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The i-Quantum Group (textbf{U}^imath (n)), II: Presenting their (varvec{q})-Schur Algebras i-量子群(textbf{U}^imath (n))， II：表示它们的(varvec{q}) -Schur代数
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-04-14 DOI: 10.1007/s10468-026-10390-9
Jian Chen, Jie Du
{"title":"The i-Quantum Group (textbf{U}^imath (n)), II: Presenting their (varvec{q})-Schur Algebras","authors":"Jian Chen,&nbsp;Jie Du","doi":"10.1007/s10468-026-10390-9","DOIUrl":"10.1007/s10468-026-10390-9","url":null,"abstract":"<div><p>Since the establishment of the quantum Schur–Weyl duality in Jimbo (Lett. Math. Phys. <b>11</b>, 247–252, 1986), the duality pair <span>((textbf{U}(mathfrak {gl}_n),varvec{mathcal {H}}(mathfrak S_r)))</span> of type <i>A</i> has been extended to the duality pairs <span>((textbf{U}^jmath (n),varvec{mathcal {H}}(B_r)))</span> and <span>((textbf{U}^imath (n),varvec{mathcal {H}}(C_r)))</span> in the Hecke algebra series in Bao and Wang (Astérisque <b>402</b>, vii+134, 2018), where <span>(textbf{U}^jmath (n),textbf{U}^imath (n))</span> are <i>i</i>-quantum groups arising from certain quantum symmetric pairs. The quantum Schur algebra associated with the pair <span>((textbf{U}(mathfrak {gl}_n),varvec{mathcal {H}}(mathfrak S_r)))</span> has a nice and simple presentation; see Doty and Giaquinto \u0000(2002). In this paper, we tackle the presentation problem for the <i>i</i>-quantum Schur algebras associated with the duality pair <span>((textbf{U}^imath (n),varvec{mathcal {H}}(C_r)))</span>. Such a <i>q</i>-Schur algebra is called the hyperoctahedral <i>q</i>-Schur algebras in Green (J. Algebra <b>192</b>, 418–438, 1997). See Bhattacharya \u0000(2026) for the <span>(textbf{U}^jmath (n))</span> case. Building on the explicit epimorphism <span>(phi _{n,r}^imath )</span> from the <i>i</i>-quantum group <span>(textbf{U}^imath (n))</span> to the hyperoctahedral <i>q</i>-Schur algebras <span>(mathcal {S}^imath (n,r))</span> (see Du and Wu, Pacific J. Math. <b>320</b>(1), 61–101, 2022), we compute the kernel of <span>(phi _{n,r}^imath )</span> in terms of generators. This results in a presentation for <span>(mathcal {S}^imath (n,r))</span> with defining relations which include not only the Doty–Giaquinto’s diagonal relations but also some tridiagonal relations.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"653 - 696"},"PeriodicalIF":0.6,"publicationDate":"2026-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-026-10390-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a Question of Auslander and Bridger on 2-Reflexive Modules 关于Auslander和Bridger关于2-自反模块的问题
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-03-30 DOI: 10.1007/s10468-026-10392-7
René Marczinzik
{"title":"On a Question of Auslander and Bridger on 2-Reflexive Modules","authors":"René Marczinzik","doi":"10.1007/s10468-026-10392-7","DOIUrl":"10.1007/s10468-026-10392-7","url":null,"abstract":"<div><p>We answer a question raised by Auslander and Bridger by showing that not every 2-reflexive module is reflexive.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"721 - 723"},"PeriodicalIF":0.6,"publicationDate":"2026-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-026-10392-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
(infty )-Dold-Kan Correspondence via Representation Theory (infty )-基于表征理论的dold - kan对应
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-03-26 DOI: 10.1007/s10468-026-10388-3
Chiara Sava
{"title":"(infty )-Dold-Kan Correspondence via Representation Theory","authors":"Chiara Sava","doi":"10.1007/s10468-026-10388-3","DOIUrl":"10.1007/s10468-026-10388-3","url":null,"abstract":"<div><p>We give a purely derivator-theoretical reformulation and proof of a classic result of Happel and Ladkani, showing that it occurs uniformly across stable derivators and it is then independent of coefficients. The resulting equivalence provides a bridge between homotopy theory and representation theory: indeed, our result is a derivator-theoretic version of the <span>(infty )</span>-Dold-Kan correspondence for bounded chain complexes. Moreover, our equivalence can also be realized as an action of a spectral bimodule in the setting of universal tilting theory developed by Groth and Šťovíček.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"565 - 607"},"PeriodicalIF":0.6,"publicationDate":"2026-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-026-10388-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Higher-Order Hom-Associative Weyl Algebras 高阶共轭Weyl代数
