Pub Date : 2022-03-20DOI: 10.1142/s1005386723000081
Xiaomin Zhu, Xu Yang, Jing Chen
The spectra of generalized Cayley graphs of finite abelian groups are investigated in this paper. For a generalized Cayley graph [Formula: see text] of a finite group [Formula: see text], the canonical double covering of [Formula: see text] is the direct product [Formula: see text]. In this paper, integral generalized Cayley graphs on finite abelian groups are characterized, using the characterization of the spectra of integral Cayley graphs. As an application, the integral generalized Cayley graphs on [Formula: see text] and [Formula: see text] are investigated, where [Formula: see text] and [Formula: see text] are odd prime numbers.
{"title":"Spectra of Generalized Cayley Graphs on Finite Abelian Groups","authors":"Xiaomin Zhu, Xu Yang, Jing Chen","doi":"10.1142/s1005386723000081","DOIUrl":"https://doi.org/10.1142/s1005386723000081","url":null,"abstract":"The spectra of generalized Cayley graphs of finite abelian groups are investigated in this paper. For a generalized Cayley graph [Formula: see text] of a finite group [Formula: see text], the canonical double covering of [Formula: see text] is the direct product [Formula: see text]. In this paper, integral generalized Cayley graphs on finite abelian groups are characterized, using the characterization of the spectra of integral Cayley graphs. As an application, the integral generalized Cayley graphs on [Formula: see text] and [Formula: see text] are investigated, where [Formula: see text] and [Formula: see text] are odd prime numbers.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80399048","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 : 2022-03-20DOI: 10.1142/s100538672300010x
Jinseok Park
A set [Formula: see text] of positive integers is called a Diophantine [Formula: see text]-tuple if [Formula: see text] is a perfect square for all [Formula: see text]. Let [Formula: see text] be the Diophantine triple with [Formula: see text]. In this paper, we find the condition for the reduction of third element [Formula: see text], and using this result, we prove the extendibility of Diophantine pair [Formula: see text], where [Formula: see text] is the [Formula: see text]-th Fibonacci number.
{"title":"Complete Solution of Diophantine Pairs Induced by Some Fibonacci Formula","authors":"Jinseok Park","doi":"10.1142/s100538672300010x","DOIUrl":"https://doi.org/10.1142/s100538672300010x","url":null,"abstract":"A set [Formula: see text] of positive integers is called a Diophantine [Formula: see text]-tuple if [Formula: see text] is a perfect square for all [Formula: see text]. Let [Formula: see text] be the Diophantine triple with [Formula: see text]. In this paper, we find the condition for the reduction of third element [Formula: see text], and using this result, we prove the extendibility of Diophantine pair [Formula: see text], where [Formula: see text] is the [Formula: see text]-th Fibonacci number.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79163202","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 : 2022-03-20DOI: 10.1142/s1005386723000111
Yu‐Qin Mei, Jiaqun Wei
Extriangulated categories were introduced by Nakaoka and Palu via extracting the similarities between exact categories and triangulated categories. In this article we introduce the notion of [Formula: see text]-tilting objects in an extriangulated category, where [Formula: see text] is a proper class of [Formula: see text]-triangles. Our results extend the relative tilting theory in extriangulated categories.
{"title":"ξ -Tilting Objects in Extriangulated Categories","authors":"Yu‐Qin Mei, Jiaqun Wei","doi":"10.1142/s1005386723000111","DOIUrl":"https://doi.org/10.1142/s1005386723000111","url":null,"abstract":"Extriangulated categories were introduced by Nakaoka and Palu via extracting the similarities between exact categories and triangulated categories. In this article we introduce the notion of [Formula: see text]-tilting objects in an extriangulated category, where [Formula: see text] is a proper class of [Formula: see text]-triangles. Our results extend the relative tilting theory in extriangulated categories.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87656288","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 : 2022-03-20DOI: 10.1142/s1005386723000056
Yao Luo, Abdukadir Obul
In the Ringel–Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Gröbner–Shirshov basis and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel–Hall algebra. We aim to generalize this result to the reduced Drinfeld double Hall algebra of type [Formula: see text]. First, we compute a minimal Gröbner–Shirshov basis for the reduced Drinfeld double Hall algebra of type [Formula: see text] by proving that all possible compositions between the commutator relations are trivial. Then, by taking the corresponding irreducible monomials, we construct a PBW-type basis for the reduced Drinfeld double Hall algebra of type [Formula: see text].
