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Residually finite dimensional algebras and polynomial almost identities 剩余有限维代数与多项式几乎恒等式
arXiv: Rings and Algebras Pub Date : 2020-05-23 DOI: 10.1142/s0219498822500384
M. Larsen, A. Shalev
{"title":"Residually finite dimensional algebras and polynomial almost identities","authors":"M. Larsen, A. Shalev","doi":"10.1142/s0219498822500384","DOIUrl":"https://doi.org/10.1142/s0219498822500384","url":null,"abstract":"Let $A$ be a residually finite dimensional algebra (not necessarily associative) over a field $k$. Suppose first that $k$ is algebraically closed. We show that if $A$ satisfies a homogeneous almost identity $Q$, then $A$ has an ideal of finite codimension satisfying the identity $Q$. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra $L$ over $k$ is almost $d$-Engel, then $L$ has a nilpotent (resp. locally nilpotent) ideal of finite codimension if char $k=0$ (resp. char $k > 0$). Next, suppose that $k$ is finite (so $A$ is residually finite). We prove that, if $A$ satisfies a homogeneous probabilistic identity $Q$, then $Q$ is a coset identity of $A$. Moreover, if $Q$ is multilinear, then $Q$ is an identity of some finite index ideal of $A$. Along the way we show that, if $Qin klangle x_1,ldots,x_nrangle$ has degree $d$, and $A$ is a finite $k$-algebra such that the probability that $Q(a_1, ldots , a_n)=0$ (where $a_i in A$ are randomly chosen) is at least $1-2^{-d}$, then $Q$ is an identity of $A$. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122836194","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
Some remarks on evolution algebras corresponding to permutations 关于与置换相对应的演化代数的若干注释
arXiv: Rings and Algebras Pub Date : 2020-05-09 DOI: 10.1142/S0219498822501122
B. A. Narkuziyev
{"title":"Some remarks on evolution algebras corresponding to permutations","authors":"B. A. Narkuziyev","doi":"10.1142/S0219498822501122","DOIUrl":"https://doi.org/10.1142/S0219498822501122","url":null,"abstract":"In the present paper we describe absolute nilpotent and some idempotent elements of an n- dimensional evolution algebra corresponding to two permutations and we decompose such algebras to the direct sum of evolution algebras corresponding to cycles of the permutations.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122571828","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: 2
Deformations of relative Rota–Baxter operators on Leibniz algebras 莱布尼兹代数上相对Rota-Baxter算子的变形
arXiv: Rings and Algebras Pub Date : 2020-05-02 DOI: 10.1142/s0219887820501741
Rong Tang, Y. Sheng, Yanqiu Zhou
{"title":"Deformations of relative Rota–Baxter operators on Leibniz algebras","authors":"Rong Tang, Y. Sheng, Yanqiu Zhou","doi":"10.1142/s0219887820501741","DOIUrl":"https://doi.org/10.1142/s0219887820501741","url":null,"abstract":"In this paper, we introduce the cohomology theory of relative Rota-Baxter operators on Leibniz algebras. We use the cohomological approach to study linear and formal deformations of relative Rota-Baxter operators. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order n deformations of a relative Rota-Baxter operator are also characterized in terms of the cohomology theory.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115182061","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: 22
Idempotent systems 幂等系统
arXiv: Rings and Algebras Pub Date : 2020-04-30 DOI: 10.5802/alco.159
K. Nomura, Paul M. Terwilliger
{"title":"Idempotent systems","authors":"K. Nomura, Paul M. Terwilliger","doi":"10.5802/alco.159","DOIUrl":"https://doi.org/10.5802/alco.159","url":null,"abstract":"In this paper we introduce the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. We focus on a family of idempotent systems, said to be symmetric. A symmetric idempotent system is an abstraction of the primary module for the subconstituent algebra of a symmetric association scheme. We describe the symmetric idempotent systems in detail. We also consider a class of symmetric idempotent systems, said to be $P$-polynomial and $Q$-polynomial. In the topic of orthogonal polynomials there is an object called a Leonard system. We show that a Leonard system is essentially the same thing as a symmetric idempotent system that is $P$-polynomial and $Q$-polynomial.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133022524","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
Partial actions and cyclic kummer’s theory 部分作用与循环kummer理论
arXiv: Rings and Algebras Pub Date : 2020-04-28 DOI: 10.1142/s0219498822500323
A. Canas, V. Mar'in, H. Pinedo
{"title":"Partial actions and cyclic kummer’s theory","authors":"A. Canas, V. Mar'in, H. Pinedo","doi":"10.1142/s0219498822500323","DOIUrl":"https://doi.org/10.1142/s0219498822500323","url":null,"abstract":"We introduce a theory of cyclic Kummer extensions of commutative rings for partial Galois extensions of finite groups, extending some of the well-known results of the theory of Kummer extensions of commutative rings developed by A. Z. Borevich. In particular, we provide necessary and sufficient conditions to determine when a partial $n$-kummerian extension is equivalent to either a radical or a $I$-radical extension, for some subgroup $I$ of the cyclic group $C_n$.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125662428","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
A tensor rank theory and maximum full rank subtensors 张量秩理论与最大满秩子张量
arXiv: Rings and Algebras Pub Date : 2020-04-22 DOI: 10.4310/CMS.2021.V19.N1.A11
L. Qi, Xinzhen Zhang, Yannan Chen
{"title":"A tensor rank theory and maximum full rank subtensors","authors":"L. Qi, Xinzhen Zhang, Yannan Chen","doi":"10.4310/CMS.2021.V19.N1.A11","DOIUrl":"https://doi.org/10.4310/CMS.2021.V19.N1.A11","url":null,"abstract":"A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the max-full-rank-submatrix property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131619850","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: 2
