{"title":"The Z2-graded dimensions of the free Jordan superalgebra J(D1|D2)","authors":"Shikui Shang","doi":"10.1016/j.jpaa.2025.107879","DOIUrl":"10.1016/j.jpaa.2025.107879","url":null,"abstract":"<div><div>Let <em>k</em> be a field of characteristic 0. For a superspace <span><math><mi>V</mi><mo>=</mo><msub><mrow><mi>V</mi></mrow><mrow><mover><mrow><mn>0</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><mo>⊕</mo><msub><mrow><mi>V</mi></mrow><mrow><mover><mrow><mn>1</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub></math></span> over <em>k</em>, we call the vector <span><math><mo>(</mo><msub><mrow><mi>dim</mi></mrow><mrow><mi>k</mi></mrow></msub><mo></mo><msub><mrow><mi>V</mi></mrow><mrow><mover><mrow><mn>0</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><mo>,</mo><msub><mrow><mi>dim</mi></mrow><mrow><mi>k</mi></mrow></msub><mo></mo><msub><mrow><mi>V</mi></mrow><mrow><mover><mrow><mn>1</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub><mo>)</mo></math></span> the (<span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-)graded dimension of <em>V</em>. Let <span><math><mi>J</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> be the free Jordan superalgebra generated by <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> even generators and <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> odd generators. In this paper, we study the graded dimensions of the <em>n</em>-components of <span><math><mi>J</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> and find the connection between them and the homology of Tits-Allison-Gao Lie superalgebra of <span><math><mi>J</mi><mo>(</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> following the method given by I. Kashuba and O. Mathieu in <span><span>[15]</span></span>, where they deal with the free Jordan algebra. And, four related conjectures on the free Jordan superalgebras and related Lie superalgebras are proposed in this article.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107879"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163716","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":"Strong congruence spaces and dimension in F1-geometry","authors":"Manoel Jarra","doi":"10.1016/j.jpaa.2025.107862","DOIUrl":"10.1016/j.jpaa.2025.107862","url":null,"abstract":"<div><div>We introduce strong congruence spaces, which are topological spaces that provide a useful concept of dimension for monoid schemes. We study their properties and show that, given a toric monoid scheme over an algebraically closed basis, its strong congruence space and the complex toric variety associated to its fan have the same dimension.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107862"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378688","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":"Deformations of quasi-categories in modules","authors":"Violeta Borges Marques, Wendy Lowen, Arne Mertens","doi":"10.1016/j.jpaa.2025.107866","DOIUrl":"10.1016/j.jpaa.2025.107866","url":null,"abstract":"<div><div>The framework of templicial objects was put forth in <span><span>[30]</span></span> in order to develop higher categorical concepts in the presence of enrichment. In particular, quasi-categories in modules constitute a subclass of templicial modules which may be considered as a kind of “weak dg-categories (concentrated in homologically positive degrees)” according to <span><span>[29]</span></span>. The main goal of the present paper is to initiate the deformation theory of templicial modules. In particular, we show that quasi-categories in modules are preserved under levelwise flat infinitesimal deformation.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107866"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163055","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":"Common divisor graphs for skew braces","authors":"Silvia Properzi , Arne Van Antwerpen","doi":"10.1016/j.jpaa.2025.107876","DOIUrl":"10.1016/j.jpaa.2025.107876","url":null,"abstract":"<div><div>We introduce two common divisor graphs associated with a finite skew brace, based on its <em>λ</em>- and <em>θ</em>-orbits. We prove that the number of connected components is at most two and the diameter of a connected component is at most four. Furthermore, we investigate their relationship with isoclinism. Similarly to its group theoretic inspiration, the skew braces with a graph with two disconnected vertices are very restricted and are determined. Finally, we classify all finite skew braces with a graph with one vertex, where four infinite families arise.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107876"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163914","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":"Low degree rational curves on quasi-polarized K3 surfaces","authors":"Sławomir Rams , Matthias Schütt","doi":"10.1016/j.jpaa.2025.107904","DOIUrl":"10.1016/j.jpaa.2025.107904","url":null,"abstract":"<div><div>We prove that there are at most <span><math><mo>(</mo><mn>24</mn><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> irreducible rational curves of positive low-degree on high-degree models of K3 surfaces with at most Du Val singularities, where <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is the number of exceptional divisors on the minimal resolution. We also provide several existence results in the above setting (i.e. for rational curves on quasi-polarized K3 surfaces), which imply that for many values of <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> our bound cannot be improved.