Pub Date : 2022-12-01DOI: 10.1142/s1005386722000414
R. Andruszkiewicz, M. Woronowicz
An abelian group [Formula: see text] is called a [Formula: see text]-group if every associative ring with the additive group [Formula: see text] is filial. The filiality of a ring [Formula: see text] means that the ring [Formula: see text] is associative and all ideals of any ideal of [Formula: see text] are ideals in [Formula: see text]. In this paper, torsion-free [Formula: see text]-groups are described up to the structure of associative nil groups. It is also proved that, for torsion-free abelian groups that are not associative nil, the condition [Formula: see text] implies the indecomposability and homogeneity. The paper contains constructions of [Formula: see text] such groups of any rank from 2 to[Formula: see text] which are pairwise non-isomorphic.
{"title":"The Classification of Torsion-free TI-Groups","authors":"R. Andruszkiewicz, M. Woronowicz","doi":"10.1142/s1005386722000414","DOIUrl":"https://doi.org/10.1142/s1005386722000414","url":null,"abstract":"An abelian group [Formula: see text] is called a [Formula: see text]-group if every associative ring with the additive group [Formula: see text] is filial. The filiality of a ring [Formula: see text] means that the ring [Formula: see text] is associative and all ideals of any ideal of [Formula: see text] are ideals in [Formula: see text]. In this paper, torsion-free [Formula: see text]-groups are described up to the structure of associative nil groups. It is also proved that, for torsion-free abelian groups that are not associative nil, the condition [Formula: see text] implies the indecomposability and homogeneity. The paper contains constructions of [Formula: see text] such groups of any rank from 2 to[Formula: see text] which are pairwise non-isomorphic.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76142812","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}
Pub Date : 2022-12-01DOI: 10.1142/s1005386722000438
H. AlHussein
In this work we find the groups of Hochschild cohomologies for the Chinese monoid algebra and derive its Hilbert and Poincaré series. In order to obtain this result we construct the Anick resolution via the algebraic discrete Morse theory and Gröbner–Shirshov basis for the Chinese monoid.
{"title":"The Hochschild Cohomology of the Chinese Monoid Algebra","authors":"H. AlHussein","doi":"10.1142/s1005386722000438","DOIUrl":"https://doi.org/10.1142/s1005386722000438","url":null,"abstract":"In this work we find the groups of Hochschild cohomologies for the Chinese monoid algebra and derive its Hilbert and Poincaré series. In order to obtain this result we construct the Anick resolution via the algebraic discrete Morse theory and Gröbner–Shirshov basis for the Chinese monoid.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81609400","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}
Pub Date : 2022-12-01DOI: 10.1142/s1005386722000426
Huabo Xu
Let [Formula: see text] be a ring with an automorphism [Formula: see text] of order two. We introduce the definition of [Formula: see text]-centrosymmetric matrices. Denote by [Formula: see text] the ring of all [Formula: see text] matrices over [Formula: see text], and by [Formula: see text] the set of all [Formula: see text]-centrosymmetric [Formula: see text] matrices over [Formula: see text] for any positive integer [Formula: see text]. We show that [Formula: see text] is a separable Frobenius extension. If [Formula: see text] is commutative, then [Formula: see text] is a cellular algebra over the invariant subring [Formula: see text] of [Formula: see text].
