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Dimer Algebras, Ghor Algebras, and Cyclic Contractions 二聚体代数、古尔代数和循环收缩
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-09-07 DOI: 10.1007/s10468-023-10224-y
Charlie Beil
{"title":"Dimer Algebras, Ghor Algebras, and Cyclic Contractions","authors":"Charlie Beil","doi":"10.1007/s10468-023-10224-y","DOIUrl":"10.1007/s10468-023-10224-y","url":null,"abstract":"<div><p>A ghor algebra is the path algebra of a dimer quiver on a surface, modulo relations that come from the perfect matchings of its quiver. Such algebras arise from abelian quiver gauge theories in physics. We show that a ghor algebra <span>(Lambda )</span> on a torus is a dimer algebra (a quiver with potential) if and only if it is noetherian, and otherwise <span>(Lambda )</span> is the quotient of a dimer algebra by homotopy relations. Furthermore, we classify the simple <span>(Lambda )</span>-modules of maximal dimension and give an explicit description of the center of <span>(Lambda )</span> using a special subset of perfect matchings. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10224-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52368933","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
An Indicator Formula for the Hopf Algebra (k^{S_{n-1}}#kC_n) Hopf代数的一个指标公式 $$k^{S_{n-1}}#kC_n$$
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-09-06 DOI: 10.1007/s10468-023-10230-0
Kayla Orlinsky
{"title":"An Indicator Formula for the Hopf Algebra (k^{S_{n-1}}#kC_n)","authors":"Kayla Orlinsky","doi":"10.1007/s10468-023-10230-0","DOIUrl":"10.1007/s10468-023-10230-0","url":null,"abstract":"<div><p>The semisimple bismash product Hopf algebra <span>(J_n=k^{S_{n-1}}#kC_n)</span> for an algebraically closed field <i>k</i> is constructed using the matched pair actions of <span>(C_n)</span> and <span>(S_{n-1})</span> on each other. In this work, we reinterpret these actions and use an understanding of the involutions of <span>(S_{n-1})</span> to derive a new Froebnius-Schur indicator formula for irreps of <span>(J_n)</span> and show that for <i>n</i> odd, all indicators of <span>(J_n)</span> are nonnegative. We also derive a variety of counting formulas including Theorem 6.2.2 which fully describes the indicators of all 2-dimensional irreps of <span>(J_n)</span> and Theorem 6.1.2 which fully describes the indicators of all odd-dimensional irreps of <span>(J_n)</span> and use these formulas to show that nonzero indicators become rare for large <i>n</i>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10230-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44116669","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 Drinfeld-Type Presentation of the Orthosymplectic Yangians 正辛杨岩系的旱田型表现
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-08-18 DOI: 10.1007/s10468-023-10227-9
A. I. Molev
{"title":"A Drinfeld-Type Presentation of the Orthosymplectic Yangians","authors":"A. I. Molev","doi":"10.1007/s10468-023-10227-9","DOIUrl":"10.1007/s10468-023-10227-9","url":null,"abstract":"<div><p>We use the Gauss decomposition of the generator matrix in the <i>R</i>-matrix presentation of the Yangian for the orthosymplectic Lie superalgebra <span>(mathfrak {osp}_{N|2m})</span> to produce its Drinfeld-type presentation. The results rely on a super-version of the embedding theorem which allows one to identify a subalgebra in the <i>R</i>-matrix presentation which is isomorphic to the Yangian associated with <span>(mathfrak {osp}_{N|2m-2})</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10227-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45744008","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 Uniqueness Property of (tau )-Exceptional Sequences $$tau $$ -例外序列的唯一性
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-08-16 DOI: 10.1007/s10468-023-10226-w
Eric J. Hanson, Hugh Thomas
