Journal of AlgebraPub Date : 2025-03-14DOI: 10.1016/j.jalgebra.2025.02.039
Evgeny Feigin
{"title":"Birational maps, PBW degenerate flags and poset polytopes","authors":"Evgeny Feigin","doi":"10.1016/j.jalgebra.2025.02.039","DOIUrl":"10.1016/j.jalgebra.2025.02.039","url":null,"abstract":"<div><div>We extend the results on the graph closures of the birational maps between projective spaces and Grassmannians to the case of PBW degenerate flag varieties. The advantage of the PBW degenerate flags (as opposed to their classical analogues) is the existence of a large group of symmetries for the graph closures. We discuss the combinatorial, algebraic and geometric sides of the picture. In particular, we show that toric degenerations of Borovik, Sturmfels and Sverrisdóttir are still available in the general settings. We also derive a description of the graph closures for flag varieties in terms of quiver representations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"674 ","pages":"Pages 235-256"},"PeriodicalIF":0.8,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685269","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}
Journal of AlgebraPub Date : 2025-03-14DOI: 10.1016/j.jalgebra.2025.02.049
Xin Huang
{"title":"On stable equivalences with endopermutation source and Külshammer–Puig classes","authors":"Xin Huang","doi":"10.1016/j.jalgebra.2025.02.049","DOIUrl":"10.1016/j.jalgebra.2025.02.049","url":null,"abstract":"<div><div>We give a new proof, by using the terminology and notation in the textbook <span><span>[9]</span></span>, to a result, due to Puig, stating that a stable equivalence of Morita type between two block algebras of finite groups induced by a bimodule with an endopermutation source preserves Külshammer–Puig classes.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"671 ","pages":"Pages 235-242"},"PeriodicalIF":0.8,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685192","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}
Journal of AlgebraPub Date : 2025-03-14DOI: 10.1016/j.jalgebra.2025.02.038
Yaohui Xue , Kaiming Zhao
{"title":"Representations of the Fermion-Virasoro superalgebras","authors":"Yaohui Xue , Kaiming Zhao","doi":"10.1016/j.jalgebra.2025.02.038","DOIUrl":"10.1016/j.jalgebra.2025.02.038","url":null,"abstract":"<div><div>Let <span><math><mi>δ</mi><mo>=</mo><mn>0</mn></math></span> or <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. In this paper we introduce two infinite-dimensional Lie superalgebras: the Fermion superalgebra <span><math><mi>F</mi><mo>(</mo><mi>δ</mi><mo>)</mo></math></span> and the Fermion-Virasoro superalgebra <span><math><mi>S</mi><mo>(</mo><mi>δ</mi><mo>)</mo></math></span>. We classify all simple smooth <span><math><mi>F</mi><mo>(</mo><mi>δ</mi><mo>)</mo></math></span>-modules, all simple weight <span><math><mi>F</mi><mo>(</mo><mi>δ</mi><mo>)</mo></math></span>-modules, all simple smooth <span><math><mi>S</mi><mo>(</mo><mi>δ</mi><mo>)</mo></math></span>-modules of nonzero level, and all simple Harish-Chandra <span><math><mi>S</mi><mo>(</mo><mi>δ</mi><mo>)</mo></math></span>-modules. We also classify all <span><math><mi>S</mi><mo>(</mo><mi>δ</mi><mo>)</mo></math></span>-modules structures on free <span><math><mi>C</mi><mo>[</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>]</mo></math></span>-modules of rank 2 with non-trivial even and odd spaces (four different classes) where <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is a degree 0 element in the graded superalgebra <span><math><mi>S</mi><mo>(</mo><mi>δ</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"675 ","pages":"Pages 133-155"},"PeriodicalIF":0.8,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143767492","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}
Journal of AlgebraPub Date : 2025-03-14DOI: 10.1016/j.jalgebra.2025.03.012
Giacomo Tendas
{"title":"Dualities in the theory of accessible categories","authors":"Giacomo Tendas","doi":"10.1016/j.jalgebra.2025.03.012","DOIUrl":"10.1016/j.jalgebra.2025.03.012","url":null,"abstract":"<div><div>Through the notion of <em>weakly sound</em> class of weights, we recover many known dualities involving accessible categories with a chosen class of limits, as instances of a general duality theorem. These include the Gabriel–Ulmer duality for locally finitely presentable categories, Diers duality for locally finitely multipresentable categories, and the Makkai–Paré duality for finitely accessible categories. In doing so, we extend these to the enriched setting, provide a more formal and unifying approach to the theory, and also discuss new dualities that arise as a consequence of our main theorem.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"674 ","pages":"Pages 29-49"},"PeriodicalIF":0.8,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644008","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}
