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Symmetries of Algebras Captured by Actions of Weak Hopf Algebras 弱Hopf代数作用俘获的代数的对称性
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-11-13 DOI: 10.1007/s10468-024-10295-5
Fabio Calderón, Hongdi Huang, Elizabeth Wicks, Robert Won
{"title":"Symmetries of Algebras Captured by Actions of Weak Hopf Algebras","authors":"Fabio Calderón,&nbsp;Hongdi Huang,&nbsp;Elizabeth Wicks,&nbsp;Robert Won","doi":"10.1007/s10468-024-10295-5","DOIUrl":"10.1007/s10468-024-10295-5","url":null,"abstract":"<div><p>In this paper, we present a generalization of well-established results regarding symmetries of <span>(Bbbk )</span>-algebras, where <span>(Bbbk )</span> is a field. Traditionally, for a <span>(Bbbk )</span>-algebra <i>A</i>, the group of <span>(Bbbk )</span>-algebra automorphisms of <i>A</i> captures the symmetries of <i>A</i> via group actions. Similarly, the Lie algebra of derivations of <i>A</i> captures the symmetries of <i>A</i> via Lie algebra actions. In this paper, given a category <span>(mathcal {C})</span> whose objects possess <span>(Bbbk )</span>-linear monoidal categories of modules, we introduce an objec <span>(operatorname {Sym}_{mathcal {C}}(A))</span> that captures the symmetries of <i>A</i> via actions of objects in <span>(mathcal {C})</span>. Our study encompasses various categories whose objects include groupoids, Lie algebroids, and more generally, cocommutative weak Hopf algebras. Notably, we demonstrate that for a positively graded non-connected <span>(Bbbk )</span>-algebra <i>A</i>, some of its symmetries are naturally captured within the weak Hopf framework.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2217 - 2266"},"PeriodicalIF":0.5,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994464","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
Equational Quantum Quasigroups 方程量子拟群
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-11-11 DOI: 10.1007/s10468-024-10300-x
Jonathan D. H. Smith
{"title":"Equational Quantum Quasigroups","authors":"Jonathan D. H. Smith","doi":"10.1007/s10468-024-10300-x","DOIUrl":"10.1007/s10468-024-10300-x","url":null,"abstract":"<div><p>As a self-dual framework to unify the study of quasigroups and Hopf algebras, quantum quasigroups are defined using a quantum analogue of the combinatorial approach to classical quasigroups, merely requiring invertibility of the left and right composites. In this paper, quantum quasigroups are redefined with a quantum analogue of the equational approach to classical quasigroups. Here, the left and right composites of auxiliary quantum quasigroups participate in diagrams whose commutativity witnesses the required invertibilities. Whenever the original and two auxiliary quantum quasigroups appear on an equal footing, the triality symmetry of the language of equational quasigroups is replicated. In particular, the problem arises as to when this triality emerges in the Hopf algebra context.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2355 - 2387"},"PeriodicalIF":0.5,"publicationDate":"2024-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994358","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
On J-folded Alcove Paths and Combinatorial Representations of Affine Hecke Algebras 仿射Hecke代数的j折叠凹形路径与组合表示
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-11-05 DOI: 10.1007/s10468-024-10293-7
Jérémie Guilhot, Eloise Little, James Parkinson
{"title":"On J-folded Alcove Paths and Combinatorial Representations of Affine Hecke Algebras","authors":"Jérémie Guilhot,&nbsp;Eloise Little,&nbsp;James Parkinson","doi":"10.1007/s10468-024-10293-7","DOIUrl":"10.1007/s10468-024-10293-7","url":null,"abstract":"<div><p>We introduce the combinatorial model of <i>J</i>-folded alcove paths in an affine Weyl group and construct representations of affine Hecke algebras using this model. We study boundedness of these representations, and we state conjectures linking our combinatorial formulae to Kazhdan-Lusztig theory and Opdam’s Plancherel Theorem.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2131 - 2185"},"PeriodicalIF":0.5,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10293-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994378","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
Generalised Lat-Igusa-Todorov Algebras and Morita Contexts 广义latu - igusa - todorov代数与Morita上下文
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-10-14 DOI: 10.1007/s10468-024-10289-3
Marcelo Lanzilotta, José Vivero
