Journal of AlgebraPub Date : 2025-02-18DOI: 10.1016/j.jalgebra.2025.02.012
Nicolas Gilliers
{"title":"Post-Hopf algebras and non-commutative probability theory","authors":"Nicolas Gilliers","doi":"10.1016/j.jalgebra.2025.02.012","DOIUrl":"10.1016/j.jalgebra.2025.02.012","url":null,"abstract":"<div><div>We study <span><math><mi>O</mi></math></span> operators and post-Lie products over the same Lie algebra compatible in a certain sense. We prove that the group product corresponding to the formal integration of the Lie algebra, which is adjacent to the sum of two compatible post-Lie products, can be factorized in a way that is reminiscent of the classical Semenov-Tian-Shanskii factorization. In the second part, we explore applications in non-commutative probability. We introduce new transforms that facilitate the computation of conditionally free and conditionally monotone multiplicative convolutions involving operator-valued non-commutative distributions.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"671 ","pages":"Pages 1-60"},"PeriodicalIF":0.8,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143487987","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}
Journal of AlgebraPub Date : 2025-02-17DOI: 10.1016/j.jalgebra.2025.02.013
Anand Pillay , Philipp Rothmaler
{"title":"Bass modules and embeddings into free modules","authors":"Anand Pillay , Philipp Rothmaler","doi":"10.1016/j.jalgebra.2025.02.013","DOIUrl":"10.1016/j.jalgebra.2025.02.013","url":null,"abstract":"<div><div>We show that the free module of infinite rank <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>(</mo><mi>κ</mi><mo>)</mo></mrow></msup></math></span> purely embeds every <em>κ</em>-generated flat left <em>R</em>-module iff <em>R</em> is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory <em>T</em> of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>(</mo><mi>κ</mi><mo>)</mo></mrow></msup></math></span> whose projectivity is equivalent to left perfectness, which allows to add a ‘stronger’ equivalent condition: <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>(</mo><mi>κ</mi><mo>)</mo></mrow></msup></math></span> purely embeds every <em>κ</em>-generated flat left <em>R</em>-module which is a model of <em>T</em>.</div><div>We extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a ‘Bass theory’ of pure-projective modules. We put this new theory to use by, among other things, reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"670 ","pages":"Pages 1-12"},"PeriodicalIF":0.8,"publicationDate":"2025-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143454942","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}
Journal of AlgebraPub Date : 2025-02-11DOI: 10.1016/j.jalgebra.2025.02.010
Robert Underwood
{"title":"Hopf orders in K[Cp3] over a local field","authors":"Robert Underwood","doi":"10.1016/j.jalgebra.2025.02.010","DOIUrl":"10.1016/j.jalgebra.2025.02.010","url":null,"abstract":"<div><div>Let <em>K</em> be a field of characteristic <em>p</em> or 0 that is complete with respect to a discrete valuation. Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> denote the cyclic group of order <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>3</mn></math></span>. In the characteristic <em>p</em> case, we construct a new collection of Hopf orders in <span><math><mi>K</mi><mo>[</mo><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub><mo>]</mo></math></span>. When the characteristic of <em>K</em> is 0 and <em>K</em> contains a primitive <em>p</em>th root of unity, we show that Hopf orders in <span><math><mi>K</mi><mo>[</mo><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>]</mo></math></span> can be translated from truncated exponential form to Gauss sum form and vice versa. We investigate the analogous translation problem in the <span><math><msub><mrow><mi>C</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span> case when the characteristic of <em>K</em> is 0 and <em>K</em> contains a primitive <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>rd root of unity.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"669 ","pages":"Pages 370-400"},"PeriodicalIF":0.8,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419033","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}
Journal of AlgebraPub Date : 2025-02-11DOI: 10.1016/j.jalgebra.2025.02.011
Zhiguang Cui , Li Ren , Chao Yang