{"title":"A Uniqueness Property of (tau )-Exceptional Sequences","authors":"Eric J. Hanson,&nbsp;Hugh Thomas","doi":"10.1007/s10468-023-10226-w","DOIUrl":"10.1007/s10468-023-10226-w","url":null,"abstract":"<div><p>Recently, Buan and Marsh showed that if two complete <span>(tau )</span>-exceptional sequences agree in all but at most one term, then they must agree everywhere, provided the algebra is <span>(tau )</span>-tilting finite. They conjectured that the result holds without that assumption. We prove their conjecture. Along the way, we also show that the dimension vectors of the modules in a <span>(tau )</span>-exceptional sequence are linearly independent.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48208544","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
Identities of Inverse Chevalley Type for the Graded Characters of Level-Zero Demazure Submodules over Quantum Affine Algebras of Type C C型量子仿射代数上零能级Demazure子模的梯度特征的逆Chevalley型恒等式
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-08-15 DOI: 10.1007/s10468-023-10221-1
Takafumi Kouno, Satoshi Naito, Daniel Orr
{"title":"Identities of Inverse Chevalley Type for the Graded Characters of Level-Zero Demazure Submodules over Quantum Affine Algebras of Type C","authors":"Takafumi Kouno,&nbsp;Satoshi Naito,&nbsp;Daniel Orr","doi":"10.1007/s10468-023-10221-1","DOIUrl":"10.1007/s10468-023-10221-1","url":null,"abstract":"<div><p>We provide identities of inverse Chevalley type for the graded characters of level-zero Demazure submodules of extremal weight modules over a quantum affine algebra of type <i>C</i>. These identities express the product <span>(e^{mu } text {gch} ~V_{x}^{-}(lambda ))</span> of the (one-dimensional) character <span>(e^{mu })</span>, where <span>(mu )</span> is a (not necessarily dominant) minuscule weight, with the graded character gch<span>(V_{x}^{-}(lambda ))</span> of the level-zero Demazure submodule <span>(V_{x}^{-}(lambda ))</span> over the quantum affine algebra <span>(U_{textsf{q}}(mathfrak {g}_{textrm{af}}))</span> as an explicit finite linear combination of the graded characters of level-zero Demazure submodules. These identities immediately imply the corresponding inverse Chevalley formulas for the torus-equivariant <i>K</i>-group of the semi-infinite flag manifold <span>(textbf{Q}_{G})</span> associated to a connected, simply-connected and simple algebraic group <i>G</i> of type <i>C</i>. Also, we derive cancellation-free identities from the identities above of inverse Chevalley type in the case that <span>(mu )</span> is a standard basis element <span>({varepsilon }_{k})</span> in the weight lattice <i>P</i> of <i>G</i>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10221-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44412270","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
Quantization of Deformed Cluster Poisson Varieties 变形簇Poisson变种的量化
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-08-09 DOI: 10.1007/s10468-023-10209-x
Man-Wai Mandy Cheung, Juan Bosco Frías-Medina, Timothy Magee
{"title":"Quantization of Deformed Cluster Poisson Varieties","authors":"Man-Wai Mandy Cheung,&nbsp;Juan Bosco Frías-Medina,&nbsp;Timothy Magee","doi":"10.1007/s10468-023-10209-x","DOIUrl":"10.1007/s10468-023-10209-x","url":null,"abstract":"<div><p>Fock and Goncharov described a quantization of cluster <span>(mathcal {X})</span>-varieties (also known as <i>cluster Poisson varieties</i>) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér. <b>42</b>(6), 865–930 2009). Meanwhile, families of deformations of cluster <span>(mathcal {X})</span>-varieties were introduced in Bossinger et al. (Compos. Math. <b>156</b>(10), 2149–2206, 2020). In this paper we show that the two constructions are compatible– we extend the Fock-Goncharov quantization of <span>(mathcal {X})</span>-varieties to the families of Bossinger et al. (Compos. Math. <b>156</b>(10), 2149–2206, 2020). As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of <span>(mathcal {A})</span>-varieties (Berenstein and Zelevinsky, Adv. Math. <b>195</b>(2), 405–455, 2005). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in Lee, et al. (Proc. Natl. Acad. Sci. <b>111</b>(27), 9712–9716, 2014), we compute a counter-example to quantum positivity of the quantum theta basis.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45363995","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 Constructing Quasi Modules for Quantum Vertex Algebras from Twisted Yangians 关于由扭Yangian构造量子点代数拟模的一个注记