Journal of AlgebraPub Date : 2025-03-14DOI: 10.1016/j.jalgebra.2025.02.048
Robert Boltje, Nariel Monteiro
{"title":"The ring of perfect p-permutation bimodules for blocks with cyclic defect groups","authors":"Robert Boltje, Nariel Monteiro","doi":"10.1016/j.jalgebra.2025.02.048","DOIUrl":"10.1016/j.jalgebra.2025.02.048","url":null,"abstract":"<div><div>Let <em>B</em> be a block algebra of a group algebra <em>FG</em> of a finite group <em>G</em> over a field <em>F</em> of characteristic <span><math><mi>p</mi><mo>></mo><mn>0</mn></math></span>. This paper studies ring theoretic properties of the representation ring <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>Δ</mi></mrow></msup><mo>(</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> of perfect <em>p</em>-permutation <span><math><mo>(</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span>-bimodules and properties of the <em>k</em>-algebra <span><math><mi>k</mi><msub><mrow><mo>⊗</mo></mrow><mrow><mi>Z</mi></mrow></msub><msup><mrow><mi>T</mi></mrow><mrow><mi>Δ</mi></mrow></msup><mo>(</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span>, for a field <em>k</em>. We show that if the Cartan matrix of <em>B</em> has 1 as an elementary divisor then <span><math><mo>[</mo><mi>B</mi><mo>]</mo></math></span> is not primitive in <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>Δ</mi></mrow></msup><mo>(</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span>. If <em>B</em> has cyclic defect groups we determine a primitive decomposition of <span><math><mo>[</mo><mi>B</mi><mo>]</mo></math></span> in <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>Δ</mi></mrow></msup><mo>(</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span>. Moreover, if <em>k</em> is a field of characteristic different from <em>p</em> and <em>B</em> has cyclic defect groups of order <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> we describe <span><math><mi>k</mi><msub><mrow><mo>⊗</mo></mrow><mrow><mi>Z</mi></mrow></msub><msup><mrow><mi>T</mi></mrow><mrow><mi>Δ</mi></mrow></msup><mo>(</mo><mi>B</mi><mo>,</mo><mi>B</mi><mo>)</mo></math></span> explicitly as a direct product of a matrix algebra and <em>n</em> group algebras.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"675 ","pages":"Pages 1-22"},"PeriodicalIF":0.8,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143686072","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}
Journal of AlgebraPub Date : 2025-03-14DOI: 10.1016/j.jalgebra.2025.02.041
Fabio Ferrari Ruffino
{"title":"Graded identities of the Virasoro algebra","authors":"Fabio Ferrari Ruffino","doi":"10.1016/j.jalgebra.2025.02.041","DOIUrl":"10.1016/j.jalgebra.2025.02.041","url":null,"abstract":"<div><div>We study the graded polynomial identities satisfied by the Virasoro algebra over an infinite field, completing the analysis realised by Fidelis et al. (2024) <span><span>[3]</span></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"674 ","pages":"Pages 155-170"},"PeriodicalIF":0.8,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685265","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}
Journal of AlgebraPub Date : 2025-03-14DOI: 10.1016/j.jalgebra.2025.03.015
Iritan Ferreira dos Santos , Alexey M. Kuz'min , Artem Lopatin
{"title":"Novikov algebras in low dimension: Identities, images and codimensions","authors":"Iritan Ferreira dos Santos , Alexey M. Kuz'min , Artem Lopatin","doi":"10.1016/j.jalgebra.2025.03.015","DOIUrl":"10.1016/j.jalgebra.2025.03.015","url":null,"abstract":"<div><div>Polynomial identities of two-dimensional Novikov algebras are studied over the complex field <span><math><mi>C</mi></math></span>. We determine minimal generating sets for the T-ideals of the polynomial identities and linear bases for the corresponding relatively free algebras. As a consequence, we establish that polynomial identities separate two-dimensional Novikov algebras, which are not associative. Namely, any two-dimensional Novikov algebras, which are not associative, are isomorphic if and only if they satisfy the same polynomial identities. Moreover, we obtain the codimension sequences of all these algebras. In particular, every two-dimensional Novikov algebra has at most linear growth of its codimension sequence. We explicitly describe multilinear images of every two-dimensional Novikov algebra. In particular, we show that these images are vector spaces.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"674 ","pages":"Pages 1-28"},"PeriodicalIF":0.8,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644007","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}
