In this paper we introduce the notion of $(b,c)$-polar elements in an associative ring $R$. Necessary and sufficient conditions of an element $ain R$ to be $(b,c)$-polar are investigated. We show that an element $ain R$ is $(b,c)$-polar if and only if $a$ is $(b,c)$-invertible. In particular the $(b,c)$-polarity is a generalization of the polarity along an element introduced by Song, Zhu and Mosi'c [14] if $b=c$, and the polarity introduced by Koliha and Patricio [10]. Further characterizations are obtained in the Banach space context.
{"title":"New characterization of $(b,c)$-inverses through polarity","authors":"Btissam Laghmam, Hassane Zguitti","doi":"arxiv-2409.11987","DOIUrl":"https://doi.org/arxiv-2409.11987","url":null,"abstract":"In this paper we introduce the notion of $(b,c)$-polar elements in an\u0000associative ring $R$. Necessary and sufficient conditions of an element $ain\u0000R$ to be $(b,c)$-polar are investigated. We show that an element $ain R$ is\u0000$(b,c)$-polar if and only if $a$ is $(b,c)$-invertible. In particular the\u0000$(b,c)$-polarity is a generalization of the polarity along an element\u0000introduced by Song, Zhu and Mosi'c [14] if $b=c$, and the polarity introduced\u0000by Koliha and Patricio [10]. Further characterizations are obtained in the\u0000Banach space context.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247460","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}
Let $S/R$ be a Frobenius extension with $_RS_R$ centrally projective over $R$. We show that if $_Romega$ is a Wakamatsu tilting module then so is $_SSotimes_Romega$, and the natural ring homomorphism from the endomorphism ring of $_Romega$ to the endomorphism ring of $_SSotimes_Romega$ is a Frobenius extension in addition that pd$(omega_T)$ is finite, where $T$ is the endomorphism ring of $_Romega$. We also obtain that the relative $n$-torsionfreeness of modules is preserved under Frobenius extensions. Furthermore, we give an application, which shows that the generalized G-dimension with respect to a Wakamatsu module is invariant under Frobenius extensions.
{"title":"Relative torsionfreeness and Frobenius extensions","authors":"Yanhong Bao, Jiafeng Lü, Zhibing Zhao","doi":"arxiv-2409.11892","DOIUrl":"https://doi.org/arxiv-2409.11892","url":null,"abstract":"Let $S/R$ be a Frobenius extension with $_RS_R$ centrally projective over\u0000$R$. We show that if $_Romega$ is a Wakamatsu tilting module then so is\u0000$_SSotimes_Romega$, and the natural ring homomorphism from the endomorphism\u0000ring of $_Romega$ to the endomorphism ring of $_SSotimes_Romega$ is a\u0000Frobenius extension in addition that pd$(omega_T)$ is finite, where $T$ is the\u0000endomorphism ring of $_Romega$. We also obtain that the relative\u0000$n$-torsionfreeness of modules is preserved under Frobenius extensions.\u0000Furthermore, we give an application, which shows that the generalized\u0000G-dimension with respect to a Wakamatsu module is invariant under Frobenius\u0000extensions.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247462","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}
We prove that, unlike in the case of paths, the signature matrix of a membrane does not satisfy any algebraic relations. We derive novel closed-form expressions for the signatures of polynomial membranes and piecewise bilinear interpolations for arbitrary $2$-parameter data in $d$-dimensional space. We show that these two families of membranes admit the same set of signature matrices and scrutinize the corresponding affine variety.
{"title":"Signature matrices of membranes","authors":"Felix Lotter, Leonard Schmitz","doi":"arxiv-2409.11996","DOIUrl":"https://doi.org/arxiv-2409.11996","url":null,"abstract":"We prove that, unlike in the case of paths, the signature matrix of a\u0000membrane does not satisfy any algebraic relations. We derive novel closed-form\u0000expressions for the signatures of polynomial membranes and piecewise bilinear\u0000interpolations for arbitrary $2$-parameter data in $d$-dimensional space. We\u0000show that these two families of membranes admit the same set of signature\u0000matrices and scrutinize the corresponding affine variety.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247463","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}
We introduce an operation on skew-symmetric matrices over $mathbb{Z}/ellmathbb{Z}$ called switching, and also define a class of skew-symmetric matrices over $mathbb{Z}/ellmathbb{Z}$ referred to as modular Eulerian matrices. We then show that these are closely related to the graded module categories over skew polynomial algebras at $ell$-th roots of unity. As an application, we study the point simplicial complexes of skew polynomial algebras at cube roots of unity.
