Journal of Algebra最新文献

{"title":"Residual intersections and Schubert varieties","authors":"Sara Angela Filippini ,&nbsp;Xianglong Ni ,&nbsp;Jacinta Torres ,&nbsp;Jerzy Weyman","doi":"10.1016/j.jalgebra.2025.07.034","DOIUrl":"10.1016/j.jalgebra.2025.07.034","url":null,"abstract":"<div><div>Inspired by the work of Ulrich <span><span>[26]</span></span> and Huneke–Ulrich <span><span>[25]</span></span>, we describe a pattern to show that the ideals of certain opposite embedded Schubert varieties (defined by this pattern) arise by taking residual intersections of two (geometrically linked) opposite Schubert varieties which we call <em>Ulrich pair</em>. This pattern is uniform for the ADE types. Some of the free resolutions of the Schubert varieties in question are important for the structure of finite free resolutions. Our proof is representation theoretical and uniform for our pattern, however it is possible to derive our results using case-by-case analysis and the aid of a computer.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 62-85"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831468","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":"Cleft extensions of rings and singularity categories","authors":"Panagiotis Kostas","doi":"10.1016/j.jalgebra.2025.07.044","DOIUrl":"10.1016/j.jalgebra.2025.07.044","url":null,"abstract":"<div><div>This paper provides a systematic treatment of Gorenstein homological aspects for cleft extensions of rings. In particular, we investigate Goresnteinness, Gorenstein projective modules and singularity categories in the context of cleft extensions of rings. This setting includes triangular matrix rings, trivial extension rings and tensor rings, among others. Under certain conditions, we prove singular equivalences between the algebras in a cleft extension, unifying an abundance of known results. Moreover, we compare the big singularity categories of cleft extensions of rings in the sense of Krause.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 160-224"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831471","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":"Edge-transitive cubic graphs: analysis, cataloguing and enumeration","authors":"Marston Conder ,&nbsp;Primož Potočnik","doi":"10.1016/j.jalgebra.2025.07.035","DOIUrl":"10.1016/j.jalgebra.2025.07.035","url":null,"abstract":"<div><div>This paper deals with finite cubic (3-regular) graphs whose automorphism group acts transitively on the edges of the graph. Such graphs split into two broad classes, namely arc-transitive and semisymmetric cubic graphs, and then these divide respectively into 7 types (according to a classification by Djoković and Miller (1980) <span><span>[17]</span></span>) and 15 types (according to a classification by Goldschmidt (1980) <span><span>[23]</span></span>), in terms of certain group amalgams. Such graphs of small order were previously known up to orders 2048 and 768, respectively, and we have extended each of the two lists of all such graphs up to order 10000. Before describing how we did that, we carry out an analysis of the 22 amalgams, to show which of the finitely-presented groups associated with the 15 Goldschmidt amalgams can be faithfully embedded in one or more of the other 21 (as subgroups of finite index), complementing what is already known about such embeddings of the 7 Djoković-Miller groups in each other. We also give an example of a graph of each of the 22 types, and in most cases, describe the smallest such graph, and we then use regular coverings to prove that there are infinitely many examples of each type. Finally, we discuss the asymptotic enumeration of the graph orders, proving that if <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> is the number of cubic edge-transitive graphs of type <span><math><mi>C</mi></math></span> on at most <em>n</em> vertices, then there exist positive real constants <em>a</em> and <em>b</em> and a positive integer <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> such that <span><math><msup><mrow><mi>n</mi></mrow><mrow><mi>a</mi><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>≤</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≤</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>b</mi><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> for all <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 703-737"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865363","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":"Skew Laurent series ring over a Dedekind domain","authors":"Daniel Vitas","doi":"10.1016/j.jalgebra.2025.07.022","DOIUrl":"10.1016/j.jalgebra.2025.07.022","url":null,"abstract":"<div><div>We show that the formal skew Laurent series ring <span><math><mi>R</mi><mo>=</mo><mi>D</mi><mo>(</mo><mo>(</mo><mi>x</mi><mo>;</mo><mi>σ</mi><mo>)</mo><mo>)</mo></math></span> over a commutative Dedekind domain <em>D</em> with an automorphism <em>σ</em> is a noncommutative Dedekind domain. If <em>σ</em> acts trivially on the ideal class group of <em>D</em>, then <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, the Grothendieck group of <em>R</em>, is isomorphic to <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>D</mi><mo>)</mo></math></span>. Furthermore, we determine the Krull dimension, the global dimension, the general linear rank, and the stable rank of <em>R</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 313-336"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144831473","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":"N-fold module factorizations: Triangle equivalences and recollements","authors":"Yongliang Sun ,&nbsp;Yaohua