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Trivial extensions of monomial algebras are symmetric fractional Brauer configuration algebras of type S 单项式代数的三维扩展是 S 型对称分数布劳尔构型代数
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-05 DOI: arxiv-2408.02537
Yuming Liu, Bohan Xing
{"title":"Trivial extensions of monomial algebras are symmetric fractional Brauer configuration algebras of type S","authors":"Yuming Liu, Bohan Xing","doi":"arxiv-2408.02537","DOIUrl":"https://doi.org/arxiv-2408.02537","url":null,"abstract":"By giving some equivalent definitions of fractional Brauer configuration\u0000algebras of type S in some special cases, we construct a fractional Brauer\u0000configuration from any monomial algebra. We show that this algebra is\u0000isomorphic to the trivial extension of the given monomial algebra. Moreover, we\u0000show that there exists a one-to-one correspondence between the isomorphism\u0000classes of monomial algebras and the equivalence classes of pairs consisting of\u0000a symmetric fractional Brauer configuration algebra of type S with trivial\u0000degree function and a given admissible cut over it.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935665","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
On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra 论后李代数的普遍包络代数的子邻接霍普夫代数
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-02 DOI: arxiv-2408.01345
Yunnan Li
{"title":"On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra","authors":"Yunnan Li","doi":"arxiv-2408.01345","DOIUrl":"https://doi.org/arxiv-2408.01345","url":null,"abstract":"Recently the notion of post-Hopf algebra was introduced, with the universal\u0000enveloping algebra of a post-Lie algebra as the fundamental example. A novel\u0000property is that any cocommutative post-Hopf algebra gives rise to a\u0000sub-adjacent Hopf algebra with a generalized Grossman-Larson product. By\u0000twisting the post-Hopf product, we provide a combinatorial antipode formula for\u0000the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie\u0000algebra. Relating to such a sub-adjacent Hopf algebra, we also obtain a closed\u0000inverse formula for the Oudom-Guin isomorphism in the context of post-Lie\u0000algebras. Especially as a byproduct, we derive a cancellation-free antipode\u0000formula for the Grossman-Larson Hopf algebra of ordered trees through a\u0000concrete tree-grafting expression.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935664","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
Countably Generated Matrix Algebras 可数生成矩阵代数
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-02 DOI: arxiv-2408.01034
Arvid Siqveland
{"title":"Countably Generated Matrix Algebras","authors":"Arvid Siqveland","doi":"arxiv-2408.01034","DOIUrl":"https://doi.org/arxiv-2408.01034","url":null,"abstract":"We define the completion of an associative algebra $A$ in a set\u0000$M={M_1,dots,M_r}$ of $r$ right $A$-modules in such a way that if $mathfrak\u0000asubseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the\u0000(right) module $A/mathfrak a$ is $hat A^Msimeq hat A^{mathfrak a}.$ This\u0000works by defining $hat A^M$ as a formal algebra determined up to a computation\u0000in a category called GMMP-algebras. From deformation theory we get that the\u0000computation results in a formal algebra which is the prorepresenting hull of\u0000the noncommutative deformation functor, and this hull is unique up to\u0000isomorphism.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935668","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
Factorization of a prime matrix in even blocks 素数矩阵的偶数块因式分解
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-01 DOI: arxiv-2408.00627
Haoming Wang
{"title":"Factorization of a prime matrix in even blocks","authors":"Haoming Wang","doi":"arxiv-2408.00627","DOIUrl":"https://doi.org/arxiv-2408.00627","url":null,"abstract":"In this paper, a matrix is said to be prime if the row and column of this\u0000matrix are both prime numbers. We establish various necessary and sufficient\u0000conditions for developing matrices into the sum of tensor products of prime\u0000matrices. For example, if the diagonal of a matrix blocked evenly are pairwise\u0000commutative, it yields such a decomposition. The computational complexity of\u0000multiplication of these algorithms is shown to be $O(n^{5/2})$. In the section\u00005, a decomposition is proved to hold if and only if every even natural number\u0000greater than 2 is the sum of two prime numbers.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885193","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