{"title":"Combinatorics of graded module categories over skew polynomial algebras at roots of unity","authors":"Akihiro Higashitani, Kenta Ueyama","doi":"arxiv-2409.10904","DOIUrl":"https://doi.org/arxiv-2409.10904","url":null,"abstract":"We introduce an operation on skew-symmetric matrices over\u0000$mathbb{Z}/ellmathbb{Z}$ called switching, and also define a class of\u0000skew-symmetric matrices over $mathbb{Z}/ellmathbb{Z}$ referred to as modular\u0000Eulerian matrices. We then show that these are closely related to the graded\u0000module categories over skew polynomial algebras at $ell$-th roots of unity. As\u0000an application, we study the point simplicial complexes of skew polynomial\u0000algebras at cube roots of unity.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"196 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247464","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}
We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type. The new contribution is a proof of this conjecture for cluster algebras of type $mathbb{D}$ and an algorithm for the exceptional types. For the type $mathbb{D}$ cases, our approach involves geometric model provided by discs with a puncture. By removing the puncture or changing the puncture to an unmarked boundary component, this also yields an alternative proof for the denominator conjecture of cluster algebras of type $mathbb{A}$ and $mathbb{C}$ respectively.
{"title":"On denominator conjecture for cluster algebras of finite type","authors":"Changjian Fu, Shengfei Geng","doi":"arxiv-2409.10914","DOIUrl":"https://doi.org/arxiv-2409.10914","url":null,"abstract":"We continue our investigation on denominator conjecture of Fomin and\u0000Zelevinsky for cluster algebras via geometric models initialed in cite{FG22}.\u0000In this paper, we confirm the denominator conjecture for cluster algebras of\u0000finite type. The new contribution is a proof of this conjecture for cluster\u0000algebras of type $mathbb{D}$ and an algorithm for the exceptional types. For\u0000the type $mathbb{D}$ cases, our approach involves geometric model provided by\u0000discs with a puncture. By removing the puncture or changing the puncture to an\u0000unmarked boundary component, this also yields an alternative proof for the\u0000denominator conjecture of cluster algebras of type $mathbb{A}$ and\u0000$mathbb{C}$ respectively.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247406","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}
In this short paper, we establish the local Noetherian property for the linear categories of Brauer, partition algebras, and other related categories of diagram algebras with no restrictions on their various parameters.
{"title":"Noetherianity of Diagram Algebras","authors":"Anthony Muljat, Khoa Ta","doi":"arxiv-2409.10885","DOIUrl":"https://doi.org/arxiv-2409.10885","url":null,"abstract":"In this short paper, we establish the local Noetherian property for the\u0000linear categories of Brauer, partition algebras, and other related categories\u0000of diagram algebras with no restrictions on their various parameters.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247408","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}
A Novikov conformal algebra is a conformal algebra such that its coefficient algebra is right-symmetric and left commutative (i.e., it is an ``ordinary'' Novikov algebra). We prove that every Novikov conformal algebra with a uniformly bounded locality function on a set of generators can be embedded into a commutative conformal algebra with a derivation. In particular, every finitely generated Novikov conformal algebra has a commutative conformal differential envelope. For infinitely generated algebras this statement is not true in general.
{"title":"Differential envelopes of Novikov conformal algebras","authors":"P. S. Kolesnikov, A. A. Nesterenko","doi":"arxiv-2409.10029","DOIUrl":"https://doi.org/arxiv-2409.10029","url":null,"abstract":"A Novikov conformal algebra is a conformal algebra such that its coefficient\u0000algebra is right-symmetric and left commutative (i.e., it is an ``ordinary''\u0000Novikov algebra). We prove that every Novikov conformal algebra with a\u0000uniformly bounded locality function on a set of generators can be embedded into\u0000a commutative conformal algebra with a derivation. In particular, every\u0000finitely generated Novikov conformal algebra has a commutative conformal\u0000differential envelope. For infinitely generated algebras this statement is not\u0000true in general.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"196 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247412","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}
Let $M_{1,2}(F)$ be the algebra of $3 times 3$ matrices with orthosymplectic superinvolution $*$ over a field $F$ of characteristic zero. We study the $*$-identities of this algebra through the representation theory of the group $mathbb{H}_n = (mathbb{Z}_2 times mathbb{Z}_2) sim S_n$. We decompose the space of multilinear $*$-identities of degree $n$ into the sum of irreducibles under the $mathbb{H}_n$-action in order to study the irreducible characters appearing in this decomposition with non-zero multiplicity. Moreover, by using the representation theory of the general linear group, we determine all the $*$-polynomial identities of $M_{1,2}(F)$ up to degree $3$.
