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Topologically Semiperfect Topological Rings 拓扑半完全拓扑环
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-07-10 DOI: 10.1007/s10468-023-10217-x
Leonid Positselski, Jan Šťovíček
{"title":"Topologically Semiperfect Topological Rings","authors":"Leonid Positselski,&nbsp;Jan Šťovíček","doi":"10.1007/s10468-023-10217-x","DOIUrl":"10.1007/s10468-023-10217-x","url":null,"abstract":"<div><p>We define topologically semiperfect (complete, separated, right linear) topological rings and characterize them by equivalent conditions. We show that the endomorphism ring of a module, endowed with the finite topology, is topologically semiperfect if and only if the module is decomposable as an (infinite) direct sum of modules with local endomorphism rings. Then we study structural properties of topologically semiperfect topological rings and prove that their topological Jacobson radicals are strongly closed and the related topological quotient rings are topologically semisimple. For the endomorphism ring of a direct sum of modules with local endomorphism rings, the topological Jacobson radical is described explicitly as the set of all matrices of nonisomorphisms. Furthermore, we prove that, over a topologically semiperfect topological ring, all finitely generated discrete modules have projective covers in the category of modules, while all lattice-finite contramodules have projective covers in both the categories of modules and contramodules. We also show that the topological Jacobson radical of a topologically semiperfect topological ring is equal to the closure of the abstract Jacobson radical, and present a counterexample demonstrating that the topological Jacobson radical can be strictly larger than the abstract one. Finally, we discuss the problem of lifting idempotents modulo the topological Jacobson radical and the structure of projective contramodules for topologically semiperfect topological rings.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-023-10217-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46266389","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
Epimorphic Quantum Subgroups and Coalgebra Codominions 外纯量子子群与协代数共域
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-07-08 DOI: 10.1007/s10468-023-10219-9
Alexandru Chirvasitu
{"title":"Epimorphic Quantum Subgroups and Coalgebra Codominions","authors":"Alexandru Chirvasitu","doi":"10.1007/s10468-023-10219-9","DOIUrl":"10.1007/s10468-023-10219-9","url":null,"abstract":"<div><p>We prove a number of results concerning monomorphisms, epimorphisms, dominions and codominions in categories of coalgebras. Examples include: (a) representation-theoretic characterizations of monomorphisms in all of these categories that when the Hopf algebras in question are commutative specialize back to the familiar necessary and sufficient conditions (due to Bien-Borel) that a linear algebraic subgroup be epimorphically embedded; (b) the fact that a morphism in the category of (cocommutative) coalgebras, (cocommutative) bialgebras, and a host of categories of Hopf algebras has the same codominion in any of these categories which contain it; (c) the invariance of the Hopf algebra or bialgebra (co)dominion construction under field extension, again mimicking the well-known corresponding algebraic-group result; (d) the fact that surjections of coalgebras, bialgebras or Hopf algebras are regular epimorphisms (i.e. coequalizers) provided the codomain is cosemisimple; (e) in particular, the fact that embeddings of compact quantum groups are equalizers in the category thereof, generalizing analogous results on (plain) compact groups; (f) coalgebra-limit preservation results for scalar-extension functors (e.g. extending scalars along a field extension <span>(Bbbk le Bbbk ')</span> is a right adjoint on the category of <span>(Bbbk )</span>-coalgebras).</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47951585","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
Stable Unital Bases, Hyperfocal Subalgebras and Basic Morita Equivalences 稳定酉基、超焦子代数和基本Morita等价
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-07-03 DOI: 10.1007/s10468-023-10216-y
Tiberiu Coconeţ, Constantin-Cosmin Todea
