{"title":"A Survey of Racks and Quandles: Some Recent Developments","authors":"M. Elhamdadi","doi":"10.1142/S1005386720000425","DOIUrl":"https://doi.org/10.1142/S1005386720000425","url":null,"abstract":"This short survey contains some recent developments of the algebraic theory of racks and quandles. We report on some elements of representation theory of quandles and ring theoretic approach to quandles.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123758843","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":"Almost pi algebras are pi","authors":"M. Larsen, A. Shalev","doi":"10.1090/proc/15818","DOIUrl":"https://doi.org/10.1090/proc/15818","url":null,"abstract":"We define the notion of an almost polynomial identity of an associative algebra $R$, and show that its existence implies the existence of an actual polynomial identity of $R$. A similar result is also obtained for Lie algebras and Jordan algebras. We also prove related quantitative results for simple and semisimple algebras.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"38 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133318757","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":"Splitting quaternion algebras defined over a finite field extension","authors":"K. Becher, Fatma Kader Bingol, D. Leep","doi":"10.1142/s021949882250061x","DOIUrl":"https://doi.org/10.1142/s021949882250061x","url":null,"abstract":"We study systems of quadratic forms over fields and their isotropy over 2-extensions. We apply this to obtain particular splitting fields for quaternion algebras defined over a finite field extension. As a consequence, we obtain that every central simple algebra of degree 16 is split by a 2-extension of degree at most 2^{16}.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131446969","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":"Rota-Baxter operators and non-skew-symmetric solutions of the classical Yang–Baxter equation on quadratic Lie algebras","authors":"M. Goncharov","doi":"10.33048/semi.2019.16.149","DOIUrl":"https://doi.org/10.33048/semi.2019.16.149","url":null,"abstract":"We study possible connections between Rota-Baxter operators of non-zero weight and non-skew-symmetric solutions of the classical Yang-Baxter equation on finite-dimensional quadratic Lie algebras. The particular attention is made to the case when for a solution $r$ the element $r+tau(r)$ is $L$-invariant.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115577247","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":"Convolutions on the complex torus","authors":"M. Leitner","doi":"10.1142/S1793042121500391","DOIUrl":"https://doi.org/10.1142/S1793042121500391","url":null,"abstract":"\"Quasi-elliptic\" functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated and given by Eisenstein series. A related structure has appeared recently in the computation of Feynman integrals. The two approaches are related by a sequence of polynomials with interlacing zeroes.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132928154","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":"Comparison of two notions of weak crossed product","authors":"J. Guccione, J. Guccione","doi":"10.1142/s0219498821500596","DOIUrl":"https://doi.org/10.1142/s0219498821500596","url":null,"abstract":"We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in cite{BB} with the notion of crossed product with a weak Hopf algebra introduced in~cite{AG}","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134497527","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 Peterson's open problem and representations of general linear groups","authors":"Dang Vo Phuc","doi":"10.4134/JKMS.j200219","DOIUrl":"https://doi.org/10.4134/JKMS.j200219","url":null,"abstract":"Fix $mathbb Z/2$ is the prime field of two elements and write $mathcal A_2$ for the mod $2$ Steenrod algebra. Denote by $GL_d:= GL(d, mathbb Z/2)$ the general linear group of rank $d$ over $mathbb Z/2$ and by $mathscr P_d$ the polynomial algebra $mathbb Z/2[x_1, x_2, ldots, x_d]$ as an unstable left $mathcal A_2$-module on $d$ generators of degree one. We study the Peterson hit problem of finding a minimal set of generators for $mathcal A_2$-module $mathscr P_d.$ Equivalently, we need to determine a basis for the $mathbb Z/2$-vector space $$Qmathscr P_d := mathbb Z/2otimes_{mathcal A_2} mathscr P_d cong mathscr P_d/mathcal A_2^+mathscr P_d$$ in each degree $ngeq 1.$ Note that this space is a representation of $GL_d$ over $mathbb Z/2.$ The problem is still open for $dgeq 5.$ \u0000In this paper, we explicitly compute the hit problem in the $5$-variable case and in generic degree $n = r(2^t -1) + 2^ts$ with $r = d = 5, s =8$ and $t$ an arbitrary positive integer. This result is used to prove that the fifth Singer algebraic transfer detects the element $h_1g_1in {rm Ext}_{mathcal {A}_2}^{5, 26}(mathbb Z/2, mathbb Z/2),$ the $E_2$-term of Adams spectral sequence of spheres. As an easy consequence, we see that it is an isomorphism in the bidegree $(5, 13.2^t)$ for $t= 1.$ Here the Singer algebraic transfer is a homomorphism from $GL_d$-coinvariants of certain subspaces of $Qmathscr P_d$ to the cohomology groups of the Steenrod algebra, ${rm Ext}_{mathcal A_2}^{d, *}(mathbb Z/2, mathbb Z/2)$.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134004800","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":"Another characterization of congruence distributive varieties","authors":"P. Lipparini","doi":"10.1556/012.2020.57.3.1471","DOIUrl":"https://doi.org/10.1556/012.2020.57.3.1471","url":null,"abstract":"We provide a Maltsev characterization of congruence distributive varieties by showing that a variety $mathcal {V}$ is congruence distributive if and only if the congruence identity $alpha cap (beta circ gamma circ beta ) subseteq alpha beta circ gamma circ alpha beta circ gamma dots $ ($k$ factors) holds in $mathcal {V}$, for some natural number $k$.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"120 10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122706578","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":"The images of noncommutative polynomials evaluated on the quaternion algebra","authors":"S. Malev","doi":"10.1142/s0219498821500742","DOIUrl":"https://doi.org/10.1142/s0219498821500742","url":null,"abstract":"Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is a vector space. In this paper we settle the analogous conjecture for a quaternion algebra.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123138536","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":"Local nearrings on finite non-abelian $2$-generated $p$-groups","authors":"Iryna Raievska, Maryna Raievska","doi":"10.15330/CMP.12.1.199-207","DOIUrl":"https://doi.org/10.15330/CMP.12.1.199-207","url":null,"abstract":"Finite non-abelian non-metacyclic $2$-generated $p$-groups (${p>2}$) of nilpotency class $2$ with cyclic commutator subgroup which are the additive groups of local nearrings are described. It is shown that the subgroup of all non-invertible elements is of index $p$ in its additive group.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126603826","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}