Journal of Algebra最新文献

{"title":"The 2 × 2 upper triangular matrix algebra and its generalized polynomial identities","authors":"Fabrizio Martino ,&nbsp;Carla Rizzo","doi":"10.1016/j.jalgebra.2024.12.005","DOIUrl":"10.1016/j.jalgebra.2024.12.005","url":null,"abstract":"<div><div>Let <span><math><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> be the algebra of <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> upper triangular matrices over a field <em>F</em> of characteristic zero. Here we study the generalized polynomial identities of <span><math><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, i.e., identical relations holding for <span><math><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> regarded as <span><math><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-algebra. We determine the generator of the <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span>-ideal of generalized polynomial identities of <span><math><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and compute the exact values of the corresponding sequence of generalized codimensions. Moreover, we give a complete description of the space of multilinear generalized identities in <em>n</em> variables in the language of Young diagrams through the representation theory of the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Finally, we prove that, unlike the ordinary case, the generalized variety of <span><math><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-algebras generated by <span><math><mi>U</mi><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> has no almost polynomial growth; nevertheless, we exhibit two distinct generalized varieties of almost polynomial growth.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 308-330"},"PeriodicalIF":0.8,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152712","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":"Some characterizations of fundamental graded algebras","authors":"Elena Pascucci","doi":"10.1016/j.jalgebra.2024.12.007","DOIUrl":"10.1016/j.jalgebra.2024.12.007","url":null,"abstract":"<div><div>One of the basic notions in the theory of varieties of algebras in characteristic zero developed by Kemer <span><span>[20]</span></span> was that of fundamental algebras. They are used as a main tool in the solution of Specht's Problem. The aim of this paper is to extend this concept to algebras with a <em>G</em>-graded structure, where <em>G</em> is a finite group, and to develop the corresponding theory. Furthermore, we explore the connection between fundamental <em>G</em>-graded algebras and generators of affine varieties of <em>G</em>-graded PI algebras which are minimal with respect to their <em>G</em>-graded exponent. In some important cases, we provide necessary and sufficient conditions so that subalgebras of these generators are fundamental. Finally, for abelian groups, we give a characterization in terms of the representation theory of the group <span><math><mi>G</mi><mo>≀</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 607-632"},"PeriodicalIF":0.8,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152658","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":"Norm one tori and Hasse norm principle, III: Degree 16 case","authors":"Akinari Hoshi ,&nbsp;Kazuki Kanai ,&nbsp;Aiichi Yamasaki","doi":"10.1016/j.jalgebra.2024.10.053","DOIUrl":"10.1016/j.jalgebra.2024.10.053","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;k&lt;/em&gt; be a field, &lt;em&gt;T&lt;/em&gt; be an algebraic &lt;em&gt;k&lt;/em&gt;-torus, &lt;em&gt;X&lt;/em&gt; be a smooth &lt;em&gt;k&lt;/em&gt;-compactification of &lt;em&gt;T&lt;/em&gt; and &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;Pic&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; be the Picard group of &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;×&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; is a fixed separable closure of &lt;em&gt;k&lt;/em&gt;. Hoshi, Kanai and Yamasaki &lt;span&gt;&lt;span&gt;[30]&lt;/span&gt;&lt;/span&gt;, &lt;span&gt;&lt;span&gt;[31]&lt;/span&gt;&lt;/span&gt; determined &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;Pic&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for norm one tori &lt;span&gt;&lt;math&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and gave a necessary and sufficient condition for the Hasse norm principle for extensions &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; of number fields with &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;15&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we treat the case where &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. Among 1954 transitive subgroups &lt;span&gt;&lt;math&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;1954&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; up to conjugacy, we determine 1101 (resp. 774, 31, 37, 1, 1, 9) cases with &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;Pic&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⊕&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⊕&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⊕&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⊕&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;) where &lt;em&gt;G&lt;/em&gt; is the Galois group of the Galois closure &lt;span&gt;&lt;","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 794-820"},"PeriodicalIF":0.8,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152668","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":"Game theory of undirected graphical models","authors":"Irem Portakal ,&nbsp;Javier