Journal of Algebra最新文献

{"title":"Modules induced from large subalgebras of the Lie algebra of differential operators of degree at most one","authors":"Matthew Ondrus","doi":"10.1016/j.jalgebra.2025.01.008","DOIUrl":"10.1016/j.jalgebra.2025.01.008","url":null,"abstract":"<div><div>We study the representation theory of the Lie algebra <span><math><mi>D</mi></math></span> of differential operators of degree at most one on the algebra of complex Laurent polynomials. We define a natural family of subalgebras of <span><math><mi>D</mi></math></span>, which we call polynomial subalgebras. After classifying the one-dimensional modules for polynomial subalgebras, we use these one-dimensional modules to construct corresponding induced modules for the full Lie algebra <span><math><mi>D</mi></math></span>. These induced modules are frequently simple and generalize a family of recently discovered simple modules. In order to understand these induced modules in certain complicated cases, we take advantage of some general results on tensor products of modules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 533-571"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164536","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}

{"title":"On integral images of Curtis homomorphisms for GLn and Un","authors":"Tzu-Jan Li","doi":"10.1016/j.jalgebra.2024.12.037","DOIUrl":"10.1016/j.jalgebra.2024.12.037","url":null,"abstract":"<div><div>For <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> defined over a finite field of characteristic <em>p</em>, we refine a result of Bonnafé and Kessar on the saturatedness of the Curtis homomorphism <span><math><msup><mrow><mi>Cur</mi></mrow><mrow><mi>G</mi></mrow></msup></math></span> by describing the image of <span><math><msup><mrow><mi>Cur</mi></mrow><mrow><mi>G</mi></mrow></msup></math></span> over <span><math><mover><mrow><mi>Z</mi></mrow><mo>‾</mo></mover><mo>[</mo><mn>1</mn><mo>/</mo><mi>p</mi><mo>]</mo></math></span> via a system of linear conditions.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"669 ","pages":"Pages 50-57"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387249","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}

{"title":"A combinatorial generalisation of rank two complex reflection groups via generators and relations","authors":"Igor Haladjian","doi":"10.1016/j.jalgebra.2024.12.032","DOIUrl":"10.1016/j.jalgebra.2024.12.032","url":null,"abstract":"<div><div>Complex reflection groups of rank two are precisely the finite groups in the family of groups that we call <em>J</em>-reflection groups. These groups are particular cases of <em>J</em>-groups as defined by Achar &amp; Aubert in 2008. The family of <em>J</em>-reflection groups generalise both complex reflection groups of rank two and toric reflection groups, a family of groups defined and studied by Gobet. We give uniform presentations by generators and relations of <em>J</em>-reflection groups, which coincide with the presentations given by Broué, Malle and Rouquier when the groups are finite. In particular, these presentations provide uniform presentations for complex reflection groups of rank two where the generators are reflections (however the proof uses the classification of irreducible complex reflection groups). Moreover, we show that the center of <em>J</em>-reflection groups is cyclic, generalising what happens for irreducible complex reflection groups of rank two and toric reflection groups. Finally, we classify <em>J</em>-reflection groups up to reflection isomorphisms.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 308-347"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164539","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}

