Journal of AlgebraPub Date : 2025-07-17DOI: 10.1016/j.jalgebra.2025.06.048
Erlend D. Børve , Jacob Fjeld Grevstad , Endre S. Rundsveen
{"title":"τ-tilting finiteness and g-tameness: Incidence algebras of posets and concealed algebras","authors":"Erlend D. Børve , Jacob Fjeld Grevstad , Endre S. Rundsveen","doi":"10.1016/j.jalgebra.2025.06.048","DOIUrl":"10.1016/j.jalgebra.2025.06.048","url":null,"abstract":"<div><div>We prove that any <em>τ</em>-tilting finite incidence algebra of a finite poset is representation-finite, and that any <strong>g</strong>-tame incidence algebra of a finite simply connected poset is tame. As the converse of these assertions are known to hold, we obtain characterizations of <em>τ</em>-tilting finite incidence algebras and <strong>g</strong>-tame simply connected incidence algebras. Both results are proved using the theory of concealed algebras. The former will be deduced from the fact that tame concealed algebras are <em>τ</em>-tilting infinite, and to prove the latter, we show that wild concealed algebras are not <strong>g</strong>-tame. We conjecture that any incidence algebra of a finite poset is wild if and only if it is not <strong>g</strong>-tame, and prove a result showing that there are relatively few possible counterexamples. In the appendix, we determine the representation type of a <em>τ</em>-tilting reduction of a concealed algebra of hyperbolic type.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 354-393"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144721588","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 : 2025-07-17DOI: 10.1016/j.jalgebra.2025.06.038
Yikun Qiao
{"title":"Geometric invariant theory for graded additive groups","authors":"Yikun Qiao","doi":"10.1016/j.jalgebra.2025.06.038","DOIUrl":"10.1016/j.jalgebra.2025.06.038","url":null,"abstract":"<div><div>We consider geometric invariant theory for <em>graded additive groups</em>, groups of the form <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>a</mi></mrow><mrow><mi>r</mi></mrow></msubsup><msub><mrow><mo>⋊</mo></mrow><mrow><mi>w</mi></mrow></msub><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> such that the <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>-action on <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>a</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span> is a scalar multiplication with weight <span><math><mi>w</mi><mo>∈</mo><msub><mrow><mi>N</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>. We provide an algorithm of equivariant birational modifications, such that we can apply the geometric invariant theory of Bérczi-Doran-Hawes-Kirwan. In particular, the geometric <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>a</mi></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>-quotient exists. This complements Bérczi-Doran-Hawes-Kirwan, in the special case of one grading weight.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"683 ","pages":"Pages 803-833"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686381","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 : 2025-07-17DOI: 10.1016/j.jalgebra.2025.06.039
Archita Gupta, Pooja Singla
{"title":"On Gelfand pairs and degenerate Gelfand-Graev modules of general linear groups of degree two over principal ideal local rings of finite length","authors":"Archita Gupta, Pooja Singla","doi":"10.1016/j.jalgebra.2025.06.039","DOIUrl":"10.1016/j.jalgebra.2025.06.039","url":null,"abstract":"<div><div>Let <em>R</em> be a principal ideal local ring of finite length with a finite residue field of odd characteristic. Denote by <span><math><mi>G</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> the general linear group of degree two over <em>R</em>, and by <span><math><mi>B</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> the Borel subgroup of <span><math><mi>G</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> consisting of upper triangular matrices. In this article, we prove that the pair <span><math><mo>(</mo><mi>G</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>,</mo><mi>B</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo></math></span> is a strong Gelfand pair. We also investigate the decomposition of the degenerate Gelfand-Graev (DGG) modules of <span><math><mi>G</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. It is known that the non-degenerate Gelfand Graev module (also called non-degenerate Whittaker model) of <span><math><mi>G</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is multiplicity-free. We characterize the DGG-modules where the multiplicities are independent of the cardinality of the residue field. We provide a complete decomposition of all DGG-modules of <span><math><mi>G</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> for <em>R</em> of length at most four.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 78-108"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686803","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 : 2025-07-17DOI: 10.1016/j.jalgebra.2025.07.012
Nao Komiyama , Takeshi Shinohara
{"title":"Shuffle product of desingularized multiple zeta functions at integer points","authors":"Nao Komiyama , Takeshi Shinohara","doi":"10.1016/j.jalgebra.2025.07.012","DOIUrl":"10.1016/j.jalgebra.2025.07.012","url":null,"abstract":"<div><div>In this paper, we investigate the “shuffle-type” formula for special values of desingularized multiple zeta functions at integer points. It is proved by giving an iterated integral/differential expression for the desingularized multiple zeta functions at integer points.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 394-440"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144738551","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 : 2025-07-17DOI: 10.1016/j.jalgebra.2025.06.044
