This work extends the idea of matched pairs presented by Majid in cite{Majid} and Takeuchi in cite{Takeuchi} for the context of weak bialgebras and weak Hopf algebras. We introduce, also inspired by partial matched pairs cite{matchedpair}, the notion of weak matched pairs and establish conditions for a subspace of the smash product be a weak bialgebra/Hopf algebra. Further, some new examples of (co)actions of weak bialgebras over algebras and some results about integral elements are presented.
{"title":"Weak Hopf algebras arising from weak matched pairs","authors":"Graziela Fonseca, Grasiela Martini, Leonardo Silva","doi":"arxiv-2408.05181","DOIUrl":"https://doi.org/arxiv-2408.05181","url":null,"abstract":"This work extends the idea of matched pairs presented by Majid in\u0000cite{Majid} and Takeuchi in cite{Takeuchi} for the context of weak bialgebras\u0000and weak Hopf algebras. We introduce, also inspired by partial matched pairs\u0000cite{matchedpair}, the notion of weak matched pairs and establish conditions\u0000for a subspace of the smash product be a weak bialgebra/Hopf algebra. Further,\u0000some new examples of (co)actions of weak bialgebras over algebras and some\u0000results about integral elements are presented.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935874","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}
This work is concerned with approximability (via Neeman) and Rouquier dimension for triangulated categories associated to noncommutative algebras over schemes. Amongst other things, we establish that the category of perfect complexes of a coherent algebra over a separated Noetherian scheme is strongly generated if, and only if, there exists an affine open cover where the algebra has finite global dimension. As a consequence, we solve an open problem posed by Neeman. Further, as a first application, we study the existence of generators and behaviour under the derived pushforward for Azumaya algebras.
{"title":"Approximability and Rouquier dimension for noncommuative algebras over schemes","authors":"Timothy De Deyn, Pat Lank, Kabeer Manali Rahul","doi":"arxiv-2408.04561","DOIUrl":"https://doi.org/arxiv-2408.04561","url":null,"abstract":"This work is concerned with approximability (via Neeman) and Rouquier\u0000dimension for triangulated categories associated to noncommutative algebras\u0000over schemes. Amongst other things, we establish that the category of perfect\u0000complexes of a coherent algebra over a separated Noetherian scheme is strongly\u0000generated if, and only if, there exists an affine open cover where the algebra\u0000has finite global dimension. As a consequence, we solve an open problem posed\u0000by Neeman. Further, as a first application, we study the existence of\u0000generators and behaviour under the derived pushforward for Azumaya algebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935658","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 develop a new cohomology theory for finite-dimensional left-symmetric color algebras and their finite-dimensional bimodules, establishing a connection between Lie color cohomology and left-symmetric color cohomology. We prove that the cohomology of a left-symmetric color algebra $A$ with coefficients in a bimodule $V$ can be computed by a lower degree cohomology of the corresponding Lie color algebra with coefficients in Hom$(A,V)$, generalizing a result of Dzhumadil'daev in right-symmetric cohomology. We also explore the varieties of two-dimensional and three-dimensional left-symmetric color algebras.
{"title":"Cohomology of left-symmetric color algebras","authors":"Yin Chen, Runxuan Zhang","doi":"arxiv-2408.04033","DOIUrl":"https://doi.org/arxiv-2408.04033","url":null,"abstract":"We develop a new cohomology theory for finite-dimensional left-symmetric\u0000color algebras and their finite-dimensional bimodules, establishing a\u0000connection between Lie color cohomology and left-symmetric color cohomology. We\u0000prove that the cohomology of a left-symmetric color algebra $A$ with\u0000coefficients in a bimodule $V$ can be computed by a lower degree cohomology of\u0000the corresponding Lie color algebra with coefficients in Hom$(A,V)$,\u0000generalizing a result of Dzhumadil'daev in right-symmetric cohomology. We also\u0000explore the varieties of two-dimensional and three-dimensional left-symmetric\u0000color algebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935669","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 study, it is proven that the universal equivalence of general linear groups (admitting the inverse-transpose automorphism) of orders greater than $2$, over local, not necessarily commutative rings with $1/2$, is equivalent to the coincidence of the orders of the groups and the universal equivalence of the corresponding rings.
