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The Functor (K_{0}^{operatorname {gr}}) is Full and only Weakly Faithful 函数$K_｛0｝^｛operatorname｛gr｝｝$是满的并且只有弱忠实
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-01-25 DOI: 10.1007/s10468-023-10199-w
Lia Vaš
{"title":"The Functor (K_{0}^{operatorname {gr}}) is Full and only Weakly Faithful","authors":"Lia Vaš","doi":"10.1007/s10468-023-10199-w","DOIUrl":"10.1007/s10468-023-10199-w","url":null,"abstract":"<div><p>The Graded Classification Conjecture states that the pointed <span>(K_{0}^{operatorname {gr}})</span>-group is a complete invariant of the Leavitt path algebras of finite graphs when these algebras are considered with their natural grading by <span>(mathbb {Z})</span>. The strong version of this conjecture states that the functor <span>(K_{0}^{operatorname {gr}})</span> is full and faithful when considered on the category of Leavitt path algebras of finite graphs and their graded homomorphisms modulo conjugations by invertible elements of the zero components. We show that the functor <span>(K_{0}^{operatorname {gr}})</span> is full for the unital Leavitt path algebras of countable graphs and that it is faithful (modulo specified conjugations) only in a certain weaker sense.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49466201","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
Non-Commutative Resolutions for the Discriminant of the Complex Reflection Group G(m, p, 2) 复反射群G（m，p，2）判别式的非交换分辨率
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-01-19 DOI: 10.1007/s10468-022-10193-8
Simon May
{"title":"Non-Commutative Resolutions for the Discriminant of the Complex Reflection Group G(m, p, 2)","authors":"Simon May","doi":"10.1007/s10468-022-10193-8","DOIUrl":"10.1007/s10468-022-10193-8","url":null,"abstract":"<div><p>We show that for the family of complex reflection groups <i>G</i> = <i>G</i>(<i>m</i>, <i>p</i>,2) appearing in the Shephard–Todd classification, the endomorphism ring of the reduced hyperplane arrangement <i>A</i>(<i>G</i>) is a non-commutative resolution for the coordinate ring of the discriminant Δ of <i>G</i>. This furthers the work of Buchweitz, Faber and Ingalls who showed that this result holds for any true reflection group. In particular, we construct a matrix factorization for Δ from <i>A</i>(<i>G</i>) and decompose it using data from the irreducible representations of <i>G</i>. For <i>G</i>(<i>m</i>, <i>p</i>,2) we give a full decomposition of this matrix factorization, including for each irreducible representation a corresponding maximal Cohen–Macaulay module. The decomposition concludes that the endomorphism ring of the reduced hyperplane arrangement <i>A</i>(<i>G</i>) will be a non-commutative resolution. For the groups <i>G</i>(<i>m</i>,1,2), the coordinate rings of their respective discriminants are all isomorphic to each other. We also calculate and compare the Lusztig algebra for these groups.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-022-10193-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42339374","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
Injective and Tilting Resolutions and a Kazhdan-Lusztig Theory for the General Linear and Symplectic Group 一般线性群和辛群的内射分解和倾斜分解及Kazhdan-Lusztig理论
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-01-11 DOI: 10.1007/s10468-022-10197-4
Rudolf Tange
{"title":"Injective and Tilting Resolutions and a Kazhdan-Lusztig Theory for the General Linear and Symplectic Group","authors":"Rudolf Tange","doi":"10.1007/s10468-022-10197-4","DOIUrl":"10.1007/s10468-022-10197-4","url":null,"abstract":"<div><p>Let <i>k</i> be an algebraically closed field of characteristic <i>p</i> &gt; 0 and let <i>G</i> be a symplectic or general linear group over <i>k</i>. We consider induced modules for <i>G</i> under the assumption that <i>p</i> is bigger than the greatest hook length in the partitions involved. We give explicit constructions of left resolutions of induced modules by tilting modules. Furthermore, we give injective resolutions for induced modules in certain truncated categories. We show that the multiplicities of the indecomposable tilting and injective modules in these resolutions are the coefficients of certain Kazhdan-Lusztig polynomials. We also show that our truncated categories have a Kazhdan-Lusztig theory in the sense of Cline, Parshall and Scott. This builds further on work of Cox-De Visscher and Brundan-Stroppel.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-022-10197-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42853552","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 Nonvanishing First Hochschild Cohomology of Twisted Finite Simple Group Algebras 扭曲有限单群代数的非消失第一Hochschild上同调
