{"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":"23 1","pages":""},"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}
{"title":"Brauer–Clifford Group of Lie–Rinehart Algebras","authors":"T. Guédénon","doi":"10.1142/s1005386722000086","DOIUrl":"https://doi.org/10.1142/s1005386722000086","url":null,"abstract":"In this paper we define the notion of Brauer–Clifford group for [Formula: see text]-Azumaya algebras when [Formula: see text] is a commutative algebra and[Formula: see text] is a [Formula: see text]-Lie algebra over a commutative ring [Formula: see text]. This is the situation that arises in applications having connections to differential geometry. This Brauer–Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"42 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81091275","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}
{"title":"On Existence of Primitive Normal Elements of Cubic Form over Finite Fields","authors":"H. Hazarika, D. Basnet","doi":"10.1142/s1005386722000128","DOIUrl":"https://doi.org/10.1142/s1005386722000128","url":null,"abstract":"For a prime [Formula: see text]and a positive integer[Formula: see text], let [Formula: see text] and [Formula: see text] be the extension field of [Formula: see text]. We derive a sufficient condition for the existence of a primitive element [Formula: see text] in[Formula: see text] such that [Formula: see text] is also a primitive element of [Formula: see text], a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is a primitive element of [Formula: see text], and a sufficient condition for the existence of a primitive normal element [Formula: see text] in [Formula: see text] over [Formula: see text] such that [Formula: see text] is also a primitive normal element of [Formula: see text] over [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"10 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87906851","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}
{"title":"Open Frobenius Cluster-Tilted Algebras","authors":"Viviana Gubitosi","doi":"10.1142/S1005386722000025","DOIUrl":"https://doi.org/10.1142/S1005386722000025","url":null,"abstract":"In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra [Formula: see text] such that [Formula: see text] has non-trivial open Frobenius structures.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"76 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75266380","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}
{"title":"Classical Radicals of Invariants of Algebraic Skew Derivations","authors":"Jeffrey Bergen, Piotr Grzeszczuk","doi":"10.1142/s1005386722000050","DOIUrl":"https://doi.org/10.1142/s1005386722000050","url":null,"abstract":"Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"121 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84921128","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}
{"title":"Some Results on Noetherian Warfield Domains","authors":"Kui Hu, J. Lim, D. Zhou","doi":"10.1142/s1005386722000062","DOIUrl":"https://doi.org/10.1142/s1005386722000062","url":null,"abstract":"Let [Formula: see text] be a domain. In this paper, we show that if [Formula: see text] is one-dimensional, then [Formula: see text] is a Noetherian Warfield domain if and only if every maximal ideal of [Formula: see text] is 2-generated and for every maximal ideal[Formula: see text] of [Formula: see text], [Formula: see text] is divisorial in the ring [Formula: see text]. We also prove that a Noetherian domain [Formula: see text] is a Noetherian Warfield domain if and only if for every maximal ideal [Formula: see text] of [Formula: see text], [Formula: see text] can be generated by two elements. Finally, we give a sufficient condition under which all ideals of [Formula: see text] are strongly Gorenstein projective.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"6 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87328773","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}
{"title":"On Finite Local Rings with Clique Number Four","authors":"Qiong Liu, Tongsuo Wu, Jin Guo","doi":"10.1142/s1005386722000037","DOIUrl":"https://doi.org/10.1142/s1005386722000037","url":null,"abstract":"We study the algebraic structure of rings [Formula: see text] whose zero-divisor graph [Formula: see text]has clique number four. Furthermore, we give complete characterizations of all the finite commutative local rings with clique number 4.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"343 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83454635","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}
{"title":"Three Ideals of Lie Superalgebras","authors":"Xiaodong Zhao, Liangyun Chen","doi":"10.1142/s1005386722000116","DOIUrl":"https://doi.org/10.1142/s1005386722000116","url":null,"abstract":"We define perfect ideals, near perfect ideals and upper bounded ideals of a finite-dimensional Lie superalgebra, and study the properties of these three kinds of ideals through their relevant sequences. We prove that a Lie superalgebra is solvable if and only if its maximal perfect ideal is zero, or its quotient superalgebra by the maximal perfect ideal is solvable. We also show that a Lie superalgebra is nilpotent if and only if its maximal near perfect ideal is zero. Moreover, we prove that a nilpotent Lie superalgebra has only one upper bounded ideal, which is the nilpotent Lie superalgebra itself.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"18 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78505296","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}
{"title":"Coleman Automorphisms of Extensions of Finite Characteristically Simple Groups by Some Finite Groups","authors":"J. Hai, Lele Zhao","doi":"10.1142/s1005386721000444","DOIUrl":"https://doi.org/10.1142/s1005386721000444","url":null,"abstract":"Let [Formula: see text] be an extension of a finite characteristically simple group by an abelian group or a finite simple group. It is shown that every Coleman automorphism of [Formula: see text] is an inner automorphism. Interest in such automorphisms arises from the study of the normalizer problem for integral group rings.","PeriodicalId":50958,"journal":{"name":"Algebra Colloquium","volume":"12 Suppl 2 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76563576","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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