Pub Date : 2026-04-01Epub Date: 2025-11-20DOI: 10.1016/j.jalgebra.2025.11.007
Hongju Zhao, Qiang Mu
We study toroidal vertex algebras and their modules over a general field of prime characteristic, and provide a conceptual construction of modular toroidal vertex algebras and their modules. As an example, we consider the toroidal vertex algebra associated with a toroidal Lie algebra and further construct a family of its quotients.
{"title":"Modular toroidal vertex algebras and their modules","authors":"Hongju Zhao, Qiang Mu","doi":"10.1016/j.jalgebra.2025.11.007","DOIUrl":"10.1016/j.jalgebra.2025.11.007","url":null,"abstract":"<div><div>We study toroidal vertex algebras and their modules over a general field of prime characteristic, and provide a conceptual construction of modular toroidal vertex algebras and their modules. As an example, we consider the toroidal vertex algebra associated with a toroidal Lie algebra and further construct a family of its quotients.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 88-127"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616558","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 : 2026-04-01Epub Date: 2025-12-02DOI: 10.1016/j.jalgebra.2025.10.057
Nicolás Andruskiewitsch , David Jaklitsch , Van C. Nguyen , Amrei Oswald , Julia Plavnik , Anne V. Shepler , Xingting Wang
We use deformation sequences of (Hopf) algebras, extending the results of Negron and Pevtsova, to show that bosonizations of some suitable braided Hopf algebras by some suitable finite-dimensional Hopf algebras have finitely generated cohomology. In fact, our results are shown in more generality for smash products. As applications, we prove the bosonizations of some Nichols algebras (such as Nichols algebras of diagonal type, the restricted Jordan plane, Nichols algebras of direct sums of Jordan blocks plus points labeled with 1), by some suitable finite-dimensional Hopf algebras, have finitely generated cohomology, recovering some known results as well as providing new examples.
{"title":"On the finite generation of the cohomology of bosonizations","authors":"Nicolás Andruskiewitsch , David Jaklitsch , Van C. Nguyen , Amrei Oswald , Julia Plavnik , Anne V. Shepler , Xingting Wang","doi":"10.1016/j.jalgebra.2025.10.057","DOIUrl":"10.1016/j.jalgebra.2025.10.057","url":null,"abstract":"<div><div>We use deformation sequences of (Hopf) algebras, extending the results of Negron and Pevtsova, to show that bosonizations of some suitable braided Hopf algebras by some suitable finite-dimensional Hopf algebras have finitely generated cohomology. In fact, our results are shown in more generality for smash products. As applications, we prove the bosonizations of some Nichols algebras (such as Nichols algebras of diagonal type, the restricted Jordan plane, Nichols algebras of direct sums of Jordan blocks plus points labeled with 1), by some suitable finite-dimensional Hopf algebras, have finitely generated cohomology, recovering some known results as well as providing new examples.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 741-780"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145733651","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 : 2026-04-01Epub Date: 2025-12-04DOI: 10.1016/j.jalgebra.2025.12.002
Hiroyuki Minamoto
The aim of this short note is to prove the formula of the Hilbert series of preprojective algebras in arbitrary characteristic by making effective use of the formulas of the Hilbert series of differential graded (dg) algebras with Adams grading. We also compute the Hilbert series of quiver Heisenberg algebras, a special class of central extensions of preprojective algebras.
{"title":"The Hilbert series of preprojective algebras","authors":"Hiroyuki Minamoto","doi":"10.1016/j.jalgebra.2025.12.002","DOIUrl":"10.1016/j.jalgebra.2025.12.002","url":null,"abstract":"<div><div>The aim of this short note is to prove the formula of the Hilbert series of preprojective algebras in arbitrary characteristic by making effective use of the formulas of the Hilbert series of differential graded (dg) algebras with Adams grading. We also compute the Hilbert series of quiver Heisenberg algebras, a special class of central extensions of preprojective algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 730-740"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145733653","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 : 2026-04-01Epub Date: 2025-12-05DOI: 10.1016/j.jalgebra.2025.11.029
Jadyn V. Breland, Sam K. Miller
We define the notion of a Brauer pair of a chain complex, extending the notion of a Brauer pair of a p-permutation module introduced by Boltje and Perepelitsky. In fact, the Brauer pairs of a splendid Rickard equivalence C coincide with the set of Brauer pairs of the corresponding p-permutation equivalence induced by C. As a result, we derive structural results for splendid Rickard equivalences that correspond to known structural properties for p-permutation equivalences. In particular, we show splendid Rickard equivalences induce local splendid Rickard equivalences between normalizer block algebras as well as centralizer block algebras.
