Journal of Algebra最新文献

{"title":"Groupoid graded semisimple rings","authors":"Zaqueu Cristiano ,&nbsp;Wellington Marques de Souza ,&nbsp;Javier Sánchez","doi":"10.1016/j.jalgebra.2025.08.020","DOIUrl":"10.1016/j.jalgebra.2025.08.020","url":null,"abstract":"<div><div>We develop the theory of groupoid graded semisimple rings. Our rings are neither unital nor one-sided artinian. Instead, they exhibit a strong version of having local units and being locally artinian, and we call them <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-artinian. One of our main results is a groupoid graded version of the Wedderburn-Artin Theorem, where we characterize groupoid graded semisimple rings as direct sums of graded simple <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-artinian rings and we exhibit the structure of this latter class of rings. In this direction, we also prove a groupoid graded version of Jacobson-Chevalley density theorem. We need to define and study properties of groupoid gradings on matrix rings (possibly of infinite size) over groupoid graded rings, and specially over groupoid graded division rings. Because of that, we study groupoid graded division rings and their graded modules. We consider a natural notion of freeness for groupoid graded modules that, when specialized to group graded rings, gives the usual one, and show that for a groupoid graded division ring all graded modules are free (in this sense). Contrary to the group graded case, there are groupoid graded rings for which all graded modules are free according to our definition, but they are not graded division rings. We exhibit an easy example of this kind of rings and characterize such class among groupoid graded semisimple rings. We also relate groupoid graded semisimple rings with the notion of semisimple category defined by B. Mitchell. For that, we show the link between functors from a preadditive category to abelian groups and graded modules over the groupoid graded ring associated to this category, generalizing a result of P. Gabriel. We characterize simple artinian categories and categories for which every functor from them to abelian groups is free in the sense of B. Mitchell.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 1-116"},"PeriodicalIF":0.8,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145048628","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":"Every finitely generated abelian group is the class group of a generalized cluster algebra","authors":"Mara Pompili","doi":"10.1016/j.jalgebra.2025.08.017","DOIUrl":"10.1016/j.jalgebra.2025.08.017","url":null,"abstract":"<div><div>We determine the class group of those generalized cluster algebras that are Krull domains. In particular, this provides a criterion for determining whether or not a generalized cluster algebra is a UFD. In fact, any finitely generated abelian group can be realized as the class group of a generalized cluster algebra. Additionally, we show that generalized cluster algebras are FF-domains and that their cluster variables are strong atoms. Finally, we examine the factorization and ring-theoretic properties of Laurent phenomenon algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 566-594"},"PeriodicalIF":0.8,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145026720","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":"Hook theorem for superalgebras with superinvolution or graded involution","authors":"Irina Sviridova ,&nbsp;Renata A. Silva","doi":"10.1016/j.jalgebra.2025.08.019","DOIUrl":"10.1016/j.jalgebra.2025.08.019","url":null,"abstract":"<div><div>We consider a superalgebra with a superinvolution or graded involution # over a field <em>F</em> of characteristic zero and assume that it is a <em>PI</em>-algebra. S.A. Amitsur and A. Regev have proved in 1982 the celebrated hook theorem for ordinary polynomial identities. In this paper, we present the proof of a version of the hook theorem for the case of multilinear #-superidentities. This theorem provides important combinatorial characteristics of identities in the language of symmetric group representations. Furthermore, we present an analogue of Amitsur identities for #-superalgebras, which are polynomial interpretations of the mentioned combinatorial characteristics, as a consequence of the hook theorem.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 117-150"},"PeriodicalIF":0.8,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145107399","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 toric degeneration of Kronecker moduli spaces","authors":"E. Kalashnikov","doi":"10.1016/j.jalgebra.2025.08.015","DOIUrl":"10.1016/j.jalgebra.2025.08.015","url":null,"abstract":"<div><div>In this paper, we show that there is a finite SAGBI basis of the coordinate ring of a Kronecker quiver moduli space, indexed by primitive semi-standard tableaux pairs. This induces a toric degeneration of the Kronecker moduli space to a normal toric variety, a generalization of the toric degeneration of the Grassmannian to the Gelfand–Cetlin polytope constructed by Gonciulea–Lakshmibai <span><span>[13]</span></span>. The moment polytope of the degenerate toric variety can be described as the intersection of two Gelfand–Cetlin polytopes. We explain when this can be generalized to degenerations coming from matching fields.