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-03-25 DOI: 10.1007/s10468-026-10391-8
Per Bäck
{"title":"The Higher-Order Hom-Associative Weyl Algebras","authors":"Per Bäck","doi":"10.1007/s10468-026-10391-8","DOIUrl":"10.1007/s10468-026-10391-8","url":null,"abstract":"<div><p>We show that the higher-order Weyl algebras over a field of characteristic zero, which are formally rigid as associative algebras, can be formally deformed in a nontrivial way as hom-associative algebras. We also show that these hom-associative Weyl algebras arise naturally as hom-associative iterated differential polynomial rings, that they contain no zero divisors, are power-associative only when associative, and that they are simple. We then determine their commuters, nuclei, centers, and derivations. Last, we classify all hom-associative Weyl algebras up to isomorphism and conjecture that all nonzero homomorphisms between any two isomorphic hom-associative Weyl algebras are isomorphisms. The latter conjecture turns out to be stably equivalent to the Dixmier Conjecture, and hence also to the Jacobian Conjecture.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"697 - 719"},"PeriodicalIF":0.6,"publicationDate":"2026-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-026-10391-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Acyclic Toric Sheaves 无环环面轮轴
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-03-23 DOI: 10.1007/s10468-026-10383-8
Klaus Altmann, Andreas Hochenegger, Frederik Witt
{"title":"Acyclic Toric Sheaves","authors":"Klaus Altmann,&nbsp;Andreas Hochenegger,&nbsp;Frederik Witt","doi":"10.1007/s10468-026-10383-8","DOIUrl":"10.1007/s10468-026-10383-8","url":null,"abstract":"<div><p>Let <span>(mathcal {E})</span> be a torus-linearised reflexive sheaf over a smooth projective toric variety. Generalising a theorem of Perlman and Smith, we prove an explicit sufficient condition for <span>(mathcal {E})</span> to be acyclic via Weil decorations.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"411 - 422"},"PeriodicalIF":0.6,"publicationDate":"2026-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-026-10383-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752377","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Non-factorizable Ribbon Hopf Algebras 不可因子带状Hopf代数
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-03-18 DOI: 10.1007/s10468-026-10382-9
Quentin Faes, Maksymilian Manko
{"title":"Non-factorizable Ribbon Hopf Algebras","authors":"Quentin Faes,&nbsp;Maksymilian Manko","doi":"10.1007/s10468-026-10382-9","DOIUrl":"10.1007/s10468-026-10382-9","url":null,"abstract":"<div><p>Building on the work of Nenciu we provide examples of non-factorizable ribbon Hopf algebras, and introduce a stronger notion of non-factorizability. These algebras are designed to provide invariants of 4-dimensional 2-handlebodies up to 2-deformations. We prove that some of the invariants derived from these examples are invariants dependent only on the boundary or on the presentation of the fundamental group of the 2-handlebody.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"353 - 409"},"PeriodicalIF":0.6,"publicationDate":"2026-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Quantum Cluster Algebra Structure on the Semi-Derived Hall Algebra 半派生霍尔代数上的量子簇代数结构
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-03-13 DOI: 10.1007/s10468-026-10389-2
Alessandro Contu