{"title":"Stable Unital Bases, Hyperfocal Subalgebras and Basic Morita Equivalences","authors":"Tiberiu Coconeţ,&nbsp;Constantin-Cosmin Todea","doi":"10.1007/s10468-023-10216-y","DOIUrl":"10.1007/s10468-023-10216-y","url":null,"abstract":"<div><p>We investigate Conjecture 1.5 introduced by Barker and Gelvin (J. Gr. Theory <b>25</b>, 973–995 2022), which says that any source algebra of a <i>p</i>-block (<i>p</i> is a prime) of a finite group has the unit group containing a basis stabilized by the left and right actions of the defect group. We will reduce this conjecture to a similar statement about the bases of the hyperfocal subalgebras in the source algebras. We will also show that such unital bases of source algebras of two <i>p</i>-blocks, stabilized by the left and right actions of the defect group, are transported through basic Morita equivalences.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49266622","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 Generalization of the Correspondences Between Quasi-Hereditary Algebras and Directed Bocses 拟遗传代数与有向Boces对应关系的推广
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-07-03 DOI: 10.1007/s10468-023-10212-2
Yuichiro Goto
{"title":"A Generalization of the Correspondences Between Quasi-Hereditary Algebras and Directed Bocses","authors":"Yuichiro Goto","doi":"10.1007/s10468-023-10212-2","DOIUrl":"10.1007/s10468-023-10212-2","url":null,"abstract":"<div><p>Quasi-hereditary algebras were introduced by Cline, Parshall and Scott to study the highest weight categories in Lie theory. On the other hand, bocses were introduced in the context of Drozd’s tame and wild dichotomy theorem. Koenig, Külshammer and Ovsienko connected the two areas by giving equivalences between the categories of <span>(Delta )</span>-filtered modules over quasi-hereditary algebras and those of modules over directed bocses. In this article, we extend this result to <span>(overline{Delta })</span>-filtered algebras. We face two problems when proving a similar theorem for <span>(overline{Delta })</span>-filtered algebras. The first one is that the <span>(textrm{Ext})</span>-algebra of proper standard modules may be infinite dimensional. The second one is that the underlying algebra <i>B</i> of the bocs <span>(mathcal {B})</span> induced from a <span>(overline{Delta })</span>-filtered algebra may be infinite dimensional. We give solutions for these problems and show the relationship between the categories of <span>(overline{Delta })</span>-filtered modules over <span>(overline{Delta })</span>-filtered algebras and those of modules over some class of bocses.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46515339","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
Finite GK-Dimensional Nichols Algebras Over the Infinite Dihedral Group 无限二面体群上的有限GK维Nichols代数
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-06-29 DOI: 10.1007/s10468-023-10213-1
Yongliang Zhang
{"title":"Finite GK-Dimensional Nichols Algebras Over the Infinite Dihedral Group","authors":"Yongliang Zhang","doi":"10.1007/s10468-023-10213-1","DOIUrl":"10.1007/s10468-023-10213-1","url":null,"abstract":"<div><p>We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over <span>(mathbb {D}_{infty })</span>, the infinite dihedral group.We find all the irreducible Yetter-Drinfeld modules <i>V</i> over <span>(mathbb {D}_{infty })</span>, and determine which Nichols algebras <span>(mathcal {B}(V))</span> of <i>V</i> are finite GK-dimensional.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49523451","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
Modular Products and Modules for Finite Groups 模积与有限群的模
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-06-26 DOI: 10.1007/s10468-023-10210-4
John F. R. Duncan, Jeffrey A. Harvey, Brandon C. Rayhaun
{"title":"Modular Products and Modules for Finite Groups","authors":"John F. R. Duncan,&nbsp;Jeffrey A. Harvey,&nbsp;Brandon C. Rayhaun","doi":"10.1007/s10468-023-10210-4","DOIUrl":"10.1007/s10468-023-10210-4","url":null,"abstract":"<div><p>Motivated by the appearance of penumbral moonshine, and by evidence that penumbral moonshine enjoys an extensive relationship to generalized monstrous moonshine via infinite products, we establish a general construction in this work which uses singular theta lifts and a concrete construction at the level of modules for a finite group to translate between moonshine in weight one-half and moonshine in weight zero. This construction serves as a foundation for a companion paper in which we explore the connection between penumbral Thompson moonshine and a special case of generalized monstrous moonshine in detail.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46964779","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 the Ideals of Ultragraph Leavitt Path Algebras 关于超图莱维特路径代数的理想
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-06-24 DOI: 10.1007/s10468-023-10206-0
T. T. H. Duyen, D. Gonçalves, T. G. Nam
