Pub Date : 2025-12-31DOI: 10.1016/j.jalgebra.2025.12.016
Sagar Saha, K.V. Krishna
In this work, we provide the first example of an infinite family of branch groups in the class of non-contracting self-similar groups. We show that these groups are super strongly fractal, not regular branch, and of exponential growth. Further, we prove that these groups do not have the congruence subgroup property by explicitly calculating the structure of their rigid kernels. This class of groups is also the first example of branch groups with non-torsion rigid kernels. As a consequence of these results, we also determine the Hausdorff dimension of these groups.
{"title":"A class of non-contracting branch groups with non-torsion rigid kernels","authors":"Sagar Saha, K.V. Krishna","doi":"10.1016/j.jalgebra.2025.12.016","DOIUrl":"10.1016/j.jalgebra.2025.12.016","url":null,"abstract":"<div><div>In this work, we provide the first example of an infinite family of branch groups in the class of non-contracting self-similar groups. We show that these groups are super strongly fractal, not regular branch, and of exponential growth. Further, we prove that these groups do not have the congruence subgroup property by explicitly calculating the structure of their rigid kernels. This class of groups is also the first example of branch groups with non-torsion rigid kernels. As a consequence of these results, we also determine the Hausdorff dimension of these groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 379-400"},"PeriodicalIF":0.8,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145939132","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 : 2025-12-16DOI: 10.1016/j.jalgebra.2025.12.010
Candace Bethea, Thomas Brazelton
Recall that a non-singular planar quartic is a canonically embedded non-hyperelliptic curve of genus three. We say such a curve is symmetric if it admits non-trivial automorphisms. The classification of (necessarily finite) groups appearing as automorphism groups of non-singular curves of genus three dates back to the last decade of the 19th century. As these groups act on the quartic via projective linear transformations, they induce symmetries on the 28 bitangents. Given such an automorphism group , we leverage tools from equivariant homotopy theory to prove that the G-orbits of the bitangents are independent of the choice of C, and we compute them for all twelve types of smooth symmetric planar quartic curves. We further observe that techniques deriving from equivariant homotopy theory directly reveal patterns which are not obvious from a classical moduli perspective.
{"title":"Bitangents to symmetric quartics","authors":"Candace Bethea, Thomas Brazelton","doi":"10.1016/j.jalgebra.2025.12.010","DOIUrl":"10.1016/j.jalgebra.2025.12.010","url":null,"abstract":"<div><div>Recall that a non-singular planar quartic is a canonically embedded non-hyperelliptic curve of genus three. We say such a curve is <em>symmetric</em> if it admits non-trivial automorphisms. The classification of (necessarily finite) groups appearing as automorphism groups of non-singular curves of genus three dates back to the last decade of the 19th century. As these groups act on the quartic via projective linear transformations, they induce symmetries on the 28 bitangents. Given such an automorphism group <span><math><mi>G</mi><mo>=</mo><mtext>Aut</mtext><mo>(</mo><mi>C</mi><mo>)</mo></math></span>, we leverage tools from equivariant homotopy theory to prove that the <em>G</em>-orbits of the bitangents are independent of the choice of <em>C</em>, and we compute them for all twelve types of smooth symmetric planar quartic curves. We further observe that techniques deriving from equivariant homotopy theory directly reveal patterns which are not obvious from a classical moduli perspective.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"694 ","pages":"Pages 397-426"},"PeriodicalIF":0.8,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146171943","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 : 2025-12-16DOI: 10.1016/j.jalgebra.2025.12.011
Nengqun Li , Yuming Liu
In 2017, Green and Schroll introduced a generalization of Brauer graph algebras which they call Brauer configuration algebras. In the present paper, we further generalize Brauer configuration algebras to fractional Brauer configuration algebras by generalizing Brauer configurations to fractional Brauer configurations. The fractional Brauer configuration algebras are locally bounded but neither finite-dimensional nor symmetric in general. We show that if the fractional Brauer configuration is of type S (resp. of type MS), then the corresponding fractional Brauer configuration algebra is a locally bounded Frobenius algebra (resp. a locally bounded special multiserial Frobenius algebra). Moreover, we show that over an algebraically closed field, the class of finite-dimensional indecomposable representation-finite fractional Brauer configuration algebras in type S coincides with the class of basic indecomposable finite-dimensional standard representation-finite self-injective algebras.