Sendra–Arranz","doi":"10.1016/j.jalgebra.2024.12.004","DOIUrl":"10.1016/j.jalgebra.2024.12.004","url":null,"abstract":"<div><div>An <em>n</em>-player game <em>X</em> in normal form can be modeled via undirected discrete graphical models where the discrete random variables represent the players and their state spaces are the set of pure strategies. There exists an edge between the vertices of the graphical model whenever there is a dependency between the associated players. We study the Spohn conditional independence (CI) variety <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>C</mi></mrow></msub></math></span>, which is the intersection of the independence model <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> with the Spohn variety of the game <em>X</em>. We prove a conjecture by the first author and Sturmfels that <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>X</mi><mo>,</mo><mi>C</mi></mrow></msub></math></span> is of codimension <em>n</em> in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>C</mi></mrow></msub></math></span> for a generic game <em>X</em> with binary choices. We show that the set of totally mixed CI equilibria i.e. the restriction of the Spohn CI variety to the open probability simplex is a smooth semialgebraic manifold for a generic game <em>X</em> with binary choices. If the undirected graph is a disjoint union of cliques, we analyze certain algebro-geometric features of Spohn CI varieties and prove affine universality theorems.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 574-606"},"PeriodicalIF":0.8,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152667","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":"On generalized covering and avoidance properties of finite groups and saturated fusion systems","authors":"Shengmin Zhang,&nbsp;Zhencai Shen","doi":"10.1016/j.jalgebra.2024.11.030","DOIUrl":"10.1016/j.jalgebra.2024.11.030","url":null,"abstract":"<div><div>A subgroup <em>A</em> of a finite group <em>G</em> is said to be a <em>CAP</em>-subgroup of <em>G</em>, if for any chief factor <span><math><mi>H</mi><mo>/</mo><mi>K</mi></math></span> of <em>G</em>, either <span><math><mi>A</mi><mi>H</mi><mo>=</mo><mi>A</mi><mi>K</mi></math></span> or <span><math><mi>A</mi><mo>∩</mo><mi>H</mi><mo>=</mo><mi>A</mi><mo>∩</mo><mi>K</mi></math></span>. Let <em>p</em> be a prime, <em>S</em> be a <em>p</em>-group and <span><math><mi>F</mi></math></span> be a saturated fusion system over <em>S</em>. Then <span><math><mi>F</mi></math></span> is said to be supersolvable, if there exists a series of <em>S</em>, namely <span><math><mn>1</mn><mo>=</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>⩽</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⩽</mo><mo>⋯</mo><mo>⩽</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>S</mi></math></span>, such that <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is cyclic, and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is strongly <span><math><mi>F</mi></math></span>-closed for any <span><math><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi></math></span>. In this paper, we first introduce the concept of strong <em>p</em>-<em>CAP</em>-subgroups, and investigate the structure of finite groups under the assumptions that some subgroups of <em>G</em> are partial <em>CAP</em>-subgroups or strong (<em>p</em>)-<em>CAP</em>-subgroups of <em>G</em>, and obtain some criteria for a group <em>G</em> to be <em>p</em>-supersolvable. After that, we investigate the characterizations for supersolvability of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> under the assumptions that some subgroups of <em>G</em> are partial <em>CAP</em>-subgroups or strong (<em>p</em>)-<em>CAP</em>-subgroups of <em>G</em>, and obtain some criteria for a fusion system <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> to be supersolvable. The above results improve some known results and develop some new results about <em>CAP</em>-subgroups from fusion systems.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 149-168"},"PeriodicalIF":0.8,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152717","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":"Real Nullstellensatz for 2-step nilpotent Lie algebras","authors":"Philipp Schmitt ,&nbsp;Matthias Schötz","doi":"10.1016/j.jalgebra.2024.12.001","DOIUrl":"10.1016/j.jalgebra.2024.12.001","url":null,"abstract":"<div><div>We prove a noncommutative real Nullstellensatz for 2-step nilpotent Lie algebras that extends the classical, commutative real Nullstellensatz as follows: Instead of the real polynomial algebra <span><math><mi>R</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>d</mi></mrow></msub><mo>]</mo></math></span> we consider the universal enveloping ^⁎-algebra of a 2-step nilpotent real Lie algebra (i.e. the universal enveloping algebra of its complexification with the canonical ^⁎-involution). Evaluation at points of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> is then generalized to evaluation through integrable ^⁎-representations, which in this case are equivalent to filtered ^⁎-algebra morphisms from the universal enveloping ^⁎-algebra to a Weyl algebra. Our Nullstellensatz characterizes the common kernels of a set of such ^⁎-algebra morphisms as the real ideals of the universal enveloping ^⁎-algebra.