Journal of Algebra最新文献

{"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}

{"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}

{"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}

{"title":"On the ascent of atomicity to monoid algebras","authors":"Felix Gotti,&nbsp;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}

{"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}

{"title":"Narrow and wide regular subalgebras of semisimple Lie algebras","authors":"Andrew Douglas ,&nbsp;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}

{"title":"Partial actions of groups on generalized matrix rings","authors":"Dirceu Bagio ,&nbsp;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}

{"title":"On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field","authors":"Thiago Castilho de Mello ,&nbsp;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}

{"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}

{"title":"Counting maximal isotropic subbundles of orthogonal bundles over a curve","authors":"Daewoong Cheong ,&nbsp;Insong Choe ,&nbsp;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}