/j.jpaa.2024.107847","url":null,"abstract":"<div><div>The theorem of Duistermaat and Van der Kallen from 1998 proved the first case of the Mathieu conjecture. Using the theory of Mathieu-Zhao spaces, we can reformulate this theorem as Ker <em>L</em> is a Mathieu-Zhao space where <em>L</em> is the linear map <span><math><mi>L</mi><mo>:</mo><mi>C</mi><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>]</mo><mo>→</mo><mi>C</mi><mo>,</mo><mspace></mspace><mi>f</mi><mo>↦</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. In this paper, we generalize this result (for <span><math><mi>n</mi><mo>=</mo><mn>1</mn></math></span>) to all non-trivial linear maps <span><math><mi>L</mi><mo>:</mo><mi>C</mi><mo>[</mo><mi>X</mi><mo>,</mo><msup><mrow><mi>X</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>→</mo><mi>C</mi></math></span> such that <span><math><mo>{</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>|</mo><mo>|</mo><mi>n</mi><mo>|</mo><mo>≥</mo><mi>N</mi><mo>}</mo><mo>⊂</mo><mi>Ker</mi><mspace></mspace><mi>L</mi></math></span> for some <span><math><mi>N</mi><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107847"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107848
Benjamin Steinberg
This paper aims to use topological methods to compute Ext between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a maximal subgroup, via a spectral sequence that collapses on the -page over fields of good characteristic. As an application, we determine the global dimension of the algebra of the monoid of all affine transformations of a vector space over a finite field. We provide a topological characterization of when a monoid homomorphism induces a homological epimorphism of monoid algebras and apply it to semidirect products. Topology is used to construct projective resolutions of modules inflated from the group completion for sufficiently nice monoids. A sequel paper will use these results to study the representation theory Hsiao's monoid of ordered G-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product ).
{"title":"Topology and monoid representations I: Foundations","authors":"Benjamin Steinberg","doi":"10.1016/j.jpaa.2024.107848","DOIUrl":"10.1016/j.jpaa.2024.107848","url":null,"abstract":"<div><div>This paper aims to use topological methods to compute Ext between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a maximal subgroup, via a spectral sequence that collapses on the <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-page over fields of good characteristic. As an application, we determine the global dimension of the algebra of the monoid of all affine transformations of a vector space over a finite field. We provide a topological characterization of when a monoid homomorphism induces a homological epimorphism of monoid algebras and apply it to semidirect products. Topology is used to construct projective resolutions of modules inflated from the group completion for sufficiently nice monoids. A sequel paper will use these results to study the representation theory Hsiao's monoid of ordered <em>G</em>-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product <span><math><mi>G</mi><mo>≀</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107848"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107859
Simon Henry , Félix Loubaton
We construct a left semi-model category of “marked strict ∞-categories” for which the fibrant objects are those whose marked arrows satisfy natural closure properties and are invertible up to higher marked arrows. The canonical model structure on strict ∞-categories can be recovered as a left Bousfield localization of this model structure. We show that an appropriate extension of the Street nerve to the marked setting produces a Quillen adjunction between our model category and the Verity model structure for complicial sets, generalizing previous results by the second named author. Finally, we use this model structure to study, in the setting of strict ∞-categories, the idea that, because they are two different “truncation functors” taking an to an -category, there are two non-equivalent definitions for the -category of -categories as a limit of the -categories of -categories. We show that in fact there seem to be at least three non-equivalent ways of constructing an -category of -categories.
