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Quantum super-symmetries (I): Quantum Grassmann super-algebras and a quantum Deligne-Morgan-Manin de Rham complex 量子超对称(I)：量子Grassmann超代数和量子delign - morgan - manin de Rham复合体
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-22 DOI: 10.1016/j.jalgebra.2025.09.009
Ge Feng , Naihong Hu , Marc Rosso
{"title":"Quantum super-symmetries (I): Quantum Grassmann super-algebras and a quantum Deligne-Morgan-Manin de Rham complex","authors":"Ge Feng ,&nbsp;Naihong Hu ,&nbsp;Marc Rosso","doi":"10.1016/j.jalgebra.2025.09.009","DOIUrl":"10.1016/j.jalgebra.2025.09.009","url":null,"abstract":"<div><div>We introduce the quantum Manin <span><math><mo>(</mo><mi>m</mi><mo>|</mo><mi>n</mi><mo>)</mo></math></span>-superspace <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>m</mi><mo>|</mo><mi>n</mi></mrow></msubsup></math></span> equipped with a super ⋆-product, and dually, the quantum Grassmann super-algebra <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>m</mi><mo>|</mo><mi>n</mi><mo>)</mo></math></span> equipped with the quantum divided power super-structure. The quantum (restricted) Grassmann superalgebra <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> and its Manin dual <span><math><msubsup><mrow><mi>Ω</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>!</mo></mrow></msubsup></math></span> are made into <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mrow><mi>gl</mi></mrow><mo>(</mo><mi>m</mi><mo>|</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>-module superalgebras, either for <em>q</em> generic, or for <em>q</em> root of unity, via quantum (super) differential operators. We give an explicit realization model for certain simple <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mrow><mi>gl</mi></mrow><mo>(</mo><mi>m</mi><mo>|</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>-modules and their dimension-formulae, and construct a quantum super de Rham cochain complex of infinite length, which is a quantized version of a classical analogue due to Manin, Deligne-Morgan in the framework of super-symmetry on supermanifolds in gauge field theory.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 492-531"},"PeriodicalIF":0.8,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145156574","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}
引用次数: 0
From classes in the Weyl group to strata 从Weyl组的阶级到阶层
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-18 DOI: 10.1016/j.jalgebra.2025.08.036
G. Lusztig
{"title":"From classes in the Weyl group to strata","authors":"G. Lusztig","doi":"10.1016/j.jalgebra.2025.08.036","DOIUrl":"10.1016/j.jalgebra.2025.08.036","url":null,"abstract":"<div><div>In a 2015 paper we have defined a map from the set of conjugacy classes in the Weyl group W to the set of irreducible representations of W (its image parametrizes the set of strata of a reductive group with Weyl group W). In this paper we provide evidence that this map makes sense even when W is replaced by a noncrystallographic finite Coxeter group.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 655-672"},"PeriodicalIF":0.8,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145156573","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}
引用次数: 0
Lusztig varieties for regular elements 正则元素的Lusztig变种
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-18 DOI: 10.1016/j.jalgebra.2025.09.008
Xuhua He, Ruben La
{"title":"Lusztig varieties for regular elements","authors":"Xuhua He,&nbsp;Ruben La","doi":"10.1016/j.jalgebra.2025.09.008","DOIUrl":"10.1016/j.jalgebra.2025.09.008","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> be a connected reductive group over an algebraically closed field. Let <em>B</em> be a Borel subgroup of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <em>W</em> be the associated Weyl group. We show that for any <span><math><mi>w</mi><mo>∈</mo><mi>W</mi></math></span> that is not contained in any standard parabolic subgroup of <em>W</em>, the intersection of the Bruhat cell <em>BwB</em> with any regular conjugacy class of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is always irreducible. We then prove that the associated Lusztig varieties are irreducible. This extends the previous work of Kim <span><span>[7]</span></span> on the regular semisimple and regular unipotent elements. The irreducibility result of Lusztig varieties will be used in an upcoming work in the study of affine Lusztig varieties.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"686 ","pages":"Pages 845-853"},"PeriodicalIF":0.8,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145118078","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}
引用次数: 0
Towards the classification of exceptional scattered polynomials 异常分散多项式的分类研究
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-18 DOI: 10.1016/j.jalgebra.2025.08.037
Daniele Bartoli , Massimo Giulietti , Giovanni Zini