Zhang","doi":"10.1016/j.jalgebra.2025.07.024","DOIUrl":"10.1016/j.jalgebra.2025.07.024","url":null,"abstract":"<div><div>As an extension of Eisenbud's matrix factorization into the non-commutative realm, X.W. Chen introduced the concept of module factorizations over an arbitrary ring. A theorem of Chen establishes triangle equivalences between the stable category of module factorizations with Gorenstein projective components and the stable category of Gorenstein projective modules over a quotient ring. In this paper, we introduce <em>n</em>-fold module factorizations, which generalize both the commutative <em>n</em>-fold matrix factorizations and the non-commutative module factorizations. To adapt triangle equivalences in module factorizations to <em>n</em>-fold module factorizations, we identify suitable subcategories of module factorizations and rings for the <em>n</em>-analogue. We further provide the <em>n</em>-analogue of Chen's theorem on triangle equivalences. Additionally, we study recollements involving the stable categories of higher-fold module factorizations, revealing intriguing recollements within the stable categories of Gorenstein modules of specific matrix subrings.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 1-25"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144826627","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":"Relative modality of elements in generalized Takiff Lie algebras","authors":"Hugo Mathevet","doi":"10.1016/j.jalgebra.2025.07.036","DOIUrl":"10.1016/j.jalgebra.2025.07.036","url":null,"abstract":"<div><div>Given a natural number <em>m</em> and a finite dimensional complex Lie algebra <span><math><mi>g</mi></math></span>, the <span><math><msup><mrow><mi>m</mi></mrow><mrow><mi>t</mi><mi>h</mi></mrow></msup></math></span> generalized Takiff Lie algebra of <span><math><mi>g</mi></math></span> is the Lie algebra <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>:</mo><mo>=</mo><mi>g</mi><mo>⊗</mo><mi>C</mi><mo>[</mo><mi>T</mi><mo>]</mo><mo>/</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. For <span><math><mi>n</mi><mo>≥</mo><mi>m</mi></math></span>, we define the <span><math><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-modality of an adjoint orbit <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> in <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> to be the minimum codimension of an adjoint orbit in the pullback of <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> in <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>.</div><div>In this paper, we study these invariants in generalized Takiff Lie algebras associated to a quadratic Lie algebra <span><math><mi>g</mi></math></span>. We show that these invariants satisfy some concavity and hereditary properties from which we deduce that <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mi>m</mi><mo>)</mo><mi>χ</mi><mo>(</mo><mi>g</mi><mo>)</mo></math></span> is a lower bound, where <span><math><mi>χ</mi><mo>(</mo><mi>g</mi><mo>)</mo></math></span> is the index of <span><math><mi>g</mi></math></span>. We prove that this lower bound is in fact an equality for a dense set of orbits, and that if <span><math><mi>g</mi></math></span> is reductive, it is always an equality when <span><math><mi>m</mi><mo>=</mo><mn>0</mn></math></span> (and also some special orbits). We conjecture that equality holds for all <em>m</em>, <em>n</em> when <span><math><mi>g</mi></math></span> is reductive.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 755-774"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865365","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":"Local Euler obstructions of reflexive projective varieties","authors":"Xiping Zhang","doi":"10.1016/j.jalgebra.2025.07.031","DOIUrl":"10.1016/j.jalgebra.2025.07.031","url":null,"abstract":"<div><div>The local Euler obstruction and the polar multiplicities are key ingredients in the study of the local topology of stratified spaces. Despite their importance, in general it's very difficult to compute them. In this paper we introduce the concept of reflexive projective varieties. These are stratified projective varieties with certain dimension constraints on their dual varieties. We prove that for such varieties, their local Euler obstructions and polar multiplicities are completely determined by their Chern-Schwartz-MacPherson classes. Explicit formulas are presented, based on which we also propose an algorithm to compute such geometric invariants using the input of the characteristic classes. These characteristic classes are easier to compute in practice and may be carried out by computer algebra. In the case of reflexive group orbits, the formula and the algorithm are further refined.</div><div>Our method is purely algebraic and works for arbitrary algebraically closed field of characteristic 0. As examples we compute the local Euler obstructions of ordinary determinantal varieties to illustrate our method.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 496-522"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865984","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":"The homological spectrum and nilpotence theorems for Lie superalgebra representations","authors":"Matthew H. Hamil ,&nbsp;Daniel K. Nakano","doi":"10.1016/j.jalgebra.2025.07.027","DOIUrl":"10.1016/j.jalgebra.2025.07.027","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In the study of cohomology of finite group schemes it is well known that nilpotence theorems play a key role in determining the spectrum of the cohomology ring. Balmer recently showed that there is a more general notion of a nilpotence theorem for tensor triangulated categories through the use of homological residue fields and the connection with the homological spectrum. The homological spectrum (like the theory of &lt;em&gt;π&lt;/em&gt;-points) can be viewed as a topological space that provides an important realization of the Balmer spectrum.