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-01-11 DOI: 10.1007/s10468-022-10195-6
William Murphy
{"title":"The Nonvanishing First Hochschild Cohomology of Twisted Finite Simple Group Algebras","authors":"William Murphy","doi":"10.1007/s10468-022-10195-6","DOIUrl":"10.1007/s10468-022-10195-6","url":null,"abstract":"<div><p>Let <i>G</i> be a finite simple group and <i>k</i> be an algebraically closed field of prime characteristic dividing the order of <i>G</i>. We show that for all 2-cocycles <i>α</i> ∈ <i>Z</i><sup>2</sup>(<i>G</i>;<i>k</i><sup>×</sup>), the first Hochschild cohomology group of the twisted group algebra <i>H</i><i>H</i><sup>1</sup>(<i>k</i><sub><i>α</i></sub><i>G</i>) is nonzero.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49628284","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
External Vertices for Crystals of Affine Type A 仿射A型晶体的外部顶点
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2022-12-24 DOI: 10.1007/s10468-022-10194-7
Ola Amara-Omari, Mary Schaps
{"title":"External Vertices for Crystals of Affine Type A","authors":"Ola Amara-Omari,&nbsp;Mary Schaps","doi":"10.1007/s10468-022-10194-7","DOIUrl":"10.1007/s10468-022-10194-7","url":null,"abstract":"<div><p>We demonstrate that for a fixed dominant integral weight and fixed defect <i>d</i>, there are only a finite number of Morita equivalence classes of blocks of cyclotomic Hecke algebras, by combining some combinatorics with the Chuang-Rouquier categorification of integrable highest weight modules over Kac-Moody algebras of affine type A. This is an extension of a proof for symmetric groups of a conjecture known as Donovan’s conjecture. We fix a dominant integral weight Λ. The blocks of cyclotomic Hecke algebras <span>(H^{Lambda }_{n})</span> for the given Λ correspond to the weights <i>P</i>(Λ) of a highest weight representation with highest weight Λ. We connect these weights into a graph we call the reduced crystal <span>(widehat {P}({Lambda }))</span>, in which vertices are connected by <i>i</i>-strings. We define the hub of a weight and show that a vertex is <i>i</i>-external for a residue <i>i</i> if the defect is less than the absolute value of the <i>i</i>-component of the hub. We demonstrate the existence of a bound on the degree after which all vertices of a given defect <i>d</i> are <i>i</i>-external in at least one <i>i</i>-string, lying at the high degree end of the <i>i</i>-string. For <i>e</i> = 2, we calculate an approximation to this bound.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43217279","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 Minimal Tilting Complexes in Highest Weight Categories 关于最高权范畴中的极小倾斜复形
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2022-12-22 DOI: 10.1007/s10468-022-10188-5
Jonathan Gruber
{"title":"On Minimal Tilting Complexes in Highest Weight Categories","authors":"Jonathan Gruber","doi":"10.1007/s10468-022-10188-5","DOIUrl":"10.1007/s10468-022-10188-5","url":null,"abstract":"<div><p>We explain the construction of <i>minimal tilting complexes</i> for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49151379","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 Group Invariants Determined by Modular Group Algebras: Even Versus Odd Characteristic 由模群代数确定的群不变量:偶与奇特征
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2022-12-21 DOI: 10.1007/s10468-022-10182-x
Diego García-Lucas, Ángel del Río, Mima Stanojkovski
