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A note on the Loewy lengths of baby Verma modules for modular Lie algebras 关于模态李代数婴孩韦尔马模块的洛伊长度的说明
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.023
Yiyang Li , Bin Shu , Yufeng Yao
{"title":"A note on the Loewy lengths of baby Verma modules for modular Lie algebras","authors":"Yiyang Li ,&nbsp;Bin Shu ,&nbsp;Yufeng Yao","doi":"10.1016/j.jalgebra.2024.09.023","DOIUrl":"10.1016/j.jalgebra.2024.09.023","url":null,"abstract":"<div><div>Let <em>G</em> be a connected reductive algebraic group over an algebraically closed field <strong>k</strong> of prime characteristic <em>p</em>, and <span><math><mi>g</mi><mo>=</mo><mtext>Lie</mtext><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. We study the representations of the reductive Lie algebra <span><math><mi>g</mi></math></span> with <em>p</em>-character <em>χ</em> of standard Levi-form in this note. We obtain similar results about the translation functor and the wall-crossing functor of simple modules parallelling to the representations of algebraic groups (cf. <span><span>[10, II.Lem.7.20]</span></span>). Moreover, we get the Loewy lengths of baby Verma modules provided the Vogan Conjecture holds.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142526774","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}
引用次数: 0
The finite presentation problem for Noetherian algebras 诺特代数的有限呈现问题
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.019
Be'eri Greenfeld
{"title":"The finite presentation problem for Noetherian algebras","authors":"Be'eri Greenfeld","doi":"10.1016/j.jalgebra.2024.09.019","DOIUrl":"10.1016/j.jalgebra.2024.09.019","url":null,"abstract":"<div><div>We prove that there exists an affine Noetherian algebra over a field of characteristic zero which is not finitely presented. Specifically, we prove that the Medvedev–Passman–Resco–Small algebra, recently shown to form a counterexample to the stability problem for Noetherian algebras, is not finitely presented. This answers a question due to Bergman and Irving in characteristic zero, a case explicitly left open by Resco and Small in 1993, who gave a counterexample to this question in positive characteristic.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554246","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}
引用次数: 0
The Steinberg tensor product theorem for general linear group schemes in the Verlinde category 韦林德范畴中一般线性群方案的斯坦伯格张量积定理
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.10.003
Arun S. Kannan
{"title":"The Steinberg tensor product theorem for general linear group schemes in the Verlinde category","authors":"Arun S. Kannan","doi":"10.1016/j.jalgebra.2024.10.003","DOIUrl":"10.1016/j.jalgebra.2024.10.003","url":null,"abstract":"<div><div>The Steinberg tensor product theorem is a fundamental result in the modular representation theory of reductive algebraic groups. It describes any finite-dimensional simple module of highest weight <em>λ</em> over such a group as the tensor product of Frobenius twists of simple modules with highest weights the weights appearing in a <em>p</em>-adic decomposition of <em>λ</em>, thereby reducing the character problem to a finite collection of weights. In recent years this theorem has been extended to various quasi-reductive supergroup schemes. In this paper, we prove the analogous result for the general linear group scheme <span><math><mi>G</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> for any object <em>X</em> in the Verlinde category <span><math><msub><mrow><mi>Ver</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444751","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}
引用次数: 0
Thin monodromy in Sp(4) and Sp(6) Sp(4) 和 Sp(6) 中的薄单色性
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.022
Jitendra Bajpai , Daniele Dona , Martin Nitsche
{"title":"Thin monodromy in Sp(4) and Sp(6)","authors":"Jitendra Bajpai ,&nbsp;Daniele Dona ,&nbsp;Martin Nitsche","doi":"10.1016/j.jalgebra.2024.09.022","DOIUrl":"10.1016/j.jalgebra.2024.09.022","url":null,"abstract":"<div><div>We explore the thinness of hypergeometric groups of type <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>4</mn><mo>)</mo></math></span> and <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>6</mn><mo>)</mo></math></span> by applying a new approach of computer-assisted ping-pong. We prove the thinness of 17 hypergeometric groups with maximally unipotent monodromy in <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>6</mn><mo>)</mo></math></span>, completing the classification of all 40 such groups into arithmetic and thin cases.