Structure spaces and allied problems on a class of rings of measurable functions 一类可测函数环上的结构空间及相关问题
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-01 DOI: arxiv-2408.00505
Soumajit Dey, Sudip Kumar Acharyya, Dhananjoy Mandal
{"title":"Structure spaces and allied problems on a class of rings of measurable functions","authors":"Soumajit Dey, Sudip Kumar Acharyya, Dhananjoy Mandal","doi":"arxiv-2408.00505","DOIUrl":"https://doi.org/arxiv-2408.00505","url":null,"abstract":"A ring $S(X,mathcal{A})$ of real valued $mathcal{A}$-measurable functions\u0000defined over a measurable space $(X,mathcal{A})$ is called a $chi$-ring if\u0000for each $Ein mathcal{A} $, the characteristic function $chi_{E}in\u0000S(X,mathcal{A})$. The set $mathcal{U}_X$ of all $mathcal{A}$-ultrafilters on\u0000$X$ with the Stone topology $tau$ is seen to be homeomorphic to an appropriate\u0000quotient space of the set $mathcal{M}_X$ of all maximal ideals in\u0000$S(X,mathcal{A})$ equipped with the hull-kernel topology $tau_S$. It is\u0000realized that $(mathcal{U}_X,tau)$ is homeomorphic to\u0000$(mathcal{M}_S,tau_S)$ if and only if $S(X,mathcal{A})$ is a Gelfand ring.\u0000It is further observed that $S(X,mathcal{A})$ is a Von-Neumann regular ring if\u0000and only if each ideal in this ring is a $mathcal{Z}_S$-ideal and\u0000$S(X,mathcal{A})$ is Gelfand when and only when every maximal ideal in it is a\u0000$mathcal{Z}_S$-ideal. A pair of topologies $u_mu$-topology and\u0000$m_mu$-topology, are introduced on the set $S(X,mathcal{A})$ and a few\u0000properties are studied.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885099","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
Simplicity of $*$-algebras of non-Hausdorff $mathbb{Z}_2$-multispinal groupoids 非豪斯多夫$mathbb{Z}_2$多核群集的$*$数组的简单性
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-01 DOI: arxiv-2408.00442
C. Farsi, N. S. Larsen, J. Packer, N. Thiem
{"title":"Simplicity of $*$-algebras of non-Hausdorff $mathbb{Z}_2$-multispinal groupoids","authors":"C. Farsi, N. S. Larsen, J. Packer, N. Thiem","doi":"arxiv-2408.00442","DOIUrl":"https://doi.org/arxiv-2408.00442","url":null,"abstract":"We study simplicity of $C^*$-algebras arising from self-similar groups of\u0000$mathbb{Z}_2$-multispinal type, a generalization of the Grigorchuk case whose\u0000simplicity was first proved by L. Clark, R. Exel, E. Pardo, C. Starling, and A.\u0000Sims in 2019, and we prove results generalizing theirs. Our first main result\u0000is a sufficient condition for simplicity of the Steinberg algebra satisfying\u0000conditions modeled on the behavior of the groupoid associated to the first\u0000Grigorchuk group. This closely resembles conditions found by B. Steinberg and\u0000N. Szak'acs. As a key ingredient we identify an infinite family of\u0000$2-(2q-1,q-1,q/2-1)$-designs, where $q$ is a positive even integer. We then\u0000deduce the simplicity of the associated $C^*$-algebra, which is our second main\u0000result. Results of similar type were considered by B. Steinberg and N.\u0000Szak'acs in 2021, and later by K. Yoshida, but their methods did not follow\u0000the original methods of the five authors.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"99 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885100","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
Diameters of endomorphism monoids of chains 链的内态单元直径
arXiv - MATH - Rings and Algebras Pub Date : 2024-08-01 DOI: arxiv-2408.00416
James East, Victoria Gould, Craig Miller, Thomas Quinn-Gregson
{"title":"Diameters of endomorphism monoids of chains","authors":"James East, Victoria Gould, Craig Miller, Thomas Quinn-Gregson","doi":"arxiv-2408.00416","DOIUrl":"https://doi.org/arxiv-2408.00416","url":null,"abstract":"The left and right diameters of a monoid are topological invariants defined\u0000in terms of suprema of lengths of derivation sequences with respect to finite\u0000generating sets for the universal left or right congruences. We compute these\u0000parameters for the endomorphism monoid $End(C)$ of a chain $C$. Specifically,\u0000if $C$ is infinite then the left diameter of $End(C)$ is 2, while the right\u0000diameter is either 2 or 3, with the latter equal to 2 precisely when $C$ is a\u0000quotient of $C{setminus}{z}$ for some endpoint $z$. If $C$ is finite then so\u0000is $End(C),$ in which case the left and right diameters are 1 (if $C$ is\u0000non-trivial) or 0.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885178","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