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107904"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143394580","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 finite spaces of non-trivial proper subgroups of finite groups","authors":"Lingling Han, Tao Zheng","doi":"10.1016/j.jpaa.2025.107894","DOIUrl":"10.1016/j.jpaa.2025.107894","url":null,"abstract":"<div><div>In this paper, we investigate the homotopy properties of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, the finite topological space consisting of all non-trivial proper subgroups of a finite group <em>G</em>. For some classes of groups <em>G</em>, we give the relations between the contractibility of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the algebraic properties of <em>G</em>, which is inspired by the study of R. E. Stong on <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107894"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143162792","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":"Δ-locally nilpotent algebras, their ideal structure and simplicity criteria","authors":"V.V. Bavula","doi":"10.1016/j.jpaa.2024.107861","DOIUrl":"10.1016/j.jpaa.2024.107861","url":null,"abstract":"<div><div>The class of Δ-locally nilpotent algebras introduced in the paper is a wide generalization of the algebras of differential operators on commutative algebras. Examples include all the rings <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> of differential operators on commutative algebras in arbitrary characteristic, all subalgebras of <span><math><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> that contain the algebra <em>A</em>, the universal enveloping algebras of nilpotent, solvable and semi-simple Lie algebras, the Poisson universal enveloping algebra of an arbitrary Poisson algebra, iterated Ore extensions <span><math><mi>A</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>;</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, certain generalized Weyl algebras, and others.</div><div>In <span><span>[8]</span></span>, simplicity criteria are given for the algebras differential operators on commutative algebras. To find the simplicity criterion was a long standing problem from 60'th. The aim of the paper is to describe the ideal structure of Δ-locally nilpotent algebras and as a corollary to give simplicity criteria for them. These results are generalizations of the results of <span><span>[8]</span></span>. Examples are considered.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107861"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163664","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":"Classification of G2-orbits for pairs of octonions","authors":"Artem Lopatin , Alexandr N. Zubkov","doi":"10.1016/j.jpaa.2025.107875","DOIUrl":"10.1016/j.jpaa.2025.107875","url":null,"abstract":"<div><div>Over an algebraically closed field, we described a minimal set of representatives for <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-orbits on the set <span><math><msup><mrow><mi>O</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of pairs of octonions.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107875"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163666","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":"Semi-galois Categories I: The classical Eilenberg variety theory","authors":"Takeo Uramoto","doi":"10.1016/j.jpaa.2025.107863","DOIUrl":"10.1016/j.jpaa.2025.107863","url":null,"abstract":"<div><div>This paper is an extended version of our proceedings paper <span><span>[49]</span></span> announced at LICS'16; in order to complement it, this version is written from a different viewpoint including topos-theoretic aspects of <span><span>[49]</span></span> that were not discussed there. Technically, this paper introduces and studies the class of <em>semi-galois categories</em>, which extend galois categories and are dual to profinite monoids in the same way as galois categories are dual to profinite groups; the study on this class of categories is aimed at providing an axiomatic reformulation of <em>Eilenberg's theory of varieties of regular languages</em>—a branch in formal language theory that has been developed since the mid 1960s and particularly concerns systematic classification of regular languages, finite monoids and deterministic finite automata. In this paper, detailed proofs of our central results announced at LICS'16 are presented, together with topos-theoretic considerations. The main results include (I) a proof of the duality theorem between profinite monoids and semi-galois categories, extending the duality theorem between profinite groups and galois categories; based on these results on semi-galois categories we then discuss (II) a reinterpretation of Eilenberg's theory from a viewpoint of the duality theorem; in relation with this reinterpretation of the theory, (III) we also give a purely topos-theoretic characterization of classifying topoi <span><math><mi>B</mi><mi>M</mi></math></span> of profinite monoids <em>M</em> among general coherent topoi, which is a topos-theoretic application of (I). This characterization states that a topos <span><math><mi>E</mi></math></span> is equivalent to the classifying topos <span><math><mi>B</mi><mi>M</mi></math></span> of some profinite monoid <em>M</em> if and only if <span><math><mi>E</mi></math></span> is (i) coherent, (ii) noetherian, and (iii) has a surjective coherent point <span><math><mi>p</mi><mo>:</mo><mrow><mi>Sets</mi></mrow><mo>→</mo><mi>E</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107863"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163676","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":"Toric splittings","authors":"Anargyros Katsabekis, Apostolos Thoma","doi":"10.1016/j.jpaa.2025.107870","DOIUrl":"10.1016/j.jpaa.2025.107870","url":null,"abstract":"<div><div>The toric ideal <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> is splittable if it has a toric splitting; namely, if there exist toric ideals <span><math><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> such that <span><math><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>+</mo><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub></math></span> and <span><math><msub><mrow><mi>I</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>≠</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>A</mi></mrow></msub></math></span> for all <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>2</mn></math></span>. We provide a necessary and sufficient condition for a toric ideal to be splittable in terms of <em>A</em>, and we apply it to prove or disprove that certain classes of toric ideals are splittable.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 2","pages":"Article 107870"},"PeriodicalIF":0.7,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163677","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}
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