{"title":"An inductive model structure for strict ∞-categories","authors":"Simon Henry , Félix Loubaton","doi":"10.1016/j.jpaa.2024.107859","DOIUrl":"10.1016/j.jpaa.2024.107859","url":null,"abstract":"<div><div>We construct a left semi-model category of “marked strict ∞-categories” for which the fibrant objects are those whose marked arrows satisfy natural closure properties and are invertible up to higher marked arrows. The canonical model structure on strict ∞-categories can be recovered as a left Bousfield localization of this model structure. We show that an appropriate extension of the Street nerve to the marked setting produces a Quillen adjunction between our model category and the Verity model structure for complicial sets, generalizing previous results by the second named author. Finally, we use this model structure to study, in the setting of strict ∞-categories, the idea that, because they are two different “truncation functors” taking an <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mi>n</mi><mo>)</mo></math></span> to an <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-category, there are two non-equivalent definitions for the <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-category of <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>-categories as a limit of the <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-categories of <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-categories. We show that in fact there seem to be at least three non-equivalent ways of constructing an <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mn>1</mn><mo>)</mo></math></span>-category of <span><math><mo>(</mo><mo>∞</mo><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>-categories.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107859"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107851
Matthias Schötz
The following representation theorem is proven: A partially ordered commutative ring is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space X if and only if is archimedean and localizable. Here we assume that the positive cone of is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on X is one that is defined on a dense open subset of X. A partially ordered commutative ring is archimedean if the underlying additive partially ordered abelian group is archimedean, and is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the σ-bounded case, lattice-ordered commutative rings (f-rings), partially ordered fields, and commutative operator algebras.
{"title":"Rings of almost everywhere defined functions","authors":"Matthias Schötz","doi":"10.1016/j.jpaa.2024.107851","DOIUrl":"10.1016/j.jpaa.2024.107851","url":null,"abstract":"<div><div>The following representation theorem is proven: A partially ordered commutative ring <span><math><mi>R</mi></math></span> is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space <em>X</em> if and only if <span><math><mi>R</mi></math></span> is archimedean and localizable. Here we assume that the positive cone of <span><math><mi>R</mi></math></span> is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on <em>X</em> is one that is defined on a dense open subset of <em>X</em>. A partially ordered commutative ring <span><math><mi>R</mi></math></span> is archimedean if the underlying additive partially ordered abelian group is archimedean, and <span><math><mi>R</mi></math></span> is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the <em>σ</em>-bounded case, lattice-ordered commutative rings (<em>f</em>-rings), partially ordered fields, and commutative operator algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107851"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2025.107865
Taku Suzuki
In this paper, we investigate Fano manifolds whose Chern characters satisfy some positivity conditions. We prove that such manifolds admit long chains of higher order minimal families of rational curves and are covered by higher rational varieties.
{"title":"Fano manifolds whose Chern characters satisfy some positivity conditions","authors":"Taku Suzuki","doi":"10.1016/j.jpaa.2025.107865","DOIUrl":"10.1016/j.jpaa.2025.107865","url":null,"abstract":"<div><div>In this paper, we investigate Fano manifolds whose Chern characters satisfy some positivity conditions. We prove that such manifolds admit long chains of higher order minimal families of rational curves and are covered by higher rational varieties.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107865"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2025.107873
Đặng Võ Phúc
This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/about/policies/article-withdrawal).
This article has been retracted at the request of the Editors-in-Chief.
Many of the results in this paper are false, including the main result (Theorem 1.3), whose proof has a mistake, and several of its consequences are asserted without proof. In a letter to the Editors providing comments to the paper, a colleague pointed out that a proof is only provided for the cases and , asserting that the main result is false for and also incomplete for . No proof is given for .
A detailed assessment by independent referees created reasonable doubt that the author's results were correct.
The Corrigendum to this article does not provide any proofs, but points out an “inaccuracy” in a different part of the paper (Cor. 1.11).
{"title":"Retraction notice to “On the fifth Singer algebraic transfer in a generic family of internal degree characterized by μ(n)=4” [J. Pure Appl. Algebra 228 (2024) 107658]","authors":"Đặng Võ Phúc","doi":"10.1016/j.jpaa.2025.107873","DOIUrl":"10.1016/j.jpaa.2025.107873","url":null,"abstract":"<div><div>This article has been retracted: please see Elsevier Policy on Article Withdrawal (<span><span>https://www.elsevier.com/about/policies/article-withdrawal</span><svg><path></path></svg></span>).</div><div>This article has been retracted at the request of the Editors-in-Chief.</div><div>Many of the results in this paper are false, including the main result (Theorem 1.3), whose proof has a mistake, and several of its consequences are asserted without proof. In a letter to the Editors providing comments to the paper, a colleague pointed out that a proof is only provided for the cases <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>8</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mo>(</mo><mn>10</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, asserting that the main result is false for <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>s</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>8</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and also incomplete for <span><math><mo>(</mo><mn>10</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. No proof is given for <span><math><mi>s</mi><mo>></mo><mn>2</mn></math></span>.</div><div>A detailed assessment by independent referees created reasonable doubt that the author's results were correct.</div><div>The Corrigendum to this article does not provide any proofs, but points out an “inaccuracy” in a different part of the paper (Cor. 1.11).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107873"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143099043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107857
Claudio Quadrelli
Let p be a prime. Following Snopce-Tanushevski, a pro-p group G is called Frattini-resistant if the function , from the poset of all closed topologically finitely generated subgroups of G into itself, is a poset embedding. We prove that for an oriented right-angled Artin pro-p group (oriented pro-p RAAG) G associated to a finite directed graph the following four conditions are equivalent: the associated directed graph is of elementary type; G is Frattini-resistant; every topologically finitely generated closed subgroup of G is an oriented pro-p RAAG; G is the maximal pro-p Galois group of a field containing a root of 1 of order p. Also, we conjecture that in the -cohomology of a Frattini-resistant pro-p group there are no essential triple Massey products.
