Pub Date : 2024-10-30DOI: 10.1016/j.jalgebra.2024.10.029
Jianzhi Han , Yukun Xiao , Shun Xu
<div><div>For any vertex operator algebra <em>V</em>, finite automorphism <em>g</em> of <em>V</em> of order <em>T</em> and <span><math><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>/</mo><mi>T</mi><mo>)</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>, we construct a family of associative algebras <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span>-bimodules <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> from the point of view of representation theory. We prove that the algebra <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> is identical to the algebra <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> constructed by Dong, Li and Mason, and that the bimodule <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> is identical to <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> which was constructed by Dong and Jiang. We also prove that the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span>-bimodule <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> is isomorphic to <span><math><mi>U</mi><msub><mrow><mo>(</mo><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow></msub><mo>/</mo><mi>U</mi><msubsup><mrow><mo>(</mo><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow><mrow><mo>−</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>/</mo><mi>T</mi></mrow></msubsup></math></span>, where <span><math><mi>U</mi><msub><mrow><mo>(</mo><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msub></math></span> is the subspace of degree <em>k</em> of the <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mi>T</mi><mo>)</mo><mi>Z</mi></math></span>-graded universal enveloping algebra <span><math><mi>U</mi><mo>(</mo><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo><mo>)</mo></math></span> of <em>V</em> with re
对于任意顶点算子代数 V、V 的阶数为 T 的有限自变量 g 以及 m,n∈(1/T)Z+，我们从表示论的角度构建了关联代数 Ag,n(V)族和 Ag,n(V)-Ag,m(V)- 双模块 Ag,n,m(V)。我们证明了代数 Ag,n(V) 与董(Dong)、李(Li)和梅森(Mason)构造的代数 Ag,n(V) 完全相同，而双模 Ag,n,m(V) 与董(Dong)和蒋(Jiang)构造的双模 Ag,n,m(V) 完全相同。我们还证明了 Ag,n(V)-Ag,m(V)-双模块 Ag,n,m(V) 与 U(V[g])n-m/U(V[g])n-m-1/T 同构，其中 U(V[g])k 是 V 的 (1/T)Z 阶通用包络代数 U(V[g]) 关于 g 的 k 度子空间，U(V[g])kl 是 U(V[g])k 的某个子空间。我们将证明所有这些双模子 Ag,n,m(V) 都可以用更简单的方法定义。
{"title":"Twisted bimodules and universal enveloping algebras associated to VOAs","authors":"Jianzhi Han , Yukun Xiao , Shun Xu","doi":"10.1016/j.jalgebra.2024.10.029","DOIUrl":"10.1016/j.jalgebra.2024.10.029","url":null,"abstract":"<div><div>For any vertex operator algebra <em>V</em>, finite automorphism <em>g</em> of <em>V</em> of order <em>T</em> and <span><math><mi>m</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>/</mo><mi>T</mi><mo>)</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>+</mo></mrow></msub></math></span>, we construct a family of associative algebras <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span>-bimodules <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> from the point of view of representation theory. We prove that the algebra <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> is identical to the algebra <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> constructed by Dong, Li and Mason, and that the bimodule <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> is identical to <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> which was constructed by Dong and Jiang. We also prove that the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span>-bimodule <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>V</mi><mo>)</mo></math></span> is isomorphic to <span><math><mi>U</mi><msub><mrow><mo>(</mo><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow></msub><mo>/</mo><mi>U</mi><msubsup><mrow><mo>(</mo><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow><mrow><mo>−</mo><mi>m</mi><mo>−</mo><mn>1</mn><mo>/</mo><mi>T</mi></mrow></msubsup></math></span>, where <span><math><mi>U</mi><msub><mrow><mo>(</mo><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msub></math></span> is the subspace of degree <em>k</em> of the <span><math><mo>(</mo><mn>1</mn><mo>/</mo><mi>T</mi><mo>)</mo><mi>Z</mi></math></span>-graded universal enveloping algebra <span><math><mi>U</mi><mo>(</mo><mi>V</mi><mo>[</mo><mi>g</mi><mo>]</mo><mo>)</mo></math></span> of <em>V</em> with re","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592837","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}
Pub Date : 2024-10-24DOI: 10.1016/j.jalgebra.2024.10.022
Chao-Ping Dong , Yongzhi Luan , Haojun Xu
The idea of using Dirac cohomology to study branching laws was initiated by Huang, et al. (2013) [11]. One of their results says that the Dirac cohomology of π completely determines , where π is any irreducible unitarizable highest weight module. This paper aims to develop this idea for the exceptional Lie groups and : we recover the K-spectrum of the Wallach modules from their Dirac cohomology.