{"title":"Generalized Centrosymmetric Matrix Algebras Induced by Automorphisms","authors":"Huabo Xu","doi":"10.1142/s1005386722000426","DOIUrl":"https://doi.org/10.1142/s1005386722000426","url":null,"abstract":"Let [Formula: see text] be a ring with an automorphism [Formula: see text] of order two. We introduce the definition of [Formula: see text]-centrosymmetric matrices. Denote by [Formula: see text] the ring of all [Formula: see text] matrices over [Formula: see text], and by [Formula: see text] the set of all [Formula: see text]-centrosymmetric [Formula: see text] matrices over [Formula: see text] for any positive integer [Formula: see text]. We show that [Formula: see text] is a separable Frobenius extension. If [Formula: see text] is commutative, then [Formula: see text] is a cellular algebra over the invariant subring [Formula: see text] of [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72455878","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}
Pub Date : 2022-12-01DOI: 10.1142/s1005386722000499
Shiqi Xing, Xiaoqiang Luo, Kui Hu
We study some homological properties of Gorenstein [Formula: see text]-injective modules, and we prove (1) a ring [Formula: see text]is not necessarily coherent if every Gorenstein [Formula: see text]-injective [Formula: see text]-module is injective, and (2) a ring [Formula: see text] is not necessarily coherent if every Gorenstein injective [Formula: see text]-module is injective. In addition, we characterize [Formula: see text]-Noetherian rings in terms of Gorenstein [Formula: see text]-injective modules, and we prove that a ring [Formula: see text] is [Formula: see text]-Noetherian if and only if every GV-torsion-free FP-injective [Formula: see text]-module is Gorenstein [Formula: see text]-injective, if and only if any direct sum of GV-torsion-free FP-injective [Formula: see text]-modules is Gorenstein [Formula: see text]-injective.
{"title":"Gorenstein FP∞-Injective Modules and w-Noetherian Rings","authors":"Shiqi Xing, Xiaoqiang Luo, Kui Hu","doi":"10.1142/s1005386722000499","DOIUrl":"https://doi.org/10.1142/s1005386722000499","url":null,"abstract":"We study some homological properties of Gorenstein [Formula: see text]-injective modules, and we prove (1) a ring [Formula: see text]is not necessarily coherent if every Gorenstein [Formula: see text]-injective [Formula: see text]-module is injective, and (2) a ring [Formula: see text] is not necessarily coherent if every Gorenstein injective [Formula: see text]-module is injective. In addition, we characterize [Formula: see text]-Noetherian rings in terms of Gorenstein [Formula: see text]-injective modules, and we prove that a ring [Formula: see text] is [Formula: see text]-Noetherian if and only if every GV-torsion-free FP-injective [Formula: see text]-module is Gorenstein [Formula: see text]-injective, if and only if any direct sum of GV-torsion-free FP-injective [Formula: see text]-modules is Gorenstein [Formula: see text]-injective.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82488085","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}
Pub Date : 2022-12-01DOI: 10.1142/s1005386722000505
Xue Wang, Jin-Xin Zhou
Let [Formula: see text] be a prime. In this paper, a complete classification of edge-transitive [Formula: see text]-covers of a cubic symmetric graph of order [Formula: see text] is given for the case when [Formula: see text] is a two-generator 2-group whose derived subgroup is either isomorphic to [Formula: see text] or generated by at most two elements. As an application, it is shown that 11 is the smallest value of [Formula: see text] for which there exist infinitely many cubic semisymmetric graphs with order of the form [Formula: see text].
{"title":"A Note on Two-Generator 2-Group Covers of Cubic Symmetric Graphs of Order 2p","authors":"Xue Wang, Jin-Xin Zhou","doi":"10.1142/s1005386722000505","DOIUrl":"https://doi.org/10.1142/s1005386722000505","url":null,"abstract":"Let [Formula: see text] be a prime. In this paper, a complete classification of edge-transitive [Formula: see text]-covers of a cubic symmetric graph of order [Formula: see text] is given for the case when [Formula: see text] is a two-generator 2-group whose derived subgroup is either isomorphic to [Formula: see text] or generated by at most two elements. As an application, it is shown that 11 is the smallest value of [Formula: see text] for which there exist infinitely many cubic semisymmetric graphs with order of the form [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76670700","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}
Pub Date : 2022-12-01DOI: 10.1142/s1005386722000451
T. Hibi, H. Mahmood
Let [Formula: see text] be a simplicial complex on [Formula: see text]. The [Formula: see text]-complex of [Formula: see text] is the simplicial complex [Formula: see text] on [Formula: see text] for which the facet ideal of [Formula: see text] is equal to the Stanley–Reisner ideal of [Formula: see text]. Furthermore, for each [Formula: see text], we introduce the [Formula: see text]th [Formula: see text]-complex [Formula: see text], which is inductively defined as [Formula: see text] by setting [Formula: see text]. One can set [Formula: see text]. The [Formula: see text]-number of [Formula: see text] is the smallest integer [Formula: see text] for which [Formula: see text]. In the present paper we are especially interested in the [Formula: see text]-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the [Formula: see text]-number of the finite graph [Formula: see text] on [Formula: see text], which is the disjoint union of the complete graphs [Formula: see text] on [Formula: see text] and [Formula: see text] on [Formula: see text] for [Formula: see text] and [Formula: see text] with [Formula: see text], is equal to [Formula: see text]. As a corollary, we find that the [Formula: see text]-number of the complete bipartite graph [Formula: see text] on [Formula: see text] is also equal to [Formula: see text].