{"title":"Universal equivalence of general linear groups over local rings with 1/2","authors":"Galina Kaleeva","doi":"arxiv-2408.04079","DOIUrl":"https://doi.org/arxiv-2408.04079","url":null,"abstract":"In this study, it is proven that the universal equivalence of general linear\u0000groups (admitting the inverse-transpose automorphism) of orders greater than\u0000$2$, over local, not necessarily commutative rings with $1/2$, is equivalent to\u0000the coincidence of the orders of the groups and the universal equivalence of\u0000the corresponding rings.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935659","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}
Vasileios Aravantinos-Sotiropoulos, Christina Vasilakopoulou
In this work, we continue the investigation of certain enrichments of dual algebraic structures in monoidal double categories, that was initiated in [Vas19]. First, we re-visit monads and comonads in double categories and establish a tensored and cotensored enrichment of the former in the latter, under general conditions. These include monoidal closedness and local presentability of the double category, notions that are proposed as tools required for our main results, but are of interest in their own right. The natural next step involves categories of the newly introduced modules for monads and comodules for comonads in double categories. After we study their main categorical properties, we establish a tensored and cotensored enrichment of modules in comodules, as well as an enriched fibration structure that involves (co)modules over (co)monads in double categories. Applying this abstract double categorical framework to the setting of V-matrices produces an enrichment of the category of V-enriched modules (fibred over V-categories) in V-enriched comodules (opfibred over V-cocategories), which is the many-object generalization of the respective result for modules (over algebras) and comodules (over coalgebras) in monoidal categories.
{"title":"Enriched duality in double categories II: modules and comodules","authors":"Vasileios Aravantinos-Sotiropoulos, Christina Vasilakopoulou","doi":"arxiv-2408.03180","DOIUrl":"https://doi.org/arxiv-2408.03180","url":null,"abstract":"In this work, we continue the investigation of certain enrichments of dual\u0000algebraic structures in monoidal double categories, that was initiated in\u0000[Vas19]. First, we re-visit monads and comonads in double categories and\u0000establish a tensored and cotensored enrichment of the former in the latter,\u0000under general conditions. These include monoidal closedness and local\u0000presentability of the double category, notions that are proposed as tools\u0000required for our main results, but are of interest in their own right. The\u0000natural next step involves categories of the newly introduced modules for\u0000monads and comodules for comonads in double categories. After we study their\u0000main categorical properties, we establish a tensored and cotensored enrichment\u0000of modules in comodules, as well as an enriched fibration structure that\u0000involves (co)modules over (co)monads in double categories. Applying this\u0000abstract double categorical framework to the setting of V-matrices produces an\u0000enrichment of the category of V-enriched modules (fibred over V-categories) in\u0000V-enriched comodules (opfibred over V-cocategories), which is the many-object\u0000generalization of the respective result for modules (over algebras) and\u0000comodules (over coalgebras) in monoidal categories.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935663","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 introduce a new Topology related to special elements in a noncummutative rings. Consider a ring $R$, we denote by $textrm{Id}(R)$ the set of all idempotent elements in $R$. Let $a$ is an element of $R$. The element absorb Topology related to $a$ is defined as $tau_a:={ Isubseteq R | Ia subseteq I} subseteq P(R)$. Since this topology is obtained from act of ring, it explains Some of algebraic properties of ring in Topological language .In a special case when $e$ ia an idempotent element, $tau_e:={ Isubseteq R | Ie subseteq I} subseteq P(R)$. We present Topological description of the pierce decomposition $ R=Reoplus R(1-e)$.
{"title":"Element absorb Topology on rings","authors":"Ali Shahidikia","doi":"arxiv-2408.03300","DOIUrl":"https://doi.org/arxiv-2408.03300","url":null,"abstract":"In this paper, we introduce a new Topology related to special elements in a\u0000noncummutative rings. Consider a ring $R$, we denote by $textrm{Id}(R)$ the\u0000set of all idempotent elements in $R$. Let $a$ is an element of $R$. The\u0000element absorb Topology related to $a$ is defined as $tau_a:={ Isubseteq R |\u0000Ia subseteq I} subseteq P(R)$. Since this topology is obtained from act of\u0000ring, it explains Some of algebraic properties of ring in Topological language\u0000.In a special case when $e$ ia an idempotent element, $tau_e:={ Isubseteq R\u0000| Ie subseteq I} subseteq P(R)$. We present Topological description of the\u0000pierce decomposition $ R=Reoplus R(1-e)$.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935661","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}
The so-called Sasaki projection was introduced by U. Sasaki on the lattice L(H) of closed linear subspaces of a Hilbert space H as a projection of L(H) onto a certain sublattice of L(H). Since L(H) is an orthomodular lattice, the Sasaki projection and its dual can serve as the logical connectives conjunction and implication within the logic of quantum mechanics. It was shown by the authors in a previous paper that these operations form a so-called adjoint pair. The natural question arises if this result can be extended also to lattices with a unary operation which need not be orthomodular or to other algebras with two binary and one unary operation. To show that this is possible is the aim of the present paper. We determine a variety of lattices with a unary operation where the Sasaki operations form an adjoint pair and we continue with so-called $lambda$-lattices and certain classes of semirings. We show that the Sasaki operations have a deeper sense than originally assumed by their author and can be applied also outside the lattices of closed linear subspaces of a Hilbert space.