{"title":"PBW-Basis of Reduced Drinfeld Double Hall Algebra of Type An via Gröbner–Shirshov Basis","authors":"Yao Luo, Abdukadir Obul","doi":"10.1142/s1005386723000056","DOIUrl":"https://doi.org/10.1142/s1005386723000056","url":null,"abstract":"In the Ringel–Hall algebra of Dynkin type, the set of all commutator relations between the isoclasses of indecomposable representations forms a minimal Gröbner–Shirshov basis and the set of the corresponding irreducible elements forms a PBW-type basis of the Ringel–Hall algebra. We aim to generalize this result to the reduced Drinfeld double Hall algebra of type [Formula: see text]. First, we compute a minimal Gröbner–Shirshov basis for the reduced Drinfeld double Hall algebra of type [Formula: see text] by proving that all possible compositions between the commutator relations are trivial. Then, by taking the corresponding irreducible monomials, we construct a PBW-type basis for the reduced Drinfeld double Hall algebra of type [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82777419","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 : 2022-03-20DOI: 10.1142/s100538672300007x
Dongping Hou
Jordan D-bialgebras were introduced by Zhelyabin. In this paper, we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan–Manin triples called double constructions of pseudo-euclidean Jordan algebras. We also show that a Jordan D-bialgebra leads to the Jordan Yang–Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang–Baxter equation corresponds to an antisymmetric bilinear form, which we call a Jordan symplectic form on Jordan algebras. Furthermore, there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.
Zhelyabin引入Jordan d双代数。本文利用Jordan代数正则表示的对偶表示的新概念，提出了一种研究Jordan d -双代数的新方法。基于李双代数与Manin三元组之间的本质联系，我们给出了一类特殊的Jordan - Manin三元组与伪欧几里德Jordan代数的双重构造之间的等价性的显式证明。我们还证明了在共边条件下Jordan d双代数可以得到Jordan Yang-Baxter方程，并且Jordan Yang-Baxter方程的反对称非退化解对应于一个反对称双线性形式，我们称之为Jordan代数上的Jordan辛形式。此外，在具有约当辛形式的约当代数上存在一种新的代数结构，称为前约当代数。
{"title":"A New Approach to Jordan D-Bialgebras via Jordan–Manin Triples","authors":"Dongping Hou","doi":"10.1142/s100538672300007x","DOIUrl":"https://doi.org/10.1142/s100538672300007x","url":null,"abstract":"Jordan D-bialgebras were introduced by Zhelyabin. In this paper, we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan–Manin triples called double constructions of pseudo-euclidean Jordan algebras. We also show that a Jordan D-bialgebra leads to the Jordan Yang–Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang–Baxter equation corresponds to an antisymmetric bilinear form, which we call a Jordan symplectic form on Jordan algebras. Furthermore, there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90749836","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 : 2022-03-20DOI: 10.1142/s1005386723000020
Zhongkui Liu, Li Wang
For a local commutative Gorenstein ring [Formula: see text], Enochs et al. in [Gorenstein projective resolvents, Comm. Algebra 44 (2016) 3989–4000] defined a functor [Formula: see text] and showed that this functor can be computed by taking a totally acyclic complex arising from a projective coresolution of the first component or a totally acyclic complex arising from a projective resolution of the second component. In order to define the functor [Formula: see text] over general rings, we introduce the right Gorenstein projective dimension of an [Formula: see text]-module [Formula: see text], [Formula: see text], via Gorenstein projective coresolutions, and give some equivalent characterizations for the finiteness of [Formula: see text]. Then over a general ring [Formula: see text] we define a co-Tate homology group [Formula: see text] for [Formula: see text]-modules [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], and prove that [Formula: see text] can be computed by complete projective coresolutions of the first variable or by complete projective resolutions of the second variable.
{"title":"Gorenstein Projective Coresolutions and Co-Tate Homology Functors","authors":"Zhongkui Liu, Li Wang","doi":"10.1142/s1005386723000020","DOIUrl":"https://doi.org/10.1142/s1005386723000020","url":null,"abstract":"For a local commutative Gorenstein ring [Formula: see text], Enochs et al. in [Gorenstein projective resolvents, Comm. Algebra 44 (2016) 3989–4000] defined a functor [Formula: see text] and showed that this functor can be computed by taking a totally acyclic complex arising from a projective coresolution of the first component or a totally acyclic complex arising from a projective resolution of the second component. In order to define the functor [Formula: see text] over general rings, we introduce the right Gorenstein projective dimension of an [Formula: see text]-module [Formula: see text], [Formula: see text], via Gorenstein projective coresolutions, and give some equivalent characterizations for the finiteness of [Formula: see text]. Then over a general ring [Formula: see text] we define a co-Tate homology group [Formula: see text] for [Formula: see text]-modules [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], and prove that [Formula: see text] can be computed by complete projective coresolutions of the first variable or by complete projective resolutions of the second variable.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74239526","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 : 2022-03-20DOI: 10.1142/s1005386723000135
Jiangtao Shi
We obtain a complete characterization of the structure of a finite group [Formula: see text] in which every maximal subgroup is nilpotent or a TI-subgroup or has order [Formula: see text] for any fixed prime divisor [Formula: see text] of [Formula: see text]. Moreover, we show that there exists at most one prime divisor [Formula: see text] of [Formula: see text] such that [Formula: see text] is neither [Formula: see text]-nilpotent nor [Formula: see text]-closed.
