Pub Date : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.017
Dominik Krasula
For Dynkin and Euclidean quivers, it is shown that Gabriel-Roiter measures of thin representations equal the induced chain length functions on the corresponding system of subquivers. This allows a combinatorial procedure to find GR filtrations of thin representations, showing that GR measures of thin representations are field-independent. It is proved that an indecomposable filtration of a thin representation is a GR filtration for a suitable choice of a length function on the category of finite-dimensional representations.
对于代金和欧几里得四元组，研究表明薄表示的加布里埃尔-罗伊特度量等于相应子四元组系统上的诱导链长函数。这样就可以通过组合程序找到薄表示的 GR filtrations，从而证明薄表示的 GR 度量与场无关。研究证明，对于有限维表征类别上长度函数的合适选择，薄表征的不可分解滤过是一个 GR 滤过。
{"title":"Generalised Gabriel-Roiter measure and thin representations","authors":"Dominik Krasula","doi":"10.1016/j.jalgebra.2024.09.017","DOIUrl":"10.1016/j.jalgebra.2024.09.017","url":null,"abstract":"<div><div>For Dynkin and Euclidean quivers, it is shown that Gabriel-Roiter measures of thin representations equal the induced chain length functions on the corresponding system of subquivers. This allows a combinatorial procedure to find GR filtrations of thin representations, showing that GR measures of thin representations are field-independent. It is proved that an indecomposable filtration of a thin representation is a GR filtration for a suitable choice of a length function on the category of finite-dimensional representations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.020
Chenwei Ruan
This paper is about the positive part of the q-deformed enveloping algebra . The algebra admits an embedding, due to Rosso, into a q-shuffle algebra . The underlying vector space of is the free algebra on two generators . Therefore, the algebra has a basis consisting of the words in . Let U denote the image of under the Rosso embedding. In our first main result, we find all the words in that are contained in U. One type of solution is called alternating. The alternating words have been studied by Terwilliger. There is another type of solution, which we call doubly alternating. In our second main result, we display many commutator relations involving the doubly alternating words. In our third main result, we describe how the doubly alternating words are related to the alternating words.
{"title":"Doubly alternating words in the positive part of Uq(slˆ2)","authors":"Chenwei Ruan","doi":"10.1016/j.jalgebra.2024.09.020","DOIUrl":"10.1016/j.jalgebra.2024.09.020","url":null,"abstract":"<div><div>This paper is about the positive part <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> of the <em>q</em>-deformed enveloping algebra <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><mi>sl</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>. The algebra <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> admits an embedding, due to Rosso, into a <em>q</em>-shuffle algebra <figure><img></figure>. The underlying vector space of <figure><img></figure> is the free algebra on two generators <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span>. Therefore, the algebra <figure><img></figure> has a basis consisting of the words in <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span>. Let <em>U</em> denote the image of <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> under the Rosso embedding. In our first main result, we find all the words in <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> that are contained in <em>U</em>. One type of solution is called alternating. The alternating words have been studied by Terwilliger. There is another type of solution, which we call doubly alternating. In our second main result, we display many commutator relations involving the doubly alternating words. In our third main result, we describe how the doubly alternating words are related to the alternating words.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.013
Calvin Pfeifer
Throughout, let K be an algebraically closed field of characteristic 0. We provide a generic classification of locally free representations of Geiß-Leclerc-Schröer's algebras associated to affine Cartan matrices C with minimal symmetrizer D and acyclic orientation Ω. Affine GLS algebras are “smooth” degenerations of tame hereditary algebras and as such their representation theory is presumably still tractable. Indeed, we observe several “tame” phenomena of affine GLS algebras even though they are in general representation wild. For the GLS algebras of type we achieve a classification of all stable representations. For general GLS algebras of affine type, we construct a 1-parameter family of representations stable with respect to the defect. Our construction is based on a generalized one-point extension technique. This confirms in particular τ-tilted versions of the second Brauer-Thrall Conjecture recently raised by Mousavand and Schroll-Treffinger-Valdivieso for the class of GLS algebras. Finally, we show that generically every locally free H-module is isomorphic to a direct sum of τ-rigid modules and modules from our 1-parameter family. This generalizes Kac's canonical decomposition from the symmetric to the symmetrizable case in affine types and we obtain such a decomposition by folding the canonical decomposition of dimension vectors over path algebras. As a corollary we obtain that affine GLS algebras are E-tame in the sense of Derksen-Fei and Asai-Iyama.
