Pub Date : 2026-01-08DOI: 10.1016/j.jalgebra.2025.12.021
Gerhard Hiss , Rafał Lutowski
We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a closed flat manifold to be an -manifold.
{"title":"The eigenvalue one property of finite groups, I","authors":"Gerhard Hiss , Rafał Lutowski","doi":"10.1016/j.jalgebra.2025.12.021","DOIUrl":"10.1016/j.jalgebra.2025.12.021","url":null,"abstract":"<div><div>We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a closed flat manifold to be an <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-manifold.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 592-626"},"PeriodicalIF":0.8,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978453","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}
Pub Date : 2026-01-07DOI: 10.1016/j.jalgebra.2026.01.001
Hongjia Chen , Han Dai , Xingpeng Liu , Qi Zhao
We establish a necessary and sufficient condition for the tensor product to be cyclic (i.e., generated by the tensor product of the highest weight vectors), where denotes the evaluation module of obtained by the Verma module of via the evaluation homomorphism. When W is cyclic, its generators and relations can be described. Moreover, by extending it, we define a class of highest weight modules, all of which belong to the category . Additionally, we determine the simplicity of these modules and offer a cyclicity criterion for their tensor products.
{"title":"Tensor products of infinite-dimensional evaluation modules over the Yangian Y(sl2)","authors":"Hongjia Chen , Han Dai , Xingpeng Liu , Qi Zhao","doi":"10.1016/j.jalgebra.2026.01.001","DOIUrl":"10.1016/j.jalgebra.2026.01.001","url":null,"abstract":"<div><div>We establish a necessary and sufficient condition for the tensor product <span><math><mi>W</mi><mo>=</mo><mi>M</mi><msub><mrow><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>⊗</mo><mo>⋯</mo><mo>⊗</mo><mi>M</mi><msub><mrow><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></msub></math></span> to be cyclic (i.e., generated by the tensor product of the highest weight vectors), where <span><math><mi>M</mi><msub><mrow><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math></span> denotes the evaluation module of <span><math><mi>Y</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> obtained by the Verma module <span><math><mi>M</mi><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> of <span><math><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> via the evaluation homomorphism. When <em>W</em> is cyclic, its generators and relations can be described. Moreover, by extending it, we define a class of highest weight modules, all of which belong to the category <span><math><mi>O</mi><mo>(</mo><mi>Y</mi><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo></math></span>. Additionally, we determine the simplicity of these modules and offer a cyclicity criterion for their tensor products.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 627-652"},"PeriodicalIF":0.8,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978454","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}
Pub Date : 2026-01-07DOI: 10.1016/j.jalgebra.2025.12.022
Alessandro Ardizzoni , Claudia Menini , Paolo Saracco
The Hopf envelope of a bialgebra is the free Hopf algebra generated by the given bialgebra. Its existence, as well as that of the cofree Hopf algebra, is a well-known fact in Hopf algebra theory, but their construction is not particularly handy or friendly. In this note, we offer a novel realisation of the Hopf envelope and of the cofree Hopf algebra of a finite-dimensional bialgebra as a particular quotient and sub-bialgebra, respectively, of the bialgebra itself. Our construction can also be extended to the infinite-dimensional case, provided that the bialgebra satisfies additional conditions, such as being right perfect as an algebra or admitting a n-antipode, the latter being a notion hereby introduced. Remarkably, the machinery we develop also allows us to give a new description of the Hopf envelope of a commutative bialgebra and of the cofree cocommutative Hopf algebra of a cocommutative bialgebra.
{"title":"On the Hopf envelope of finite-dimensional bialgebras","authors":"Alessandro Ardizzoni , Claudia Menini , Paolo Saracco","doi":"10.1016/j.jalgebra.2025.12.022","DOIUrl":"10.1016/j.jalgebra.2025.12.022","url":null,"abstract":"<div><div>The Hopf envelope of a bialgebra is the free Hopf algebra generated by the given bialgebra. Its existence, as well as that of the cofree Hopf algebra, is a well-known fact in Hopf algebra theory, but their construction is not particularly handy or friendly. In this note, we offer a novel realisation of the Hopf envelope and of the cofree Hopf algebra of a finite-dimensional bialgebra as a particular quotient and sub-bialgebra, respectively, of the bialgebra itself. Our construction can also be extended to the infinite-dimensional case, provided that the bialgebra satisfies additional conditions, such as being right perfect as an algebra or admitting a <em>n</em>-antipode, the latter being a notion hereby introduced. Remarkably, the machinery we develop also allows us to give a new description of the Hopf envelope of a commutative bialgebra and of the cofree cocommutative Hopf algebra of a cocommutative bialgebra.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 653-705"},"PeriodicalIF":0.8,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978455","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}
Pub Date : 2026-01-06DOI: 10.1016/j.jalgebra.2025.12.020
Cristina Acciarri , Robert M. Guralnick , Evgeny Khukhro , Pavel Shumyatsky
For a group A acting by automorphisms on a group G, let denote the set of commutators , where and , so that is the subgroup generated by . We prove that if A is a π-group of automorphisms of a π-soluble finite group G such that any subset of generates a subgroup that can be generated by r elements, then the rank of is bounded in terms of r. Examples show that such a result does not hold without the assumption of π-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow p-subgroups of p-soluble groups.
