Journal of Algebra最新文献

{"title":"G-tables and the Poisson structure of the even cohomology of cotangent bundle of the Heisenberg Lie group","authors":"Leandro Cagliero , Gonzalo Gutierrez","doi":"10.1016/j.jalgebra.2025.02.040","DOIUrl":"10.1016/j.jalgebra.2025.02.040","url":null,"abstract":"<div><div>In the first part of the paper, we define the concept of a <em>G</em>-table of a <em>G</em>-(co)algebra and we compute the <em>G</em>-table of some <em>G</em>-(co)algebras (here, a <em>G</em>-algebra is an algebra on which <em>G</em> acts, semisimply, by algebra automorphisms). The <em>G</em>-table of a <em>G</em>-algebra <span><math><mi>A</mi></math></span> is a set of scalars that provides precise and concise information about both the algebra structure and the <em>G</em>-module structure of <span><math><mi>A</mi></math></span>. In particular, the ordinary multiplication table of <span><math><mi>A</mi></math></span> can be derived from the <em>G</em>-table of <span><math><mi>A</mi></math></span>. Using the <em>G</em>-table of a <em>G</em>-algebra <span><math><mi>A</mi></math></span>, we define an associated plain algebra <span><math><mi>P</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> and present some basic functoriality results related to <em>P</em>.</div><div>Obtaining the <em>G</em>-table of a given <em>G</em>-algebra <span><math><mi>A</mi></math></span> requires significant work, but the result is a very powerful tool, as shown in the second part of the paper. Here, we compute the <span><math><mrow><mi>SL</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>K</mi><mo>)</mo></math></span>-tables of the Poisson algebra structure of the even-degree part of the cohomology associated to the cotangent bundle of the 3-dimensional Heisenberg Lie group with Lie algebra <span><math><mi>h</mi></math></span>, that is <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>E</mi></mrow><mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo>(</mo><mi>h</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mi>H</mi></mrow><mrow><mi>E</mi></mrow><mrow><mo>•</mo></mrow></msubsup><mo>(</mo><mi>h</mi><mo>,</mo><msup><mrow><mo>⋀</mo></mrow><mrow><mo>•</mo></mrow></msup><mi>h</mi><mo>)</mo></math></span>. This Poisson <span><math><mrow><mi>SL</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>K</mi><mo>)</mo></math></span>-algebra has dimension 18. From these <span><math><mrow><mi>SL</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>K</mi><mo>)</mo></math></span>-tables we deduce that the underlying Lie algebra of <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>E</mi></mrow><mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo>(</mo><mi>h</mi><mo>)</mo></math></span> is isomorphic to <span><math><mrow><mi>gl</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo><mo>⋉</mo><mrow><mi>gl</mi></mrow><msub><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo></mrow><mrow><mi>a</mi><mi>b</mi></mrow></msub></math></span> with the first factor acting on the second (abelian) factor by the adjoint representation. It is notable that the Lie algebra structure on <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>E</mi></mrow><mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo>(</mo><mi>h</mi><mo>)</mo></math></span> contains a semisimple Lie subalgebra (in this case <","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"674 ","pages":"Pages 205-234"},"PeriodicalIF":0.8,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685267","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":"Radicals in polynomial rings skewed by an endomorphism","authors":"Nam Kyun Kim ,&nbsp;Pace P. Nielsen ,&nbsp;Michał Ziembowski","doi":"10.1016/j.jalgebra.2025.02.044","DOIUrl":"10.1016/j.jalgebra.2025.02.044","url":null,"abstract":"<div><div>Given a ring <em>R</em>, radicals of the polynomial ring <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, and even of the skew polynomial ring <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>;</mo><mi>σ</mi><mo>]</mo></math></span> skewed by an endomorphism <em>σ</em> on <em>R</em>, have been studied and described for many different radicals. Often, in those descriptions, the ring <em>R</em> was assumed to be unital, or the endomorphism <em>σ</em> was assumed to be an automorphism. Here we systematically study what happens when such assumptions are dropped, and generalize to even more radicals. Our results reveal three key properties that, when present, allow a simple description of a radical of <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>;</mo><mi>σ</mi><mo>]</mo></math></span>. Moreover, multiple examples are provided showing that when <em>σ</em> is injective, but not surjective, wild growth patterns may occur.</div><div>We also answer an open question in the literature, by showing that the Levitzki radical of the skew Laurent polynomial ring <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>;</mo><mi>σ</mi><mo>]</mo></math></span> (when <em>σ</em> is an automorphism) is not naively describable in terms of the Levitzki radicals of <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>;</mo><mi>σ</mi><mo>]</mo></math></span> and <span><math><mi>R</mi><mo>[</mo><mi>x</mi><mo>;</mo><msup><mrow><mi>σ</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"674 ","pages":"Pages 117-142"},"PeriodicalIF":0.8,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685263","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":"Affine flag varieties of type D","authors":"Quanyong Chen,&nbsp;Zhaobing Fan,&nbsp;Qi Wang","doi":"10.1016/j.jalgebra.2025.02.043","DOIUrl":"10.1016/j.jalgebra.2025.02.043","url":null,"abstract":"<div><div>The Hecke algebras and quantum group of affine type <em>A</em> admit geometric realizations in terms of complete flags and partial flags over a local field, respectively. Subsequently, it is demonstrated that the quantum group associated to partial flag varieties of affine type <em>C</em> is a coideal subalgebra of quantum group of affine type <em>A</em>. In this paper, we establish a lattice presentation of the complete (partial) flag varieties of affine type <em>D</em>. Additionally, we determine the structures of convolution algebra associated to complete flag varieties of affine type <em>D</em>, which is isomorphic to the (extended) affine Hecke algebra. We also show that there exists a monomial basis and a canonical basis of the convolution algebra, and establish the positivity properties of the canonical basis with respect to multiplication.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"674 ","pages":"Pages 257-275"},"PeriodicalIF":0.8,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143685271","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":"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}

{"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}

{"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}

{"title":"The ring of perfect p-permutation bimodules for blocks with cyclic defect groups","authors":"Robert Boltje,&nbsp;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>&gt;</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}

{"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}

{"title":"Novikov algebras in low dimension: Identities, images and codimensions","authors":"Iritan Ferreira dos Santos ,&nbsp;Alexey M. Kuz'min ,&nbsp;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}

{"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}