{"title":"Fractional Brauer configuration algebras I: Definitions and examples","authors":"Nengqun Li , Yuming Liu","doi":"10.1016/j.jalgebra.2025.12.011","DOIUrl":"10.1016/j.jalgebra.2025.12.011","url":null,"abstract":"<div><div>In 2017, Green and Schroll introduced a generalization of Brauer graph algebras which they call Brauer configuration algebras. In the present paper, we further generalize Brauer configuration algebras to fractional Brauer configuration algebras by generalizing Brauer configurations to fractional Brauer configurations. The fractional Brauer configuration algebras are locally bounded but neither finite-dimensional nor symmetric in general. We show that if the fractional Brauer configuration is of type S (resp. of type MS), then the corresponding fractional Brauer configuration algebra is a locally bounded Frobenius algebra (resp. a locally bounded special multiserial Frobenius algebra). Moreover, we show that over an algebraically closed field, the class of finite-dimensional indecomposable representation-finite fractional Brauer configuration algebras in type S coincides with the class of basic indecomposable finite-dimensional standard representation-finite self-injective algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 336-378"},"PeriodicalIF":0.8,"publicationDate":"2025-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145837887","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 : 2025-12-12DOI: 10.1016/j.jalgebra.2025.12.013
W. Liu, G.-S. Zhou
Restricted Lie algebras of dimension up to 3 over algebraically closed fields of positive characteristic were classified by Wang and his collaborators in [25], [19]. In this paper, we obtain a classification of restricted Lie algebras of dimension 4 over such fields.
{"title":"Classification of restricted Lie algebras of dimension 4","authors":"W. Liu, G.-S. Zhou","doi":"10.1016/j.jalgebra.2025.12.013","DOIUrl":"10.1016/j.jalgebra.2025.12.013","url":null,"abstract":"<div><div>Restricted Lie algebras of dimension up to 3 over algebraically closed fields of positive characteristic were classified by Wang and his collaborators in <span><span>[25]</span></span>, <span><span>[19]</span></span>. In this paper, we obtain a classification of restricted Lie algebras of dimension 4 over such fields.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 271-316"},"PeriodicalIF":0.8,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798824","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 : 2025-12-12DOI: 10.1016/j.jalgebra.2025.12.012
Xiangqian Guo , Xuewen Liu , Junzhou Qin
In this paper, we study the higher rank polynomial modules for the quantum group , improving Bavula's description of irreducible nonweight modules using the terminology of generalized Weyl algebras, constructing examples of irreducible polynomial modules of arbitrary finite rank, describing irreducible polynomial modules in terms of Smith normal form, and presenting properties of dual polynomial modules. Then we consider the tensor products of irreducible polynomial modules with finite-dimensional simple modules, obtaining a direct sum decomposition formula, similar to classical Clebsch-Gordan formula, under some conditions. These results are generalizations of our previous results for the rank-1 polynomial modules.
{"title":"Higher rank polynomial modules over Uq(sl2)","authors":"Xiangqian Guo , Xuewen Liu , Junzhou Qin","doi":"10.1016/j.jalgebra.2025.12.012","DOIUrl":"10.1016/j.jalgebra.2025.12.012","url":null,"abstract":"<div><div>In this paper, we study the higher rank polynomial modules for the quantum group <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>, improving Bavula's description of irreducible nonweight modules using the terminology of generalized Weyl algebras, constructing examples of irreducible polynomial modules of arbitrary finite rank, describing irreducible polynomial modules in terms of Smith normal form, and presenting properties of dual polynomial modules. Then we consider the tensor products of irreducible polynomial modules with finite-dimensional simple modules, obtaining a direct sum decomposition formula, similar to classical Clebsch-Gordan formula, under some conditions. These results are generalizations of our previous results for the rank-1 polynomial modules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 238-270"},"PeriodicalIF":0.8,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798813","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 : 2025-12-12DOI: 10.1016/j.jalgebra.2025.12.014
Li Wang , Jiaqun Wei
Let be an extriangulated length category. We introduce the notation of Gabriel-Roiter measure with respect to Θ and extend Gabriel's main property to this setting. Using this measure, when satisfies some reasonable conditions, we prove that has an infinite number of pairwise nonisomorphic indecomposable objects if and only if it has indecomposable objects of arbitrarily large length. That is, the first Brauer-Thrall conjecture holds.