{"title":"The NF-Number of a Simplicial Complex","authors":"T. Hibi, H. Mahmood","doi":"10.1142/s1005386722000451","DOIUrl":"https://doi.org/10.1142/s1005386722000451","url":null,"abstract":"Let [Formula: see text] be a simplicial complex on [Formula: see text]. The [Formula: see text]-complex of [Formula: see text] is the simplicial complex [Formula: see text] on [Formula: see text] for which the facet ideal of [Formula: see text] is equal to the Stanley–Reisner ideal of [Formula: see text]. Furthermore, for each [Formula: see text], we introduce the [Formula: see text]th [Formula: see text]-complex [Formula: see text], which is inductively defined as [Formula: see text] by setting [Formula: see text]. One can set [Formula: see text]. The [Formula: see text]-number of [Formula: see text] is the smallest integer [Formula: see text] for which [Formula: see text]. In the present paper we are especially interested in the [Formula: see text]-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the [Formula: see text]-number of the finite graph [Formula: see text] on [Formula: see text], which is the disjoint union of the complete graphs [Formula: see text] on [Formula: see text] and [Formula: see text] on [Formula: see text] for [Formula: see text] and [Formula: see text] with [Formula: see text], is equal to [Formula: see text]. As a corollary, we find that the [Formula: see text]-number of the complete bipartite graph [Formula: see text] on [Formula: see text] is also equal to [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80947487","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}
Pub Date : 2022-12-01DOI: 10.1142/s1005386722000463
A. Iacob
We introduce a generalization of the Gorenstein injective modules: the Gorenstein [Formula: see text]-injective modules (denoted by [Formula: see text]). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor [Formula: see text], with [Formula: see text] any [Formula: see text]-injective module. Thus, [Formula: see text] is the class of classical Gorenstein injective modules, and [Formula: see text] is the class of Ding injective modules. We prove that over any ring [Formula: see text], for any [Formula: see text], the class [Formula: see text] is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For [Formula: see text] we show that [Formula: see text] (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring [Formula: see text] is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein [Formula: see text]-projectives (denoted by [Formula: see text]). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for any[Formula: see text] the class [Formula: see text] is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.
{"title":"Generalized Gorenstein Modules","authors":"A. Iacob","doi":"10.1142/s1005386722000463","DOIUrl":"https://doi.org/10.1142/s1005386722000463","url":null,"abstract":"We introduce a generalization of the Gorenstein injective modules: the Gorenstein [Formula: see text]-injective modules (denoted by [Formula: see text]). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor [Formula: see text], with [Formula: see text] any [Formula: see text]-injective module. Thus, [Formula: see text] is the class of classical Gorenstein injective modules, and [Formula: see text] is the class of Ding injective modules. We prove that over any ring [Formula: see text], for any [Formula: see text], the class [Formula: see text] is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For [Formula: see text] we show that [Formula: see text] (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring [Formula: see text] is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein [Formula: see text]-projectives (denoted by [Formula: see text]). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for any[Formula: see text] the class [Formula: see text] is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77184977","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}
Pub Date : 2022-12-01DOI: 10.1142/s1005386722000475
L. Xia, Qianqian Cai, Jiao Zhang
Let [Formula: see text] be a finite dimensional complex simple Lie algebra with Cartan subalgebra [Formula: see text]. Then [Formula: see text] has a [Formula: see text]-module structure if and only if [Formula: see text] is of type [Formula: see text] or of type [Formula: see text]; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over [Formula: see text] have been classified. In this paper, we prove that the quantum group [Formula: see text] has no rank one polynomial module.