{"title":"A generic classification of locally free representations of affine GLS algebras","authors":"Calvin Pfeifer","doi":"10.1016/j.jalgebra.2024.09.013","DOIUrl":"10.1016/j.jalgebra.2024.09.013","url":null,"abstract":"<div><div>Throughout, let <em>K</em> be an algebraically closed field of characteristic 0. We provide a generic classification of locally free representations of Geiß-Leclerc-Schröer's algebras <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>,</mo><mi>D</mi><mo>,</mo><mi>Ω</mi><mo>)</mo></math></span> associated to affine Cartan matrices <em>C</em> with minimal symmetrizer <em>D</em> and acyclic orientation Ω. Affine GLS algebras are “smooth” degenerations of tame hereditary algebras and as such their representation theory is presumably still tractable. Indeed, we observe several “tame” phenomena of affine GLS algebras even though they are in general representation wild. For the GLS algebras of type <span><math><msub><mrow><mover><mrow><mi>BC</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mn>1</mn></mrow></msub></math></span> we achieve a classification of all stable representations. For general GLS algebras of affine type, we construct a 1-parameter family of representations stable with respect to the defect. Our construction is based on a generalized one-point extension technique. This confirms in particular <em>τ</em>-tilted versions of the second Brauer-Thrall Conjecture recently raised by Mousavand and Schroll-Treffinger-Valdivieso for the class of GLS algebras. Finally, we show that generically every locally free <em>H</em>-module is isomorphic to a direct sum of <em>τ</em>-rigid modules and modules from our 1-parameter family. This generalizes Kac's canonical decomposition from the symmetric to the symmetrizable case in affine types and we obtain such a decomposition by folding the canonical decomposition of dimension vectors over path algebras. As a corollary we obtain that affine GLS algebras are <em>E</em>-tame in the sense of Derksen-Fei and Asai-Iyama.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-11DOI: 10.1016/j.jalgebra.2024.10.002
Felix Gotti, Henrick Rabinovitz
A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the question of whether the fact that a torsion-free monoid M and an integral domain R are both atomic implies that the monoid algebra of M over R is also atomic. In general this is not true, and the first negative answer to this question was given by Roitman in 1993: he constructed an atomic integral domain whose polynomial extension is not atomic. More recently, Coykendall and the first author constructed finite-rank torsion-free atomic monoids whose monoid algebras over certain finite fields are not atomic. Still, the ascent of atomicity from finite-rank torsion-free monoids to their corresponding monoid algebras over fields of characteristic zero is an open problem. Coykendall and the first author also constructed an infinite-rank torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. As the primary result of this paper, we construct a rank-one torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. To do so, we introduce and study a methodological construction inside the class of rank-one torsion-free monoids that we call lifting, which consists in embedding a given monoid into another monoid that is often more tractable from the arithmetic viewpoint. For instance, we prove here that the embedding in the lifting construction preserves the ascending chain condition on principal ideals and the existence of maximal common divisors.
