{"title":"A review of Lie 2-algebras","authors":"H. Lang, Zhangju Liu","doi":"10.11845/sxjz.2020003a.","DOIUrl":"https://doi.org/10.11845/sxjz.2020003a.","url":null,"abstract":"We first recall two equivalent definitions of Lie $2$-algebras, categorification of Lie algebras and $2$-term $L_infty$-algebras. Then we present four different kinds of Lie $2$-algebras from $2$-plectic manifolds, Courant algebroids, homotopy Poisson manifolds and affine multivector fields on a Lie groupoid respectively. Moreover, we recall the cohomology theory of Lie $2$-algebras and analyze its lower degree cases. The integration of strict Lie $2$-algebras to strict Lie $2$-groups is also discussed.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"261 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122689800","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}
{"title":"Lie automorphisms of incidence algebras","authors":"'. Z. Fornaroli, M. Khrypchenko, E. A. Santulo","doi":"10.1090/proc/15786","DOIUrl":"https://doi.org/10.1090/proc/15786","url":null,"abstract":"Let $X$ be a finite connected poset and $K$ a field. We give a full description of Lie automorphisms of the incidence algebra $I(X,K)$. In particular, we show that they are in general not proper.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124050604","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}
{"title":"Minimal Hopf-Galois Structures on Separable Field Extensions","authors":"Tony Ezome, C. Greither","doi":"10.1285/I15900932V41N1P55","DOIUrl":"https://doi.org/10.1285/I15900932V41N1P55","url":null,"abstract":"In Hopf-Galois theory, every $H$-Hopf-Galois structure on a field extension $K/k$ gives rise to an injective map $mathcal{F}$ from the set of $k$-sub-Hopf algebras of $H$ into the intermediate fields of $K/k$. Recent papers on the failure of the surjectivity of $mathcal{F}$ reveal that there exist many Hopf-Galois structures for which there are many more subfields than sub-Hopf algebras. This paper surveys and illustrates group-theoretical methods to determine $H$-Hopf-Galois structures on finite separable extensions in the extreme situation when $H$ has only two sub-Hopf algebras.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"269 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123304916","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}
{"title":"Locally-zero Groupoids and the Center of Bin(X).","authors":"Hiba F. Fayoumi","doi":"10.4134/CKMS.2011.26.2.163","DOIUrl":"https://doi.org/10.4134/CKMS.2011.26.2.163","url":null,"abstract":"In this paper we introduce the notion of the center $ZBin(X)$ in the semigroup $Bin(X)$ of all binary systems on a set $X$, and show that if $(X,bullet)in ZBin(X)$, then $xnot=y$ implies ${x,y}={xbullet y,ybullet x}$.Moreover, we show that a groupoid $(X,bullet )in ZBin(X)$ if and only if it is a locally-zero groupoid.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121022684","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}
{"title":"Jordan Algebras of Symmetric Matrices","authors":"A. Bik, Henrik Eisenmann, B. Sturmfels","doi":"10.4418/2021.76.2.3","DOIUrl":"https://doi.org/10.4418/2021.76.2.3","url":null,"abstract":"We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"215 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122433747","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}
{"title":"$frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras","authors":"B. Ferreira, I. Kaygorodov, V. Lopatkin","doi":"10.1007/s13398-021-01088-2","DOIUrl":"https://doi.org/10.1007/s13398-021-01088-2","url":null,"abstract":"","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"253 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123054351","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}
{"title":"Maximum number of solutions of $x^q = x$ in a finite noncommutative algebra","authors":"Vineeth Chintala","doi":"10.1142/S0219498812500892","DOIUrl":"https://doi.org/10.1142/S0219498812500892","url":null,"abstract":"We obtain a bound on the number of solutions of $x^q=x$ in a finite noncommutative algebra over a field with $q$ elements. Furthermore, we completely characterize those rings for which this maximum number is attained.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123063136","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}
{"title":"On graded $ s $-prime submodules","authors":"Hicham Saber, T. Alraqad, R. Abu-Dawwas","doi":"10.3934/math.2021152","DOIUrl":"https://doi.org/10.3934/math.2021152","url":null,"abstract":"In this article, we introduce the concepts of graded $s$-prime submodules which is a generalization of graded prime submodules. We study the behavior of this notion with respect to graded homomorphisms, localization of graded modules, direct product, and idealization. We succeeded to prove the existence of graded $s$-prime submodules in the case of graded-Noetherian modules. Also, we provide some sufficient conditions for the existence of such objects in the general case, as well as, in the particular case of grading by $mathbb{Z}$, a finite group, or a polycyclic-by-finite group, in addition to crossed product grading.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132081357","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}
{"title":"Quotients of the Booleanization of an Inverse Semigroup","authors":"G. Kudryavtseva","doi":"10.1007/978-981-33-4842-4_6","DOIUrl":"https://doi.org/10.1007/978-981-33-4842-4_6","url":null,"abstract":"","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114304595","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}
{"title":"Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture","authors":"A. Kanel-Belov, S. Malev, L. Rowen, R. Yavich","doi":"10.3842/sigma.2020.071","DOIUrl":"https://doi.org/10.3842/sigma.2020.071","url":null,"abstract":"Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set ${rm sl}_n(K)$ of matrices of trace 0, or all of $M_n(K)$. This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for $n=2$ in Section 2, some decisive results for $n=3$ in Section 3, and partial information for $ngeq 3$ in Section 4, also for non-multilinear polynomials. In addition we consider the case of $K$ not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131615694","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}
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