{"title":"Towards the classification of exceptional scattered polynomials","authors":"Daniele Bartoli ,&nbsp;Massimo Giulietti ,&nbsp;Giovanni Zini","doi":"10.1016/j.jalgebra.2025.08.037","DOIUrl":"10.1016/j.jalgebra.2025.08.037","url":null,"abstract":"<div><div>Scattered polynomials over finite fields attracted an increasing attention in the last years. One of the reasons is their deep connection with Maximum Rank Distance (MRD) codes. Known classification results for exceptional scattered polynomials, i.e. polynomials which are scattered over infinite field extensions, are limited to the cases where their index <em>ℓ</em> is small, or a prime number larger than the <em>q</em>-degree <em>k</em> of the polynomial, or an integer smaller than <em>k</em> in the case where <em>k</em> is a prime. In this paper we completely classify exceptional scattered polynomials when the maximum between <em>ℓ</em> and <em>k</em> is odd, and give partial results when it is even, extending a result of Ferraguti and Micheli in 2021.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 229-243"},"PeriodicalIF":0.8,"publicationDate":"2025-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109746","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}
引用次数: 0
A strong version of the Hilbert Nullstellensatz for slice regular polynomials in several quaternionic variables 在几个四元元变量中切片正则多项式的希尔伯特零定理的一个强版本
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-17 DOI: 10.1016/j.jalgebra.2025.08.039
Anna Gori , Giulia Sarfatti , Fabio Vlacci
{"title":"A strong version of the Hilbert Nullstellensatz for slice regular polynomials in several quaternionic variables","authors":"Anna Gori ,&nbsp;Giulia Sarfatti ,&nbsp;Fabio Vlacci","doi":"10.1016/j.jalgebra.2025.08.039","DOIUrl":"10.1016/j.jalgebra.2025.08.039","url":null,"abstract":"<div><div>In this paper we prove a strong version of the Hilbert Nullstellensatz in the ring <span><math><mi>H</mi><mo>[</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span> of slice regular polynomials in several quaternionic variables. Our proof deeply depends on a detailed analysis of the common zeros of slice regular polynomials which belong to an ideal in <span><math><mi>H</mi><mo>[</mo><msub><mrow><mi>q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>. This study motivates the introduction of a new notion of algebraic set in the quaternionic setting, which allows us to define a Zariski-type topology on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 269-291"},"PeriodicalIF":0.8,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119619","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}
引用次数: 0
Singular Cayley graphs and p-blocks of finite groups 有限群的奇异Cayley图与p块
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-17 DOI: 10.1016/j.jalgebra.2025.08.038
Mahdi Ebrahimi
{"title":"Singular Cayley graphs and p-blocks of finite groups","authors":"Mahdi Ebrahimi","doi":"10.1016/j.jalgebra.2025.08.038","DOIUrl":"10.1016/j.jalgebra.2025.08.038","url":null,"abstract":"<div><div>For a simple finite graph Γ, the multiplicity of the eigenvalue 0 of the adjacency matrix of Γ is called the nullity of Γ. The energy of Γ is defined as the sum of the absolute values of its eigenvalues. In this paper, we apply the block theory of finite groups to study the Cayley graph <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> defined on a finite group <em>G</em> in which its connecting set consists of <em>p</em>-singular elements of <em>G</em>. We use this Cayley graph to investigate several methods for constructing singular graphs. Then we assume that <em>G</em> is <em>p</em>-solvable and obtain some nice restrictions on the structure of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. We first achieve an explicit formula for the nullity of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In addition, we find a lower bound for the energy of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Finally, we prove that the diameter of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is at most <span><math><mo>|</mo><mi>G</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>p</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 477-491"},"PeriodicalIF":0.8,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145156575","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}
引用次数: 0
Symbolic summation in multivariate difference fields 多元差分域的符号求和
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-17 DOI: 10.1016/j.jalgebra.2025.08.043
Lixin Du , Yarong Wei
