Journal of Algebra最新文献

{"title":"Integral braces and flat affine manifolds associated with finite L-algebras","authors":"Wolfgang Rump","doi":"10.1016/j.jalgebra.2025.07.011","DOIUrl":"10.1016/j.jalgebra.2025.07.011","url":null,"abstract":"<div><div>The structure group <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> of a finite non-degenerate involutive solution <span><math><mo>(</mo><mi>X</mi><mo>;</mo><mi>S</mi><mo>)</mo></math></span> to the set-theoretic Yang-Baxter equation is a cofinite integral brace, or equivalently, a crystallographic group with an affine structure. Furthermore, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> is the structure group of a finite <em>L</em>-algebra associated with the solution <span><math><mo>(</mo><mi>X</mi><mo>;</mo><mi>S</mi><mo>)</mo></math></span>. As is well known, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> is torsion-free, hence the fundamental group of a complete flat affine manifold. It is proved that conversely, a wide class of finite <em>L</em>-algebras have an associated integral brace with a torsion-free affine crystallographic adjoint group. The braces arising from finite <em>L</em>-algebras are Jacobson radicals of rings with a natural coalgebra structure. A slight extension of our construction yields the braces (alias pregroups) recently found in connection with the Hopf algebra of rooted trees in the sense of Connes and Kreimer.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"683 ","pages":"Pages 734-760"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144653995","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":"Interpolation of open-closed TQFTs","authors":"Barthélémy Neyra","doi":"10.1016/j.jalgebra.2025.06.036","DOIUrl":"10.1016/j.jalgebra.2025.06.036","url":null,"abstract":"<div><div>For any symmetric monoidal category <span><math><mi>C</mi></math></span>, Lauda and Pfeiffer showed the equivalence between the <span><math><mi>C</mi></math></span>-valued open-closed 2-dimensional TQFTs and the so-called knowledgeable Frobenius algebras (kFAs) in <span><math><mi>C</mi></math></span>. A kFA in the category of finite-dimensional vector spaces over a field <span><math><mi>K</mi></math></span> provides a sequence of scalars indexed by the set <span><math><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> of diffeomorphism classes of connected endocobordisms of the empty set, given by evaluation of the associated TQFT on each such cobordism class. More generally, from an arbitrary sequence <span><math><mi>χ</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mrow><mi>χ</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>w</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>g</mi><mo>,</mo><mi>w</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span>, we show how to build a symmetric monoidal category <span><math><mo>〈</mo><msub><mrow><mi>T</mi></mrow><mrow><mtext>kFA</mtext></mrow></msub><mo>|</mo><mi>χ</mi><mo>〉</mo></math></span>, with unit object <strong>1</strong> satisfying <span><math><mtext>End</mtext><mo>(</mo><mtext>1</mtext><mo>)</mo><mo>≅</mo><mi>K</mi></math></span>, generated by a kFA affording this sequence. We then determine which sequences <em>χ</em> produce semisimple abelian categories <span><math><mo>〈</mo><msub><mrow><mi>T</mi></mrow><mrow><mtext>kFA</mtext></mrow></msub><mo>|</mo><mi>χ</mi><mo>〉</mo></math></span> with finite-dimensional hom-spaces. These categories generalise results of Deligne concerning the interpolation of families of categories of representations such as <span><math><mtext>Rep</mtext><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></math></span>, <span><math><mtext>Rep</mtext><mo>(</mo><msub><mrow><mi>O</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></math></span>, and <span><math><mtext>Rep</mtext><mo>(</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>≀</mo><mi>P</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 1-36"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680479","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":"Closedness of the singular locus and generation for derived categories","authors":"Souvik Dey ,&nbsp;Pat Lank","doi":"10.1016/j.jalgebra.2025.07.007","DOIUrl":"10.1016/j.jalgebra.2025.07.007","url":null,"abstract":"<div><div>This work is concerned with a relationship regarding the closedness of the singular locus of a Noetherian scheme and existence of classical generators in its category of coherent sheaves, associated bounded derived category, and singularity category. Particularly, we extend an observation initially made by Iyengar and Takahashi in the affine context to the global setting. Furthermore, we furnish an example a Noetherian scheme whose bounded derived category admits a classical generator, yet not every finite scheme over it exhibits the same property.