Pub Date : 2025-12-01Epub Date: 2025-11-07DOI: 10.1016/j.jpaa.2025.108127
Dmitriy Voloshyn
We study birational quasi-isomorphisms between normal Noetherian domains endowed with cluster structures of geometric type. We prove an analogue of the Starfish lemma that allows one to transfer various cluster and algebraic properties of one variety onto another. In particular, we develop tools for proving that an upper cluster algebra equals the given commutative ring.
{"title":"Starfish lemma via birational quasi-isomorphisms","authors":"Dmitriy Voloshyn","doi":"10.1016/j.jpaa.2025.108127","DOIUrl":"10.1016/j.jpaa.2025.108127","url":null,"abstract":"<div><div>We study birational quasi-isomorphisms between normal Noetherian domains endowed with cluster structures of geometric type. We prove an analogue of the Starfish lemma that allows one to transfer various cluster and algebraic properties of one variety onto another. In particular, we develop tools for proving that an upper cluster algebra equals the given commutative ring.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108127"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528934","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 : 2025-12-01Epub Date: 2025-11-10DOI: 10.1016/j.jpaa.2025.108130
Abhishek Das , Santosha Pattanayak
We consider typical finite dimensional complex irreducible representations of a basic classical Lie superalgebra, and give a sufficient condition on when unique factorization of finite tensor products of such representations hold. We also prove unique factorization of tensor products of singly atypical finite dimensional irreducible modules for , , and under some assumptions. This result is a Lie superalgebra analogue of Rajan's fundamental result [10] on unique factorization of tensor products for finite dimensional complex simple Lie algebras.
{"title":"On tensor products of representations of Lie superalgebras","authors":"Abhishek Das , Santosha Pattanayak","doi":"10.1016/j.jpaa.2025.108130","DOIUrl":"10.1016/j.jpaa.2025.108130","url":null,"abstract":"<div><div>We consider typical finite dimensional complex irreducible representations of a basic classical Lie superalgebra, and give a sufficient condition on when unique factorization of finite tensor products of such representations hold. We also prove unique factorization of tensor products of singly atypical finite dimensional irreducible modules for <span><math><mrow><mi>sl</mi></mrow><mo>(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>, <span><math><mrow><mi>osp</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mi>n</mi><mo>)</mo></math></span>, <span><math><mi>G</mi><mo>(</mo><mn>3</mn><mo>)</mo></math></span> and <span><math><mi>F</mi><mo>(</mo><mn>4</mn><mo>)</mo></math></span> under some assumptions. This result is a Lie superalgebra analogue of Rajan's fundamental result <span><span>[10]</span></span> on unique factorization of tensor products for finite dimensional complex simple Lie algebras.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108130"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145579725","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 : 2025-12-01Epub Date: 2025-12-02DOI: 10.1016/j.jpaa.2025.108138
Matthew B. Day , Trevor Nakamura
We consider normal subgroups N of the braid group such that the quotient is an extension of the symmetric group by an abelian group. We show that, if , then there are exactly 8 commensurability classes of such subgroups. We define a Specht subgroup to be a subgroup of this form that is maximal in its commensurability class. We give descriptions of the Specht subgroups in terms of winding numbers and in terms of infinite generating sets. The quotient of the pure braid group by a Specht subgroup is a module over the symmetric group. We show that the modules arising this way are closely related to Specht modules for the partitions and , working over the integers. We compute the second cohomology of the symmetric group with coefficients in both of these Specht modules, working over an arbitrary commutative ring. Finally, we determine which of the extensions of the symmetric group arising from Specht subgroups are split extensions.
{"title":"Quotients of the braid group that are extensions of the symmetric group","authors":"Matthew B. Day , Trevor Nakamura","doi":"10.1016/j.jpaa.2025.108138","DOIUrl":"10.1016/j.jpaa.2025.108138","url":null,"abstract":"<div><div>We consider normal subgroups <em>N</em> of the braid group <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> such that the quotient <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>N</mi></math></span> is an extension of the symmetric group by an abelian group. We show that, if <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, then there are exactly 8 commensurability classes of such subgroups. We define a <em>Specht subgroup</em> to be a subgroup of this form that is maximal in its commensurability class. We give descriptions of the Specht subgroups in terms of winding numbers and in terms of infinite generating sets. The quotient of the pure braid group by a Specht subgroup is a module over the symmetric group. We show that the modules arising this way are closely related to Specht modules for the partitions <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, working over the integers. We compute the second cohomology of the symmetric group with coefficients in both of these Specht modules, working over an arbitrary commutative ring. Finally, we determine which of the extensions of the symmetric group arising from Specht subgroups are split extensions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108138"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145747650","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 : 2025-12-01Epub Date: 2025-12-02DOI: 10.1016/j.jpaa.2025.108144
Cheng Meng
In this paper, we prove that if P is a homogeneous prime ideal inside a standard graded polynomial ring S with , and for , adjoining s general linear forms to the prime ideal changes the -th Hilbert coefficient by 1, then . This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring.
