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Noncommutative Kn"orrer's periodicity theorem and noncommutative quadric hypersurfaces 非交换Kn / orver的周期定理与非交换二次超曲面
arXiv: Rings and Algebras Pub Date : 2019-05-29 DOI: 10.2140/ant.2022.16.467
I. Mori, Kenta Ueyama
{"title":"Noncommutative Kn\"orrer's periodicity theorem and noncommutative quadric hypersurfaces","authors":"I. Mori, Kenta Ueyama","doi":"10.2140/ant.2022.16.467","DOIUrl":"https://doi.org/10.2140/ant.2022.16.467","url":null,"abstract":"Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Knorrer's periodicity theorem is a powerful tool to study Cohen-Macaulay representation theory for hypersurfaces, so, in this paper, we show a noncommutative version of graded Knorrer periodicity theorem. Moreover, under high rank property defined in this paper, we show that computing the stable category of graded maximal Cohen-Macaulay modules over a noncommutative smooth quadric hypersurface up to 6 variables can be reduced to one or two variables cases. We also give a complete classification of the stable category of graded maximal Cohen-Macaulay modules over a smooth quadric hypersurface in a skew $mathbb P^{n-1}$ up to 6 variables without high rank property using graphical methods.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134150330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 12
Equivariant one-parameter formal deformations of Hom-Leibniz algebras homl - leibniz代数的等变单参数形式变形
arXiv: Rings and Algebras Pub Date : 2019-05-19 DOI: 10.1142/s0219199720500820
G. Mukherjee, R. Saha
{"title":"Equivariant one-parameter formal deformations of Hom-Leibniz algebras","authors":"G. Mukherjee, R. Saha","doi":"10.1142/s0219199720500820","DOIUrl":"https://doi.org/10.1142/s0219199720500820","url":null,"abstract":"We introduce an equivariant $1$-parameter formal deformation theory of Hom-Leibniz algebras equipped with finite group actions on Hom-Leibniz algebras. We define a suitable equivariant deformation cohomology which controls deformations.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116666925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Degenerations of nilpotent algebras 幂零代数的退化
arXiv: Rings and Algebras Pub Date : 2019-05-14 DOI: 10.1016/j.jpaa.2021.106850
Amir Fernández Ouaridi, I. Kaygorodov, M. Khrypchenko, Y. Volkov
{"title":"Degenerations of nilpotent algebras","authors":"Amir Fernández Ouaridi, I. Kaygorodov, M. Khrypchenko, Y. Volkov","doi":"10.1016/j.jpaa.2021.106850","DOIUrl":"https://doi.org/10.1016/j.jpaa.2021.106850","url":null,"abstract":"","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"118557059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 36
Locally solvable and solvable-by-finite maximal subgroups of $GL_n(D)$. $GL_n(D)$的局部可解和有限极大子群可解。
arXiv: Rings and Algebras Pub Date : 2019-03-23 DOI: 10.5565/PUBLMAT6612203
H. Khanh, B. X. Hai
{"title":"Locally solvable and solvable-by-finite maximal subgroups of $GL_n(D)$.","authors":"H. Khanh, B. X. Hai","doi":"10.5565/PUBLMAT6612203","DOIUrl":"https://doi.org/10.5565/PUBLMAT6612203","url":null,"abstract":"This paper aims at studying solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup of the general skew linear group $GL_n(D)$ over a division ring $D$. It turns out that in the case $D$ is non-commutative, if such kinds of maximal subgroups exist, then either it is abelian or $[D:F]<infty$. If $F$ is an infinite field and $ngeq 5$, then every locally solvable maximal subgroup of a normal subgroup of $GL_n(F)$ is abelian.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"155 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114796248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Research topics in finite groups and vertex algebras 有限群与顶点代数的研究课题
arXiv: Rings and Algebras Pub Date : 2019-03-21 DOI: 10.1090/CONM/753/15167
R. Griess
{"title":"Research topics in finite groups and vertex algebras","authors":"R. Griess","doi":"10.1090/CONM/753/15167","DOIUrl":"https://doi.org/10.1090/CONM/753/15167","url":null,"abstract":"We suggest a few projects for studying vertex algebras with emphasis on finite group viewpoints.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114768541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Crossed modules for Hom–Lie antialgebras homlie反代数的交叉模
arXiv: Rings and Algebras Pub Date : 2019-03-21 DOI: 10.1142/S0219498822501353
Zhang Tao, Heyu Zhang
