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Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations 椭圆量子环状代数、Z-代数结构与表示
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-02-13 DOI: 10.1007/s10468-024-10251-3
Hitoshi Konno, Kazuyuki Oshima
{"title":"Elliptic Quantum Toroidal Algebras, Z-algebra Structure and Representations","authors":"Hitoshi Konno,&nbsp;Kazuyuki Oshima","doi":"10.1007/s10468-024-10251-3","DOIUrl":"10.1007/s10468-024-10251-3","url":null,"abstract":"<div><p>We introduce a new elliptic quantum toroidal algebra <span>(U_{q,kappa ,p}({mathfrak {g}}_{tor}))</span> associated with an arbitrary toroidal algebra <span>({mathfrak {g}}_{tor})</span>. We show that <span>(U_{q,kappa ,p}({mathfrak {g}}_{tor}))</span> contains two elliptic quantum algebras associated with a corresponding affine Lie algebra <span>(widehat{mathfrak {g}})</span> as subalgebras. They are analogue of the horizontal and the vertical subalgebras in the quantum toroidal algebra <span>(U_{q,kappa }({mathfrak {g}}_{tor}))</span>. A Hopf algebroid structure is introduced as a co-algebra structure of <span>(U_{q,kappa ,p}({mathfrak {g}}_{tor}))</span> using the Drinfeld comultiplication. We also investigate the <i>Z</i>-algebra structure of <span>(U_{q,kappa ,p}({mathfrak {g}}_{tor}))</span> and show that the <i>Z</i>-algebra governs the irreducibility of the level <span>((k (ne 0),l))</span>-infinite dimensional <span>(U_{q,kappa ,p}({mathfrak {g}}_{tor}))</span>-modules in the same way as in the elliptic quantum group <span>(U_{q,p}(widehat{mathfrak {g}}))</span>. As an example, we construct the level (1, <i>l</i>) irreducible representation of <span>(U_{q,kappa ,p}({mathfrak {g}}_{tor}))</span> for the simply laced <span>({mathfrak {g}}_{tor})</span>. We also construct the level (0, 1) representation of <span>(U_{q,kappa ,p}({mathfrak {gl}}_{N,tor}))</span> and discuss a conjecture on its geometric interpretation as an action on the torus equivariant elliptic cohomology of the affine <span>(A_{N-1})</span> quiver variety.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139772620","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
Presentations of Braid Groups of Type A Arising from ((m+2))-angulations of Regular Polygons 由正多边形的 $$(m+2)$$ -angulation 产生的 A 型辫状花序群的演示文稿
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-02-02 DOI: 10.1007/s10468-024-10257-x
Davide Morigi
{"title":"Presentations of Braid Groups of Type A Arising from ((m+2))-angulations of Regular Polygons","authors":"Davide Morigi","doi":"10.1007/s10468-024-10257-x","DOIUrl":"10.1007/s10468-024-10257-x","url":null,"abstract":"<div><p>Coloured quiver mutation, introduced by Buan, A.B., Thomas, H (Adv. Math. <b>222</b>(3), 971–995 2009), gives a combinatorial interpretation of tilting in higher cluster categories. In type <i>A</i> work of Baur, K., Marsh, B. (Trans. Am. Math. Soc. <b>360</b>(11), 5789-5803 2008) shows that <i>m</i>-coloured quivers and <i>m</i>-coloured quiver mutations have a nice geometrical description, given in terms of <span>((m+2))</span>-angulations of a regular polygon, and rotations of an <i>m</i>-diagonal. In this paper, using such correspondence, we describe presentations of braid groups of type <i>A</i> arising from coloured quivers of mutation type <i>A</i>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10257-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139663400","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
Unique Factorization for Tensor Products of Parabolic Verma Modules 抛物线维尔马模块张量乘的唯一因式分解
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-02-01 DOI: 10.1007/s10468-024-10254-0
K. N. Raghavan, V. Sathish Kumar, R. Venkatesh, Sankaran Viswanath
{"title":"Unique Factorization for Tensor Products of Parabolic Verma Modules","authors":"K. N. Raghavan,&nbsp;V. Sathish Kumar,&nbsp;R. Venkatesh,&nbsp;Sankaran Viswanath","doi":"10.1007/s10468-024-10254-0","DOIUrl":"10.1007/s10468-024-10254-0","url":null,"abstract":"<div><p>Let <span>(mathfrak g)</span> be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra <span>(mathfrak h)</span>. We prove a unique factorization property for tensor products of parabolic Verma modules. More generally, we prove unique factorization for products of characters of parabolic Verma modules when restricted to certain subalgebras of <span>(mathfrak h)</span>. These include fixed point subalgebras of <span>(mathfrak h)</span> under subgroups of diagram automorphisms of <span>(mathfrak g)</span> and twisted graph automorphisms in the affine case.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139663407","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