Categorical properties and homological conjectures for bounded extensions of algebras 代数有界扩展的分类属性和同调猜想
arXiv - MATH - Rings and Algebras Pub Date : 2024-07-31 DOI: arxiv-2407.21480
Yongyun Qin, Xiaoxiao Xu, Jinbi Zhang, Guodong Zhou
{"title":"Categorical properties and homological conjectures for bounded extensions of algebras","authors":"Yongyun Qin, Xiaoxiao Xu, Jinbi Zhang, Guodong Zhou","doi":"arxiv-2407.21480","DOIUrl":"https://doi.org/arxiv-2407.21480","url":null,"abstract":"An extension $Bsubset A$ of finite dimensional algebras is bounded if the\u0000$B$-$B$-bimodule $A/B$ is $B$-tensor nilpotent, its projective dimension is\u0000finite and $mathrm{Tor}_i^B(A/B, (A/B)^{otimes_B j})=0$ for all $i, jgeq 1$.\u0000We show that for a bounded extension $Bsubset A$, the algebras $A$ and $B$ are\u0000singularly equivalent of Morita type with level. Additively, under some\u0000conditions, their stable categories of Gorenstein projective modules and\u0000Gorenstein defect categories are equivalent, respectively. Some homological\u0000conjectures are also investigated for bounded extensions, including\u0000Auslander-Reiten conjecture, finististic dimension conjecture, Fg condition,\u0000Han's conjecture, and Keller's conjecture. Applications to trivial extensions\u0000and triangular matrix algebras are given. In course of proofs, we give some handy criteria for a functor between module\u0000categories induces triangle functors between stable categories of Gorenstein\u0000projective modules and Gorenstein defect categories, which generalise some\u0000known criteria and hence, might be of independent interest.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141871572","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
On the asymptotic behaviour of the graded-star-codimension sequence of upper triangular matrices 论上三角矩阵的梯度-星-维序列的渐近行为
arXiv - MATH - Rings and Algebras Pub Date : 2024-07-31 DOI: arxiv-2408.00087
Diogo Diniz, Felipe Yukihide Yasumura
{"title":"On the asymptotic behaviour of the graded-star-codimension sequence of upper triangular matrices","authors":"Diogo Diniz, Felipe Yukihide Yasumura","doi":"arxiv-2408.00087","DOIUrl":"https://doi.org/arxiv-2408.00087","url":null,"abstract":"We study the algebra of upper triangular matrices endowed with a group\u0000grading and a homogeneous involution over an infinite field. We compute the\u0000asymptotic behaviour of its (graded) star-codimension sequence. It turns out\u0000that the asymptotic growth of the sequence is independent of the grading and\u0000the involution under consideration, depending solely on the size of the matrix\u0000algebra. This independence of the group grading also applies to the graded\u0000codimension sequence of the associative algebra of upper triangular matrices.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885027","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
Constructing Multiresolution Analysis via Wavelet Packets on Sobolev Space in Local Fields 通过小波包在局部域的索波列夫空间上构建多分辨率分析
arXiv - MATH - Rings and Algebras Pub Date : 2024-07-31 DOI: arxiv-2408.00028
Manish Kumar
{"title":"Constructing Multiresolution Analysis via Wavelet Packets on Sobolev Space in Local Fields","authors":"Manish Kumar","doi":"arxiv-2408.00028","DOIUrl":"https://doi.org/arxiv-2408.00028","url":null,"abstract":"We define Sobolev spaces $H^{mathfrak{s}}(K_q)$ over a local field $K_q$ of\u0000finite characteristic $p>0$, where $q=p^c$ for a prime $p$ and $cin\u0000mathbb{N}$. This paper introduces novel fractal functions, such as the\u0000Weierstrass type and 3-adic Cantor type, as intriguing examples within these\u0000spaces and a few others. Employing prime elements, we develop a\u0000Multi-Resolution Analysis (MRA) and examine wavelet expansions, focusing on the\u0000orthogonality of both basic and fractal wavelet packets at various scales. We\u0000utilize convolution theory to construct Haar wavelet packets and demonstrate\u0000the orthogonality of all discussed wavelet packets within\u0000$H^{mathfrak{s}}(K_q)$, enhancing the analytical capabilities of these Sobolev\u0000spaces.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885028","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
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