如果每个非可逆元素都因数化为不可还原元素（也称为原子），那么一个交换可取消单元就是原子的；如果一个积分域的乘法单元是原子的，那么这个积分域就是原子的。早在八十年代，吉尔默就提出过这样一个问题：一个无扭单元 M 和一个积分域 R 都是原子的，这是否意味着 M 在 R 上的单元代数 R[M] 也是原子的？一般来说，这是不正确的。罗伊特曼在 1993 年给出了这个问题的第一个否定答案：他构造了一个原子积分域，其多项式扩展不是原子的。最近，Coykendall 和第一作者构造了有限秩无扭原子单体，其在某些有限域上的单体代数不是原子的。不过，从有限秩无扭单体到它们在特征为零的域上的相应单体代数的原子性上升，仍然是一个悬而未决的问题。Coykendall 和第一作者还构造了一个无穷级无扭原子单体，它的单体代数（在任意积分域上）不是原子的。作为本文的主要成果，我们构建了一个秩一的无扭原子单体，其单体代数（在任意积分域上）不是原子的。为此，我们在秩一无扭单元类中引入并研究了一种方法论构造，我们称之为提升，即把给定的单元嵌入到另一个单元中，从算术的角度来看，这通常更容易实现。例如，我们在这里证明了提升构造中的嵌入保留了主理想的升链条件和最大公有除数的存在。
{"title":"On the ascent of atomicity to monoid algebras","authors":"Felix Gotti, Henrick Rabinovitz","doi":"10.1016/j.jalgebra.2024.10.002","DOIUrl":"10.1016/j.jalgebra.2024.10.002","url":null,"abstract":"<div><div>A commutative cancellative monoid is atomic if every non-invertible element factors into irreducibles (also called atoms), while an integral domain is atomic if its multiplicative monoid is atomic. Back in the eighties, Gilmer posed the question of whether the fact that a torsion-free monoid <em>M</em> and an integral domain <em>R</em> are both atomic implies that the monoid algebra <span><math><mi>R</mi><mo>[</mo><mi>M</mi><mo>]</mo></math></span> of <em>M</em> over <em>R</em> is also atomic. In general this is not true, and the first negative answer to this question was given by Roitman in 1993: he constructed an atomic integral domain whose polynomial extension is not atomic. More recently, Coykendall and the first author constructed finite-rank torsion-free atomic monoids whose monoid algebras over certain finite fields are not atomic. Still, the ascent of atomicity from finite-rank torsion-free monoids to their corresponding monoid algebras over fields of characteristic zero is an open problem. Coykendall and the first author also constructed an infinite-rank torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. As the primary result of this paper, we construct a rank-one torsion-free atomic monoid whose monoid algebras (over any integral domain) are not atomic. To do so, we introduce and study a methodological construction inside the class of rank-one torsion-free monoids that we call lifting, which consists in embedding a given monoid into another monoid that is often more tractable from the arithmetic viewpoint. For instance, we prove here that the embedding in the lifting construction preserves the ascending chain condition on principal ideals and the existence of maximal common divisors.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.016
Pedro J. Chocano
Given a regular representation of a finite group G and a positive integer number n, we construct a (finite) topological space X such that its group of homotopy classes of self-homotopy equivalences and its group of homeomorphisms are isomorphic to G, and the action of G on the n-th homology group is the regular representation. We also discuss other representations.
给定有限群 G 的正则表达式和正整数 n，我们构建一个（有限）拓扑空间 X，使其自同调等价类群 E(X) 和同构群 Aut(X) 与 G 同构，并且 G 对 n 次同调群 Hn(X) 的作用就是正则表达式。我们还讨论了其他表示。
{"title":"Realizing regular representations of finite groups","authors":"Pedro J. Chocano","doi":"10.1016/j.jalgebra.2024.09.016","DOIUrl":"10.1016/j.jalgebra.2024.09.016","url":null,"abstract":"<div><div>Given a regular representation of a finite group <em>G</em> and a positive integer number <em>n</em>, we construct a (finite) topological space <em>X</em> such that its group of homotopy classes of self-homotopy equivalences <span><math><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and its group of homeomorphisms <span><math><mi>A</mi><mi>u</mi><mi>t</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> are isomorphic to <em>G</em>, and the action of <em>G</em> on the <em>n</em>-th homology group <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is the regular representation. We also discuss other representations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.027
Andrew Douglas , Joe Repka
A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restrictions of all non-trivial simple modules to the subalgebra have proper decompositions. A semisimple Lie algebra is regular extreme if any regular subalgebra of the semisimple Lie algebra is either narrow or wide. We determine necessary and sufficient conditions for a simple module of a semisimple Lie algebra to remain indecomposable when restricted to a regular subalgebra. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras [10]. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Then, we show that all simple Lie algebras are regular extreme. Finally, we show that no non-simple, semisimple Lie algebra is regular extreme.