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 64-77"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680481","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":"Remarks on the sections of universal hyperelliptic curves","authors":"Tatsunari Watanabe","doi":"10.1016/j.jalgebra.2025.06.040","DOIUrl":"10.1016/j.jalgebra.2025.06.040","url":null,"abstract":"<div><div>In this paper, we study the obstruction for the sections of the universal hyperelliptic curves of genus <span><math><mi>g</mi><mo>≥</mo><mn>3</mn></math></span>. The obstruction of our interest comes from the relative completion of the hyperelliptic mapping class groups and the Lie algebra of the unipotent completion of the fundamental group of the configuration space of a compact oriented surface. Using the obstruction, we prove that the Birman exact sequence for the hyperelliptic mapping class groups does not split for <span><math><mi>g</mi><mo>≥</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"683 ","pages":"Pages 761-802"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144653996","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 greatest semilattice decomposition of subsemigroups of regular rings","authors":"Xavier Mary","doi":"10.1016/j.jalgebra.2025.06.043","DOIUrl":"10.1016/j.jalgebra.2025.06.043","url":null,"abstract":"<div><div>Combining arguments issued from semigroup theory, ring theory and lattice theory, we build up on a study of the idempotent-generated subsemigroup of regular separative rings by Hannah and O'Meara <span><span>[25]</span></span> to completely characterize the greatest semilattice decomposition of certain subsemigroups of regular rings. In particular, we prove that the greatest homomorphic image of a unit-regular ring is given by the additive semilattice of principal ideals of the ring. Many examples are given.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 37-63"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680480","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":"Groups with finitely many isomorphism classes of non-pronormal subgroups","authors":"Fausto De Mari","doi":"10.1016/j.jalgebra.2025.07.004","DOIUrl":"10.1016/j.jalgebra.2025.07.004","url":null,"abstract":"<div><div>A subgroup <em>H</em> of a group <em>G</em> is said to be <em>pronormal</em> if <em>H</em> and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>g</mi></mrow></msup></math></span> are conjugate in <span><math><mo>〈</mo><mi>H</mi><mo>,</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>g</mi></mrow></msup><mo>〉</mo></math></span> for every element <em>g</em> of <em>G</em>. The behaviour of pronormal subgroups in finite or infinite groups has been often investigated and, in particular, the structure of (generalized) soluble groups in which all subgroups are pronormal is known. Here it is proved that any (generalized) soluble group in which non-pronormal subgroups fall into finitely many isomorphism classes either is minimax or a group in which all subgroups are pronormal.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"683 ","pages":"Pages 719-733"},"PeriodicalIF":0.8,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144653994","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":"Non-weight modules over the algebra HW(b)","authors":"Yan Liu,&nbsp;Yao Ma,&nbsp;Liangyun Chen","doi":"10.1016/j.jalgebra.2025.07.003","DOIUrl":"10.1016/j.jalgebra.2025.07.003","url":null,"abstract":"<div><div>For the parameter <span><math><mi>b</mi><mo>∈</mo><mi>C</mi></math></span>, let <span><math><mrow><mi>HW</mi></mrow><mo>(</mo><mi>b</mi><mo>)</mo></math></span> be the semidirect product of the Witt algebra and the loop Heisenberg Lie algebra. In this paper, we study some non-weight modules over <span><math><mrow><mi>HW</mi></mrow><mo>(</mo><mi>b</mi><mo>)</mo></math></span>, specifically focusing on restricted modules, <span><math><mi>U</mi><mo>(</mo><mi>C</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span>-free modules of rank 1 and the tensor product of both. We prove that these three classes of non-weight <span><math><mrow><mi>HW</mi></mrow><mo>(</mo><mi>b</mi><mo>)</mo></math></span>-modules are pairwise non-isomorphic. Finally, we transform some tensor product modules over <span><math><mrow><mi>HW</mi></mrow><mo>(</mo><mi>b</mi><mo>)</mo></math></span> into induced modules from modules of its certain subalgebras for the case <span><math><mi>b</mi><mo>≠</mo><mo>±</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"683 ","pages":"Pages 604-632"},"PeriodicalIF":0.8,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632795","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":"An interconnection between pre-Lie rings, braces and associative rings","authors":"Agata Smoktunowicz","doi":"10.1016/j.jalgebra.2025.07.002","DOIUrl":"10.1016/j.jalgebra.2025.07.002","url":null,"abstract":"<div><div>Let <em>A</em> be a brace of cardinality <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for some prime number <em>p</em>. Denote <span><math><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>)</mo><mo>=</mo><mo>{</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo>:</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msup><mi>a</mi><mo>=</mo><mn>0</mn><mo>}</mo></math></span>. Suppose that for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></math></span> and all <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>A</mi></math></span> we