{"title":"Finite Groups in Which Every Maximal Subgroup Is Nilpotent or a TI-Subgroup or Has Order p′","authors":"Jiangtao Shi","doi":"10.1142/s1005386723000135","DOIUrl":"https://doi.org/10.1142/s1005386723000135","url":null,"abstract":"We obtain a complete characterization of the structure of a finite group [Formula: see text] in which every maximal subgroup is nilpotent or a TI-subgroup or has order [Formula: see text] for any fixed prime divisor [Formula: see text] of [Formula: see text]. Moreover, we show that there exists at most one prime divisor [Formula: see text] of [Formula: see text] such that [Formula: see text] is neither [Formula: see text]-nilpotent nor [Formula: see text]-closed.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87034594","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 : 2022-03-20DOI: 10.1142/s1005386723000032
T. Kwak, Yang Lee, Zhelin Piao, Yeonsook Seo
The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements. Such rings shall be called right exp-DR. We investigate the structures of group rings, right quotient rings, matrix rings and (skew) polynomial rings, through the study of right exp-DR rings. In addition, we provide a method of constructing finite non-abelian [Formula: see text]-groups for any prime [Formula: see text].
{"title":"Duo Property Applied to Powers and Regular Elements","authors":"T. Kwak, Yang Lee, Zhelin Piao, Yeonsook Seo","doi":"10.1142/s1005386723000032","DOIUrl":"https://doi.org/10.1142/s1005386723000032","url":null,"abstract":"The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements. Such rings shall be called right exp-DR. We investigate the structures of group rings, right quotient rings, matrix rings and (skew) polynomial rings, through the study of right exp-DR rings. In addition, we provide a method of constructing finite non-abelian [Formula: see text]-groups for any prime [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80494796","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 : 2022-03-20DOI: 10.1142/s1005386723000068
Y. Rao, Zhongkui Liu, Xiaoyan Yang, Wenjing Chen
This paper introduces the notion of depth with respect to ideals for unbounded DG-modules, and gives a reduction formula and the local nature of this depth. As applications, we provide several bounds of the depth in special cases, and recover and generalize the known results about the depth of complexes. In addition, the width with respect to ideals for unbounded DG-modules is investigated and the depth and width formulas for DG-modules are generalized.
{"title":"Depth and Width for Unbounded DG-Modules","authors":"Y. Rao, Zhongkui Liu, Xiaoyan Yang, Wenjing Chen","doi":"10.1142/s1005386723000068","DOIUrl":"https://doi.org/10.1142/s1005386723000068","url":null,"abstract":"This paper introduces the notion of depth with respect to ideals for unbounded DG-modules, and gives a reduction formula and the local nature of this depth. As applications, we provide several bounds of the depth in special cases, and recover and generalize the known results about the depth of complexes. In addition, the width with respect to ideals for unbounded DG-modules is investigated and the depth and width formulas for DG-modules are generalized.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84429355","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 : 2022-03-20DOI: 10.1142/s1005386723000044
J. Moori
We aim to study maximal pairwise commuting sets of 3-transpositions (transvections) of the simple unitary group [Formula: see text] over [Formula: see text], and to construct designs from these sets. Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions. Let [Formula: see text]. It is well known that [Formula: see text] is a 3-transposition group with the set [Formula: see text], the conjugacy class consisting of its transvections, as the set of 3-transpositions. Let [Formula: see text] be a set of basic transpositions in [Formula: see text]. We give general descriptions of [Formula: see text] and [Formula: see text]- [Formula: see text] designs [Formula: see text], with [Formula: see text] and [Formula: see text]. The parameters [Formula: see text], [Formula: see text] and further properties of [Formula: see text] are determined. We also, as examples, apply the method to the unitary simple groups [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].
{"title":"Basic 3-Transpositions of Unitary Group Un (2 )","authors":"J. Moori","doi":"10.1142/s1005386723000044","DOIUrl":"https://doi.org/10.1142/s1005386723000044","url":null,"abstract":"We aim to study maximal pairwise commuting sets of 3-transpositions (transvections) of the simple unitary group [Formula: see text] over [Formula: see text], and to construct designs from these sets. Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions. Let [Formula: see text]. It is well known that [Formula: see text] is a 3-transposition group with the set [Formula: see text], the conjugacy class consisting of its transvections, as the set of 3-transpositions. Let [Formula: see text] be a set of basic transpositions in [Formula: see text]. We give general descriptions of [Formula: see text] and [Formula: see text]- [Formula: see text] designs [Formula: see text], with [Formula: see text] and [Formula: see text]. The parameters [Formula: see text], [Formula: see text] and further properties of [Formula: see text] are determined. We also, as examples, apply the method to the unitary simple groups [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75174662","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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