{"title":"Computing supersingular endomorphism rings using inseparable endomorphisms","authors":"Jenny Fuselier ,&nbsp;Annamaria Iezzi ,&nbsp;Mark Kozek ,&nbsp;Travis Morrison ,&nbsp;Changningphaabi Namoijam","doi":"10.1016/j.jalgebra.2025.01.012","DOIUrl":"10.1016/j.jalgebra.2025.01.012","url":null,"abstract":"<div><div>We give an algorithm for computing an inseparable endomorphism of a supersingular elliptic curve <em>E</em> defined over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>, which, conditional on GRH, runs in expected <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>p</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>p</mi><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> bit operations and requires <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>p</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> storage. This matches the time and storage complexity of the best conditional algorithms for computing a nontrivial supersingular endomorphism, such as those of Eisenträger–Hallgren–Leonardi–Morrison–Park and Delfs–Galbraith. Unlike these prior algorithms, which require two paths from <em>E</em> to a curve defined over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, the algorithm we introduce only requires one; thus when combined with the algorithm of Corte-Real Santos–Costello–Shi, our algorithm will be faster in practice. Moreover, our algorithm produces endomorphisms with predictable discriminants, enabling us to prove properties about the orders they generate. With two calls to our algorithm, we can provably compute a Bass suborder of <span><math><mi>End</mi><mo>(</mo><mi>E</mi><mo>)</mo></math></span>. This result is then used in an algorithm for computing a basis for <span><math><mi>End</mi><mo>(</mo><mi>E</mi><mo>)</mo></math></span> with the same time complexity, assuming GRH. We also argue that <span><math><mi>End</mi><mo>(</mo><mi>E</mi><mo>)</mo></math></span> can be computed using <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span> calls to our algorithm along with polynomial overhead, conditional on a heuristic assumption about the distribution of the discriminants of these endomorphisms. Conditional on GRH and this additional heuristic, this yields a <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>p</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>log</mi><mo>⁡</mo><mi>p</mi><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> algorithm for computing <span><math><mi>End</mi><mo>(</mo><mi>E</mi><mo>)</mo></math></span> requiring <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>p</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> storage.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 145-189"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163260","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}

{"title":"AI-rings on almost completely decomposable Abelian groups","authors":"Ekaterina Kompantseva ,&nbsp;Askar Tuganbaev","doi":"10.1016/j.jalgebra.2025.01.007","DOIUrl":"10.1016/j.jalgebra.2025.01.007","url":null,"abstract":"<div><div>We consider the class <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> of reduced abelian block-rigid <em>CRQ</em>-groups of ring type. An <span>absolute ideal</span> of an abelian group <em>G</em> is a subgroup of <em>G</em> which is an ideal in any ring on <em>G</em>. A ring <em>R</em> is called an <em>AI</em><strong>-ring</strong> if any ideal of <em>R</em> is an absolute ideal of its additive group. An abelian group <em>G</em> is called an <em>RAI</em><strong>-group</strong> if there exists at least one <em>AI</em>-ring on <em>G</em>. It is proved that any group in <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is an <em>RAI</em>-group. Thus, in the class <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, we solve Problem 93 in the monograph Fuchs (1973) <span><span>[9]</span></span>. We classify <em>AI</em>-rings on groups in the class <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 1-19"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164154","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}

{"title":"Generalized Verma modules induced from infinite dimensional simple Uq(sl2)-modules","authors":"Limeng Xia","doi":"10.1016/j.jalgebra.2024.12.035","DOIUrl":"10.1016/j.jalgebra.2024.12.035","url":null,"abstract":"<div><div>Let <span><math><mi>g</mi></math></span> be a simple complex finite dimensional Lie algebra. For generic <span><math><mi>q</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, let <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>g</mi><mo>)</mo></math></span> be the quantum group with standard generators <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>±</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup></math></span>. Fixing an index <span><math><msub><mrow><mi>i</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, let <span><math><mi>P</mi></math></span> be the subalgebra generated by all <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>±</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup></math></span> and <span><math><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msub></math></span>, which has sub-quotient algebras isomorphic to <span><math><msub><mrow><mi>U</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. With each generalized Verma <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>g</mi><mo>)</mo></math></span>-module <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> induced from a simple <span><math><msub><mrow><mi>U</mi></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msub></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-module <em>V</em>, we associate a Verma module <span><math><mi>M</mi><mo>(</mo><mi>V</mi><mo>)</mo></math></span>. In this paper, for infinite dimensional <em>V</em> we prove that <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>P</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> is simple if and only if <span><math><mi>M</mi><mo>(</mo><mi>V</mi><mo>)</mo></math></span> is simple.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 20-45"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164153","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}