</div><div>In addition, we establish the thinness of an additional 46 hypergeometric groups in <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>6</mn><mo>)</mo></math></span>, and of three hypergeometric groups in <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>4</mn><mo>)</mo></math></span>, completing the classification of all <span><math><mrow><mi>Sp</mi></mrow><mo>(</mo><mn>4</mn><mo>)</mo></math></span> hypergeometric groups. To the best of our knowledge, this article produces the first 63 examples in the cyclotomic family of Zariski dense non-arithmetic hypergeometric monodromy groups of real rank three.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442412","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}
引用次数: 0
Towards a classification of simple partial comodules of Hopf algebras 霍普夫数组的简单部分组合数的分类
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.10.005
Eliezer Batista , William Hautekiet , Paolo Saracco , Joost Vercruysse
{"title":"Towards a classification of simple partial comodules of Hopf algebras","authors":"Eliezer Batista ,&nbsp;William Hautekiet ,&nbsp;Paolo Saracco ,&nbsp;Joost Vercruysse","doi":"10.1016/j.jalgebra.2024.10.005","DOIUrl":"10.1016/j.jalgebra.2024.10.005","url":null,"abstract":"<div><div>Making the first steps towards a classification of simple partial comodules, we give a general construction for partial comodules of a Hopf algebra <em>H</em> using central idempotents in right coideal subalgebras and show that any 1-dimensional partial comodule is of that form. We conjecture that in fact all finite-dimensional simple partial <em>H</em>-comodules arise this way. For <span><math><mi>H</mi><mo>=</mo><mi>k</mi><mi>G</mi></math></span> for some finite group <em>G</em>, we give conditions for the constructed partial comodule to be simple, and we determine when two of them are isomorphic. If <span><math><mi>H</mi><mo>=</mo><mi>k</mi><msup><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, then our construction recovers the work of M. Dokuchaev and N. Zhukavets <span><span>[12]</span></span>. We also study the partial modules and comodules of the non-commutative non-cocommutative Kac-Paljutkin algebra <span><math><mi>A</mi></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529839","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}
引用次数: 0
Smooth symmetric systems over a finite field and applications 有限域上的光滑对称系统及其应用
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.011
Nardo Giménez , Guillermo Matera , Mariana Pérez , Melina Privitelli
{"title":"Smooth symmetric systems over a finite field and applications","authors":"Nardo Giménez ,&nbsp;Guillermo Matera ,&nbsp;Mariana Pérez ,&nbsp;Melina Privitelli","doi":"10.1016/j.jalgebra.2024.09.011","DOIUrl":"10.1016/j.jalgebra.2024.09.011","url":null,"abstract":"<div><div>We study the set of common <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span>–rational solutions of “smooth” systems of multivariate symmetric polynomials with coefficients in a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span>. We show that, under certain conditions, the set of common solutions of such polynomial systems over the algebraic closure of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span> has a “good” geometric behavior. This allows us to obtain precise estimates on the corresponding number of common <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span>–rational solutions. In the case of hypersurfaces we are able to strengthen the results. We illustrate the interest of these estimates through their application to certain classical combinatorial problems over finite fields.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529841","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}
引用次数: 0
Clifford's theorem for bricks 砖块的克利福德定理
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.021
Yuta Kozakai , Arashi Sakai
{"title":"Clifford's theorem for bricks","authors":"Yuta Kozakai ,&nbsp;Arashi Sakai","doi":"10.1016/j.jalgebra.2024.09.021","DOIUrl":"10.1016/j.jalgebra.2024.09.021","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group, <em>N</em> a normal subgroup of <em>G</em>, and <em>k</em> a field of characteristic <span><math><mi>p</mi><mo>&gt;</mo><mn>0</mn></math></span>. In this paper, we formulate the brick version of Clifford's theorem under suitable assumptions and prove it by using the theory of wide subcategories. As an application of our theorem, we consider the restrictions of semibricks and two-term simple-minded collections under the assumption that the index of the normal subgroup <em>N</em> in <em>G</em> is a <em>p</em>-power.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444749","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}