本文证明了如果P是标准渐变多项式环S内的齐次素理想，且当S≤d时，与该素理想相邻的S种一般线性形式使(d - S)-希尔伯特系数改变1，则深度(S/P)= S - 1。这个判据也告诉我们多项式环内素数理想的一般初始理想的可能约束。
{"title":"Restrictions on Hilbert coefficients give depths of graded domains","authors":"Cheng Meng","doi":"10.1016/j.jpaa.2025.108144","DOIUrl":"10.1016/j.jpaa.2025.108144","url":null,"abstract":"<div><div>In this paper, we prove that if <em>P</em> is a homogeneous prime ideal inside a standard graded polynomial ring <em>S</em> with <span><math><mi>dim</mi><mo></mo><mo>(</mo><mi>S</mi><mo>/</mo><mi>P</mi><mo>)</mo><mo>=</mo><mi>d</mi></math></span>, and for <span><math><mi>s</mi><mo>≤</mo><mi>d</mi></math></span>, adjoining <em>s</em> general linear forms to the prime ideal changes the <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mi>s</mi><mo>)</mo></math></span>-th Hilbert coefficient by 1, then <span><math><mtext>depth</mtext><mo>(</mo><mi>S</mi><mo>/</mo><mi>P</mi><mo>)</mo><mo>=</mo><mi>s</mi><mo>−</mo><mn>1</mn></math></span>. This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108144"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693706","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 : 2025-12-01Epub Date: 2025-12-02DOI: 10.1016/j.jpaa.2025.108143
Onofrio M. Di Vincenzo , Vincenzo C. Nardozza
Let F be a field of characteristic zero and let E be the Grassmann algebra of an infinite-dimensional F-vector space. We consider a class of solvable nonabelian finite-dimensional Lie algebras acting on E by derivations, and completely describe the differential polynomial identities satisfied by E. The corresponding -cocharacter and differential codimension sequences are computed. Finally, we prove that the differential exponent exists and equals the ordinary exponent of E.
{"title":"Differential identities of the Grassmann algebra","authors":"Onofrio M. Di Vincenzo , Vincenzo C. Nardozza","doi":"10.1016/j.jpaa.2025.108143","DOIUrl":"10.1016/j.jpaa.2025.108143","url":null,"abstract":"<div><div>Let <em>F</em> be a field of characteristic zero and let <em>E</em> be the Grassmann algebra of an infinite-dimensional <em>F</em>-vector space. We consider a class of solvable nonabelian finite-dimensional Lie algebras acting on <em>E</em> by derivations, and completely describe the differential polynomial identities satisfied by <em>E</em>. The corresponding <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-cocharacter and differential codimension sequences are computed. Finally, we prove that the differential exponent exists and equals the ordinary exponent of <em>E</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108143"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693705","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 : 2025-12-01Epub Date: 2025-12-02DOI: 10.1016/j.jpaa.2025.108141
Xiao-Wu Chen , Ren Wang
Let be a field of characteristic p and G be a cyclic p-group which acts on a finite acyclic quiver Q. The folding process associates a Cartan triple to the action. We establish a Morita equivalence between the skew group algebra of the preprojective algebra of Q and the generalized preprojective algebra associated to the Cartan triple in the sense of Geiss, Leclerc and Schröer. The Morita equivalence induces an isomorphism between certain ideal monoids of these preprojective algebras, which is compatible with the embedding of Weyl groups appearing in the folding process.
{"title":"Preprojective algebras, skew group algebras and Morita equivalences","authors":"Xiao-Wu Chen , Ren Wang","doi":"10.1016/j.jpaa.2025.108141","DOIUrl":"10.1016/j.jpaa.2025.108141","url":null,"abstract":"<div><div>Let <span><math><mi>K</mi></math></span> be a field of characteristic <em>p</em> and <em>G</em> be a cyclic <em>p</em>-group which acts on a finite acyclic quiver <em>Q</em>. The folding process associates a Cartan triple to the action. We establish a Morita equivalence between the skew group algebra of the preprojective algebra of <em>Q</em> and the generalized preprojective algebra associated to the Cartan triple in the sense of Geiss, Leclerc and Schröer. The Morita equivalence induces an isomorphism between certain ideal monoids of these preprojective algebras, which is compatible with the embedding of Weyl groups appearing in the folding process.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108141"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145693703","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 : 2025-12-01Epub Date: 2025-11-03DOI: 10.1016/j.jpaa.2025.108123
Ruipeng Zhu
This paper investigates the homological properties of the faithfully flat Hopf Galois extension . It establishes that when B is a noetherian affine PI algebra and A is AS Gorenstein, B inherits the AS Gorenstein property. Furthermore, we demonstrate that injective dimension serves as a monoidal invariant for AS Gorenstein Hopf algebras. Specifically, if two such Hopf algebras have equivalent monoidal categories of comodules, then their injective dimensions are equal.