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 749-774"},"PeriodicalIF":0.8,"publicationDate":"2025-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145045407","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":"Elementary construction of canonical bases, foldings, and piecewise linear bijections","authors":"Toshiaki Shoji ,&nbsp;Zhiping Zhou","doi":"10.1016/j.jalgebra.2025.08.013","DOIUrl":"10.1016/j.jalgebra.2025.08.013","url":null,"abstract":"<div><div>Let <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>−</mo></mrow></msubsup></math></span> be the negative half of a quantum group of finite type. We construct the canonical basis of <span><math><msubsup><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>−</mo></mrow></msubsup></math></span> by applying the folding theory of quantum groups and piecewise linear parametrization of canonical basis. Our construction is elementary, in the sense that we don't appeal to Lusztig's geometric theory of canonical bases, nor to Kashiwara's theory of crystal bases.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 481-502"},"PeriodicalIF":0.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019379","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":"Integer-valued polynomials on subsets of quaternion algebras","authors":"Nicholas J. Werner","doi":"10.1016/j.jalgebra.2025.08.011","DOIUrl":"10.1016/j.jalgebra.2025.08.011","url":null,"abstract":"<div><div>Let <em>R</em> be either the ring of Lipschitz quaternions, or the ring of Hurwitz quaternions. Then, <em>R</em> is a subring of the division ring <span><math><mi>D</mi></math></span> of rational quaternions. For <span><math><mi>S</mi><mo>⊆</mo><mi>R</mi></math></span>, we study the collection <span><math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>f</mi><mo>∈</mo><mi>D</mi><mo>[</mo><mi>x</mi><mo>]</mo><mo>|</mo><mi>f</mi><mo>(</mo><mi>S</mi><mo>)</mo><mo>⊆</mo><mi>R</mi><mo>}</mo></math></span> of polynomials that are integer-valued on <em>S</em>. The set <span><math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> is always a left <em>R</em>-submodule of <span><math><mi>D</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, but need not be a subring of <span><math><mi>D</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>. We say that <em>S</em> is a ringset of <em>R</em> if <span><math><mrow><mi>Int</mi></mrow><mo>(</mo><mi>S</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> is a subring of <span><math><mi>D</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>. In this paper, we give a complete classification of the finite subsets of <em>R</em> that are ringsets.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 195-219"},"PeriodicalIF":0.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144922395","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":"Star-algebras and almost polynomial growth of central polynomials","authors":"A. Giambruno ,&nbsp;D. La Mattina ,&nbsp;C. Polcino Milies","doi":"10.1016/j.jalgebra.2025.08.012","DOIUrl":"10.1016/j.jalgebra.2025.08.012","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;A&lt;/em&gt; be an algebra with involution ⁎ over a field of characteristic zero. There are three numerical sequences attached to the ⁎-polynomial identities &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Id&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; satisfied by &lt;em&gt;A&lt;/em&gt;: the sequence of codimensions &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, the sequence of central codimensions &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and the sequence of proper central codimensions &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;They give a measure of the ⁎-polynomial identities, the central ⁎-polynomials and the proper central ⁎-polynomials of the algebra &lt;em&gt;A&lt;/em&gt;.&lt;/div&gt;&lt;div&gt;It has been proved (&lt;span&gt;&lt;span&gt;[9]&lt;/span&gt;&lt;/span&gt;, &lt;span&gt;&lt;span&gt;[18]&lt;/span&gt;&lt;/span&gt;) that when &lt;em&gt;A&lt;/em&gt; satisfies a non-trivial identity, the three sequences either grow exponentially or are polynomially bounded. Now, an algebra &lt;em&gt;A&lt;/em&gt; has almost polynomial growth of the codimensions if &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; grows exponentially and for any algebra &lt;em&gt;B&lt;/em&gt; such that &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Id&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;⊋&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Id&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is polynomially bounded. Similarly we have the definitions of almost polynomial growth of the central codimensions &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and of the proper central codimensions &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;We aim to classify, up to ⁎-PI-equivalence, the algebras with almost polynomial growth of one of the above codimensions. This has already been done in &lt;span&gt;&lt;span&gt;[8]&lt;/span&gt;&lt;/span&gt; regarding the codimensions &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;Here we classify, up to ⁎-PI-equivalence, the algebras having almost polynomial growth of the central","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 398-416"},"PeriodicalIF":0.