have<span><span><span><math><mi>a</mi><mo>⁎</mo><mo>(</mo><mi>a</mi><mo>⁎</mo><mo>(</mo><mo>⋯</mo><mo>⁎</mo><mi>a</mi><mo>⁎</mo><mi>b</mi><mo>)</mo><mo>)</mo><mo>∈</mo><mi>p</mi><mi>A</mi><mo>,</mo><mi>a</mi><mo>⁎</mo><mo>(</mo><mi>a</mi><mo>⁎</mo><mo>(</mo><mo>⋯</mo><mo>⁎</mo><mi>a</mi><mo>⁎</mo><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>)</mo><mo>)</mo><mo>)</mo><mo>∈</mo><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span></span></span> where <em>a</em> appears less than <span><math><mfrac><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> times in this expression. Let <em>k</em> be such that <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi><mo>(</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mi>A</mi><mo>=</mo><mn>0</mn></math></span>. It is shown that the brace <span><math><mi>A</mi><mo>/</mo><mi>a</mi><mi>n</mi><mi>n</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>4</mn><mi>k</mi></mrow></msup><mo>)</mo></math></span> is obtained from a left nilpotent pre-Lie ring by a formula which depends only on the additive group of brace <em>A</em>. We also obtain some applications of this result.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"683 ","pages":"Pages 576-603"},"PeriodicalIF":0.8,"publicationDate":"2025-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632252","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":"Fundamental polytope for the isometry group of an alcove","authors":"Lucas Seco ,&nbsp;Arthur Garnier ,&nbsp;Karl-Hermann Neeb","doi":"10.1016/j.jalgebra.2025.06.035","DOIUrl":"10.1016/j.jalgebra.2025.06.035","url":null,"abstract":"<div><div>A fundamental alcove <span><math><mi>A</mi></math></span> is a tile in a paving of a vector space <em>V</em> by an affine reflection group <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>aff</mi></mrow></msub></math></span>. Its geometry encodes essential features of <span><math><msub><mrow><mi>W</mi></mrow><mrow><mi>aff</mi></mrow></msub></math></span>, such as its affine Dynkin diagram <span><math><mover><mrow><mi>D</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> and fundamental group Ω. In this article we investigate its full isometry group <span><math><mi>Aut</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. It is well known that the isometry group of a regular polyhedron is generated by hyperplane reflections on its faces. Being a simplex, an alcove <span><math><mi>A</mi></math></span> is the simplest of polyhedra, nevertheless it is seldom a regular one. In our first main result we show that <span><math><mi>Aut</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is isomorphic to <span><math><mi>Aut</mi><mo>(</mo><mover><mrow><mi>D</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo></math></span>. Building on this connection, we establish that <span><math><mi>Aut</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is an abstract Coxeter group, with generators given by affine isometric involutions of the ambient space. Although these involutions are seldom reflections, our second main result leverages them to construct, by slicing the Komrakov–Premet fundamental polytope <span><math><mi>K</mi></math></span> for the action of Ω, a family of fundamental polytopes for the action of <span><math><mi>Aut</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> on <span><math><mi>A</mi></math></span>, whose vertices are contained in the vertices of <span><math><mi>K</mi></math></span> and whose faces are parametrized by the so-called balanced minuscule roots, which we introduce here. In an appendix, we discuss some related negative results on stratified centralizers and equivariant triangulations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"683 ","pages":"Pages 633-671"},"PeriodicalIF":0.8,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632251","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":"Entwined comodules and contramodules over coalgebras with several objects: Frobenius, separability and Maschke theorems","authors":"Abhishek Banerjee ,&nbsp;Surjeet Kour","doi":"10.1016/j.jalgebra.2025.07.001","DOIUrl":"10.1016/j.jalgebra.2025.07.001","url":null,"abstract":"<div><div>We study module like objects over categorical quotients of algebras by the action of coalgebras with several objects. These take the form of “entwined comodules” and “entwined contramodules” over a triple <span><math><mo>(</mo><mi>C</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>ψ</mi><mo>)</mo></math></span>, where <em>A</em> is an algebra, <span><math><mi>C</mi></math></span> is a coalgebra with several objects and <em>ψ</em> is a collection of maps that “entwines” <span><math><mi>C</mi></math></span> with <em>A</em>. Our objective is to prove Frobenius, separability and Maschke type theorems for functors between categories of entwined comodules and entwined contramodules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"683 ","pages":"Pages 533-575"},"PeriodicalIF":0.8,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144614581","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}