The length and depth of associative algebras 结合代数的长度和深度
arXiv: Rings and Algebras Pub Date : 2020-04-15 DOI: 10.2140/PJM.2021.311.197
Damian Sercombe, A. Shalev
{"title":"The length and depth of associative algebras","authors":"Damian Sercombe, A. Shalev","doi":"10.2140/PJM.2021.311.197","DOIUrl":"https://doi.org/10.2140/PJM.2021.311.197","url":null,"abstract":"Recently there has been considerable interest in studying the length and the depth of finite groups, algebraic groups and Lie groups. In this paper we introduce and study similar notions for algebras. Let $k$ be a field and let $A$ be an associative, not necessarily unital, algebra over $k$. An unrefinable chain of $A$ is a chain of subalgebras $A=A_0>A_1>...>A_t=0$ for some integer $t$ where each $A_i$ is a maximal subalgebra of $A_{i-1}$. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of $A$. It turns out that finite length, finite depth and finite dimension are equivalent properties for $A$. For $A$ finite dimensional, we give a formula for the length of $A$, we bound the depth of $A$, and we study when the length of $A$ equals its dimension and its depth respectively. Finally, we investigate under what circumstances the dimension of $A$ is bounded above by a function of its length, or its depth, or its length minus its depth.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"111 8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115522230","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
Local and $2$-local automorphisms of simple generalized Witt algebras 简单广义Witt代数的局部和$2$-局部自同构
arXiv: Rings and Algebras Pub Date : 2020-03-31 DOI: 10.4310/ARKIV.2021.V59.N1.A1
Yang Chen, K. Zhao, Yueqiang Zhao
{"title":"Local and $2$-local automorphisms of simple generalized Witt algebras","authors":"Yang Chen, K. Zhao, Yueqiang Zhao","doi":"10.4310/ARKIV.2021.V59.N1.A1","DOIUrl":"https://doi.org/10.4310/ARKIV.2021.V59.N1.A1","url":null,"abstract":"In this paper, we prove that every invertible $2$-local or local automorphism of a simple generalized Witt algebra over any field of characteristic $0$ is an automorphism. In particular, every $2$-local or local automorphism of Witt algebras $W_n$ is an automorphism for all $nin mathbb{N}$. But some simple generalized Witt algebras indeed have $2$-local (and local) automorphisms that are not automorphisms.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"88 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114506852","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
Symmetric polynomials in Leibniz algebras and their inner automorphisms 莱布尼兹代数中的对称多项式及其内自同构
arXiv: Rings and Algebras Pub Date : 2020-03-29 DOI: 10.3906/mat-2006-44
Şehmus Fındık, Z. Özkurt
{"title":"Symmetric polynomials in Leibniz algebras and their inner automorphisms","authors":"Şehmus Fındık, Z. Özkurt","doi":"10.3906/mat-2006-44","DOIUrl":"https://doi.org/10.3906/mat-2006-44","url":null,"abstract":"Let $L_n$ be the free metabelian Leibniz algebra generated by the set $X_n={x_1,ldots,x_n}$ over a field $K$ of characteristic zero. This is the free algebra of rank $n$ in the variety of solvable of class $2$ Leibniz algebras. We call an element $s(X_n)in L_n$ symmetric if $s(x_{sigma(1)},ldots,x_{sigma(n)})=s(x_1,ldots,x_n)$ for each permutation $sigma$ of ${1,ldots,n}$. The set $L_n^{S_n}$ of symmetric polynomials of $L_n$ is the algebra of invariants of the symmetric group $S_n$. Let $K[X_n]$ be the usual polynomial algebra with indeterminates from $X_n$. The description of the algebra $K[X_n]^{S_n}$ is well known, and the algebra $(L_n')^{S_n}$ in the commutator ideal $L_n'$ is a right $K[X_n]^{S_n}$-module. We give explicit forms of elements of the $K[X_n]^{S_n}$-module $(L_n')^{S_n}$. Additionally, we determine the description of the group ${rm Inn}(L_{n}^{S_n})$ of inner automorphisms of the algebra $L_n^{S_n}$. The findings can be considered as a generalization of the recent results obtained for the free metabelian Lie algebra of rank $n$.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124087059","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: 2
The talented monoid of a directed graph with applications to graph algebras 有向图的天才单峰及其在图代数中的应用
arXiv: Rings and Algebras Pub Date : 2020-03-22 DOI: 10.4171/rmi/1277
L. Cordeiro, D. Gonccalves, R. Hazrat
{"title":"The talented monoid of a directed graph with applications to graph algebras","authors":"L. Cordeiro, D. Gonccalves, R. Hazrat","doi":"10.4171/rmi/1277","DOIUrl":"https://doi.org/10.4171/rmi/1277","url":null,"abstract":"It is a conjecture that for the class of Leavitt path algebras associated to finite directed graphs, their graded Grothendieck groups $K_0^{mathrm{gr}}$ are a complete invariant. For a Leavitt path algebra $L_{mathsf k}(E)$, with coefficient in a field ${mathsf k}$, the monoid of the positive cone of $K_0^{mathrm{gr}}(L_{mathsf k}(E))$ can be described completely in terms of the graph $E$. In this note we further investigate the structure of this \"talented monoid\", showing how it captures intrinsic properties of the graph and hence the structure of its associated Leavitt path algebras. In particular, for the class of strongly connected graphs, we show that the notion of the period of a graph can be completely described via the talented monoid. As an application, we will give a finer characterisation of the purely infinite simple Leavitt path algebras in terms of properties of the associated graph. We show that graded isomorphism of algebras preserve the period of the graphs, and obtain results giving more evidence to the graded classification conjecture.","PeriodicalId":233322,"journal":{"name":"arXiv: Rings and Algebras","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116411107","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: 13
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