{"title":"Brauer pairs for splendid Rickard equivalences","authors":"Jadyn V. Breland, Sam K. Miller","doi":"10.1016/j.jalgebra.2025.11.029","DOIUrl":"10.1016/j.jalgebra.2025.11.029","url":null,"abstract":"<div><div>We define the notion of a Brauer pair of a chain complex, extending the notion of a Brauer pair of a <em>p</em>-permutation module introduced by Boltje and Perepelitsky. In fact, the Brauer pairs of a splendid Rickard equivalence <em>C</em> coincide with the set of Brauer pairs of the corresponding <em>p</em>-permutation equivalence <span><math><mi>Λ</mi><mo>(</mo><mi>C</mi><mo>)</mo></math></span> induced by <em>C</em>. As a result, we derive structural results for splendid Rickard equivalences that correspond to known structural properties for <em>p</em>-permutation equivalences. In particular, we show splendid Rickard equivalences induce local splendid Rickard equivalences between normalizer block algebras as well as centralizer block algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 694-729"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145733654","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 : 2026-04-01Epub Date: 2025-12-04DOI: 10.1016/j.jalgebra.2025.11.024
F.J. Lobillo , Paolo Santonastaso , John Sheekey
We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods lead to the construction of new (not necessarily associative) division algebras and maximum rank distance (MRD) codes over both finite and infinite division rings. In particular, we construct new non-associative division algebras whose right nucleus is a central simple algebra having degree greater than 1. Over finite fields, we obtain new semifields and MRD codes for infinitely many choices of parameters. These families extend and contain many of the best previously known constructions.
{"title":"Quotients of skew polynomial rings: New constructions of division algebras and MRD codes","authors":"F.J. Lobillo , Paolo Santonastaso , John Sheekey","doi":"10.1016/j.jalgebra.2025.11.024","DOIUrl":"10.1016/j.jalgebra.2025.11.024","url":null,"abstract":"<div><div>We achieve new results on skew polynomial rings and their quotients, including the first explicit example of a skew polynomial ring where the ratio of the degree of a skew polynomial to the degree of its bound is not extremal. These methods lead to the construction of new (not necessarily associative) division algebras and maximum rank distance (MRD) codes over both finite and infinite division rings. In particular, we construct new non-associative division algebras whose right nucleus is a central simple algebra having degree greater than 1. Over finite fields, we obtain new semifields and MRD codes for infinitely many choices of parameters. These families extend and contain many of the best previously known constructions.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 648-693"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145733655","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 : 2026-04-01Epub Date: 2025-12-04DOI: 10.1016/j.jalgebra.2025.11.020
Kayo Masuda
Let X be a factorial complex affine variety of dimension ≥3 with an algebraic action of the additive group . Let be the algebraic quotient morphism where we assume Y is an affine variety. When π is faithfully flat, we investigate π by -equivariant affine modifications and give criteria for π to be a trivial -bundle. For a smooth acyclic fourfold X with a free -action and a -equivariant -fibration where acts trivially on , we give a criterion for the algebraic quotient Y to be isomorphic to with f as a coordinate. Together with a criterion for to be a trivial -bundle, we obtain a sufficient condition for .
{"title":"Faithfully flat quotient morphisms by Ga-actions on factorial affine varieties","authors":"Kayo Masuda","doi":"10.1016/j.jalgebra.2025.11.020","DOIUrl":"10.1016/j.jalgebra.2025.11.020","url":null,"abstract":"<div><div>Let <em>X</em> be a factorial complex affine variety of dimension ≥3 with an algebraic action of the additive group <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span>. Let <span><math><mi>π</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></math></span> be the algebraic quotient morphism where we assume <em>Y</em> is an affine variety. When <em>π</em> is faithfully flat, we investigate <em>π</em> by <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span>-equivariant affine modifications and give criteria for <em>π</em> to be a trivial <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-bundle. For a smooth acyclic fourfold <em>X</em> with a free <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span>-action and a <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span>-equivariant <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>-fibration <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> where <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>a</mi></mrow></msub></math></span> acts trivially on <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>, we give a criterion for the algebraic quotient <em>Y</em> to be isomorphic to <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> with <em>f</em> as a coordinate. Together with a criterion for <span><math><mi>π</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></math></span> to be a trivial <span><math><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-bundle, we obtain a sufficient condition for <span><math><mi>X</mi><mo>≅</mo><mi>Y</mi><mo>×</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>≅</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 577-596"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145733657","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 : 2026-04-01Epub 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":"2026-04-01","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 : 2026-04-01Epub Date: 2025-11-20DOI: 10.1016/j.jalgebra.2025.10.053
Ishan Banerjee , Peter Huxford
We prove for and that the level m congruence subgroup of the braid group associated to the integral Burau representation is generated by mth powers of half-twists and the braid Torelli group. This solves a problem of Margalit, generalizing work of Assion, Brendle–Margalit, Nakamura, Stylianakis and Wajnryb.