{"title":"A Quantum Cluster Algebra Structure on the Semi-Derived Hall Algebra","authors":"Alessandro Contu","doi":"10.1007/s10468-026-10389-2","DOIUrl":"10.1007/s10468-026-10389-2","url":null,"abstract":"<div><p>Using Hernandez–Leclerc’s isomorphism between the derived Hall algebra of a representation-finite quiver <i>Q</i> and the quantum Grothendieck ring of the quantum loop algebra of the Dynkin type of <i>Q</i>, we lift the (quantum) cluster algebra structure of the quantum Grothendieck ring to the semi-derived Hall algebra, introduced by Gorsky, of the category of bounded complexes of projective modules over the path algebra of <i>Q</i>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"609 - 651"},"PeriodicalIF":0.6,"publicationDate":"2026-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Duality Properties for Induced and Coinduced Representations in Positive Characteristic 正特征中诱导表示和共诱导表示的对偶性质
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-03-11 DOI: 10.1007/s10468-026-10384-7
Sophie Chemla
{"title":"Duality Properties for Induced and Coinduced Representations in Positive Characteristic","authors":"Sophie Chemla","doi":"10.1007/s10468-026-10384-7","DOIUrl":"10.1007/s10468-026-10384-7","url":null,"abstract":"<div><p>Let <i>k</i> be a field of positive characteristic <span>(p&gt;2)</span>. Generalizing a result of Farnsteiner and Strade (Math. Z., <b>206</b>, 153–168, 1991), we study the links between coinduced representations and induced representations in the case of restricted Lie superalgebras. As a corollary, we prove a duality property concerning the kernel of coinduced representations of Lie <i>k</i>-superalgebras. This property was already proved by Duflo (Invent. Math., <b>67</b>, 385–393, 1982) for Lie algebras in any characteristic under more restrictive finiteness conditions. It was then generalized to Lie superalgebras in characteristic 0 in previous works Chemla (PhD, 1990; Annales de l’Institut Fourier, <b>44</b>, 1067–1090, 1994 and Mathematische Annalen, <b>297</b>, 371–382, 1994).</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"423 - 442"},"PeriodicalIF":0.6,"publicationDate":"2026-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-026-10384-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
1-Cycle Deformations for Yetter-Drinfeld Coalgebras 叶特-德林菲尔德代数的1环变形
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-03-11 DOI: 10.1007/s10468-026-10377-6
Daniel Bulacu, Blas Torrecillas
{"title":"1-Cycle Deformations for Yetter-Drinfeld Coalgebras","authors":"Daniel Bulacu,&nbsp;Blas Torrecillas","doi":"10.1007/s10468-026-10377-6","DOIUrl":"10.1007/s10468-026-10377-6","url":null,"abstract":"<div><p>For a quasi-Hopf algebra <i>H</i>, we study two types of 1-cycle deformations for a coalgebra <i>C</i> within the category of Yetter-Drinfeld modules over <i>H</i>, <span>({}_H^H{mathcal YD})</span>. The two deformations produce <i>C</i>-comodule structures in <span>({}_H^H{mathcal YD})</span> and new coalgebra structures on <i>C</i> in <span>({}_H^H{mathcal YD})</span>, respectively. We show that the isomorphism types of these structures are described by a 1-homology <span>({mathcal H}^1_H(C, H_0))</span> that we will introduce. Then we apply our results to the so called symplectic fermion quasi-Hopf algebras, algebras recently introduced by Farsad, Gainutdinov and Runkel.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"279 - 329"},"PeriodicalIF":0.6,"publicationDate":"2026-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-026-10377-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Invertible exterior powers 可逆外幂
IF 0.6 4区 数学
Algebras and Representation Theory Pub Date : 2026-03-09 DOI: 10.1007/s10468-025-10374-1
Kevin Coulembier
{"title":"Invertible exterior powers","authors":"Kevin Coulembier","doi":"10.1007/s10468-025-10374-1","DOIUrl":"10.1007/s10468-025-10374-1","url":null,"abstract":"<div><p>We present a proof of the fact that in a symmetric monoidal category over a field of characteristic zero, objects with an invertible exterior power are rigid. As an application we prove two recent conjectures on dimensions in symmetric monoidal categories by Baez, Moeller and Trimble and further conjectures by Baez and Trimble.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"29 2","pages":"221 - 230"},"PeriodicalIF":0.6,"publicationDate":"2026-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10374-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147752378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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