{"title":"Duality and Galois correspondence for vertex superalgebras","authors":"Zhiguang Cui , Li Ren , Chao Yang","doi":"10.1016/j.jalgebra.2025.02.011","DOIUrl":"10.1016/j.jalgebra.2025.02.011","url":null,"abstract":"<div><div>Let <em>V</em> be a simple vertex superalgebra of countable dimension, <em>G</em> a finite automorphism group of <em>V</em> and <em>T</em> a positive integer. Let <span><math><mi>S</mi></math></span> be a finite <em>G</em>-stable set of inequivalent irreducible <span><math><mo>(</mo><mi>V</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span>-modules. Then there is a finite dimensional semisimple associative algebra <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> for a suitable 2-cocycle <em>α</em> naturally determined by the <em>G</em>-action on <span><math><mi>S</mi></math></span> such that the actions of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> and <span><math><msup><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msup></math></span> on the direct sum of <span><math><mo>(</mo><mi>V</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span>-modules in <span><math><mi>S</mi></math></span> form a Schur-Weyl type duality. As applications, we establish the following results: (1) Every irreducible <em>g</em>-twisted <em>V</em>-module is a completely reducible <span><math><msup><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msup></math></span>-module for arbitrary <span><math><mi>g</mi><mo>∈</mo><mi>G</mi></math></span> and irreducible <span><math><msup><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msup></math></span>-modules appearing in different <em>G</em>-orbits are inequivalent; (2) The quantum Galois correspondence theorem in the context of vertex superalgebras is proved.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"669 ","pages":"Pages 353-369"},"PeriodicalIF":0.8,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419032","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}
Journal of AlgebraPub Date : 2025-02-07DOI: 10.1016/j.jalgebra.2025.02.008
André Dosea, Cleto B. Miranda-Neto
{"title":"On Huneke's conjecture about associated primes of local cohomology modules","authors":"André Dosea, Cleto B. Miranda-Neto","doi":"10.1016/j.jalgebra.2025.02.008","DOIUrl":"10.1016/j.jalgebra.2025.02.008","url":null,"abstract":"<div><div>A conjecture raised in 1990 by C. Huneke predicts that, for a <em>d</em>-dimensional Noetherian local ring <em>R</em>, local cohomology modules of finitely generated <em>R</em>-modules have finitely many associated primes. Although counterexamples do exist, the conjecture has been confirmed in several cases, for instance if <span><math><mi>d</mi><mo>≤</mo><mn>3</mn></math></span>, and witnessed some progress in special cases for higher <em>d</em>. In this paper, assuming that <em>R</em> is a factorial domain in the main results, we establish the case <span><math><mi>d</mi><mo>=</mo><mn>4</mn></math></span>, and under certain additional conditions also the case <span><math><mi>d</mi><mo>=</mo><mn>5</mn></math></span>. Finally, when <em>R</em> is regular and contains a field, we apply the Hartshorne-Lichtenbaum vanishing theorem as a tool to deal with the case <span><math><mi>d</mi><mo>=</mo><mn>6</mn></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"669 ","pages":"Pages 143-158"},"PeriodicalIF":0.8,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387258","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}
Journal of AlgebraPub Date : 2025-02-07DOI: 10.1016/j.jalgebra.2025.02.007
J. Brox , X. García-Martínez , M. Mancini , T. Van der Linden , C. Vienne
{"title":"Weak representability of actions of non-associative algebras","authors":"J. Brox , X. García-Martínez , M. Mancini , T. Van der Linden , C. Vienne","doi":"10.1016/j.jalgebra.2025.02.007","DOIUrl":"10.1016/j.jalgebra.2025.02.007","url":null,"abstract":"<div><div>We study the categorical-algebraic condition that <em>internal actions are weakly representable</em> (WRA) in the context of varieties of (non-associative) algebras over a field.</div><div>Our first aim is to give a complete characterization of action accessible, operadic quadratic varieties of non-associative algebras which satisfy an identity of degree two and to study the representability of actions for them. Here we prove that the varieties of two-step nilpotent (anti-)commutative algebras and that of commutative associative algebras are weakly action representable, and we explain that the condition (WRA) is closely connected to the existence of a so-called <em>amalgam</em>.</div><div>Our second aim is to work towards the construction, still within the context of algebras over a field, of a weakly representing object <span><math><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> for the actions on (or split extensions of) an object <em>X</em>. We actually obtain a <em>partial</em> algebra <span><math><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, which we call <em>external weak actor</em> of <em>X</em>, together with a monomorphism of functors <span><math><mi>SplExt</mi><mo>(</mo><mo>−</mo><mo>,</mo><mi>X</mi><mo>)</mo><mo>↣</mo><mi>Hom</mi><mo>(</mo><mi>U</mi><mo>(</mo><mo>−</mo><mo>)</mo><mo>,</mo><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>)</mo></math></span>, which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the <em>universal strict general actor</em> <span><math><mi>USGA</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and that of <span><math><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> are described in detail. We end with some open questions.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"669 ","pages":"Pages 401-444"},"PeriodicalIF":0.8,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453762","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}
Journal of AlgebraPub Date : 2025-02-06DOI: 10.1016/j.jalgebra.2025.02.009