{"title":"Dirac cohomology, branching laws and Wallach modules","authors":"Chao-Ping Dong , Yongzhi Luan , Haojun Xu","doi":"10.1016/j.jalgebra.2024.10.022","DOIUrl":"10.1016/j.jalgebra.2024.10.022","url":null,"abstract":"<div><div>The idea of using Dirac cohomology to study branching laws was initiated by Huang, et al. (2013) <span><span>[11]</span></span>. One of their results says that the Dirac cohomology of <em>π</em> completely determines <span><math><mi>π</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>K</mi></mrow></msub></math></span>, where <em>π</em> is any irreducible unitarizable highest weight <span><math><mo>(</mo><mi>g</mi><mo>,</mo><mi>K</mi><mo>)</mo></math></span> module. This paper aims to develop this idea for the exceptional Lie groups <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>6</mn><mo>(</mo><mo>−</mo><mn>14</mn><mo>)</mo></mrow></msub></math></span> and <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>7</mn><mo>(</mo><mo>−</mo><mn>25</mn><mo>)</mo></mrow></msub></math></span>: we recover the <em>K</em>-spectrum of the Wallach modules from their Dirac cohomology.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554703","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}
Pub Date : 2024-10-24DOI: 10.1016/j.jalgebra.2024.10.021
Monica Nevins , Susanne Pumplün
We determine and explicitly parametrize the isomorphism classes of nonassociative quaternion algebras over a field of characteristic different from two, as well as the isomorphism classes of nonassociative cyclic algebras of odd prime degree m when the base field contains a primitive mth root of unity. In the course of doing so, we prove that any two such algebras can be isomorphic only if the cyclic field extension and the chosen generator of the Galois group are the same. As an application, we give a parametrization of nonassociative cyclic algebras of prime degree over a local nonarchimedean field F, which is entirely explicit under mild hypotheses on the residual characteristic. In particular, this gives a rich understanding of the important class of nonassociative quaternion algebras up to isomorphism over nonarchimedean local fields.
我们确定并明确参数化了特征值不同于 2 的域上的非共轭四元数代数的同构类，以及奇素数 m 的非共轭循环代数的同构类，当基域包含一个原始的第 m 个统一根时。在此过程中，我们证明了只有当循环域扩展和伽罗瓦群的所选生成子相同时，任何两个这样的代数方程才能同构。作为应用，我们给出了一个本地非archimedean 场 F 上素数级的非共轭循环代数的参数化，它在关于残差特征的温和假设下是完全显式的。特别是，这让我们对非共轭四元数代数的重要类别有了丰富的理解，直到非archimedean 局部域上的同构。
{"title":"A parametrization of nonassociative cyclic algebras of prime degree","authors":"Monica Nevins , Susanne Pumplün","doi":"10.1016/j.jalgebra.2024.10.021","DOIUrl":"10.1016/j.jalgebra.2024.10.021","url":null,"abstract":"<div><div>We determine and explicitly parametrize the isomorphism classes of nonassociative quaternion algebras over a field of characteristic different from two, as well as the isomorphism classes of nonassociative cyclic algebras of odd prime degree <em>m</em> when the base field contains a primitive <em>m</em>th root of unity. In the course of doing so, we prove that any two such algebras can be isomorphic only if the cyclic field extension and the chosen generator of the Galois group are the same. As an application, we give a parametrization of nonassociative cyclic algebras of prime degree over a local nonarchimedean field <em>F</em>, which is entirely explicit under mild hypotheses on the residual characteristic. In particular, this gives a rich understanding of the important class of nonassociative quaternion algebras up to isomorphism over nonarchimedean local fields.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554700","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}
Pub Date : 2024-10-22DOI: 10.1016/j.jalgebra.2024.09.033
Wei Hu , Xiaojuan Yin
Rigidity dimension is a new homological dimension which is intended to measure the quality of the best resolution of an algebra. In this paper, we determine the rigidity dimensions of self-injective Nakayama algebras with n simple modules and the Loewy length .
刚性维度是一种新的同调维度，用于衡量代数最佳解析的质量。在本文中，我们确定了具有 n 个简单模块和洛维长度 m⩾n 的自注入中山代数 An,m 的刚性维数。
{"title":"Rigidity dimensions of self-injective Nakayama algebras","authors":"Wei Hu , Xiaojuan Yin","doi":"10.1016/j.jalgebra.2024.09.033","DOIUrl":"10.1016/j.jalgebra.2024.09.033","url":null,"abstract":"<div><div>Rigidity dimension is a new homological dimension which is intended to measure the quality of the best resolution of an algebra. In this paper, we determine the rigidity dimensions of self-injective Nakayama algebras <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>m</mi></mrow></msub></math></span> with <em>n</em> simple modules and the Loewy length <span><math><mi>m</mi><mo>⩾</mo><mi>n</mi></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554719","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}
Pub Date : 2024-10-22DOI: 10.1016/j.jalgebra.2024.09.034
Ilya Karzhemanov
We give a geometric condition for two finite subgroups of the plane Cremona group to be conjugate. The argument is based on the properties of a “Burnside-type” ring encoding rational G-surfaces.
