Journal of Algebra最新文献

{"title":"On the automorphism group of quotient modular curves","authors":"Francesc Bars ,&nbsp;Tarun Dalal","doi":"10.1016/j.jalgebra.2025.02.037","DOIUrl":"10.1016/j.jalgebra.2025.02.037","url":null,"abstract":"<div><div>In this article, we determine the automorphism group of all the quotient modular curves of the modular curve <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>p</mi><mi>q</mi><mo>)</mo></math></span>, where <span><math><mi>p</mi><mo>,</mo><mi>q</mi></math></span> are two distinct primes. In obtaining such results, we provide different insights to compute the automorphism group for any quotient modular curve, which are very effective when the level of the curve is square-free. In particular, in the case where the level of the quotient curve is non square-free, we would mention that we present an unfamiliar automorphism of order 3 for the genus 4 curve <span><math><msubsup><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>(</mo><mn>25</mn><mo>⋅</mo><mn>11</mn><mo>)</mo></math></span> defined over <span><math><mi>Q</mi><mo>[</mo><msqrt><mrow><mn>5</mn></mrow></msqrt><mo>]</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"672 ","pages":"Pages 334-378"},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143619608","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":"Deformed mirror symmetry for punctured surfaces","authors":"Raf Bocklandt ,&nbsp;Jasper van de Kreeke","doi":"10.1016/j.jalgebra.2025.02.024","DOIUrl":"10.1016/j.jalgebra.2025.02.024","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Mirror symmetry originally envisions a correspondence between deformations of the A-side and deformations of the B-side. In this paper, we achieve an explicit correspondence in the case of punctured surfaces.&lt;/div&gt;&lt;div&gt;The starting point is the noncommutative mirror equivalence &lt;span&gt;&lt;math&gt;&lt;mi&gt;Gtl&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mi&gt;≅&lt;/mi&gt;&lt;mi&gt;mf&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Jac&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for a punctured surface &lt;em&gt;Q&lt;/em&gt;. We pick a deformation &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Gtl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; which captures a large part of the deformation theory and includes the relative Fukaya category. To find the corresponding deformation of &lt;span&gt;&lt;math&gt;&lt;mi&gt;mf&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Jac&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, we deform work of Cho-Hong-Lau which interprets mirror symmetry as Koszul duality. As result we explicitly obtain the corresponding deformation &lt;span&gt;&lt;math&gt;&lt;mi&gt;mf&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Jac&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; together with a deformed mirror functor &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Gtl&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mi&gt;mf&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Jac&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;The bottleneck is to verify that the algebra &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Jac&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; is indeed a (flat) deformation of &lt;span&gt;&lt;math&gt;&lt;mi&gt;Jac&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt;. We achieve this by deploying a result of Berger-Ginzburg-Taillefer on deformations of CY3 algebras, which however requires the relations to be homogeneous. We show how to replace this homogeneity requirement by a simple boundedness condition and obtain flatness of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Jac&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;ˇ&lt;/mo&gt;&lt;/mrow&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; for almost all &lt;em&gt;Q&lt;/em&gt;.&lt;/div&gt;&lt;div&gt;We finish the paper with examples, including a full treatment of the 3-punctured sphere and 4-punctured torus. With the help of our computations in &lt;span&gt;&lt;span&gt;[24]&lt;/span&gt;&lt;/span&gt;, we describe &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Jac&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"672 ","pages":"Pages 413-602"},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643495","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":"The graded classification conjectures hold for various finite representations of Leavitt path algebras","authors":"Wolfgang Bock ,&nbsp;Roozbeh Hazrat ,&nbsp;Alfilgen Sebandal","doi":"10.1016/j.jalgebra.2025.02.035","DOIUrl":"10.1016/j.jalgebra.2025.02.035","url":null,"abstract":"&lt;div&gt;&lt;div&gt;The Graded Classification Conjecture states that for finite directed graphs &lt;em&gt;E&lt;/em&gt; and &lt;em&gt;F&lt;/em&gt;, the associated Leavitt path algebras &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; are graded Morita equivalent, i.e., &lt;span&gt;&lt;math&gt;&lt;mi&gt;Gr-&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;≈&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;gr&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;Gr-&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, if and only if, their graded Grothendieck groups are isomorphic &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;gr&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;gr&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; as order-preserving &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-modules. Furthermore, if under this isomorphism, the class &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is sent to &lt;span&gt;&lt;math&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; then the algebras are graded isomorphic, i.e., &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;gr&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;In this note we show that, for finite graphs &lt;em&gt;E&lt;/em&gt; and &lt;em&gt;F&lt;/em&gt; with no sinks and sources, an order-preserving &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-module isomorphism &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;gr&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≅&lt;/mo&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;gr&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; gives that the categories of locally finite dimensional graded modules