{"title":"Directed graphs, Frattini-resistance, and maximal pro-p Galois groups","authors":"Claudio Quadrelli","doi":"10.1016/j.jpaa.2024.107857","DOIUrl":"10.1016/j.jpaa.2024.107857","url":null,"abstract":"<div><div>Let <em>p</em> be a prime. Following Snopce-Tanushevski, a pro-<em>p</em> group <em>G</em> is called Frattini-resistant if the function <span><math><mi>H</mi><mo>↦</mo><mi>Φ</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>, from the poset of all closed topologically finitely generated subgroups of <em>G</em> into itself, is a poset embedding. We prove that for an oriented right-angled Artin pro-<em>p</em> group (oriented pro-<em>p</em> RAAG) <em>G</em> associated to a finite directed graph the following four conditions are equivalent: the associated directed graph is of elementary type; <em>G</em> is Frattini-resistant; every topologically finitely generated closed subgroup of <em>G</em> is an oriented pro-<em>p</em> RAAG; <em>G</em> is the maximal pro-<em>p</em> Galois group of a field containing a root of 1 of order <em>p</em>. Also, we conjecture that in the <span><math><mi>Z</mi><mo>/</mo><mi>p</mi></math></span>-cohomology of a Frattini-resistant pro-<em>p</em> group there are no essential triple Massey products.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107857"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-01DOI: 10.1016/j.jpaa.2024.107843
Cindy (Sin Yi) Tsang
Let G be a group. The holomorph of G may be defined as the normalizer of the subgroup of either left or right translations in the group of all permutations of G. The multiple holomorph is in turn defined as the normalizer of the holomorph. Their quotient has been computed for various families of groups G. In this paper, we consider the case when G is centerless, and we show that must have exponent at most 2 unless G satisfies some fairly strong conditions. As applications of our main theorem, we are able to show that has order 2 for all almost simple groups G, and that has exponent at most 2 for all centerless perfect or complete groups G.
{"title":"The multiple holomorph of centerless groups","authors":"Cindy (Sin Yi) Tsang","doi":"10.1016/j.jpaa.2024.107843","DOIUrl":"10.1016/j.jpaa.2024.107843","url":null,"abstract":"<div><div>Let <em>G</em> be a group. The holomorph <span><math><mrow><mi>Hol</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em> may be defined as the normalizer of the subgroup of either left or right translations in the group of all permutations of <em>G</em>. The multiple holomorph <span><math><mrow><mi>NHol</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is in turn defined as the normalizer of the holomorph. Their quotient <span><math><mi>T</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mrow><mi>NHol</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo><mo>/</mo><mrow><mi>Hol</mi></mrow><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has been computed for various families of groups <em>G</em>. In this paper, we consider the case when <em>G</em> is centerless, and we show that <span><math><mi>T</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> must have exponent at most 2 unless <em>G</em> satisfies some fairly strong conditions. As applications of our main theorem, we are able to show that <span><math><mi>T</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has order 2 for all almost simple groups <em>G</em>, and that <span><math><mi>T</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> has exponent at most 2 for all centerless perfect or complete groups <em>G</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 1","pages":"Article 107843"},"PeriodicalIF":0.7,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143094011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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