{"title":"Symbolic summation in multivariate difference fields","authors":"Lixin Du ,&nbsp;Yarong Wei","doi":"10.1016/j.jalgebra.2025.08.043","DOIUrl":"10.1016/j.jalgebra.2025.08.043","url":null,"abstract":"<div><div>The bivariate difference fields provide an algebraic framework for sequences satisfying a recurrence of order two. Based on this, we focus on sequences satisfying a recurrence of higher order with constant coefficients, and consider the multivariate difference fields, in which the summation problem of these sequences could be transformed into solving the first order difference equations. In a special class of multivariate difference fields, we present a criterion for deciding whether the difference equation has a rational solution and present an algorithm for computing one rational solution of such a difference equation, if it exists. Moreover we get the set of all rational solutions of such an equation.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 532-565"},"PeriodicalIF":0.8,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145156572","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}
引用次数: 0
Deformations of an affine Gorenstein toric pair 仿射Gorenstein环面对的变形
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-17 DOI: 10.1016/j.jalgebra.2025.09.007
Matej Filip
{"title":"Deformations of an affine Gorenstein toric pair","authors":"Matej Filip","doi":"10.1016/j.jalgebra.2025.09.007","DOIUrl":"10.1016/j.jalgebra.2025.09.007","url":null,"abstract":"<div><div>We consider deformations of a pair <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mo>∂</mo><mi>X</mi><mo>)</mo></math></span>, where <em>X</em> is an affine toric Gorenstein variety and ∂<em>X</em> is its boundary. We compute the tangent and obstruction space for the corresponding deformation functor and for an admissible lattice degree <em>m</em> we construct the miniversal deformation of <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mo>∂</mo><mi>X</mi><mo>)</mo></math></span> in degrees <span><math><mo>−</mo><mi>k</mi><mi>m</mi></math></span>, for all <span><math><mi>k</mi><mo>∈</mo><mi>N</mi></math></span>. This in particular generalizes Altmann's construction of the miniversal deformation of an isolated Gorenstein toric singularity to an arbitrary non-isolated Gorenstein toric singularity. Moreover, we show that the irreducible components of the reduced miniversal deformation are in one to one correspondence with maximal Minkowski decompositions of the polytope <span><math><mi>P</mi><mo>∩</mo><mo>(</mo><mi>m</mi><mo>=</mo><mn>1</mn><mo>)</mo></math></span>, where <em>P</em> is the lattice polytope defining <em>X</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 419-445"},"PeriodicalIF":0.8,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119719","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}
引用次数: 0
On subruled function fields 在子函数域上
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-17 DOI: 10.1016/j.jalgebra.2025.08.033
Shashikant Mulay
{"title":"On subruled function fields","authors":"Shashikant Mulay","doi":"10.1016/j.jalgebra.2025.08.033","DOIUrl":"10.1016/j.jalgebra.2025.08.033","url":null,"abstract":"<div><div>Let <em>K</em> be a function field with ground field <em>k</em>. A well known theorem of Jack Ohm proves that if <em>k</em> is infinite and <em>K</em> is separably subruled over <em>k</em>, then <em>K</em> is separably uniruled over <em>k</em>. At the end of his proof of this important result, Ohm asks if his theorem remains valid in the case of a finite ground field <em>k</em>. In this article we present an alternative proof of Ohm's theorem that is valid for all ground field <em>k</em>, finite or infinite, thereby answering Ohm's question.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 345-351"},"PeriodicalIF":0.8,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119730","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}
引用次数: 0
On dominant ℓ-weights and maps between Weyl modules for quantum affine An 量子仿射An的显性加权和Weyl模间映射
IF 0.8 2区 数学
Journal of Algebra Pub Date : 2025-09-17 DOI: 10.1016/j.jalgebra.2025.08.042
Matheus Brito , Vyjayanthi Chari
{"title":"On dominant ℓ-weights and maps between Weyl modules for quantum affine An","authors":"Matheus Brito ,&nbsp;Vyjayanthi Chari","doi":"10.1016/j.jalgebra.2025.08.042","DOIUrl":"10.1016/j.jalgebra.2025.08.042","url":null,"abstract":"<div><div>We determine the set of dominant <em>ℓ</em>-weights in the Weyl (or standard) modules for quantum affine <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. We then prove that the space of homomorphisms between standard modules is at most one-dimensional and give a necessary and sufficient condition for equality to hold. We also describe the socle of the standard module and prove that the socle is simple for large <em>n</em>. Finally, we give applications of our results to mixed Weyl modules, calculating extensions in the category and identify new families of tensor subcategories of finite-dimensional representations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":"687 ","pages":"Pages 352-379"},"PeriodicalIF":0.8,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145119718","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}
引用次数: 0
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