Preradicals Over Some Group Algebras 某些群代数上的悖论
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-01-25 DOI: 10.1007/s10468-024-10256-y
Rogelio Fernández-Alonso, Benigno Mercado, Silvia Gavito
{"title":"Preradicals Over Some Group Algebras","authors":"Rogelio Fernández-Alonso,&nbsp;Benigno Mercado,&nbsp;Silvia Gavito","doi":"10.1007/s10468-024-10256-y","DOIUrl":"10.1007/s10468-024-10256-y","url":null,"abstract":"<div><p>For a field <span>(varvec{K})</span> and a finite group <span>(varvec{G})</span>, we study the lattice of preradicals over the group algebra <span>(varvec{KG})</span>, denoted by <span>(varvec{KG})</span>-<span>(varvec{pr})</span>. We show that if <span>(varvec{KG})</span> is a semisimple algebra, then <span>(varvec{KG})</span>-<span>(varvec{pr})</span> is completely described, and we establish conditions for counting the number of its atoms in some specific cases. If <span>(varvec{KG})</span> is an algebra of finite representation type, but not a semisimple one, we completely describe <span>(varvec{KG})</span>-<span>(varvec{pr})</span> when the characteristic of <span>(varvec{K})</span> is a prime <span>(varvec{p})</span> and <span>(varvec{G})</span> is a cyclic <span>(varvec{p})</span>-group. For group algebras of infinite representation type, we show that the lattices of preradicals over two representative families of such algebras are not sets (in which case, we say the algebras are <span>(varvec{mathfrak {p}})</span>-large). Besides, we provide new examples of <span>(varvec{mathfrak {p}})</span>-large algebras. Finally, we prove the main theorem of this paper which characterizes the representation type of group algebras <span>(varvec{KG})</span> in terms of their lattice of preradicals.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139587328","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
Publisher Correction: Quipu Quivers and Nakayama Algebras with Almost Separate Relations 出版商更正：具有几乎独立关系的奎布四元组和中山代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-01-25 DOI: 10.1007/s10468-024-10252-2
Didrik Fosse
{"title":"Publisher Correction: Quipu Quivers and Nakayama Algebras with Almost Separate Relations","authors":"Didrik Fosse","doi":"10.1007/s10468-024-10252-2","DOIUrl":"10.1007/s10468-024-10252-2","url":null,"abstract":"","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10252-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413572","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
A Quantization of the Loday-Ronco Hopf Algebra Loday-Ronco 霍普夫代数的量子化
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-01-20 DOI: 10.1007/s10468-024-10253-1
João N. Esteves
{"title":"A Quantization of the Loday-Ronco Hopf Algebra","authors":"João N. Esteves","doi":"10.1007/s10468-024-10253-1","DOIUrl":"10.1007/s10468-024-10253-1","url":null,"abstract":"<div><p>We propose a quantization algebra of the Loday-Ronco Hopf algebra <span>(k[Y^infty ])</span>, based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the Loday-Ronco Hopf algebra of planar binary trees is a space of solutions for the genus 0 version of Topological Recursion, and that an extension of the Loday Ronco Hopf algebra as to include some new graphs with loops is the correct setting to find a solution space for arbitrary genus. Here we show that this new algebra <span>(k[Y^infty ]_h)</span> is still a Hopf algebra that can be seen in some sense to be made precise in the text as a quantization of the Hopf algebra of planar binary trees, and that the solution space of Topological Recursion <span>(mathcal {A}^h_{text {TopRec}})</span> is a subalgebra of a quotient algebra <span>(mathcal {A}_{text {Reg}}^h)</span> obtained from <span>(k[Y^infty ]_h)</span> that nevertheless doesn’t inherit the Hopf algebra structure. We end the paper with a discussion on the cohomology of <span>(mathcal {A}^h_{text {TopRec}})</span> in low degree.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10253-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139509411","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