{"title":"Narrow and wide regular subalgebras of semisimple Lie algebras","authors":"Andrew Douglas , Joe Repka","doi":"10.1016/j.jalgebra.2024.09.027","DOIUrl":"10.1016/j.jalgebra.2024.09.027","url":null,"abstract":"<div><div>A subalgebra of a semisimple Lie algebra is <em>wide</em> if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. A subalgebra is <em>narrow</em> if the restrictions of all non-trivial simple modules to the subalgebra have proper decompositions. A semisimple Lie algebra is <em>regular extreme</em> if any regular subalgebra of the semisimple Lie algebra is either narrow or wide. We determine necessary and sufficient conditions for a simple module of a semisimple Lie algebra to remain indecomposable when restricted to a regular subalgebra. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras <span><span>[10]</span></span>. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Then, we show that all simple Lie algebras are regular extreme. Finally, we show that no non-simple, semisimple Lie algebra is regular extreme.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142529840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.018
Dirceu Bagio , Héctor Pinedo
Let n be a positive integer and be a generalized matrix ring. For each , let be an ideal of the ring and denote . We give sufficient conditions for the subset of R to be an ideal of R. Also, suppose that is a partial action of a group G on , for all . We construct, under certain conditions, a partial action γ of G on R such that γ restricted to coincides with . We study the relation between this construction and the notion of Morita equivalent partial group action given in [1]. Moreover, we investigate properties related to Galois theory for the extension . Some examples to illustrate the results are considered in the last part of the paper.
设 n 为正整数，R=(Mij)1≤i,j≤n 为广义矩阵环。对于每个 1≤i,j≤n，设 Ii 为环 Ri:=Mii 的理想，并表示 Iij=IiMij+MijIj 。我们给出了 R 的子集 I=(Iij)1≤i,j≤n 是 R 的理想的充分条件。同时，假设 α(i) 是一个群 G 对 Ri 的部分作用，对于所有 1≤i≤n。我们在一定条件下构造了 G 在 Ri 上的部分作用 γ，使得限制于 Ri 的 γ 与 α(i) 重合。我们将研究这一构造与 [1] 中给出的莫里塔等价部分群作用概念之间的关系。此外，我们还研究了扩展 Rγ⊂R 的伽罗瓦理论相关性质。本文的最后一部分列举了一些例子来说明这些结果。
{"title":"Partial actions of groups on generalized matrix rings","authors":"Dirceu Bagio , Héctor Pinedo","doi":"10.1016/j.jalgebra.2024.09.018","DOIUrl":"10.1016/j.jalgebra.2024.09.018","url":null,"abstract":"<div><div>Let <em>n</em> be a positive integer and <span><math><mi>R</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> be a generalized matrix ring. For each <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></math></span>, let <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> be an ideal of the ring <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>:</mo><mo>=</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>i</mi></mrow></msub></math></span> and denote <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>. We give sufficient conditions for the subset <span><math><mi>I</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math></span> of <em>R</em> to be an ideal of <em>R</em>. Also, suppose that <span><math><msup><mrow><mi>α</mi></mrow><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></msup></math></span> is a partial action of a group <span>G</span> on <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math></span>. We construct, under certain conditions, a partial action <em>γ</em> of <span>G</span> on <em>R</em> such that <em>γ</em> restricted to <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> coincides with <span><math><msup><mrow><mi>α</mi></mrow><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></msup></math></span>. We study the relation between this construction and the notion of Morita equivalent partial group action given in <span><span>[1]</span></span>. Moreover, we investigate properties related to Galois theory for the extension <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>γ</mi></mrow></msup><mo>⊂</mo><mi>R</mi></math></span>. Some examples to illustrate the results are considered in the last part of the paper.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-11DOI: 10.1016/j.jalgebra.2024.09.032
Thiago Castilho de Mello , Felipe Yukihide Yasumura
We study the graded polynomial identities with a homogeneous involution on the algebra of upper triangular matrices endowed with a fine group grading. We compute their polynomial identities and a basis of the relatively free algebra, considering an arbitrary base field. We obtain the asymptotic behaviour of the codimension sequence when the characteristic of the base field is zero. As a consequence, we compute the exponent and the second exponent of the same algebra endowed with any group grading and any homogeneous involution.