{"title":"On Group Invariants Determined by Modular Group Algebras: Even Versus Odd Characteristic","authors":"Diego García-Lucas,&nbsp;Ángel del Río,&nbsp;Mima Stanojkovski","doi":"10.1007/s10468-022-10182-x","DOIUrl":"10.1007/s10468-022-10182-x","url":null,"abstract":"<div><p>Let <i>p</i> be a an odd prime and let <i>G</i> be a finite <i>p</i>-group with cyclic commutator subgroup <span>(G^{prime })</span>. We prove that the exponent and the abelianization of the centralizer of <span>(G^{prime })</span> in <i>G</i> are determined by the group algebra of <i>G</i> over any field of characteristic <i>p</i>. If, additionally, <i>G</i> is 2-generated then almost all the numerical invariants determining <i>G</i> up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of <span>(G^{prime })</span> is determined. These claims are known to be false for <i>p</i> = 2.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-022-10182-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45961136","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
Computation of the Homology of the Complexes of Finite Verma Modules for ({K}_{4}^{prime }) ${K}_{4}^{素数}$有限Verma模复合体的同调计算
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2022-12-21 DOI: 10.1007/s10468-022-10176-9
Lucia Bagnoli
{"title":"Computation of the Homology of the Complexes of Finite Verma Modules for ({K}_{4}^{prime })","authors":"Lucia Bagnoli","doi":"10.1007/s10468-022-10176-9","DOIUrl":"10.1007/s10468-022-10176-9","url":null,"abstract":"<div><p>We compute the homology of the complexes of finite Verma modules over the annihilation superalgebra <span>(mathcal {A}({K}_{4}^{prime }))</span>, associated with the conformal superalgebra <span>({K}_{4}^{prime })</span>, obtained in Bagnoli and Caselli (J. Math. Phys. <b>63</b>, 091701, 2022). We use the computation of the homology in order to provide an explicit realization of all the irreducible quotients of finite Verma modules over <span>(mathcal {A}({K}_{4}^{prime }))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-022-10176-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42381272","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
Weyl Modules for Toroidal Lie Algebras 环面李代数的Weyl模
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2022-12-20 DOI: 10.1007/s10468-022-10187-6
Sudipta Mukherjee, Santosha Kumar Pattanayak, Sachin S. Sharma
{"title":"Weyl Modules for Toroidal Lie Algebras","authors":"Sudipta Mukherjee,&nbsp;Santosha Kumar Pattanayak,&nbsp;Sachin S. Sharma","doi":"10.1007/s10468-022-10187-6","DOIUrl":"10.1007/s10468-022-10187-6","url":null,"abstract":"<div><p>In this paper, we study Weyl modules for a toroidal Lie algebra <span>(mathcal {T})</span> with arbitrary <i>n</i> variables. Using the work of Rao (Pac. J. Math. <b>171</b>(2), 511–528 1995), we prove that the level one global Weyl modules of <span>(mathcal {T})</span> are isomorphic to suitable submodules of a Fock space representation of <span>(mathcal {T})</span> up to a twist. As an application, we compute the graded character of the level one local Weyl module of <span>(mathcal {T})</span>, thereby generalising the work of Kodera (Lett. Math. Phys. <b>110</b>(11) 3053–3080 2020).</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46848705","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
Superpotentials and Quiver Algebras for Semisimple Hopf Actions 半简单Hopf作用的超势和颤动代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2022-12-19 DOI: 10.1007/s10468-022-10165-y
Simon Crawford
{"title":"Superpotentials and Quiver Algebras for Semisimple Hopf Actions","authors":"Simon Crawford","doi":"10.1007/s10468-022-10165-y","DOIUrl":"10.1007/s10468-022-10165-y","url":null,"abstract":"<div><p>We consider the action of a semisimple Hopf algebra <i>H</i> on an <i>m</i>-Koszul Artin–Schelter regular algebra <i>A</i>. Such an algebra <i>A</i> is a derivation-quotient algebra for some twisted superpotential <i><span>w</span></i>, and we show that the homological determinant of the action of <i>H</i> on <i>A</i> can be easily calculated using <i><span>w</span></i>. Using this, we show that the smash product <i>A</i><i>#</i><i>H</i> is also a derivation-quotient algebra, and use this to explicitly determine a quiver algebra Λ to which <i>A</i><i>#</i><i>H</i> is Morita equivalent, generalising a result of Bocklandt–Schedler–Wemyss. We also show how Λ can be used to determine whether the Auslander map is an isomorphism. We compute a number of examples, and show how several results for the quantum Kleinian singularities studied by Chan–Kirkman–Walton–Zhang follow using our techniques.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-022-10165-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43288493","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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