Xing Gao , Li Guo , Zongjian Han , Yi Zhang
{"title":"Rota-Baxter operators, differential operators, pre- and Novikov structures on groups and Lie algebras","authors":"Xing Gao , Li Guo , Zongjian Han , Yi Zhang","doi":"10.1016/j.jalgebra.2025.06.044","DOIUrl":"10.1016/j.jalgebra.2025.06.044","url":null,"abstract":"<div><div>Rota-Baxter operators on various structures have found important applications in diverse areas, from renormalization of quantum field theory to Yang-Baxter equations. Relative Rota-Baxter operators on Lie algebras are closely related to pre-Lie algebras and post-Lie algebras. Some of their group counterparts have been introduced to study post-groups, skew left braces and set-theoretic solutions of Yang-Baxter equations, but searching suitable notions of relative Rota-Baxter operators on groups with weight zero and of pre-groups has been challenging and has been the focus of recent studies, by provisionally imposing an abelian condition.</div><div>Arising from the works of Balinsky-Novikov and Gelfand-Dorfman, Novikov algebras and their constructions from differential commutative algebras have led to broad applications. Finding their suitable counterparts for groups and Lie algebras has also attracted quite much recent interests.</div><div>This paper uses one-sided-inverse pairs of maps to give a perturbative approach to a general notion of relative Rota-Baxter operators and differential operators on a group and a Lie algebra with limit-weight. With the extra condition of limit-abelianess on the group or Lie algebra, we give an interpretation of relative Rota-Baxter and differential operators with weight zero. These operators motivate us to define pre-groups and Novikov groups respectively as the induced structures. The tangent maps of these operators on Lie groups are shown to give relative Rota-Baxter and differential operators with weight zero on Lie algebras. The tangent spaces of the pre-Lie and Novikov Lie groups are pre-Lie algebras and Novikov Lie algebras, fulfilling the expected property. Furthermore, limit-weighted relative Rota-Baxter operators on groups give rise to skew left braces and then set-theoretic solutions of the Yang-Baxter equation.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 109-148"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686804","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 : 2025-07-17DOI: 10.1016/j.jalgebra.2025.06.043
Xavier Mary
{"title":"On the greatest semilattice decomposition of subsemigroups of regular rings","authors":"Xavier Mary","doi":"10.1016/j.jalgebra.2025.06.043","DOIUrl":"10.1016/j.jalgebra.2025.06.043","url":null,"abstract":"<div><div>Combining arguments issued from semigroup theory, ring theory and lattice theory, we build up on a study of the idempotent-generated subsemigroup of regular separative rings by Hannah and O'Meara <span><span>[25]</span></span> to completely characterize the greatest semilattice decomposition of certain subsemigroups of regular rings. In particular, we prove that the greatest homomorphic image of a unit-regular ring is given by the additive semilattice of principal ideals of the ring. Many examples are given.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 37-63"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680480","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 : 2025-07-17DOI: 10.1016/j.jalgebra.2025.06.047
Yan-an Cai, Rencai Lü, Xinyue Wang
{"title":"Non-weight modules over the Lie (super)algebras related to the Virasoro algebra","authors":"Yan-an Cai, Rencai Lü, Xinyue Wang","doi":"10.1016/j.jalgebra.2025.06.047","DOIUrl":"10.1016/j.jalgebra.2025.06.047","url":null,"abstract":"<div><div>In this paper, we denote by <span><math><mi>L</mi></math></span> the infinite-dimensional Lie (super)algebra related to the Virasoro algebra introduced in <span><span>[4]</span></span>. We classify a class of non-weight modules over <span><math><mi>L</mi></math></span> and its universal central extension <span><math><mover><mrow><mi>L</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span> that are free of rank 1 when restricted to <span><math><mi>U</mi><mo>(</mo><mi>h</mi><mo>)</mo></math></span>, where <span><math><mi>h</mi></math></span> is a toral subalgebra of <span><math><mi>L</mi></math></span>. We also determine both the simplicity and isomorphism classes of these modules. Furthermore, we consider the tensor products of finitely many rank 1 free simple modules with an arbitrary simple restricted module over <span><math><mover><mrow><mi>L</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>. The simplicity and isomorphism classes are characterized. Finally, we apply our results to recover some known results and classify such non-weight modules over several new Lie (super)algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"683 ","pages":"Pages 834-857"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686383","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 : 2025-07-17DOI: 10.1016/j.jalgebra.2025.07.006
Uriya First , Ben Williams