{"title":"Generalised Lat-Igusa-Todorov Algebras and Morita Contexts","authors":"Marcelo Lanzilotta,&nbsp;José Vivero","doi":"10.1007/s10468-024-10289-3","DOIUrl":"10.1007/s10468-024-10289-3","url":null,"abstract":"<div><p>In this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising from a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT <span>(mathbb {K})</span>-algebra with a path algebra of a finite quiver without oriented cycles is GLIT.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2045 - 2066"},"PeriodicalIF":0.5,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10289-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994470","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
Highest Weight Modules for Affine and Loop Superalgebras of (mathfrak {osp}_{1|2}(mathbb C)) 的仿射超代数和环超代数的最大权模 (mathfrak {osp}_{1|2}(mathbb C))
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-09-28 DOI: 10.1007/s10468-024-10292-8
Fulin Chen, Xin Huang, Shaobin Tan
{"title":"Highest Weight Modules for Affine and Loop Superalgebras of (mathfrak {osp}_{1|2}(mathbb C))","authors":"Fulin Chen,&nbsp;Xin Huang,&nbsp;Shaobin Tan","doi":"10.1007/s10468-024-10292-8","DOIUrl":"10.1007/s10468-024-10292-8","url":null,"abstract":"<div><p>This paper is about the highest weight module theory for affine superalgebra <span>(widetilde{mathfrak g})</span> of <span>({mathfrak g}={mathfrak {osp}_{1|2}(mathbb C)})</span> and loop superalgebra <span>({mathfrak g}{otimes }{mathbb {C}}[t,t^{-1}])</span>. Among the main results, we obtain (i) a necessary and sufficient condition for Verma type <span>(ell )</span>-highest weight <span>(widetilde{mathfrak g})</span>-modules to be irreducible; (ii) a free field(-like) realization of all irreducible <span>(ell )</span>-highest weight <span>(widetilde{mathfrak g})</span>-modules; (iii) a character formula for all irreducible <span>(ell )</span>-highest weight <span>(widetilde{mathfrak g})</span>-modules with finite dimensional weight spaces. We also obtain three similar results for highest weight <span>({mathfrak g}{otimes }{mathbb {C}}[t,t^{-1}])</span>-modules.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2099 - 2130"},"PeriodicalIF":0.5,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995695","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
Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group 扩展双仿射编织群作用下量子环面代数的自同构
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-09-24 DOI: 10.1007/s10468-024-10291-9
Duncan Laurie
{"title":"Automorphisms of Quantum Toroidal Algebras from an Action of the Extended Double Affine Braid Group","authors":"Duncan Laurie","doi":"10.1007/s10468-024-10291-9","DOIUrl":"10.1007/s10468-024-10291-9","url":null,"abstract":"<div><p>We first construct an action of the extended double affine braid group <span>(mathcal {ddot{B}})</span> on the quantum toroidal algebra <span>(U_{q}(mathfrak {g}_{textrm{tor}}))</span> in untwisted and twisted types. As a crucial step in the proof, we obtain a finite Drinfeld new style presentation for a broad class of quantum affinizations. In the simply laced cases, using our action and certain involutions of <span>(mathcal {ddot{B}})</span> we produce automorphisms and anti-involutions of <span>(U_{q}(mathfrak {g}_{textrm{tor}}))</span> which exchange the horizontal and vertical subalgebras. Moreover, they switch the central elements <i>C</i> and <span>(k_{0}^{a_{0}}dots k_{n}^{a_{n}})</span> up to inverse. This can be viewed as the analogue, for these quantum toroidal algebras, of the duality for double affine braid groups used by Cherednik to realise the difference Fourier transform in his celebrated proof of the Macdonald evaluation conjectures. Our work generalises existing results in type <i>A</i> due to Miki which have been instrumental in the study of the structure and representation theory of <span>(U_{q}(mathfrak {sl}_{n+1,textrm{tor}}))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2067 - 2097"},"PeriodicalIF":0.5,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-024-10291-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142995845","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 Deformed Tanisaki-Garsia-Procesi Modules 变形Tanisaki-Garsia-Procesi模块
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-09-20 DOI: 10.1007/s10468-024-10288-4
Maico Freitas, Evgeny Mukhin
{"title":"The Deformed Tanisaki-Garsia-Procesi Modules","authors":"Maico Freitas,&nbsp;Evgeny Mukhin","doi":"10.1007/s10468-024-10288-4","DOIUrl":"10.1007/s10468-024-10288-4","url":null,"abstract":"<div><p>The polynomial ideals studied by A. Garsia and C. Procesi play an important role in the theory of Kostka polynomials. We give multiparameter flat deformations of these ideals and define an action of the extended affine symmetric group on the corresponding quotient algebras multiplied by the sign representation. We show that the images of these modules under the affine Schur-Weyl duality are dual to the local Weyl modules for the loop algebra <span>(mathfrak {sl}_{n+1}[t^{pm 1}])</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 6","pages":"2019 - 2044"},"PeriodicalIF":0.5,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142994937","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 Schur-Weyl Dualities for GL(m) and GL(m|n) 关于 GL(m) 和 GL(m|n) 的舒尔-韦尔对偶性的说明
