Journal of Algebra最新文献

{"title":"On the classification of unitary highest weight modules in the exceptional cases","authors":"Pavle Pandžić ,&nbsp;Ana Prlić ,&nbsp;Gordan Savin ,&nbsp;Vladimír Souček ,&nbsp;Vít Tuček","doi":"10.1016/j.jalgebra.2025.07.018","DOIUrl":"10.1016/j.jalgebra.2025.07.018","url":null,"abstract":"<div><div>In our previous paper <span><span>[12]</span></span>, we gave a complete classification of the unitary highest weight modules for the universal covers of the Lie groups <span><math><mi>S</mi><mi>p</mi><mo>(</mo><mn>2</mn><mi>n</mi><mo>,</mo><mi>R</mi><mo>)</mo><mo>,</mo><mi>S</mi><msup><mrow><mi>O</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><mn>2</mn><mi>n</mi><mo>)</mo></math></span> and <span><math><mi>S</mi><mi>U</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, using the Dirac inequality and the so called PRV product. In this paper, we complete the classification of the unitary highest weight modules for the remaining cases; i.e., universal covers of the Lie groups <span><math><mi>S</mi><msub><mrow><mi>O</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>(</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>6</mn><mo>(</mo><mo>−</mo><mn>14</mn><mo>)</mo></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>7</mn><mo>(</mo><mo>−</mo><mn>25</mn><mo>)</mo></mrow></msub></math></span>. We also describe unitary highest weight modules with given infinitesimal characters.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 524-562"},"PeriodicalIF":0.8,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772958","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":"PBW bases for ıquantum groups","authors":"Ming Lu ,&nbsp;Ruiqi Yang ,&nbsp;Weinan Zhang","doi":"10.1016/j.jalgebra.2025.07.014","DOIUrl":"10.1016/j.jalgebra.2025.07.014","url":null,"abstract":"<div><div>We establish PBW type bases for <em>ı</em>quantum groups of arbitrary finite type, using the relative braid group symmetries. Explicit formulas for root vectors are provided for <em>ı</em>quantum groups of each rank 1 type. We show that our PBW type bases give rise to integral bases for the modified <em>ı</em>quantum groups. The leading terms of our bases can be identified with the usual PBW bases in the theory of quantum groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 441-478"},"PeriodicalIF":0.8,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144749833","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":"Constructing skew bracoids via abelian maps, and solutions to the Yang-Baxter equation","authors":"Alan Koch ,&nbsp;Paul J. Truman","doi":"10.1016/j.jalgebra.2025.07.013","DOIUrl":"10.1016/j.jalgebra.2025.07.013","url":null,"abstract":"<div><div>We show how one can use the skew braces constructed using abelian maps to generate families of skew bracoids as defined by Martin-Lyons and Truman. Under certain circumstances, these bracoids give right non-degenerate solutions to the Yang-Baxter equation.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 336-353"},"PeriodicalIF":0.8,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144721546","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":"Defects in weighted graphs and commutators","authors":"Harish Kishnani,&nbsp;Amit Kulshrestha","doi":"10.1016/j.jalgebra.2025.07.009","DOIUrl":"10.1016/j.jalgebra.2025.07.009","url":null,"abstract":"<div><div>Let <em>R</em> be a commutative ring. In <span><span>[5]</span></span>, the authors introduced <em>R</em>-weighted graphs as a tool for studying commutators in groups and Lie algebras. These graphs are equivalent to a system of balance equations, and their consistent labelings correspond to solutions of this system of balance equations. In this article, we apply these ideas in the case when <em>R</em> is a field <em>F</em>. We focus on <em>F</em>-weighted graphs with four vertices and establish necessary and sufficient conditions for the existence of a consistent labeling on them. A notion of defects in weighted graphs is introduced for this purpose. We prove that defects in weighted graphs prevent Lie brackets from being surjective onto the derived Lie subalgebra. Similarly, these defects prevent certain elements in the commutator subgroup of a nilpotent group of class 2 from being a commutator. As an application of our techniques, we prove that for a Lie algebra <em>L</em> whose derived subalgebra <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is at most 3-dimensional, the Lie bracket is surjective onto <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. We provide a counterexample when <span><math><mi>dim</mi><mo>⁡</mo><mo>(</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo><mo>=</mo><mn>4</mn></math></span>. We also characterize commutators among the elements of <em>L</em>' when <span><math><mi>dim</mi><mo>⁡</mo><mo>(</mo><mi>L</mi><mo>/</mo><mi>Z</mi><mo>(</mo><mi>L</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mn>4</mn></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 213-233"},"PeriodicalIF":0.8,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686807","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 new characteristic subgroup for pushing up I","authors":"G. Parmeggiani ,&nbsp;B. Stellmacher","doi":"10.1016/j.jalgebra.2025.07.010","DOIUrl":"10.1016/j.jalgebra.2025.07.010","url":null,"abstract":"<div><div>Let <em>T</em> be a finite <em>p</em>-group. In this paper we introduce a new characteristic subgroup <span><math><mi>W</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> of <em>T</em> containing <span><math><mi>Z</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span>. This subgroup allows to give the structure of <em>p</em>-minimal finite groups <em>G</em> of characteristic <em>p</em> with <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>-factor group which satisfy <span><math><mi>Z</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>⋬</mo><mi>G</mi></math></span> and <span><math><mi>W</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>⋬</mo><mi>G</mi></math></span>. This result generalizes a classical pushing up result obtained by Baumann <span><span>[1]</span></span> (for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>) and Niles <span><span>[8]</span></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 308-335"},"PeriodicalIF":0.8,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144694953","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":"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":"D-modules on the basic affine space and large g-modules","authors":"Masatoshi Kitagawa","doi":"10.1016/j.jalgebra.2025.06.037","DOIUrl":"10.1016/j.jalgebra.2025.06.037","url":null,"abstract":"<div><div>In this paper, we consider <span><math><mi>D</mi></math></span>-modules on the basic affine space <span><math><mi>G</mi><mo>/</mo><mi>U</mi></math></span> and their global sections for a semisimple complex algebraic group <em>G</em>. Our aim is to prepare basic results about large non-irreducible modules for the branching problem and harmonic analysis of reductive Lie groups. A main tool is a formula given by Bezrukavnikov–Braverman–Positselskii. The formula is about a product of functions and their Fourier transforms on <span><math><mi>G</mi><mo>/</mo><mi>U</mi></math></span> like Capelli's identity. Using the formula, we give a generalization of the Beilinson–Bernstein correspondence.</div><div>It is also shown that the global sections of holonomic <span><math><mi>D</mi></math></span>-modules are also holonomic using the formula. As a consequence, we give a large algebra action on the <span><math><mi>u</mi></math></span>-cohomologies <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>(</mo><mi>u</mi><mo>;</mo><mi>V</mi><mo>)</mo></math></span> of a <span><math><mi>g</mi></math></span>-module <em>V</em> when <em>V</em> is realized as a holonomic <span><math><mi>D</mi></math></span>-module. We consider affinity of the supports of the <span><math><mi>t</mi></math></span>-modules <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>(</mo><mi>u</mi><mo>;</mo><mi>V</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"684 ","pages":"Pages 176-212"},"PeriodicalIF":0.8,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686806","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}