{"title":"The first Brauer-Thrall conjecture for extriangulated length categories","authors":"Li Wang , Jiaqun Wei","doi":"10.1016/j.jalgebra.2025.12.014","DOIUrl":"10.1016/j.jalgebra.2025.12.014","url":null,"abstract":"<div><div>Let <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>Θ</mi><mo>)</mo></math></span> be an extriangulated length category. We introduce the notation of Gabriel-Roiter measure with respect to Θ and extend Gabriel's main property to this setting. Using this measure, when <span><math><mo>(</mo><mi>A</mi><mo>,</mo><mi>Θ</mi><mo>)</mo></math></span> satisfies some reasonable conditions, we prove that <span><math><mi>A</mi></math></span> has an infinite number of pairwise nonisomorphic indecomposable objects if and only if it has indecomposable objects of arbitrarily large length. That is, the first Brauer-Thrall conjecture holds.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 317-335"},"PeriodicalIF":0.8,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798825","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 : 2025-12-09DOI: 10.1016/j.jalgebra.2025.12.009
Daniel Rogalski , Robert Won , James J. Zhang
The notion of a homological integral of an infinite-dimensional weak Hopf algebra is introduced. We show that the homological integral is an invertible object in the associated monoidal category. Using integrals, we prove that the Artin–Schelter property and the Van den Bergh condition are equivalent for a noetherian weak Hopf algebra, and that the antipode is automatically invertible in this case. We also prove that any weak Hopf algebra finite over an affine center is a direct sum of Artin–Schelter Gorenstein weak Hopf algebras.
引入了无穷维弱Hopf代数的同调积分的概念。证明了同调积分在相关的一元范畴中是可逆对象。利用积分证明了noether弱Hopf代数的Artin-Schelter性质和Van den Bergh条件是等价的，并且在这种情况下对映对是自动可逆的。我们还证明了在仿射中心上有限的任何弱Hopf代数是Artin-Schelter Gorenstein弱Hopf代数的直接和。
{"title":"Homological integrals for weak Hopf algebras","authors":"Daniel Rogalski , Robert Won , James J. Zhang","doi":"10.1016/j.jalgebra.2025.12.009","DOIUrl":"10.1016/j.jalgebra.2025.12.009","url":null,"abstract":"<div><div>The notion of a homological integral of an infinite-dimensional weak Hopf algebra is introduced. We show that the homological integral is an invertible object in the associated monoidal category. Using integrals, we prove that the Artin–Schelter property and the Van den Bergh condition are equivalent for a noetherian weak Hopf algebra, and that the antipode is automatically invertible in this case. We also prove that any weak Hopf algebra finite over an affine center is a direct sum of Artin–Schelter Gorenstein weak Hopf algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 173-204"},"PeriodicalIF":0.8,"publicationDate":"2025-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798741","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 : 2025-12-08DOI: 10.1016/j.jalgebra.2025.12.007
Egle Bettio
Let be the number of distinct prime divisors occurring among the conjugacy class sizes of a finite group G, and let be the maximum number of such divisors in any single class size. We prove that the inequality holds for all finite groups, with no assumption of solvability. The bound is sharp, and refines earlier partial results.