of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;L&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msu","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"672 ","pages":"Pages 303-333"},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143619643","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":"On the Kauffman bracket skein module of (S1 × S2) # (S1 × S2)","authors":"Rhea Palak Bakshi ,&nbsp;Seongjeong Kim ,&nbsp;Shangjun Shi ,&nbsp;Xiao Wang","doi":"10.1016/j.jalgebra.2025.01.028","DOIUrl":"10.1016/j.jalgebra.2025.01.028","url":null,"abstract":"<div><div>Determining the structure of the Kauffman bracket skein module of all 3-manifolds over the ring of Laurent polynomials <span><math><mi>Z</mi><mo>[</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span> is a big open problem in skein theory. Very little is known about the skein module of non-prime manifolds over this ring. In this paper, we compute the Kauffman bracket skein module of the 3-manifold <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mspace></mspace><mi>#</mi><mspace></mspace><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> over the ring <span><math><mi>Z</mi><mo>[</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span>. We do this by analysing the submodule of handle sliding relations, for which we provide a suitable basis. Along the way we compute the Kauffman bracket skein module of <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mspace></mspace><mi>#</mi><mspace></mspace><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. We also show that the skein module of <span><math><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mspace></mspace><mi>#</mi><mspace></mspace><mo>(</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> does not split into the sum of free and torsion submodules. Furthermore, we illustrate two families of torsion elements in this skein module.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"673 ","pages":"Pages 103-137"},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620734","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":"Intersection of irreducible curves and the Hermitian curve","authors":"Peter Beelen ,&nbsp;Mrinmoy Datta ,&nbsp;Maria Montanucci ,&nbsp;Jonathan Niemann","doi":"10.1016/j.jalgebra.2025.02.036","DOIUrl":"10.1016/j.jalgebra.2025.02.036","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; denote the Hermitian curve in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be an irreducible plane projective curve in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; also defined over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of degree &lt;em&gt;d&lt;/em&gt;. Can &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; intersect in exactly &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; distinct &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-rational points? Bézout's theorem immediately implies that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; intersect in at most &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; points, but equality is not guaranteed over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. In this paper we prove that for many &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, the answer to this question is affirmative. The case &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; is trivial: it is well known that any secant line of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; defined over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; intersects &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; rational points. Moreover, all possible intersections of conics and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; were classified in &lt;span&gt;&lt;span&gt;[9]&lt;/span&gt;&lt;/span&gt; and their results imply that the answer to the question above is affirmative for &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, as well. However, an exhaustive computer search quickly reveals that for &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;{&lt;/m","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"671 ","pages":"Pages 75-94"},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621167","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":"Derived equivalences via Tate resolutions","authors":"K. Ganapathy,&nbsp;Sarang Sane","doi":"10.1016/j.jalgebra.2025.02.015","DOIUrl":"10.1016/j.jalgebra.2025.02.015","url":null,"abstract":"<div><div>For any finite sequence of elements <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> in a commutative noetherian ring <em>R</em>, we show that for <span><math><mi>n</mi><mo>≫</mo><mn>0</mn></math></span>, the natural map from the Koszul complex <span><math><mi>K</mi><mo>(</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>)</mo></math></span> to the Koszul complex <span><math><mi>K</mi><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></math></span> factors through the Tate resolution on <span><math><msubsup><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>. Using this, for any resolving subcategory <span><math><mi>A</mi></math></span> of <span><math><mrow><mi>mod</mi><mi>(</mi><mtext>R</mtext><mi>)</mi></mrow></math></span> and any ideal <em>I</em> such that it has a filtration <span><math><mo>{</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></math></span> which is equivalent to the <em>I</em>-adic filtration and <span><math><msub><mrow><mtext>dim</mtext></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>/</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>&lt;</mo><mo>∞</mo></math></span>, we show a derived equivalence between the bounded derived category of finitely generated modules supported on <span><math><mi>V</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> having finite <span><math><mi>A</mi></math></span>-dimension and the bounded derived category of <span><math><mi>A</mi></math></span> with homologies supported on <span><math><mi>V</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span>. As a special case, when <em>R</em> is of prime characteristic and <em>I</em> is of finite projective dimension, we obtain a derived equivalence between the bounded derived category of finite projective dimension modules supported on <span><math><mi>V</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span> and the bounded derived category of projective modules with homologies