{"title":"Codimensions of algebras with pseudoautomorphism and their exponential growth","authors":"Elena Campedel,&nbsp;Ginevra Giordani,&nbsp;Antonio Ioppolo","doi":"10.1016/j.jalgebra.2025.01.015","DOIUrl":"10.1016/j.jalgebra.2025.01.015","url":null,"abstract":"<div><div>Let <em>F</em> be a fixed field of characteristic zero containing an element <em>i</em> such that <span><math><msup><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mo>−</mo><mn>1</mn></math></span>. In this paper we consider finite dimensional superalgebras over <em>F</em> endowed with a pseudoautomorphism <em>p</em> and we investigate the asymptotic behavior of the corresponding sequence of <em>p</em>-codimensions <span><math><msubsup><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, <span><math><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></math></span>. First we give a positive answer to a conjecture of Amitsur in this setting: the <em>p</em>-exponent <span><math><msup><mrow><mi>exp</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>⁡</mo><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>lim</mi></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mo>⁡</mo><mroot><mrow><msubsup><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></mroot></math></span> always exists and it is an integer. In the final part we characterize the algebras whose exponential growth is bounded by 2.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 75-91"},"PeriodicalIF":0.8,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164156","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}

{"title":"Characters and Sylow subgroup abelianization","authors":"Eugenio Giannelli ,&nbsp;Noelia Rizo ,&nbsp;A.A. Schaeffer Fry ,&nbsp;Carolina Vallejo","doi":"10.1016/j.jalgebra.2024.12.031","DOIUrl":"10.1016/j.jalgebra.2024.12.031","url":null,"abstract":"<div><div>We characterize when a finite group <em>G</em> possesses a Sylow 3-subgroup <em>P</em> with abelianization of order 9 in terms of the number of height zero characters lying in the principal 3-block of <em>G</em>, settling a conjecture put forward by Navarro, Sambale, and Tiep in 2018. Along the way, we show that a recent result by Laradji on the number of character of height zero in a block that lie above a given character of some normal subgroup holds, without any hypothesis on the group for blocks of maximal defect.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"667 ","pages":"Pages 824-864"},"PeriodicalIF":0.8,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143104168","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}

{"title":"Lefschetz properties of squarefree monomial ideals via Rees algebras","authors":"Thiago Holleben","doi":"10.1016/j.jalgebra.2024.12.030","DOIUrl":"10.1016/j.jalgebra.2024.12.030","url":null,"abstract":"<div><div>The theory of Rees algebras of monomial ideals has been extensively studied, and as a consequence, many (sometimes partial) equivalences between algebraic properties of monomial ideals, and combinatorial properties of simplicial complexes and hypergraphs are known. In this paper we show how this theory can be used to find interesting examples in the theory of Lefschetz properties. We explore the consequences of known results from Lefschetz properties for the Rees algebras of squarefree monomial ideals, for example in the calculation of analytic spread. In particular, we show a connection between symbolic powers and <em>f</em>-vectors of simplicial complexes. This perspective leads us to a generalization of Postnikov's “mixed Eulerian numbers”. We prove the positivity of such numbers in our setting.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 572-599"},"PeriodicalIF":0.8,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143164538","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}

{"title":"Counting in nilpotent injectors and Carter subgroups","authors":"S. Aivazidis ,&nbsp;M. Loukaki ,&nbsp;J. Shareshian","doi":"10.1016/j.jalgebra.2024.12.033","DOIUrl":"10.1016/j.jalgebra.2024.12.033","url":null,"abstract":"<div><div>We investigate number-theoretic properties of the collection of nilpotent injectors or nilpotent projectors containing certain subgroups of finite soluble (or <span><math><mi>N</mi></math></span>-constrained) groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"668 ","pages":"Pages 627-646"},"PeriodicalIF":0.8,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394435","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}