Minimal Triangular Structures on Abelian Extensions 阿贝尔扩展上的最小三角形结构
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-01-12 DOI: 10.1007/s10468-023-10250-w
Hong Fei Zhang, Kun Zhou
{"title":"Minimal Triangular Structures on Abelian Extensions","authors":"Hong Fei Zhang,&nbsp;Kun Zhou","doi":"10.1007/s10468-023-10250-w","DOIUrl":"10.1007/s10468-023-10250-w","url":null,"abstract":"<div><p>We study minimal triangular structures on abelian extensions. In particular, we construct a family of minimal triangular semisimple Hopf algebras and prove that the Hopf algebra <span>(H_{b:y})</span> in the semisimple Hopf algebras of dimension 16 classified by Y. Kashina in 2000 is minimal triangular Hopf algebra with smallest dimension among non-trivial semisimple triangular Hopf algebras (i.e. not group algebras or their dual).</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139465152","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 Note on Singularity Categories and Triangular Matrix Algebras 奇异性类别和三角矩阵代数的说明
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-01-06 DOI: 10.1007/s10468-023-10249-3
Yongyun Qin
{"title":"A Note on Singularity Categories and Triangular Matrix Algebras","authors":"Yongyun Qin","doi":"10.1007/s10468-023-10249-3","DOIUrl":"10.1007/s10468-023-10249-3","url":null,"abstract":"<div><p>Let <span>(Lambda = left[ begin{array}{cc} A &amp;{} 0 M &amp;{} B end{array}right] )</span> be an Artin algebra and <span>(_BM_A)</span> a <i>B</i>-<i>A</i>-bimodule. We prove that there is a triangle equivalence <span>(D_{sg}(Lambda ) cong D_{sg}(A)coprod D_{sg}(B))</span> between the corresponding singularity categories if <span>(_BM)</span> is semi-simple and <span>(M_A)</span> is projective. As a result, we obtain a new method for describing the singularity categories of certain bounded quiver algebras.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139373363","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
Fullness of the Kuznetsov–Polishchuk Exceptional Collection for the Spinor Tenfold 库兹涅佐夫-波利什丘克旋转体十倍异常集合的丰满度
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-12-28 DOI: 10.1007/s10468-023-10246-6
Riccardo Moschetti, Marco Rampazzo
{"title":"Fullness of the Kuznetsov–Polishchuk Exceptional Collection for the Spinor Tenfold","authors":"Riccardo Moschetti,&nbsp;Marco Rampazzo","doi":"10.1007/s10468-023-10246-6","DOIUrl":"10.1007/s10468-023-10246-6","url":null,"abstract":"<div><p>Kuznetsov and Polishchuk provided a general algorithm to construct exceptional collections of maximal length for homogeneous varieties of type <i>A</i>, <i>B</i>, <i>C</i>, <i>D</i>. We consider the case of the spinor tenfold and we prove that the corresponding collection is full, i.e. it generates the whole derived category of coherent sheaves. We also verify strongness and purity of such collection. As a step of the proof, we construct some resolutions of homogeneous vector bundles which might be of independent interest.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139065372","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
Growth Rates of the Number of Indecomposable Summands in Tensor Powers 张量幂中不可分解求和数的增长率
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-12-19 DOI: 10.1007/s10468-023-10245-7
Kevin Coulembier, Victor Ostrik, Daniel Tubbenhauer
{"title":"Growth Rates of the Number of Indecomposable Summands in Tensor Powers","authors":"Kevin Coulembier,&nbsp;Victor Ostrik,&nbsp;Daniel Tubbenhauer","doi":"10.1007/s10468-023-10245-7","DOIUrl":"10.1007/s10468-023-10245-7","url":null,"abstract":"<div><p>In this paper we study the asymptotic behavior of the number of summands in tensor products of finite dimensional representations of affine (semi)group (super)schemes and related objects.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10245-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138744864","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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