{"title":"On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field","authors":"Thiago Castilho de Mello , Felipe Yukihide Yasumura","doi":"10.1016/j.jalgebra.2024.09.032","DOIUrl":"10.1016/j.jalgebra.2024.09.032","url":null,"abstract":"<div><div>We study the graded polynomial identities with a homogeneous involution on the algebra of upper triangular matrices endowed with a fine group grading. We compute their polynomial identities and a basis of the relatively free algebra, considering an arbitrary base field. We obtain the asymptotic behaviour of the codimension sequence when the characteristic of the base field is zero. As a consequence, we compute the exponent and the second exponent of the same algebra endowed with any group grading and any homogeneous involution.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-10DOI: 10.1016/j.jalgebra.2024.08.038
Jack Barlow
Let be a prime with , and let K be a p-adic field. Let F be the function field of a curve over K. Let be the set of all divisorial discrete valuations of F. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over F. We give a positive answer to this question for a class of adjoint classical groups.
设 p∈N 是一个 p≠2 的素数，设 K 是一个 p-adic 域。让 F 是 K 上曲线的函数域，让 ΩF 是 F 的所有除法离散值的集合。在本文中，我们提出了哈塞原理对于 F 上的半简单邻接线性代数群是否成立的问题，并对一类邻接经典群给出了肯定的答案。
{"title":"A local-global principle for similarities over function fields of p-adic curves","authors":"Jack Barlow","doi":"10.1016/j.jalgebra.2024.08.038","DOIUrl":"10.1016/j.jalgebra.2024.08.038","url":null,"abstract":"<div><div>Let <span><math><mi>p</mi><mo>∈</mo><mi>N</mi></math></span> be a prime with <span><math><mi>p</mi><mo>≠</mo><mn>2</mn></math></span>, and let <em>K</em> be a <em>p</em>-adic field. Let <em>F</em> be the function field of a curve over <em>K</em>. Let <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>F</mi></mrow></msub></math></span> be the set of all divisorial discrete valuations of <em>F</em>. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over <em>F</em>. We give a positive answer to this question for a class of adjoint classical groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-10DOI: 10.1016/j.jalgebra.2024.08.037
Daewoong Cheong , Insong Choe , George H. Hitching
Let C be a smooth projective curve and V an orthogonal bundle over C. Let be the isotropic Quot scheme parameterizing degree e isotropic subsheaves of maximal rank in V. We give a closed formula for intersection numbers on components of whose generic element is saturated. As a special case, for , we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthogonal bundle in most cases when this is finite. This is an orthogonal analogue of Holla's enumeration of maximal subbundles in [16], and of the symplectic case studied in [7].
设 C 是光滑投影曲线，V 是 C 上的正交束。设 IQe(V) 是参数化 V 中最大秩的 e 等向子束的等向 Quot 方案。我们给出了 IQe(V) 中通元饱和的分量的交集数的封闭公式。作为一种特例，对于 g≥2，我们计算了一般稳定正交束的最大秩和度的各向同性子束的数量，在大多数情况下这是有限的。这是霍拉在[16]中枚举最大子束的正交类似方法，也是[7]中研究的交映情况的类似方法。
{"title":"Counting maximal isotropic subbundles of orthogonal bundles over a curve","authors":"Daewoong Cheong , Insong Choe , George H. Hitching","doi":"10.1016/j.jalgebra.2024.08.037","DOIUrl":"10.1016/j.jalgebra.2024.08.037","url":null,"abstract":"<div><div>Let <em>C</em> be a smooth projective curve and <em>V</em> an orthogonal bundle over <em>C</em>. Let <span><math><msub><mrow><mi>IQ</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> be the isotropic Quot scheme parameterizing degree <em>e</em> isotropic subsheaves of maximal rank in <em>V</em>. We give a closed formula for intersection numbers on components of <span><math><msub><mrow><mi>IQ</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> whose generic element is saturated. As a special case, for <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span>, we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthogonal bundle in most cases when this is finite. This is an orthogonal analogue of Holla's enumeration of maximal subbundles in <span><span>[16]</span></span>, and of the symplectic case studied in <span><span>[7]</span></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142444918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号