{"title":"Local–global generation property of commutators in finite π-soluble groups","authors":"Cristina Acciarri , Robert M. Guralnick , Evgeny Khukhro , Pavel Shumyatsky","doi":"10.1016/j.jalgebra.2025.12.020","DOIUrl":"10.1016/j.jalgebra.2025.12.020","url":null,"abstract":"<div><div>For a group <em>A</em> acting by automorphisms on a group <em>G</em>, let <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> denote the set of commutators <span><math><mo>[</mo><mi>g</mi><mo>,</mo><mi>a</mi><mo>]</mo><mo>=</mo><msup><mrow><mi>g</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>g</mi></mrow><mrow><mi>a</mi></mrow></msup></math></span>, where <span><math><mi>g</mi><mo>∈</mo><mi>G</mi></math></span> and <span><math><mi>a</mi><mo>∈</mo><mi>A</mi></math></span>, so that <span><math><mo>[</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo>]</mo></math></span> is the subgroup generated by <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. We prove that if <em>A</em> is a <em>π</em>-group of automorphisms of a <em>π</em>-soluble finite group <em>G</em> such that any subset of <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>)</mo></math></span> generates a subgroup that can be generated by <em>r</em> elements, then the rank of <span><math><mo>[</mo><mi>G</mi><mo>,</mo><mi>A</mi><mo>]</mo></math></span> is bounded in terms of <em>r</em>. Examples show that such a result does not hold without the assumption of <em>π</em>-solubility. Earlier we obtained this type of results for groups of coprime automorphisms and for Sylow <em>p</em>-subgroups of <em>p</em>-soluble groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 519-549"},"PeriodicalIF":0.8,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939128","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}
Pub Date : 2026-01-05DOI: 10.1016/j.jalgebra.2025.11.034
Cristina Bertone , Francesca Cioffi , Matthias Orth , Werner M. Seiler
Using techniques from the theory of marked bases, we develop new effective methods for detecting and constructing Cohen-Macaulay, Gorenstein and complete intersection homogeneous polynomial ideals over a field . Due to the functorial properties of marked bases, an elementary proof follows for the openness of the arithmetically Cohen-Macaulay, arithmetically Gorenstein and strict complete intersection -rational points loci in a Hilbert scheme with a non-constant Hilbert polynomial.
{"title":"Cohen-Macaulay, Gorenstein and complete intersection conditions by marked bases","authors":"Cristina Bertone , Francesca Cioffi , Matthias Orth , Werner M. Seiler","doi":"10.1016/j.jalgebra.2025.11.034","DOIUrl":"10.1016/j.jalgebra.2025.11.034","url":null,"abstract":"<div><div>Using techniques from the theory of marked bases, we develop new effective methods for detecting and constructing Cohen-Macaulay, Gorenstein and complete intersection homogeneous polynomial ideals over a field <span><math><mi>K</mi></math></span>. Due to the functorial properties of marked bases, an elementary proof follows for the openness of the arithmetically Cohen-Macaulay, arithmetically Gorenstein and strict complete intersection <span><math><mi>K</mi></math></span>-rational points loci in a Hilbert scheme with a non-constant Hilbert polynomial.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 550-581"},"PeriodicalIF":0.8,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939131","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}
Pub Date : 2026-01-05DOI: 10.1016/j.jalgebra.2025.11.033
Marston D.E. Conder
A graph Γ is called locally s-arc-transitive if the stabiliser in of a vertex v is transitive on the set of all r-arcs in Γ with initial vertex v, for every . A theorem by Stellmacher and van Bon (2015) states that if Γ is a connected finite locally s-arc-transitive graph in which every vertex has valency at least 3, then . This theorem complements Tutte's famous theorem for s-arc-transitive finite graphs of valency 3 (showing that ) and its extension by Weiss to s-arc-transitive finite graphs of higher valency (for which ). In the current paper, the author gives a positive answer to a question by Michael Giudici, by showing that locally 9-arc-transitive graphs are not as rare as might have been expected. Specifically, it is proved that for all but finitely many n, there exists a finite graph upon which the alternating group acts as a locally 9-arc-transitive group of automorphisms. The proof involves the construction and combination of finite quotients of an amalgamated product where A and B are vertex-stabilisers of orders 12288 and 20480 intersecting in an edge-stabiliser of order 4096.