{"title":"Crossed modules for Hom–Lie antialgebras","authors":"Zhang Tao, Heyu Zhang","doi":"10.1142/S0219498822501353","DOIUrl":"https://doi.org/10.1142/S0219498822501353","url":null,"abstract":"In this paper, we introduced the concept of crossed module for Hom-Lie antialgebras. It is proved that the category of crossed modules for Hom-Lie antialgebras and the category of $Cat^1$-Hom-Lie antialgebras are equivalent to each other. The relationship between the crossed module extension of Hom-Lie antialgebras and the third cohomology group are investigated.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127044559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Integral forms in vertex operator algebras, a survey 顶点算子代数中的积分形式，综述
arXiv: Rings and Algebras Pub Date : 2019-03-20 DOI: 10.22108/IJGT.2020.114954.1523
R. Griess
{"title":"Integral forms in vertex operator algebras, a survey","authors":"R. Griess","doi":"10.22108/IJGT.2020.114954.1523","DOIUrl":"https://doi.org/10.22108/IJGT.2020.114954.1523","url":null,"abstract":"We give a brief survey of recent work on integral forms in vertex operator algebras (VOAs).","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128428389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Directly decomposable ideals and congruence kernels of commutative semirings 交换半环的直接可分解理想与同余核
arXiv: Rings and Algebras Pub Date : 2019-01-17 DOI: 10.18514/mmn.2020.2819
I. Chajda, G. Eigenthaler, H. Langer
{"title":"Directly decomposable ideals and congruence kernels of commutative semirings","authors":"I. Chajda, G. Eigenthaler, H. Langer","doi":"10.18514/mmn.2020.2819","DOIUrl":"https://doi.org/10.18514/mmn.2020.2819","url":null,"abstract":"As pointed out in the monographs by J. S. Golan and by W. Kuich and A. Salomaa on semirings, ideals play an important role despite the fact that they need not be congruence kernels as in the case of rings. Hence, having two commutative semirings S1 and S2, one can ask whether an ideal I of their direct product S = S1 x S2 can be expressed in the form I1 x I2 where Ij is an ideal of Sj for j=1,2. Of course, the converse is elementary, namely if Ij is an ideal of Sj for j=1,2 then I1 x I2 is an ideal of S1 x S2. Having a congruence on a commutative semiring S, its 0-class is an ideal of S, but not every ideal is of this form. Hence, the lattice Id S of all ideals of S and the lattice Ker S of all congruence kernels (i.e. 0-classes of congruences) of S need not be equal. Furthermore, we show that the mapping which assigns to every congruence its kernel need not be a homomorphism from Con S onto Ker S. Moreover, the question arises when a congruence kernel of the direct product S1 x S2 of two commutative semirings can be expressed as a direct product of the corresponding kernels on the factors. In the paper we present necessary and sufficient conditions for such direct decompositions both for ideals and for congruence kernels of commutative semirings. We also provide sufficient conditions for varieties of commutative semirings to have directly decomposable kernels.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124259673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cohomology of BiHom-associative algebras 双共轭代数的上同调
arXiv: Rings and Algebras Pub Date : 2019-01-02 DOI: 10.1142/s0219498822500086
A. Das
{"title":"Cohomology of BiHom-associative algebras","authors":"A. Das","doi":"10.1142/s0219498822500086","DOIUrl":"https://doi.org/10.1142/s0219498822500086","url":null,"abstract":"Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"208 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121636079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
What is the probability that two elements of a finite ring have product zero? 一个有限环的两个元素积为0的概率是多少?
arXiv: Rings and Algebras Pub Date : 2018-10-31 DOI: 10.11113/MJFAS.V16N4.1914
S. Khasraw
{"title":"What is the probability that two elements of a finite ring have product zero?","authors":"S. Khasraw","doi":"10.11113/MJFAS.V16N4.1914","DOIUrl":"https://doi.org/10.11113/MJFAS.V16N4.1914","url":null,"abstract":"In this paper we consider the probability that two elements of a finite ring have product zero. We find bounds of this probability of a finite commutative ring with identity 1. The explicit computations for the ring $mathbf{Z}_n$, the ring of integers modulo $n$, have been obtained.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126051071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
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