{"title":"Algebras and varieties where Sasaki operations form an adjoint pair","authors":"Ivan Chajda, Helmut Länger","doi":"arxiv-2408.03432","DOIUrl":"https://doi.org/arxiv-2408.03432","url":null,"abstract":"The so-called Sasaki projection was introduced by U. Sasaki on the lattice\u0000L(H) of closed linear subspaces of a Hilbert space H as a projection of L(H)\u0000onto a certain sublattice of L(H). Since L(H) is an orthomodular lattice, the\u0000Sasaki projection and its dual can serve as the logical connectives conjunction\u0000and implication within the logic of quantum mechanics. It was shown by the\u0000authors in a previous paper that these operations form a so-called adjoint\u0000pair. The natural question arises if this result can be extended also to\u0000lattices with a unary operation which need not be orthomodular or to other\u0000algebras with two binary and one unary operation. To show that this is possible\u0000is the aim of the present paper. We determine a variety of lattices with a\u0000unary operation where the Sasaki operations form an adjoint pair and we\u0000continue with so-called $lambda$-lattices and certain classes of semirings. We\u0000show that the Sasaki operations have a deeper sense than originally assumed by\u0000their author and can be applied also outside the lattices of closed linear\u0000subspaces of a Hilbert space.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"117 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935660","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}
Be'eri Greenfeld, Carlos Gustavo Moreira, Efim Zelmanov
We characterize the complexity functions of subshifts up to asymptotic equivalence. The complexity function of every aperiodic function is non-decreasing, submultiplicative and grows at least linearly. We prove that conversely, every function satisfying these conditions is asymptotically equivalent to the complexity function of a recurrent subshift, equivalently, a recurrent infinite word. Our construction is explicit, algorithmic in nature and is philosophically based on constructing certain 'Cantor sets of integers', whose 'gaps' correspond to blocks of zeros. We also prove that every non-decreasing submultiplicative function is asymptotically equivalent, up a linear error term, to the complexity function of a minimal subshift.
{"title":"On the complexity of subshifts and infinite words","authors":"Be'eri Greenfeld, Carlos Gustavo Moreira, Efim Zelmanov","doi":"arxiv-2408.03403","DOIUrl":"https://doi.org/arxiv-2408.03403","url":null,"abstract":"We characterize the complexity functions of subshifts up to asymptotic\u0000equivalence. The complexity function of every aperiodic function is\u0000non-decreasing, submultiplicative and grows at least linearly. We prove that\u0000conversely, every function satisfying these conditions is asymptotically\u0000equivalent to the complexity function of a recurrent subshift, equivalently, a\u0000recurrent infinite word. Our construction is explicit, algorithmic in nature\u0000and is philosophically based on constructing certain 'Cantor sets of integers',\u0000whose 'gaps' correspond to blocks of zeros. We also prove that every\u0000non-decreasing submultiplicative function is asymptotically equivalent, up a\u0000linear error term, to the complexity function of a minimal subshift.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935673","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}
Zaqueu Cristiano, Wellington Marques de Souza, Javier Sánchez
We develop the theory of groupoid graded semisimple rings. Our rings are neither unital nor one-sided artinian. Instead, they exhibit a strong version of having local units and being locally artinian, and we call them $Gamma_0$-artinian. One of our main results is a groupoid graded version of the Wedderburn-Artin Theorem, where we characterize groupoid graded semisimple rings as direct sums of graded simple $Gamma_0$-artinian rings and we exhibit the structure of this latter class of rings. In this direction, we also prove a groupoid graded version of Jacobson-Chevalley density theorem. We need to define and study properties of groupoid gradings on matrix rings (possibly of infinite size) over groupoid graded rings, and specially over groupoid graded division rings. Because of that, we study groupoid graded division rings and their graded modules. We consider a natural notion of freeness for groupoid graded modules that, when specialized to group graded rings, gives the usual one, and show that for a groupoid graded division ring all graded modules are free (in this sense). Contrary to the group graded case, there are groupoid graded rings for which all graded modules are free according to our definition, but they are not graded division rings. We exhibit an easy example of this kind of rings and characterize such class among groupoid graded semisimple rings. We also relate groupoid graded semisimple rings with the notion of semisimple category defined by B. Mitchell. For that, we show the link between functors from a preadditive category to abelian groups and graded modules over the groupoid graded ring associated to this category, generalizing a result of P. Gabriel. We characterize simple artinian categories and categories for which every functor from them to abelian groups is free in the sense of B. Mitchell.