Journal of AlgebraPub Date : 2025-03-13DOI: 10.1016/j.jalgebra.2025.03.005
Yirui Xiong
{"title":"Invariant stability conditions of local P1×P1 (after Del Monte-Longhi)","authors":"Yirui Xiong","doi":"10.1016/j.jalgebra.2025.03.005","DOIUrl":"10.1016/j.jalgebra.2025.03.005","url":null,"abstract":"<div><div>Let <em>X</em> be the total space of the canonical bundle of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>, we study an invariant subspace of stability conditions on <em>X</em> under an autoequivalence of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>b</mi></mrow></msup><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. We describe the complete set of stable objects with respect to the invariant stability conditions and characterize the space of invariant stability conditions.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"671 ","pages":"Pages 189-234"},"PeriodicalIF":0.8,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643992","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}
Journal of AlgebraPub Date : 2025-03-13DOI: 10.1016/j.jalgebra.2025.03.006
Esther Banaian , Yadira Valdivieso
{"title":"Snake graphs and Caldero-Chapoton functions from triangulated orbifolds","authors":"Esther Banaian , Yadira Valdivieso","doi":"10.1016/j.jalgebra.2025.03.006","DOIUrl":"10.1016/j.jalgebra.2025.03.006","url":null,"abstract":"<div><div>Generalized cluster algebras from orbifolds were defined by Chekhov and Shapiro to give a combinatorial description of their Teichmüller spaces. One can also assign a gentle algebra to a triangulated orbifold, as in the work of Labardini-Fragoso and Mou. In this work, we show that the Caldero-Chapoton map and the snake graph expansion map agree for arcs in triangulated orbifolds and arc modules, and similarly for closed curves and certain band modules. As a consequence, we have a bijection between some indecomposable modules over a gentle algebra and cluster variables in a generalized cluster algebra where both algebras arise from the same triangulated orbifold.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"674 ","pages":"Pages 77-116"},"PeriodicalIF":0.8,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685203","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}
Journal of AlgebraPub Date : 2025-03-12DOI: 10.1016/j.jalgebra.2025.03.007
Matej Brešar
{"title":"Splittable Jordan homomorphisms and commutator ideals","authors":"Matej Brešar","doi":"10.1016/j.jalgebra.2025.03.007","DOIUrl":"10.1016/j.jalgebra.2025.03.007","url":null,"abstract":"<div><div>We define a Jordan homomorphism <em>φ</em> from a ring <em>R</em> to a ring <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> to be splittable if the ideal (of the subring generated by the image of <em>φ</em>) generated by all <span><math><mi>φ</mi><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo><mo>−</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>φ</mi><mo>(</mo><mi>y</mi><mo>)</mo></math></span>, <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>R</mi></math></span>, has trivial intersection with the ideal generated by all <span><math><mi>φ</mi><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo><mo>−</mo><mi>φ</mi><mo>(</mo><mi>y</mi><mo>)</mo><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, <span><math><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>R</mi></math></span>. Our main result states that a splittable Jordan homomorphism is the sum of a homomorphism and an antihomomorphism on the commutator ideal. As applications, we obtain results that give new insight into the question of the structure of Jordan homomorphisms on several classes of rings.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"671 ","pages":"Pages 159-180"},"PeriodicalIF":0.8,"publicationDate":"2025-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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