{"title":"The infinitude of locally 9-arc-transitive graphs","authors":"Marston D.E. Conder","doi":"10.1016/j.jalgebra.2025.11.033","DOIUrl":"10.1016/j.jalgebra.2025.11.033","url":null,"abstract":"<div><div>A graph Γ is called <em>locally s-arc-transitive</em> if the stabiliser in <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> of a vertex <em>v</em> is transitive on the set of all <em>r</em>-arcs in Γ with initial vertex <em>v</em>, for every <span><math><mi>r</mi><mo>≤</mo><mi>s</mi></math></span>. A theorem by Stellmacher and van Bon (2015) states that if Γ is a connected finite locally <em>s</em>-arc-transitive graph in which every vertex has valency at least 3, then <span><math><mi>s</mi><mo>≤</mo><mn>9</mn></math></span>. This theorem complements Tutte's famous theorem for <em>s</em>-arc-transitive finite graphs of valency 3 (showing that <span><math><mi>s</mi><mo>≤</mo><mn>5</mn></math></span>) and its extension by Weiss to <em>s</em>-arc-transitive finite graphs of higher valency (for which <span><math><mi>s</mi><mo>≤</mo><mn>7</mn></math></span>). In the current paper, the author gives a positive answer to a question by Michael Giudici, by showing that locally 9-arc-transitive graphs are not as rare as might have been expected. Specifically, it is proved that for all but finitely many <em>n</em>, there exists a finite graph upon which the alternating group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> acts as a locally 9-arc-transitive group of automorphisms. The proof involves the construction and combination of finite quotients of an amalgamated product <span><math><mi>A</mi><msub><mrow><mo>⁎</mo></mrow><mrow><mi>C</mi></mrow></msub><mi>B</mi></math></span> where <em>A</em> and <em>B</em> are vertex-stabilisers of orders 12288 and 20480 intersecting in an edge-stabiliser of order 4096.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 582-591"},"PeriodicalIF":0.8,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978452","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}
Pub Date : 2026-01-05DOI: 10.1016/j.jalgebra.2025.11.035
Josh Hall, Aparna Upadhyay
The action of the symmetric group on set partitions of a set of size 2m into m sets each of size 2 generates the Foulkes module . In this paper, we study both the ordinary and the modular structure of the twisted Foulkes module of the symmetric group , where , defined over a field. Over characteristic zero, we construct a polynomial whose coefficients are the ordinary characters of the various twisted Foulkes modules of as m and k vary. Further, when the underlying field has odd characteristic, we study the asymptotics of the non-projective part of the tensor powers of these modules by computing the gamma invariant as defined by Dave Benson and Peter Symonds.
{"title":"Ordinary and modular properties of twisted Foulkes modules","authors":"Josh Hall, Aparna Upadhyay","doi":"10.1016/j.jalgebra.2025.11.035","DOIUrl":"10.1016/j.jalgebra.2025.11.035","url":null,"abstract":"<div><div>The action of the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msub></math></span> on set partitions of a set of size 2<em>m</em> into <em>m</em> sets each of size 2 generates the Foulkes module <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow></msup></math></span>. In this paper, we study both the ordinary and the modular structure of the twisted Foulkes module <span><math><msup><mrow><mi>H</mi></mrow><mrow><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>;</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> of the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, where <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mi>m</mi><mo>+</mo><mi>k</mi></math></span>, defined over a field. Over characteristic zero, we construct a polynomial whose coefficients are the ordinary characters of the various twisted Foulkes modules of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> as <em>m</em> and <em>k</em> vary. Further, when the underlying field has odd characteristic, we study the asymptotics of the non-projective part of the tensor powers of these modules by computing the gamma invariant as defined by Dave Benson and Peter Symonds.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 484-518"},"PeriodicalIF":0.8,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939129","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}
Pub Date : 2026-01-05DOI: 10.1016/j.jalgebra.2025.12.017
Yunsong Gan , Weijun Liu , Binzhou Xia
A regular bipartite graph Γ is called semisymmetric if its full automorphism group acts transitively on the edge set but not on the vertex set. For a subgroup G of that stabilizes the biparts of Γ, we say that Γ is G-biprimitive if G acts primitively on each part. In this paper, we first provide a method to construct infinite families of biprimitive semisymmetric graphs admitting almost simple groups. With the aid of this result, a classification of G-biprimitive semisymmetric graphs is obtained for or . In pursuit of this goal, we determine all pairs of almost simple groups of the same order and all pairs of maximal subgroups of or with the same order.