{"title":"Generators for the level m congruence subgroups of braid groups","authors":"Ishan Banerjee , Peter Huxford","doi":"10.1016/j.jalgebra.2025.10.053","DOIUrl":"10.1016/j.jalgebra.2025.10.053","url":null,"abstract":"<div><div>We prove for <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>5</mn></math></span> that the level <em>m</em> congruence subgroup <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>[</mo><mi>m</mi><mo>]</mo></math></span> of the braid group <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> associated to the integral Burau representation <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>→</mo><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> is generated by <em>m</em>th powers of half-twists and the braid Torelli group. This solves a problem of Margalit, generalizing work of Assion, Brendle–Margalit, Nakamura, Stylianakis and Wajnryb.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 1-16"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145584336","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 : 2026-04-01Epub Date: 2025-12-02DOI: 10.1016/j.jalgebra.2025.10.058
Damian Sercombe
Let G be an affine algebraic group scheme over a field k. We show there exists a unipotent normal subgroup of G which contains all other such subgroups; we call it the restricted unipotent radical of G. We investigate some properties of , and study those G for which is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine k-groups.
{"title":"Unipotent normal subgroups of algebraic groups","authors":"Damian Sercombe","doi":"10.1016/j.jalgebra.2025.10.058","DOIUrl":"10.1016/j.jalgebra.2025.10.058","url":null,"abstract":"<div><div>Let <em>G</em> be an affine algebraic group scheme over a field <em>k</em>. We show there exists a unipotent normal subgroup of <em>G</em> which contains all other such subgroups; we call it the restricted unipotent radical <span><math><msub><mrow><mi>Rad</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em>. We investigate some properties of <span><math><msub><mrow><mi>Rad</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and study those <em>G</em> for which <span><math><msub><mrow><mi>Rad</mi></mrow><mrow><mi>u</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine <em>k</em>-groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 337-351"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145682382","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 : 2026-04-01Epub Date: 2025-12-05DOI: 10.1016/j.jalgebra.2025.11.031
Ruipeng Zhu
This paper shows that if H is a Hopf algebra and is a faithfully flat H-Galois extension, then B is skew Calabi–Yau provided A and H are. Specifically, for cleft extensions , the Nakayama automorphism of B can be derived from those of A and H, along with the homological determinant of the H-action on A. This finding is based on the study of the Hopf bimodule structure on .
{"title":"Skew Calabi–Yau property of faithfully flat Hopf Galois extensions","authors":"Ruipeng Zhu","doi":"10.1016/j.jalgebra.2025.11.031","DOIUrl":"10.1016/j.jalgebra.2025.11.031","url":null,"abstract":"<div><div>This paper shows that if <em>H</em> is a Hopf algebra and <span><math><mi>A</mi><mo>⊆</mo><mi>B</mi></math></span> is a faithfully flat <em>H</em>-Galois extension, then <em>B</em> is skew Calabi–Yau provided <em>A</em> and <em>H</em> are. Specifically, for cleft extensions <span><math><mi>A</mi><mo>⊆</mo><mi>B</mi></math></span>, the Nakayama automorphism of <em>B</em> can be derived from those of <em>A</em> and <em>H</em>, along with the homological determinant of the <em>H</em>-action on <em>A</em>. This finding is based on the study of the Hopf bimodule structure on <span><math><msubsup><mrow><mi>Ext</mi></mrow><mrow><msup><mrow><mi>A</mi></mrow><mrow><mi>e</mi></mrow></msup></mrow><mrow><mi>i</mi></mrow></msubsup><mo>(</mo><mi>A</mi><mo>,</mo><msup><mrow><mi>B</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>)</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"691 ","pages":"Pages 597-647"},"PeriodicalIF":0.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145733656","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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