{"title":"On the identities and cocharacters of the algebra of $3 times 3$ matrices with orthosymplectic superinvolution","authors":"Sara Accomando","doi":"arxiv-2409.10187","DOIUrl":"https://doi.org/arxiv-2409.10187","url":null,"abstract":"Let $M_{1,2}(F)$ be the algebra of $3 times 3$ matrices with orthosymplectic\u0000superinvolution $*$ over a field $F$ of characteristic zero. We study the\u0000$*$-identities of this algebra through the representation theory of the group\u0000$mathbb{H}_n = (mathbb{Z}_2 times mathbb{Z}_2) sim S_n$. We decompose the\u0000space of multilinear $*$-identities of degree $n$ into the sum of irreducibles\u0000under the $mathbb{H}_n$-action in order to study the irreducible characters\u0000appearing in this decomposition with non-zero multiplicity. Moreover, by using\u0000the representation theory of the general linear group, we determine all the\u0000$*$-polynomial identities of $M_{1,2}(F)$ up to degree $3$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247410","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}
In this paper, we prove that double extension regular algebras of type (14641) are not differentially smooth.
本文证明，type(14641) 的双外延正则表达式不具有差分光滑性。
{"title":"Smooth geometry of double extension regular algebras of type (14641)","authors":"Andrés Rubiano, Armando Reyes","doi":"arxiv-2409.10264","DOIUrl":"https://doi.org/arxiv-2409.10264","url":null,"abstract":"In this paper, we prove that double extension regular algebras of type\u0000(14641) are not differentially smooth.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247415","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}
Let $R$ be a finite non-commutative ring with $1ne 0$. By a polynomial function on $R$, we mean a function $Fcolon Rlongrightarrow R$ induced by a polynomial $f=sumlimits_{i=0}^{n}a_ix^iin R[x]$ via right substitution of the variable $x$, i.e. $F(a)=f(a)= sumlimits_{i=0}^{n}a_ia^i$ for every $ain R$. In this paper, we study the polynomial functions of the free $R$-algebra with a central basis ${1,beta_1,ldots,beta_k}$ ($kge 1$) such that $beta_ibeta_j=0$ for every $1le i,jle k$, $R[beta_1,ldots,beta_k]$. %, the ring of dual numbers over $R$ in $k$ variables. Our investigation revolves around assigning a polynomial $lambda_f(y,z)$ over $R$ in non-commutative variables $y$ and $z$ to each polynomial $f$ in $R[x]$; and describing the polynomial functions on $R[beta_1,ldots,beta_k]$ through the polynomial functions induced on $R$ by polynomials in $R[x]$ and by their assigned polynomials in the in non-commutative variables $y$ and $z$. %and analyzing the resulting polynomial functions on $R[beta_1,ldots,beta_k]$. By extending results from the commutative case to the non-commutative scenario, we demonstrate that several properties and theorems in the commutative case can be generalized to the non-commutative setting with appropriate adjustments.
{"title":"Polynomial functions on a class of finite non-commutative rings","authors":"Amr Ali Abdulkader Al-Maktry, Susan F. El-Deken","doi":"arxiv-2409.10208","DOIUrl":"https://doi.org/arxiv-2409.10208","url":null,"abstract":"Let $R$ be a finite non-commutative ring with $1ne 0$. By a polynomial\u0000function on $R$, we mean a function $Fcolon Rlongrightarrow R$ induced by a\u0000polynomial $f=sumlimits_{i=0}^{n}a_ix^iin R[x]$ via right substitution of\u0000the variable $x$, i.e. $F(a)=f(a)= sumlimits_{i=0}^{n}a_ia^i$ for every $ain R$. In this paper,\u0000we study the polynomial functions of the free $R$-algebra with a central basis\u0000${1,beta_1,ldots,beta_k}$ ($kge 1$) such that $beta_ibeta_j=0$ for\u0000every $1le i,jle k$, $R[beta_1,ldots,beta_k]$. %, the ring of dual numbers\u0000over $R$ in $k$ variables. Our investigation revolves around assigning a polynomial $lambda_f(y,z)$\u0000over $R$ in non-commutative variables $y$ and $z$ to each polynomial $f$ in\u0000$R[x]$; and describing the polynomial functions on $R[beta_1,ldots,beta_k]$\u0000through the polynomial functions induced on $R$ by polynomials in $R[x]$ and by\u0000their assigned polynomials in the in non-commutative variables $y$ and $z$.\u0000%and analyzing the resulting polynomial functions on\u0000$R[beta_1,ldots,beta_k]$. By extending results from the commutative case to the non-commutative\u0000scenario, we demonstrate that several properties and theorems in the\u0000commutative case can be generalized to the non-commutative setting with\u0000appropriate adjustments.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142247411","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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