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-07-24 DOI: 10.1007/s10468-023-10215-z
Slaven Kožić, Marina Sertić
{"title":"A Note on Constructing Quasi Modules for Quantum Vertex Algebras from Twisted Yangians","authors":"Slaven Kožić,&nbsp;Marina Sertić","doi":"10.1007/s10468-023-10215-z","DOIUrl":"10.1007/s10468-023-10215-z","url":null,"abstract":"<div><p>In this note, we consider the twisted Yangians <span>(text {Y}(mathfrak {g}_N))</span> associated with the orthogonal and symplectic Lie algebras <span>(mathfrak {g}_N=mathfrak {o}_N,mathfrak {sp}_N)</span>. First, we introduce a certain subalgebra <span>(text {A}_c(mathfrak {g}_N))</span> of the double Yangian for <span>(mathfrak {gl}_N)</span> at the level <span>(cin mathbb {C})</span>, which contains the centrally extended <span>(text {Y}(mathfrak {g}_N))</span> at the level <i>c</i> as well as its vacuum module <span>(mathcal {M}_c(mathfrak {g}_N))</span>. Next, we employ its structure to construct examples of quasi modules for the quantum affine vertex algebra <span>(mathcal {V}_c(mathfrak {gl}_N))</span> associated with the Yang <i>R</i>-matrix. Finally, we use the description of the center of <span>(mathcal {V}_c(mathfrak {gl}_N))</span> to obtain explicit formulae for families of central elements for a certain completion of <span>(text {A}_c(mathfrak {g}_N))</span> and invariants of <span>(mathcal {M}_c(mathfrak {g}_N))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46095062","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
Module Categories of Small Radical Nilpotency 小根幂零的模范畴
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-07-24 DOI: 10.1007/s10468-023-10211-3
Shiping Liu, Youqi Yin
{"title":"Module Categories of Small Radical Nilpotency","authors":"Shiping Liu,&nbsp;Youqi Yin","doi":"10.1007/s10468-023-10211-3","DOIUrl":"10.1007/s10468-023-10211-3","url":null,"abstract":"<div><p>This paper aims to initiate a study of the representation theory of representation-finite artin algebras in terms of the nilpotency of the radical of their module category. Firstly, we shall calculate this nilpotency explicitly for hereditary algebras of type <span>(mathbb {A}_n)</span> and for Nakayama algebras. Surprisingly, this nilpotency for a given algebra coincides with its Loewy length if and only if the algebra is a hereditary Nakayama algebra. Secondly, we shall find all artin algebras for which this nilpotency is equal to any given positive integer up to four and describe completely their module category.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43244892","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
Wild Local Structures of Automorphic Lie Algebras 自同构李代数的Wild局部结构
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-07-20 DOI: 10.1007/s10468-023-10208-y
Drew Damien Duffield, Vincent Knibbeler, Sara Lombardo
{"title":"Wild Local Structures of Automorphic Lie Algebras","authors":"Drew Damien Duffield,&nbsp;Vincent Knibbeler,&nbsp;Sara Lombardo","doi":"10.1007/s10468-023-10208-y","DOIUrl":"10.1007/s10468-023-10208-y","url":null,"abstract":"<div><p>We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10208-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47936091","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
Classifying Recollements of Derived Module Categories for Derived Discrete Algebras 导出离散代数的导出模范畴的分类集合
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-07-12 DOI: 10.1007/s10468-023-10220-2
Xiuli Bian
{"title":"Classifying Recollements of Derived Module Categories for Derived Discrete Algebras","authors":"Xiuli Bian","doi":"10.1007/s10468-023-10220-2","DOIUrl":"10.1007/s10468-023-10220-2","url":null,"abstract":"<div><p>We study a class of derived discrete Nakayama algebras. All indecomposable compact objects in the derived module category are determined and all recollements generated by the indecomposable compact exceptional objects are classified. It reveals that all such recollements are derived equivalent to stratifying recollements. As a byproduct, this confirms a question due to Xi for these recollements.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43022067","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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