supported on <span><math><mi>V</mi><mo>(</mo><mi>I</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"672 ","pages":"Pages 89-119"},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143619638","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 extensions of certain representations of reductive algebraic groups with Frobenius maps","authors":"Xiaoyu Chen ,&nbsp;Junbin Dong","doi":"10.1016/j.jalgebra.2025.02.028","DOIUrl":"10.1016/j.jalgebra.2025.02.028","url":null,"abstract":"<div><div>Let <strong>G</strong> be a connected reductive algebraic group defined over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with <em>q</em> elements, where <em>q</em> is a power of a prime number <em>p</em>. Let <span><math><mi>k</mi></math></span> be a field and we study the extensions of certain <span><math><mi>k</mi><mi>G</mi></math></span>-modules in this paper. We show that the extensions of any modules in <span><math><mi>O</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> by a finite-dimensional <span><math><mi>k</mi><mi>G</mi></math></span>-module is zero if <span><math><mi>char</mi><mspace></mspace><mi>k</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>char</mi><mspace></mspace><mi>k</mi><mo>≠</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi>p</mi></math></span>, where <span><math><mi>O</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the principal representation category defined in <span><span>[8]</span></span>. We determine the necessary and sufficient condition for the vanishing of extensions between naive induced modules. As an application, we give the conditions for the vanishing of extensions between simple modules in <span><math><mi>O</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for <span><math><mi>G</mi><mo>=</mo><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><mi>F</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"672 ","pages":"Pages 71-88"},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143601050","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":"Projective linear codes and their simplex complementary codes","authors":"Hao Chen,&nbsp;Conghui Xie","doi":"10.1016/j.jalgebra.2025.02.026","DOIUrl":"10.1016/j.jalgebra.2025.02.026","url":null,"abstract":"<div><div>The famous Erdös-Kleitman bound for a binary anticode <strong>C</strong> of the length <em>n</em> and the diameter <em>δ</em> asserts that<span><span><span><math><mo>|</mo><mi>C</mi><mo>|</mo><mo>≤</mo><msubsup><mrow><mi>Σ</mi></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mfrac><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>i</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>.</mo></math></span></span></span> In this paper, we give an antiGriesmer bound for <em>q</em>-ary projective linear anticodes, which is stronger than the above Erdös-Kleitman bound for binary projective linear anticodes. From known projective linear anticodes, we construct their Simplex complementary codes with optimal or almost optimal minimum distances. A complementary theorem constructing infinitely many new projective linear <span><math><mo>(</mo><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-weight code from a known <em>t</em>-weight linear code is presented. As applications, we indicate that many new optimal or almost optimal few-weight linear codes, and many infinite families of strongly regular graphs and <em>l</em>-strongly walk-regular graphs can be obtained.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"673 ","pages":"Pages 304-320"},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620617","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":"The blockwise Alperin weight conjecture and inertial blocks","authors":"Xueqin Hu,&nbsp;Yuanyang Zhou","doi":"10.1016/j.jalgebra.2025.01.030","DOIUrl":"10.1016/j.jalgebra.2025.01.030","url":null,"abstract":"<div><div>In this paper, we prove that the inductive blockwise Alperin weight condition holds for inertial blocks. As an application, we give an alternative proof to the blockwise Alperin weight conjecture for 2-blocks with abelian defect groups.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"672 ","pages":"Pages 379-399"},"PeriodicalIF":0.8,"publicationDate":"2025-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636441","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":"Representations of the small quasi-quantum group","authors":"Hua Sun ,&nbsp;Hui-Xiang Chen ,&nbsp;Yinhuo Zhang","doi":"10.1016/j.jalgebra.2025.02.025","DOIUrl":"10.1016/j.jalgebra.2025.02.025","url":null,"abstract":"<div><div>In this paper, we study the representation theory of the small quantum group <span><math><msub><mrow><mover><mrow><mi>U</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>q</mi></mrow></msub></math></span> and the small quasi-quantum group <span><math><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub></math></span>, where <em>q</em> is a primitive <em>n</em>-th root of unity and <span><math><mi>n</mi><mo>&gt;</mo><mn>2</mn></math></span> is odd. All finite dimensional indecomposable <span><math><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub></math></span>-modules are described and classified. Moreover, the decomposition rules for the tensor products of <span><math><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub></math></span>-modules are given. Finally, we describe the structures of the projective class ring <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> and the Green ring <span><math><mi>r</mi><mo>(</mo><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>. We show that <span><math><mi>r</mi><mo>(</mo><msub><mrow><mover><mrow><mi>U</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> is isomorphic to a subring of <span><math><mi>r</mi><mo>(</mo><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span>, and the stable Green rings <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><mi>U</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> and <span><math><msub><mrow><mi>r</mi></mrow><mrow><mi>s</mi><mi>t</mi></mrow></msub><mo>(</mo><msub><mrow><mover><mrow><mi>U</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> are isomorphic.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"673 ","pages":"Pages 188-221"},"PeriodicalIF":0.8,"publicationDate":"2025-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620605","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}