Sandra Rodríguez-Villalobos
{"title":"BCM-thresholds of hypersurfaces","authors":"Sandra Rodríguez-Villalobos","doi":"10.1016/j.jalgebra.2025.02.009","DOIUrl":"10.1016/j.jalgebra.2025.02.009","url":null,"abstract":"<div><div>In this paper, we use big Cohen-Macaulay algebras to define a characteristic free analog of the <em>F</em>-thresholds, which we call BCM-thresholds, in the case of principal ideals. We prove that, similarly to the case of the <em>F</em>-thresholds, the set of BCM-thresholds and the set of BCM-jumping numbers agree. We also relate some BCM-thresholds to splittings of maps from the ring to a big Cohen-Macaulay algebra.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"669 ","pages":"Pages 341-352"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419031","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}
Journal of AlgebraPub Date : 2025-02-06DOI: 10.1016/j.jalgebra.2025.01.024
Ezgi Kantarcı Oğuz , Emine Yıldırım
{"title":"Cluster expansions: T-walks, labeled posets and matrix calculations","authors":"Ezgi Kantarcı Oğuz , Emine Yıldırım","doi":"10.1016/j.jalgebra.2025.01.024","DOIUrl":"10.1016/j.jalgebra.2025.01.024","url":null,"abstract":"<div><div>We give two new combinatorial methods for computing cluster expansion formulas for arcs coming from possibly punctured surfaces. The first is by using <em>T-walks</em>, an extension of <em>T</em>-path models from <span><span>[43]</span></span>, <span><span>[44]</span></span> for unpunctured surfaces to general surfaces. To do so, we introduce a new combinatorial way to generate these paths. The second is by using order ideals of labeled posets associated to arcs. In this context, we use the methods introduced in <span><span>[25]</span></span>, <span><span>[26]</span></span> to give a quick way to calculate the expressions using 2 by 2 matrices. The techniques introduced are applicable to different settings in cluster algebras and beyond.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"669 ","pages":"Pages 183-219"},"PeriodicalIF":0.8,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387260","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}
Journal of AlgebraPub Date : 2025-02-05DOI: 10.1016/j.jalgebra.2025.02.003
Detlev W. Hoffmann , Kristýna Zemková
{"title":"Vishik equivalence and similarity of quadratic forms over fields of characteristic 2","authors":"Detlev W. Hoffmann , Kristýna Zemková","doi":"10.1016/j.jalgebra.2025.02.003","DOIUrl":"10.1016/j.jalgebra.2025.02.003","url":null,"abstract":"<div><div>An important aspect in the algebraic theory of quadratic forms is the study of equivalence relations based on algebraic-geometric properties of the associated quadrics. A well-known criterion originally proved by Vishik in characteristic zero states that two nonsingular quadratic forms of the same dimension have identical Witt indices over all field extensions if and only if their motives are isomorphic in the category of (integral or mod 2) Chow motives. In characteristic 2, it is meaningful to include singular forms. We therefore define two quadratic forms (including singular ones) of the same dimension to be Vishik-equivalent if they share the same isotropy behavior (in a suitably defined way) over all field extensions. Similar quadratic forms are always Vishik-equivalent, but the converse need not hold. We determine various classes of quadratic forms in characteristic 2 where Vishik equivalence implies similarity and give nonsingular counterexamples in all dimensions <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>≥</mo><mn>8</mn></math></span>, and also singular counterexamples in dimension 8. To construct the counterexamples, we use a generalized notion of so-called half-neighbors in characteristic 2.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"669 ","pages":"Pages 118-142"},"PeriodicalIF":0.8,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387253","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}
Journal of AlgebraPub Date : 2025-02-05DOI: 10.1016/j.jalgebra.2025.02.002
John Graf, Naihuan Jing
{"title":"Pfaffian formulation of Schur's Q-functions","authors":"John Graf, Naihuan Jing","doi":"10.1016/j.jalgebra.2025.02.002","DOIUrl":"10.1016/j.jalgebra.2025.02.002","url":null,"abstract":"<div><div>We introduce a Pfaffian formula that extends Schur's <em>Q</em>-functions <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> to be indexed by compositions <em>λ</em> with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the Young tableau and Vertex Operator constructions. With this construction, we develop a proof technique involving decomposing <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> into sums indexed by partitions with removed parts. Consequently, we are able to prove several identities of Schur's <em>Q</em>-functions using only simple algebraic methods.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"669 ","pages":"Pages 1-25"},"PeriodicalIF":0.8,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143372176","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}
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