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 850-877"},"PeriodicalIF":0.8,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152725","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":"Geometric model for weighted projective lines of type (p,q)","authors":"Jianmin Chen ,&nbsp;Shiquan Ruan ,&nbsp;Hongxia Zhang","doi":"10.1016/j.jalgebra.2024.11.023","DOIUrl":"10.1016/j.jalgebra.2024.11.023","url":null,"abstract":"<div><div>We give a geometric model for the category of coherent sheaves over the weighted projective line of type <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> in terms of an annulus with marked points on its boundary. We establish a bijection between indecomposable sheaves over the weighted projective line and certain homotopy classes of oriented curves in the annulus, and prove that the dimension of extension group between indecomposable sheaves equals to the positive intersection number between the corresponding curves.</div><div>By using the geometric model, we provide a combinatorial description for the titling graph of tilting bundles, which is a line or composed of quadrilaterals. Moreover, we obtain that the automorphism group of the coherent sheaf category is isomorphic to the mapping class group of the marked annulus, and show the compatibility of their actions on the tilting graph of coherent sheaves and on the triangulation of the geometric model respectively. A geometric description of the perpendicular category with respect to an exceptional sheaf is presented at the end of the paper.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 530-573"},"PeriodicalIF":0.8,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152659","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":"Limit groups and automorphisms of κ-existentially closed groups","authors":"Burak Kaya ,&nbsp;Mahmut Kuzucuoğlu ,&nbsp;Patrizia Longobardi ,&nbsp;Mercede Maj","doi":"10.1016/j.jalgebra.2024.12.003","DOIUrl":"10.1016/j.jalgebra.2024.12.003","url":null,"abstract":"<div><div>The structure of automorphism groups of <em>κ</em>-existentially closed groups has been studied by Kaya-Kuzucuoğlu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and <span><math><mo>|</mo><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>κ</mi></mrow></msup></math></span> whenever <em>κ</em> is an inaccessible cardinal and <em>G</em> is the unique <em>κ</em>-existentially closed group of cardinality <em>κ</em>. The cardinality of the automorphism group of a <em>κ</em>-existentially closed group of cardinality <span><math><mi>λ</mi><mo>&gt;</mo><mi>κ</mi></math></span> is asked in Kourovka Notebook Question 20.40. Here we answer positively the promised case <span><math><mi>κ</mi><mo>=</mo><mi>λ</mi></math></span> namely: If <em>G</em> is a <em>κ</em>-existentially closed group of cardinality <em>κ</em>, then <span><math><mo>|</mo><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>κ</mi></mrow></msup></math></span>. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal <em>κ</em>, there exist universal groups of cardinality <em>κ</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 840-849"},"PeriodicalIF":0.8,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152727","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 finite permutation groups of rank three","authors":"Hong Yi Huang ,&nbsp;Cai Heng Li ,&nbsp;Yan Zhou Zhu","doi":"10.1016/j.jalgebra.2024.11.028","DOIUrl":"10.1016/j.jalgebra.2024.11.028","url":null,"abstract":"<div><div>The classification of the finite primitive permutation groups of rank 3 was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general transitive setting, a classical result of Higman shows that every finite imprimitive rank 3 permutation group <em>G</em> has a unique non-trivial block system <span><math><mi>B</mi></math></span> and this provides a natural way to partition the analysis of these groups. Indeed, the induced permutation group <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>B</mi></mrow></msup></math></span> is 2-transitive and one can also show that the action induced on each block in <span><math><mi>B</mi></math></span> is also 2-transitive (and so both induced groups are either affine or almost simple). In this paper, we make progress towards a classification of the rank 3 imprimitive groups by studying the case where the induced action of <em>G</em> on a block in <span><math><mi>B</mi></math></span> is of affine type. Our main theorem divides these rank 3 groups into four classes, which are defined in terms of the kernel of the action of <em>G</em> on <span><math><mi>B</mi></math></span>. In particular, we completely determine the rank 3 semiprimitive groups for which <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>B</mi></mrow></msup></math></span> is almost simple, extending recent work of Baykalov, Devillers and Praeger. We also prove that if <em>G</em> is rank 3 semiprimitive and <span><math><msup><mrow><mi>G</mi></mrow><mrow><mi>B</mi></mrow></msup></math></span> is affine, then <em>G</em> has a regular normal subgroup which is a special <em>p</em>-group for some prime <em>p</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 703-732"},"PeriodicalIF":0.8,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152661","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":"Verbal width in arithmetic Chevalley groups","authors":"Pavel Gvozdevsky","doi":"10.1016/j.jalgebra.2024.12.002","DOIUrl":"10.1016/j.jalgebra.2024.12.002","url":null,"abstract":"<div><div>We prove that the width of any word in a simply connected Chevalley group of rank at least 2 over the ring that is a localisation of the ring of integers in a number field is bounded by a constant that depends only on the root system and on the degree of the number field.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"666 ","pages":"Pages 82-93"},"PeriodicalIF":0.8,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143152650","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}