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144931653","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":"Lascoux-type resolutions, derived categories, and flips","authors":"Qingyuan Jiang","doi":"10.1016/j.jalgebra.2025.07.053","DOIUrl":"10.1016/j.jalgebra.2025.07.053","url":null,"abstract":"<div><div>This paper introduces Lascoux-type complexes that extend the Lascoux complexes for resolving generic determinantal ideals. These Lascoux-type complexes naturally arise when analyzing the correspondences between two different types of resolutions of singularities of determinantal varieties. We also discuss the applications of these resolutions in various geometric contexts, including blowups, standard flips, virtual flips, and projectivizations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 417-453"},"PeriodicalIF":0.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010259","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 semisimple proto-Abelian categories associated to inverse monoids","authors":"Alexander Sistko","doi":"10.1016/j.jalgebra.2025.07.052","DOIUrl":"10.1016/j.jalgebra.2025.07.052","url":null,"abstract":"<div><div>Let <em>G</em> be a finite abelian group written multiplicatively, with <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>=</mo><mi>G</mi><mo>⊔</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span> the pointed abelian group formed by adjoining an absorbing element 0. There is an associated finitary, proto-abelian category <span><math><msub><mrow><mi>Vect</mi></mrow><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow></msub></math></span>, whose objects can be thought of as finite-dimensional vector spaces over <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>. The class of <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-linear monoids are then defined in terms of this category. In this paper, we study the finitary, proto-abelian category <span><math><mi>Rep</mi><mo>(</mo><mi>M</mi><mo>,</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> of finite-dimensional <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-linear representations of a <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-linear monoid <em>M</em>. Although this category is only a slight modification of the usual category of <em>M</em>-modules, it exhibits significantly different behavior for interesting classes of monoids. Assuming that the regular principal factors of <em>M</em> are objects of <span><math><mi>Rep</mi><mo>(</mo><mi>M</mi><mo>,</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span>, we develop a version of the Clifford-Munn-Ponizovskiĭ Theorem and classify the <em>M</em> for which each non-zero object of <span><math><mi>Rep</mi><mo>(</mo><mi>M</mi><mo>,</mo><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover><mo>)</mo></math></span> is a direct sum of simple objects. When <em>M</em> is the endomorphism monoid of an object in <span><math><msub><mrow><mi>Vect</mi></mrow><mrow><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow></msub></math></span>, we discuss alternate frameworks for studying its <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></math></span>-linear representations and contrast the various approaches.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 354-397"},"PeriodicalIF":0.8,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926057","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":"Classification of non-degenerate symmetric bilinear and quadratic forms in the Verlinde category Ver4+","authors":"Iz Chen ,&nbsp;Arun S. Kannan ,&nbsp;Krishna Pothapragada","doi":"10.1016/j.jalgebra.2025.08.010","DOIUrl":"10.1016/j.jalgebra.2025.08.010","url":null,"abstract":"<div><div>Although Deligne's theorem classifies all symmetric tensor categories (STCs) with moderate growth over algebraically closed fields of characteristic zero, the classification does not extend to positive characteristic. At the forefront of the study of STCs is the search for an analog to Deligne's theorem in positive characteristic, and it has become increasingly apparent that the Verlinde categories are to play a significant role. Moreover, these categories are largely unstudied, but have already shown very interesting phenomena as both a generalization of and a departure from superalgebra and supergeometry. In this paper, we study <span><math><msubsup><mrow><mi>Ver</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span>, the simplest non-trivial Verlinde category in characteristic 2. In particular, we classify all isomorphism classes of non-degenerate symmetric bilinear forms and non-degenerate quadratic forms and study the associated Witt semi-ring that arises from the addition and multiplication operations on bilinear forms.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 220-262"},"PeriodicalIF":0.8,"publicationDate":"2025-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144926060","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}