我们发展了类群分级半简单环的理论。我们的环既不是单素环，也不是单面自洽环。相反，它们展示了具有局部单元和局部自洽性的强版本，我们称之为$Gamma_0$-自洽性。我们的主要结果之一是韦德伯恩-阿尔丁定理的类群分级版本，我们把类群分级半等分描述为分级简单 $Gamma_0$-artinian 环的直接和，并展示了后一类环的结构。在这个方向上，我们还证明了雅各布森-切瓦利密度定理的一个类梯度版本。我们需要定义和研究在类群分级环上，特别是在类群分级环上的矩阵环（可能是无限大的）上的类群分级的性质。正因为如此，我们研究类群分级环及其分级模块。我们考虑了群似有级模块的自然自由度概念，当把它专门用于群似有级环时，就得到了通常的自由度概念，并证明对于群似有级分割环，所有有级模块都是自由的（在这个意义上）。与群分级的情况相反，有一些类群分级环，根据我们的定义，所有分级模块都是自由的，但它们不是分级划分环。我们展示了这类环的一个简单例子，并描述了类群分级半简单环中这类环的特征。我们还将类梯度半简单环与 B. Mitchell 定义的半简单范畴概念联系起来。为此，我们概括了 P. Gabriel 的一个结果，展示了从预增量范畴到非良性群的函子与与该范畴相关的群有级环上的有级模块之间的联系。我们描述了简单artinian范畴和B. Mitchell意义上从它们到非良性群的每个函数都是自由的范畴的特征。
{"title":"Groupoid Graded Semisimple Rings","authors":"Zaqueu Cristiano, Wellington Marques de Souza, Javier Sánchez","doi":"arxiv-2408.03141","DOIUrl":"https://doi.org/arxiv-2408.03141","url":null,"abstract":"We develop the theory of groupoid graded semisimple rings. Our rings are\u0000neither unital nor one-sided artinian. Instead, they exhibit a strong version\u0000of having local units and being locally artinian, and we call them\u0000$Gamma_0$-artinian. One of our main results is a groupoid graded version of\u0000the Wedderburn-Artin Theorem, where we characterize groupoid graded semisimple\u0000rings as direct sums of graded simple $Gamma_0$-artinian rings and we exhibit\u0000the structure of this latter class of rings. In this direction, we also prove a\u0000groupoid graded version of Jacobson-Chevalley density theorem. We need to\u0000define and study properties of groupoid gradings on matrix rings (possibly of\u0000infinite size) over groupoid graded rings, and specially over groupoid graded\u0000division rings. Because of that, we study groupoid graded division rings and\u0000their graded modules. We consider a natural notion of freeness for groupoid\u0000graded modules that, when specialized to group graded rings, gives the usual\u0000one, and show that for a groupoid graded division ring all graded modules are\u0000free (in this sense). Contrary to the group graded case, there are groupoid\u0000graded rings for which all graded modules are free according to our definition,\u0000but they are not graded division rings. We exhibit an easy example of this kind\u0000of rings and characterize such class among groupoid graded semisimple rings. We\u0000also relate groupoid graded semisimple rings with the notion of semisimple\u0000category defined by B. Mitchell. For that, we show the link between functors\u0000from a preadditive category to abelian groups and graded modules over the\u0000groupoid graded ring associated to this category, generalizing a result of P.\u0000Gabriel. We characterize simple artinian categories and categories for which\u0000every functor from them to abelian groups is free in the sense of B. Mitchell.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935662","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 develop a Galois theory of commutative rings under actions of finite inverse semigroups. We present equivalences for the definition of Galois extension as well as a Galois correspondence theorem. We also show how the theory behaves in the case of inverse semigroups with zero.
{"title":"Galois Theory under inverse semigroup actions","authors":"Wesley G. Lautenschlaeger, Thaísa Tamusiunas","doi":"arxiv-2408.02850","DOIUrl":"https://doi.org/arxiv-2408.02850","url":null,"abstract":"We develop a Galois theory of commutative rings under actions of finite\u0000inverse semigroups. We present equivalences for the definition of Galois\u0000extension as well as a Galois correspondence theorem. We also show how the\u0000theory behaves in the case of inverse semigroups with zero.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141935671","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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