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-09-17 DOI: 10.1007/s10468-024-10290-w
František Marko
{"title":"A Note on Schur-Weyl Dualities for GL(m) and GL(m|n)","authors":"František Marko","doi":"10.1007/s10468-024-10290-w","DOIUrl":"10.1007/s10468-024-10290-w","url":null,"abstract":"<div><p>We use a unified elementary approach to prove the second part of classical, mixed, super, and mixed super Schur-Weyl dualities for general linear groups and supergroups over an infinite ground field of arbitrary characteristic. These dualities describe the endomorphism algebras of the tensor space and mixed tensor space, respectively, over the group algebra of the symmetric group and the Brauer wall algebra, respectively. Our main new results are the second part of the mixed Schur-Weyl dualities and mixed super Schur-Weyl dualities over an infinite ground field of positive characteristic.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 5","pages":"1957 - 1979"},"PeriodicalIF":0.5,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142264524","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
Homologically Smooth Connected Cochain DGAs 同源光滑连接共链 DGA
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-09-09 DOI: 10.1007/s10468-024-10287-5
X.-F. Mao
{"title":"Homologically Smooth Connected Cochain DGAs","authors":"X.-F. Mao","doi":"10.1007/s10468-024-10287-5","DOIUrl":"10.1007/s10468-024-10287-5","url":null,"abstract":"<div><p>Let <span>(mathscr {A})</span> be a connected cochain DG algebra such that <span>(H(mathscr {A}))</span> is a Noetherian graded algebra. We give some criteria for <span>(mathscr {A})</span> to be homologically smooth in terms of the singularity category, the cone length of the canonical module <i>k</i> and the global dimension of <span>(mathscr {A})</span>. For any cohomologically finite DG <span>(mathscr {A})</span>-module <i>M</i>, we show that it is compact when <span>(mathscr {A})</span> is homologically smooth. If <span>(mathscr {A})</span> is in addition Gorenstein, we get </p><div><div><span>$$begin{aligned} textrm{CMreg}M = textrm{depth}_{mathscr {A}}mathscr {A} + mathrm {Ext.reg}, M&lt;infty , end{aligned}$$</span></div></div><p>where <span>(textrm{CMreg}M)</span> is the Castelnuovo-Mumford regularity of <i>M</i>, <span>(textrm{depth}_{mathscr {A}}mathscr {A})</span> is the depth of <span>(mathscr {A})</span> and <span>( mathrm {Ext.reg}, M)</span> is the Ext-regularity of <i>M</i>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 5","pages":"1931 - 1955"},"PeriodicalIF":0.5,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142226102","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
Modified Ariki-Koike Algebra and Yokonuma-Hecke like Relations 修正的有熊小池代数和横沼-赫克相似关系
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2024-09-04 DOI: 10.1007/s10468-024-10286-6
Myungho Kim, Sungsoon Kim
{"title":"Modified Ariki-Koike Algebra and Yokonuma-Hecke like Relations","authors":"Myungho Kim,&nbsp;Sungsoon Kim","doi":"10.1007/s10468-024-10286-6","DOIUrl":"10.1007/s10468-024-10286-6","url":null,"abstract":"<div><p>We find new presentations of the modified Ariki-Koike algebra (known also as Shoji’s algebra) <span>(mathcal {H}_{n,r})</span> over an integral domain <i>R</i> associated with a set of parameters <span>(q,u_1,ldots ,u_r)</span> in <i>R</i>. It turns out that the algebra <span>(mathcal {H}_{n,r})</span> has a set of generators <span>(t_1,ldots ,t_n)</span> and <span>(g_1,ldots g_{n-1})</span> subject to some defining relations similar to the relations of Yokonuma-Hecke algebra. We also obtain a presentation of <span>(mathcal {H}_{n,r})</span> which is independent of the choice of <span>(u_1,ldots u_r)</span>. As applications of the presentations, we find an explicit and direct isomorphism between the modified Ariki-Koike algebras with different choices of parameters <span>((u_1,ldots ,u_r))</span>. We also find an explicit trace form on the algebra <span>(mathcal {H}_{n,r})</span> which is symmetrizing provided the parameters <span>(u_1,ldots , u_r)</span> are invertible in <i>R</i>. We show that the symmetric group <span>(mathfrak {S}(r))</span> acts on the algebra <span>(mathcal H_{n,r})</span>, and find a basis and a set of generators of the fixed subalgebra <span>(mathcal H_{n,r}^{mathfrak {S}(r)})</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 5","pages":"1909 - 1930"},"PeriodicalIF":0.5,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204750","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
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