设[公式:见文]为具有Cartan子代数的有限维复单李代数[公式:见文]。那么，当且仅当[Formula: see text]的类型为[Formula: see text]或[Formula: see text]时，[Formula: see text]具有[Formula: see text]-模块结构;这叫做第1阶的多项式模。在量子版本中，[公式:见文本]上的1阶多项式模块已被分类。本文证明了量子群[公式:见文]不存在秩一多项式模。
{"title":"The Polynomial Modules over Quantum Group Uq(sl3)","authors":"L. Xia, Qianqian Cai, Jiao Zhang","doi":"10.1142/s1005386722000475","DOIUrl":"https://doi.org/10.1142/s1005386722000475","url":null,"abstract":"Let [Formula: see text] be a finite dimensional complex simple Lie algebra with Cartan subalgebra [Formula: see text]. Then [Formula: see text] has a [Formula: see text]-module structure if and only if [Formula: see text] is of type [Formula: see text] or of type [Formula: see text]; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over [Formula: see text] have been classified. In this paper, we prove that the quantum group [Formula: see text] has no rank one polynomial module.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77347773","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}
Pub Date : 2022-12-01DOI: 10.1142/s100538672200044x
Yu Zhang, Xiaomin Tang
We study a class of super W-algebras whose even part is the Virasoro type Lie algebra [Formula: see text]. We quantize the Lie superbialgebra of the super W-algebra by the Drinfeld twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras.
{"title":"Quantization of a Class of Super W-Agebras","authors":"Yu Zhang, Xiaomin Tang","doi":"10.1142/s100538672200044x","DOIUrl":"https://doi.org/10.1142/s100538672200044x","url":null,"abstract":"We study a class of super W-algebras whose even part is the Virasoro type Lie algebra [Formula: see text]. We quantize the Lie superbialgebra of the super W-algebra by the Drinfeld twist quantization technique and obtain a class of noncommutative and noncocommutative Hopf superalgebras.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87628961","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}
Pub Date : 2022-07-26DOI: 10.1142/s1005386722000335
Ruifang Chen, Xianhe Zhao
Let [Formula: see text] be a finite group and [Formula: see text] a normal subgroup of [Formula: see text]. Denote by [Formula: see text] the graph whose vertices are all distinct [Formula: see text]-conjugacy class sizes of non-central elements in [Formula: see text], and two vertices of [Formula: see text] are adjacent if and only if they are not coprime numbers. We prove that if the center [Formula: see text] and [Formula: see text]is [Formula: see text]-regular for [Formula: see text], then either a section of [Formula: see text]is a quasi-Frobenius group or [Formula: see text] is a complete graph with [Formula: see text] vertices.
{"title":"On Conjugacy Class Graph of Normal Subgroup","authors":"Ruifang Chen, Xianhe Zhao","doi":"10.1142/s1005386722000335","DOIUrl":"https://doi.org/10.1142/s1005386722000335","url":null,"abstract":"Let [Formula: see text] be a finite group and [Formula: see text] a normal subgroup of [Formula: see text]. Denote by [Formula: see text] the graph whose vertices are all distinct [Formula: see text]-conjugacy class sizes of non-central elements in [Formula: see text], and two vertices of [Formula: see text] are adjacent if and only if they are not coprime numbers. We prove that if the center [Formula: see text] and [Formula: see text]is [Formula: see text]-regular for [Formula: see text], then either a section of [Formula: see text]is a quasi-Frobenius group or [Formula: see text] is a complete graph with [Formula: see text] vertices.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75840670","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}
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