{"title":"On the conjugacy problem for finite groups in the plane Cremona group","authors":"Ilya Karzhemanov","doi":"10.1016/j.jalgebra.2024.09.034","DOIUrl":"10.1016/j.jalgebra.2024.09.034","url":null,"abstract":"<div><div>We give a geometric condition for two finite subgroups <span><math><mi>G</mi><mo>≃</mo><msup><mrow><mi>G</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> of the plane Cremona group <span><math><msub><mrow><mi>Cr</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> to be conjugate. The argument is based on the properties of a “Burnside-type” ring <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>G</mi><mo>)</mo></math></span> encoding rational <em>G</em>-surfaces.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554698","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}
Pub Date : 2024-10-22DOI: 10.1016/j.jalgebra.2024.10.019
Dylan Johnston
In this paper we investigate contramodules for algebraic groups. Namely, we give contramodule analogs to two century results about comodules. Firstly, we show that induction of contramodules over coordinate rings of algebraic groups is exact if and only if the associated quotient variety is affine. Secondly, we give an inverse limit theorem for constructing projective covers of simple G-modules using G-structures of projective covers of simple modules of the first Frobenius kernel, . We conclude by showing that the inverse limit theorem is a special case of a more general phenomenon between injective covers in -Comod and projective covers in -Contra.
在本文中，我们研究了代数群的反模子。也就是说，我们给出了等映模与 20 世纪关于逗点的两个结果的对应关系。首先，我们证明了代数群坐标环上的反模量归纳是精确的，当且仅当相关商综是仿射的。其次，我们给出了一个逆极限定理，利用第一弗罗本尼乌斯核的简单模块 G1 的投影盖的 G 结构，构造简单 G 模块的投影盖。最后，我们证明了逆极限定理是 k[G]-Comod 中的注入盖和 k[G]-Contra 中的投影盖之间更普遍现象的特例。
{"title":"Contramodules for algebraic groups: Induction and projective covers","authors":"Dylan Johnston","doi":"10.1016/j.jalgebra.2024.10.019","DOIUrl":"10.1016/j.jalgebra.2024.10.019","url":null,"abstract":"<div><div>In this paper we investigate contramodules for algebraic groups. Namely, we give contramodule analogs to two <span><math><msup><mrow><mn>20</mn></mrow><mrow><mtext>th</mtext></mrow></msup></math></span> century results about comodules. Firstly, we show that induction of contramodules over coordinate rings of algebraic groups is exact if and only if the associated quotient variety is affine. Secondly, we give an inverse limit theorem for constructing projective covers of simple <em>G</em>-modules using <em>G</em>-structures of projective covers of simple modules of the first Frobenius kernel, <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. We conclude by showing that the inverse limit theorem is a special case of a more general phenomenon between injective covers in <span><math><mi>k</mi><mo>[</mo><mi>G</mi><mo>]</mo></math></span>-Comod and projective covers in <span><math><mi>k</mi><mo>[</mo><mi>G</mi><mo>]</mo></math></span>-Contra.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554699","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}
Pub Date : 2024-10-22DOI: 10.1016/j.jalgebra.2024.10.016
Yael Davidov
A finite group G is said to be admissible over a field F if there exists a division algebra D central over F with a maximal subfield L such that is Galois with group G. In this paper we give a complete characterization of admissible groups over function fields of curves over equicharacteristic complete discretely valued fields with algebraically closed residue fields, such as the field .
{"title":"Admissibility over semi-global fields in the bad characteristic case","authors":"Yael Davidov","doi":"10.1016/j.jalgebra.2024.10.016","DOIUrl":"10.1016/j.jalgebra.2024.10.016","url":null,"abstract":"<div><div>A finite group <em>G</em> is said to be admissible over a field <em>F</em> if there exists a division algebra <em>D</em> central over <em>F</em> with a maximal subfield <em>L</em> such that <span><math><mi>L</mi><mo>/</mo><mi>F</mi></math></span> is Galois with group <em>G</em>. In this paper we give a complete characterization of admissible groups over function fields of curves over equicharacteristic complete discretely valued fields with algebraically closed residue fields, such as the field <span><math><mover><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>P</mi></mrow></msub></mrow><mo>‾</mo></mover><mo>(</mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo><mo>(</mo><mi>x</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554696","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}
This paper is dedicated to the study of the converse implication in Hilbert 17th problem for a general commutative ring. We introduce the notions of central and precentral prime cones which generalize the notion of central real points of irreducible real algebraic varieties. We study these families of prime cones which both live in the real spectrum of the ring and allow to state new Positivstellensätze and to obtain an equivalence in Hilbert 17th problem.