{"title":"Artin-Schelter Gorenstein property of Hopf Galois extensions","authors":"Ruipeng Zhu","doi":"10.1016/j.jpaa.2025.108123","DOIUrl":"10.1016/j.jpaa.2025.108123","url":null,"abstract":"<div><div>This paper investigates the homological properties of the faithfully flat Hopf Galois extension <span><math><mi>A</mi><mo>⊆</mo><mi>B</mi></math></span>. It establishes that when <em>B</em> is a noetherian affine PI algebra and <em>A</em> is AS Gorenstein, <em>B</em> inherits the AS Gorenstein property. Furthermore, we demonstrate that injective dimension serves as a monoidal invariant for AS Gorenstein Hopf algebras. Specifically, if two such Hopf algebras have equivalent monoidal categories of comodules, then their injective dimensions are equal.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108123"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145475353","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 : 2025-12-01Epub Date: 2025-11-05DOI: 10.1016/j.jpaa.2025.108126
Hongda Lin , Honglian Zhang
In this paper, we establish the first rigorous framework for the Drinfeld super Yangian associated with an exceptional Lie superalgebra, which lacks a classical Lie algebraic counterpart. Specifically, we systematically investigate the Drinfeld presentation and structural properties of the super Yangian associated with the exceptional Lie superalgebra . First, we introduce a Drinfeld presentation for the super Yangian associated with the exceptional Lie superalgebra , explicitly constructing its current generators and defining relations. A key innovation is the construction of a Poincaré-Birkhoff-Witt (PBW) basis using degeneration techniques from the corresponding quantum loop superalgebra. Furthermore, we demonstrate that the super Yangian possesses a Hopf superalgebra structure, explicitly providing the coproduct, counit, and antipode.
{"title":"Drinfeld super Yangian of the exceptional Lie superalgebra D(2,1;λ)","authors":"Hongda Lin , Honglian Zhang","doi":"10.1016/j.jpaa.2025.108126","DOIUrl":"10.1016/j.jpaa.2025.108126","url":null,"abstract":"<div><div>In this paper, we establish the first rigorous framework for the Drinfeld super Yangian associated with an exceptional Lie superalgebra, which lacks a classical Lie algebraic counterpart. Specifically, we systematically investigate the Drinfeld presentation and structural properties of the super Yangian associated with the exceptional Lie superalgebra <span><math><mi>D</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>;</mo><mi>λ</mi><mo>)</mo></math></span>. First, we introduce a Drinfeld presentation for the super Yangian associated with the exceptional Lie superalgebra <span><math><mi>D</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>;</mo><mi>λ</mi><mo>)</mo></math></span>, explicitly constructing its current generators and defining relations. A key innovation is the construction of a Poincaré-Birkhoff-Witt (PBW) basis using degeneration techniques from the corresponding quantum loop superalgebra. Furthermore, we demonstrate that the super Yangian possesses a Hopf superalgebra structure, explicitly providing the coproduct, counit, and antipode.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108126"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145475355","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 : 2025-12-01Epub Date: 2025-11-19DOI: 10.1016/j.jpaa.2025.108134
Carmelo Antonio Finocchiaro , K. Alan Loper
Let D be a Dedekind domain (not a field) with finite residue fields and let be the ring of integer-valued polynomials over D. We completely classify in topological terms some relevant classes of radical unitary ideals of (and of its overrings). This project strongly extends the classification given in a previous paper and regarding special unitary ideals, precisely the ones lying over a given maximal ideal of D.
{"title":"On radical unitary ideals of rings of integer-valued polynomials","authors":"Carmelo Antonio Finocchiaro , K. Alan Loper","doi":"10.1016/j.jpaa.2025.108134","DOIUrl":"10.1016/j.jpaa.2025.108134","url":null,"abstract":"<div><div>Let <em>D</em> be a Dedekind domain (not a field) with finite residue fields and let <span><math><mi>Int</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span> be the ring of integer-valued polynomials over <em>D</em>. We completely classify in topological terms some relevant classes of radical unitary ideals of <span><math><mi>Int</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span> (and of its overrings). This project strongly extends the classification given in a previous paper and regarding special unitary ideals, precisely the ones lying over a given maximal ideal of <em>D</em>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108134"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145623948","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 : 2025-12-01Epub Date: 2025-11-04DOI: 10.1016/j.jpaa.2025.108124
Jianbei An
We introduce a way to classify weight subgroups of a block. As an application we classified weight subgroups and proved the Alperin weight conjecture for quasi-isolated 2-blocks of .
{"title":"Weight subgroups of quasi-isolated 2-blocks of the Chevalley groups F4(q)","authors":"Jianbei An","doi":"10.1016/j.jpaa.2025.108124","DOIUrl":"10.1016/j.jpaa.2025.108124","url":null,"abstract":"<div><div>We introduce a way to classify weight subgroups of a block. As an application we classified weight subgroups and proved the Alperin weight conjecture for quasi-isolated 2-blocks of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 12","pages":"Article 108124"},"PeriodicalIF":0.8,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145475356","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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