{"title":"Counterexamples in involutions of Azumaya algebras","authors":"Uriya First , Ben Williams","doi":"10.1016/j.jalgebra.2025.07.006","DOIUrl":"10.1016/j.jalgebra.2025.07.006","url":null,"abstract":"<div><div>Suppose <em>A</em> is an Azumaya algebra over a ring <em>R</em> and <em>σ</em> is an involution of <em>A</em> extending an order-2 automorphism <span><math><mi>λ</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span>. We say <em>σ</em> is <em>extraordinary</em> if there does not exist a Brauer-trivial Azumaya algebra <span><math><msub><mrow><mi>End</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo></math></span> over <em>R</em> carrying an involution <em>τ</em> so that <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><msub><mrow><mi>End</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>(</mo><mi>P</mi><mo>)</mo><mo>,</mo><mi>τ</mi><mo>)</mo></math></span> become isomorphic over some faithfully flat extension of the fixed ring of <span><math><mi>λ</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></math></span>. We give, for the first time, an example of such an algebra and involution. We do this by finding suitable cohomological obstructions and showing they do not always vanish.</div><div>We also give an example of a commutative ring <em>R</em> with involution <em>λ</em> so that the scheme-theoretic fixed locus <em>Z</em> of <span><math><mi>λ</mi><mo>:</mo><mi>Spec</mi><mspace></mspace><mi>R</mi><mo>→</mo><mi>Spec</mi><mspace></mspace><mi>R</mi></math></span> is disconnected, but such that every Azumaya algebra over <em>R</em> with involution extending <em>λ</em> is either orthogonal at every point of <em>Z</em>, or symplectic at every point of <em>Z</em>. No examples of this kind were previously known.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 256-279"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144694951","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 : 2025-07-17DOI: 10.1016/j.jalgebra.2025.06.042
Per Bäck , Patrik Lundström , Johan Öinert , Johan Richter
{"title":"Ore extensions of abelian groups with operators","authors":"Per Bäck , Patrik Lundström , Johan Öinert , Johan Richter","doi":"10.1016/j.jalgebra.2025.06.042","DOIUrl":"10.1016/j.jalgebra.2025.06.042","url":null,"abstract":"<div><div>Given a set <em>A</em> and an abelian group <em>B</em> with operators in <em>A</em>, in the sense of Krull and Noether, we introduce the Ore group extension <span><math><mi>B</mi><mo>[</mo><mi>x</mi><mo>;</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>]</mo></math></span> as the additive group <span><math><mi>B</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, with <span><math><mi>A</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> as a set of operators. Here, the action of <span><math><mi>A</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> on <span><math><mi>B</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span> is defined by mimicking the multiplication used in the classical case where <em>A</em> and <em>B</em> are the same ring. We derive generalizations of Vandermonde's and Leibniz's identities for this construction, and they are then used to establish associativity criteria. Additionally, we prove a version of Hilbert's basis theorem for this structure, under the assumption that the action of <em>A</em> on <em>B</em> is what we call weakly <em>s</em>-unital. Finally, we apply these results to the case where <em>B</em> is a left module over a ring <em>A</em>, and specifically to the case where <em>A</em> and <em>B</em> coincide with a non-associative ring which is left distributive but not necessarily right distributive.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 176-194"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144902157","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 : 2025-07-17DOI: 10.1016/j.jalgebra.2025.07.008
Mindy Y. Huerta
{"title":"Reduction for (n + 2)-rigid subcategories in extriangulated categories","authors":"Mindy Y. Huerta","doi":"10.1016/j.jalgebra.2025.07.008","DOIUrl":"10.1016/j.jalgebra.2025.07.008","url":null,"abstract":"<div><div>In this work we study how to extend the concept of <em>“reduction,”</em> given for rigid and functorially finite subcategories in an extriangulated category <span><math><mi>C</mi></math></span>, to <span><math><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span>-rigid ones. We define the reduction of such subcategories as the intersection of orthogonal complements when certain orthogonal condition is satisfied and we prove that this reduction depends mainly on the subcategory itself beyond the type of extriangulated category for which belongs to. Specifically, we show that some results proven for Frobenius extriangulated categories can be carried to extriangulated categories in general. We also study some properties among we can mention: weakly idempotent completeness, existence of enough <span><math><mi>E</mi></math></span>-projectives and <span><math><mi>E</mi></math></span>-injectives, and conditions to be Frobenius. That generalization covers the usual case and, for the general case (for <span><math><mi>n</mi><mo>≥</mo><mn>0</mn></math></span>), we provide several examples. Finally, we extend a well-known result given for the reduction in stably 2-Calabi-Yau Frobenius extriangulated categories related with 2-cluster tilting subcategories.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 149-175"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686805","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}