&lt;/div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&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;mover&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&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;mover&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be a classical Lie superalgebra over &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, the authors consider the tensor triangular geometry for the stable category of finite-dimensional Lie superalgebra representations: &lt;span&gt;&lt;math&gt;&lt;mtext&gt;stab&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&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;mover&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. The localizing subcategories for the detecting subalgebra &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are classified which answers a question of Boe, Kujawa, and Nakano. As a consequence of these results, the authors prove a nilpotence theorem and determine the homological spectrum for the stable module category of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. The authors also verify Balmer's “Nerves-of-Steel” Conjecture for &lt;span&gt;&lt;math&gt;&lt;mi&gt;stab&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; is a detecting subalgebra.&lt;/div&gt;&lt;div&gt;Let &lt;em&gt;F&lt;/em&gt; (resp. &lt;em&gt;G&lt;/em&gt;) be the associated supergroup (scheme) for &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;). Under the condition that &lt;em&gt;F&lt;/em&gt; is a splitting subgroup for &lt;em&gt;G&lt;/em&gt;, the results for the detecting subalgebra can be used to prove a nilpotence theorem for &lt;span&gt;&lt;math&gt;&lt;mtext&gt;stab&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&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;mover&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, and to determine the homological spectrum in this case. Then using natural assumptions in terms of realization of supports, the authors provid","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"685 ","pages":"Pages 801-830"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144878113","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 the solvable Poisson algebras","authors":"Amir Fernández Ouaridi ,&nbsp;Bakhrom A. Omirov","doi":"10.1016/j.jalgebra.2025.07.032","DOIUrl":"10.1016/j.jalgebra.2025.07.032","url":null,"abstract":"<div><div>In this manuscript, we study nilpotent and solvable Poisson algebras of dimension <em>n</em>. In the first part, we establish classical results such as Engel's theorem and Lie's theorem for Poisson algebras, and we examine the role of idempotents in these algebras. We also address the construction of nilpotent and solvable Poisson algebras, exploring the existence of Poisson algebras associated with a fixed Lie algebra, the study of filiform Poisson algebras, and constructions involving the tensor product and generalized Jacobians. Furthermore, we show that, under mild restrictions, the solvability and nilpotency of a Poisson algebra are essentially determined by those of the Lie bracket. This motivates a deeper investigation into Poisson algebra structures on solvable Lie algebras. In the second part, we provide criteria for the non existence of Poisson algebra structures on solvable extensions of nilpotent Lie algebras by a torus. In particular, we prove that complete solvable Lie algebras do not admit a Poisson algebra structure, and other related results. Additionally, we present results and examples illustrating the diversity of Poisson algebras arising in solvable Lie algebras that are non-maximal solvable extensions of nilpotent Lie algebras and highlighting the difficulty in formulating a unified criterion for all solvable Lie algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 792-827"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144830075","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":"Classification of simple Harish-Chandra modules of the Cartan type Lie algebra S¯2","authors":"Jinxin Hu,&nbsp;Rencai Lü","doi":"10.1016/j.jalgebra.2025.07.038","DOIUrl":"10.1016/j.jalgebra.2025.07.038","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub></math></span> be the Lie algebra of vector fields on <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with constant divergence. In this paper, we classify all simple Harish-Chandra modules of <span><math><msub><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub></math></span> including bounded and non-bounded simple Harish-Chandra modules. In simple terms, any simple Harish-Chandra module of <span><math><msub><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub></math></span> is isomorphic to a tensor module <span><math><mi>F</mi><mo>(</mo><mi>P</mi><mo>,</mo><mi>M</mi><mo>)</mo></math></span> or its simple sub-quotient where <em>P</em> is a simple weight <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mo>:</mo><mo>=</mo><mi>C</mi><mo>[</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>,</mo><mfrac><mrow><mo>∂</mo></mrow><mrow><mo>∂</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>]</mo><mo>)</mo></math></span> module and <em>M</em> is a simple weight <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> module.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 828-851"},"PeriodicalIF":0.8,"publicationDate":"2025-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144830076","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}