引用次数: 0
On the twisted group ring isomorphism problem for a class of groups 关于一类群的扭曲群环同构问题
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.08.036
Sumana Hatui , Gurleen Kaur , Sahanawaj Sabnam
{"title":"On the twisted group ring isomorphism problem for a class of groups","authors":"Sumana Hatui ,&nbsp;Gurleen Kaur ,&nbsp;Sahanawaj Sabnam","doi":"10.1016/j.jalgebra.2024.08.036","DOIUrl":"10.1016/j.jalgebra.2024.08.036","url":null,"abstract":"<div><div>The twisted group ring isomorphism problem (TGRIP) is a variation of the classical group ring isomorphism problem. It asks whether the ring structure of the twisted group ring determines the group up to isomorphism. In this article, we study the TGRIP for direct product and central product of groups. We provide some criteria to answer the TGRIP for groups by answering the TGRIP for certain associated quotients. As an application of these results, we provide several examples. Finally, we answer the TGRIP for extra-special <em>p</em>-groups, and for all but five groups of order <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>5</mn></mrow></msup></math></span>, where <span><math><mi>p</mi><mo>≥</mo><mn>5</mn></math></span> is prime.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529895","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}
引用次数: 0
Presentations of mapping class groups and an application to cluster algebras from surfaces 映射类群的表述及其在曲面簇代数中的应用
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.014
Jinlei Dong, Fang Li
{"title":"Presentations of mapping class groups and an application to cluster algebras from surfaces","authors":"Jinlei Dong,&nbsp;Fang Li","doi":"10.1016/j.jalgebra.2024.09.014","DOIUrl":"10.1016/j.jalgebra.2024.09.014","url":null,"abstract":"<div><div>In this paper, we give presentations of the mapping class groups of marked surfaces stabilizing boundaries for any genus. Note that in the existing works, the mapping class groups of marked surfaces were the isotopy classes of homeomorphisms fixing boundaries pointwise. The condition for stabilizing boundaries of mapping class groups makes the requirement for mapping class groups to fix boundaries pointwise to be unnecessary.</div><div>As an application of presentations of the mapping class groups of marked surfaces stabilizing boundaries, we obtain the presentation of the cluster automorphism group of a cluster algebra from a feasible surface <span><math><mo>(</mo><mi>S</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span>.</div><div>Lastly, for the case (1) 4-punctured sphere, the cluster automorphism group of a cluster algebra from the surface is characterized. Since cluster automorphism groups of cluster algebras from those surfaces were given in <span><span>[1]</span></span> in the cases (2) the once-punctured 4-gon and (3) the twice-punctured digon, we indeed give presentations of cluster automorphism groups of cluster algebras from surfaces which are not feasible.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142526773","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}
引用次数: 0
Triviality criteria for unbounded complexes 无界复合物的三性标准
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2024-10-11 DOI: 10.1016/j.jalgebra.2024.09.028
Ioannis Emmanouil, Olympia Talelli
{"title":"Triviality criteria for unbounded complexes","authors":"Ioannis Emmanouil,&nbsp;Olympia Talelli","doi":"10.1016/j.jalgebra.2024.09.028","DOIUrl":"10.1016/j.jalgebra.2024.09.028","url":null,"abstract":"<div><div>We study properties of modules that appear as syzygies of acyclic complexes of projective, injective, flat or flat-cotorsion modules and obtain criteria for these complexes to be contractible or totally acyclic. Our results illustrate the importance of strongly fp-injective modules in the study of these properties. We examine implications of the existence of complete resolutions (in a certain weak sense) and the finiteness of the Gorenstein projective dimension of pure-projective modules. We also use the orthogonality in the homotopy category between complexes of flat modules and pure acyclic complexes of cotorsion modules, in order to study the syzygies of acyclic complexes of flat modules. Finally, we present some applications of our results to group rings, regarding complete resolutions and cohomological periodicity.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444750","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}
引用次数: 0
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