{"title":"On the Ideals of Ultragraph Leavitt Path Algebras","authors":"T. T. H. Duyen,&nbsp;D. Gonçalves,&nbsp;T. G. Nam","doi":"10.1007/s10468-023-10206-0","DOIUrl":"10.1007/s10468-023-10206-0","url":null,"abstract":"<div><p>In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for graded ideals, the graded uniqueness and Cuntz-Krieger theorems, the semiprimeness, and the semiprimitivity of ultragraph Leavitt path algebras, a complete characterization of the prime and primitive ideals of an ultragraph Leavitt path algebra. We also show that every primitive ideal of an ultragraph Leavitt path algebra is exactly the annihilator of a Chen simple module. Consequently, we prove Exel’s Effros-Hahn conjecture on primitive ideals in the ultragraph Leavitt path algebra setting (a conclusion that is also new in the context of Leavitt path algebras of graphs).</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42101336","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
Crystal Bases of Modified (imath )quantum Groups of Certain Quasi-Split Types 某些准分裂型修正$$imath $$量子群的晶体基
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-06-14 DOI: 10.1007/s10468-023-10207-z
Hideya Watanabe
{"title":"Crystal Bases of Modified (imath )quantum Groups of Certain Quasi-Split Types","authors":"Hideya Watanabe","doi":"10.1007/s10468-023-10207-z","DOIUrl":"10.1007/s10468-023-10207-z","url":null,"abstract":"<div><p>In order to see the behavior of <span>(imath )</span>canonical bases at <span>(q = infty )</span>, we introduce the notion of <span>(imath )</span>crystals associated to an <span>(imath )</span>quantum group of certain quasi-split type. The theory of <span>(imath )</span>crystals clarifies why <span>(imath )</span>canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of <span>(imath )</span>crystals whose projective limit can be thought of as the <span>(imath )</span>canonical basis of the modified <span>(imath )</span>quantum group at <span>(q = infty )</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48458798","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
From the Lattice of Torsion Classes to the Posets of Wide Subcategories and ICE-closed Subcategories 从扭转类的格到宽子范畴和ice闭子范畴的偏序集
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-06-13 DOI: 10.1007/s10468-023-10214-0
Haruhisa Enomoto
{"title":"From the Lattice of Torsion Classes to the Posets of Wide Subcategories and ICE-closed Subcategories","authors":"Haruhisa Enomoto","doi":"10.1007/s10468-023-10214-0","DOIUrl":"10.1007/s10468-023-10214-0","url":null,"abstract":"<div><p>In this paper, we compute the posets of wide subcategories and ICE-closed subcategories from the lattice of torsion classes in an abelian length category in a purely lattice-theoretical way, by using the kappa map in a completely semidistributive lattice. As for the poset of wide subcategories, we give two more simple constructions via a bijection between wide subcategories and torsion classes with canonical join representations. More precisely, for a completely semidistributive lattice, we give two poset structures on the set of elements with canonical join representations: the kappa order (defined using the extended kappa map of Barnard–Todorov–Zhu), and the core label order (generalizing the shard intersection order for congruence-uniform lattices). Then we show that these posets for the lattice of torsion classes coincide and are isomorphic to the poset of wide subcategories. As a byproduct, we give a simple description of the shard intersection order on a finite Coxeter group using the extended kappa map.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46074602","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
The Igusa-Todorov (phi )-Dimension on Morita Context Algebras 莫里塔上下文结构上的 Igusa-Todorov （phi ）-维度
IF 0.5 4区 数学
Algebras and Representation Theory Pub Date : 2023-06-13 DOI: 10.1007/s10468-023-10218-w
Marcos Barrios, Gustavo Mata
{"title":"The Igusa-Todorov (phi )-Dimension on Morita Context Algebras","authors":"Marcos Barrios,&nbsp;Gustavo Mata","doi":"10.1007/s10468-023-10218-w","DOIUrl":"10.1007/s10468-023-10218-w","url":null,"abstract":"<div><p>In this article we prove that, under certain hypotheses, Morita context algebras with zero bimodule morphisms have finite <span>(phi )</span>-dimension. For these algebras we also study the behaviour of the <span>(phi )</span>-dimension for an algebra and its opposite. In particular we show that the <span>(phi )</span>-dimension of an Artin algebra is not symmetric, i.e. there exists an Artin algebra <i>A</i> such that <span>(phi dim (A) not = phi dim (A^{op}))</span>.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136066085","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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