{"title":"Huppert's ρ − σ conjecture for conjugacy class sizes","authors":"Egle Bettio","doi":"10.1016/j.jalgebra.2025.12.007","DOIUrl":"10.1016/j.jalgebra.2025.12.007","url":null,"abstract":"<div><div>Let <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the number of distinct prime divisors occurring among the conjugacy class sizes of a finite group <em>G</em>, and let <span><math><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the maximum number of such divisors in any single class size. We prove that the inequality <span><math><mi>ρ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>3</mn><mi>σ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> holds for all finite groups, with no assumption of solvability. The bound is sharp, and refines earlier partial results.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 518-525"},"PeriodicalIF":0.8,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145733658","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 : 2025-12-08DOI: 10.1016/j.jalgebra.2025.12.006
Victor Guba
A (discrete) group is called amenable if there exists a finitely additive right-invariant probability measure on it. The question of whether Thompson's group F is amenable is a long-standing open problem. We consider the presentation of F in terms of non-spherical semigroup diagrams. There is a natural partition of F into 7 parts in terms of these diagrams. We show that for any finitely additive right-invariant probability measure on F, all but one of these sets have zero measure. This helps to clarify the structure of Følner sets in F, provided the group is amenable.
{"title":"On zero-measured subsets of Thompson's group F","authors":"Victor Guba","doi":"10.1016/j.jalgebra.2025.12.006","DOIUrl":"10.1016/j.jalgebra.2025.12.006","url":null,"abstract":"<div><div>A (discrete) group is called <em>amenable</em> if there exists a finitely additive right-invariant probability measure on it. The question of whether Thompson's group <em>F</em> is amenable is a long-standing open problem. We consider the presentation of <em>F</em> in terms of non-spherical semigroup diagrams. There is a natural partition of <em>F</em> into 7 parts in terms of these diagrams. We show that for any finitely additive right-invariant probability measure on <em>F</em>, all but one of these sets have zero measure. This helps to clarify the structure of Følner sets in <em>F</em>, provided the group is amenable.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 106-122"},"PeriodicalIF":0.8,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145750307","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 : 2025-12-08DOI: 10.1016/j.jalgebra.2025.11.026
Rebecca Goldin , Martha Precup
We study families of matrix Hessenberg schemes in the affine scheme of complex matrices, each defined over a fixed sheet in the Lie algebra . Abe, Fujita and Zeng show in [3] that such families over the regular sheet are flat, and every regular Hessenberg scheme degenerates to a regular nilpotent Hessenberg scheme. This paper explores whether flat degenerations exist outside of the regular case.
For each matrix Hessenberg scheme, we introduce a one-parameter family of matrix Hessenberg schemes that degenerates it to a specific nilpotent Hessenberg scheme. Our main theorem states that, when the family lies over the minimal sheet in , this degeneration is flat. The proof leverages commutative algebra on the polynomial ring to identify the structure of the family concretely, and we explore several applications. We conjecture that flatness holds for these families over other sheets as well.
{"title":"A flat family of matrix Hessenberg schemes over the minimal sheet","authors":"Rebecca Goldin , Martha Precup","doi":"10.1016/j.jalgebra.2025.11.026","DOIUrl":"10.1016/j.jalgebra.2025.11.026","url":null,"abstract":"<div><div>We study families of matrix Hessenberg schemes in the affine scheme of complex <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices, each defined over a fixed sheet in the Lie algebra <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span>. Abe, Fujita and Zeng show in <span><span>[3]</span></span> that such families over the regular sheet are flat, and every regular Hessenberg scheme degenerates to a regular nilpotent Hessenberg scheme. This paper explores whether flat degenerations exist outside of the regular case.</div><div>For each matrix Hessenberg scheme, we introduce a one-parameter family of matrix Hessenberg schemes that degenerates it to a specific nilpotent Hessenberg scheme. Our main theorem states that, when the family lies over the minimal sheet in <span><math><msub><mrow><mi>gl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span>, this degeneration is flat. The proof leverages commutative algebra on the polynomial ring to identify the structure of the family concretely, and we explore several applications. We conjecture that flatness holds for these families over other sheets as well.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"692 ","pages":"Pages 123-172"},"PeriodicalIF":0.8,"publicationDate":"2025-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145798710","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}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号