{"title":"On biprimitive semisymmetric graphs","authors":"Yunsong Gan , Weijun Liu , Binzhou Xia","doi":"10.1016/j.jalgebra.2025.12.017","DOIUrl":"10.1016/j.jalgebra.2025.12.017","url":null,"abstract":"<div><div>A regular bipartite graph Γ is called semisymmetric if its full automorphism group <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> acts transitively on the edge set but not on the vertex set. For a subgroup <em>G</em> of <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></math></span> that stabilizes the biparts of Γ, we say that Γ is <em>G</em>-biprimitive if <em>G</em> acts primitively on each part. In this paper, we first provide a method to construct infinite families of biprimitive semisymmetric graphs admitting almost simple groups. With the aid of this result, a classification of <em>G</em>-biprimitive semisymmetric graphs is obtained for <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In pursuit of this goal, we determine all pairs of almost simple groups of the same order and all pairs of maximal subgroups of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with the same order.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 422-462"},"PeriodicalIF":0.8,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939126","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}
Pub Date : 2026-01-02DOI: 10.1016/j.jalgebra.2025.12.019
Oksana S. Yakimova
Let be an algebraic Lie algebra of index 1, i.e., a generic Q-orbit on has codimension 1. We show that the following conditions are equivalent: is contact; a generic Q-orbit on is not conical; there is a generic stabiliser for the coadjoint action of . In addition, if is contact, then the subalgebra generated by symmetric semi-invariants of is a polynomial ring. We study also affine seaweed Lie algebras of type A and find some contact as well as non-contact examples among them.
{"title":"Contact Lie algebras, generic stabilisers, and affine seaweeds","authors":"Oksana S. Yakimova","doi":"10.1016/j.jalgebra.2025.12.019","DOIUrl":"10.1016/j.jalgebra.2025.12.019","url":null,"abstract":"<div><div>Let <span><math><mi>q</mi><mo>=</mo><mrow><mi>Lie</mi><mspace></mspace></mrow><mi>Q</mi></math></span> be an algebraic Lie algebra of index 1, i.e., a generic <em>Q</em>-orbit on <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> has codimension 1. We show that the following conditions are equivalent: <span><math><mi>q</mi></math></span> is contact; a generic <em>Q</em>-orbit on <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is not conical; there is a generic stabiliser for the coadjoint action of <span><math><mi>q</mi></math></span>. In addition, if <span><math><mi>q</mi></math></span> is contact, then the subalgebra <span><math><mi>S</mi><msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mi>si</mi></mrow></msub><mo>⊂</mo><mi>S</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> generated by symmetric semi-invariants of <span><math><mi>q</mi></math></span> is a polynomial ring. We study also affine seaweed Lie algebras of type <span>A</span> and find some contact as well as non-contact examples among them.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 401-421"},"PeriodicalIF":0.8,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939130","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}
Pub Date : 2025-12-31DOI: 10.1016/j.jalgebra.2025.12.018
Li Liang , Yajun Ma
We prove that under some mild conditions, each faithful Frobenius functor preserves and reflects right -Gorenstein objects and further preserves the right -Gorenstein dimension of unbounded complexes, where is the left part of a right periodic cotorsion pair. Consequently, such functors preserve and reflect Gorenstein flat-cotorsion modules. As an application, we show that the Gorenstein flat-cotorsion property of module factorizations and -shaped diagrams has a “local-global” principle. Finally, we study the behavior of relative stable categories, singularity categories and Gorenstein defect categories under Frobenius functors.
{"title":"Frobenius functors and Gorenstein objects with applications to the flat-cotorsion theory","authors":"Li Liang , Yajun Ma","doi":"10.1016/j.jalgebra.2025.12.018","DOIUrl":"10.1016/j.jalgebra.2025.12.018","url":null,"abstract":"<div><div>We prove that under some mild conditions, each faithful Frobenius functor preserves and reflects right <span><math><mi>U</mi></math></span>-Gorenstein objects and further preserves the right <span><math><mi>U</mi></math></span>-Gorenstein dimension of unbounded complexes, where <span><math><mi>U</mi></math></span> is the left part of a right periodic cotorsion pair. Consequently, such functors preserve and reflect Gorenstein flat-cotorsion modules. As an application, we show that the Gorenstein flat-cotorsion property of module factorizations and <span><math><mi>Q</mi></math></span>-shaped diagrams has a “local-global” principle. Finally, we study the behavior of relative stable categories, singularity categories and Gorenstein defect categories under Frobenius functors.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 463-483"},"PeriodicalIF":0.8,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939127","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}
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