{"title":"Hilbert 17th property and central cones","authors":"Goulwen Fichou , Jean-Philippe Monnier , Ronan Quarez","doi":"10.1016/j.jalgebra.2024.10.020","DOIUrl":"10.1016/j.jalgebra.2024.10.020","url":null,"abstract":"<div><div>This paper is dedicated to the study of the converse implication in Hilbert 17th problem for a general commutative ring. We introduce the notions of central and precentral prime cones which generalize the notion of central real points of irreducible real algebraic varieties. We study these families of prime cones which both live in the real spectrum of the ring and allow to state new Positivstellensätze and to obtain an equivalence in Hilbert 17th problem.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554697","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}
Pub Date : 2024-10-22DOI: 10.1016/j.jalgebra.2024.10.013
Ron M. Adin , Pál Hegedüs , Yuval Roichman
We prove that, for any integer k, the k-th root enumerator in the classical Weyl group of type D is a proper character. The proof uses higher Lie characters of type B.
我们证明，对于任意整数 k，D 型经典韦尔群中的 k-th 根列举符是一个合适的字符。证明使用的是 B 型的高等李字符。
{"title":"Higher Lie characters and root enumeration in classical Weyl groups","authors":"Ron M. Adin , Pál Hegedüs , Yuval Roichman","doi":"10.1016/j.jalgebra.2024.10.013","DOIUrl":"10.1016/j.jalgebra.2024.10.013","url":null,"abstract":"<div><div>We prove that, for any integer <em>k</em>, the <em>k</em>-th root enumerator in the classical Weyl group of type <em>D</em> is a proper character. The proof uses higher Lie characters of type <em>B</em>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592957","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}
Pub Date : 2024-10-22DOI: 10.1016/j.jalgebra.2024.09.038
Robert Boltje, Philipp Perepelitsky
We extend the notion of a p-permutation equivalence between two p-blocks A and B of finite groups G and H, from the definition in [5] to a virtual p-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of A and B are preserved, including defect groups, fusion systems, and Külshammer-Puig classes. Moreover it is shown that p-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of p-permutation equivalences between A and B is finite (possibly empty). The paper uses new methods: a consequent use of module structures on subgroups of arising from Brauer constructions which in general are not direct product subgroups, the necessary adaptation of the notion of tensor products between bimodules, and a general formula (stated in these new terms) for the Brauer construction of a tensor product of p-permutation bimodules.
我们将有限群 G 和 H 的两个 p 块 A 和 B 之间的 p 置换等价概念从 [5] 中的定义扩展到虚拟 p 置换双模块，其成分具有扭曲对角顶点。研究表明，A 和 B 的各种不变式都得到了保留，包括缺陷群、融合系统和 Külshammer-Puig 类。此外，研究还证明了 p-permutation等价具有额外的惊人性质。它们只有一个具有最大顶点的成分，而且 A 和 B 之间的 p-permutation 等价集是有限的（可能是空）。本文使用了新方法：在布劳尔构造产生的 G×H 子群上使用模块结构，而这些子群一般不是直接乘积子群；对双模之间的张量积概念进行必要的调整；以及（用这些新术语表述的）p-permutation 双模张量积的布劳尔构造通式。
{"title":"p-Permutation equivalences between blocks of group algebras","authors":"Robert Boltje, Philipp Perepelitsky","doi":"10.1016/j.jalgebra.2024.09.038","DOIUrl":"10.1016/j.jalgebra.2024.09.038","url":null,"abstract":"<div><div>We extend the notion of a <em>p-permutation equivalence</em> between two <em>p</em>-blocks <em>A</em> and <em>B</em> of finite groups <em>G</em> and <em>H</em>, from the definition in <span><span>[5]</span></span> to a virtual <em>p</em>-permutation bimodule whose components have twisted diagonal vertices. It is shown that various invariants of <em>A</em> and <em>B</em> are preserved, including defect groups, fusion systems, and Külshammer-Puig classes. Moreover it is shown that <em>p</em>-permutation equivalences have additional surprising properties. They have only one constituent with maximal vertex and the set of <em>p</em>-permutation equivalences between <em>A</em> and <em>B</em> is finite (possibly empty). The paper uses new methods: a consequent use of module structures on subgroups of <span><math><mi>G</mi><mo>×</mo><mi>H</mi></math></span> arising from Brauer constructions which in general are not direct product subgroups, the necessary adaptation of the notion of tensor products between bimodules, and a general formula (stated in these new terms) for the Brauer construction of a tensor product of <em>p</em>-permutation bimodules.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554720","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：