Journal of Algebra最新文献

英文中文

Symplectic completion over smooth affine algebras 光滑仿射代数上的辛补全

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-11-11 DOI: 10.1016/j.jalgebra.2025.10.050

Gopal Sharma, Sampat Sharma

<div><div>In this article, we prove the following results:<ul><li>(1).<div>Let R be a smooth affine algebra of dimension 3 over an algebraically closed field K with <math><mn>3</mn><mo>!</mo><mo>∈</mo><mi>K</mi></math>, then we show that <math><msub><mrow><mtext>Um</mtext></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mtext>Sp</mtext></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math> and <math><msub><mrow><mtext>Um</mtext></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>[</mo><mi>X</mi><mo>]</mo><mo>)</mo><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mtext>Sp</mtext></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>[</mo><mi>X</mi><mo>]</mo><mo>)</mo></math>.</div></li><li>(2).<div>We also show that if R is a smooth affine algebra of dimension 4 over an algebraically closed field K with <math><mn>4</mn><mo>!</mo><mo>∈</mo><mi>K</mi></math>, and assume that <math><msub><mrow><mtext>W</mtext></mrow><mrow><mi>E</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math> is divisible, then <math><msub><mrow><mtext>Um</mtext></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mtext>SL</mtext></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math>. As a consequence it is shown that if R is a smooth affine algebra of dimension 4 over an algebraically closed field K with <math><mn>4</mn><mo>!</mo><mo>∈</mo><mi>K</mi></math>, and assume that <math><msub><mrow><mtext>W</mtext></mrow><mrow><mi>E</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math> is divisible, then <math><msub><mrow><mtext>Um</mtext></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mtext>Sp</mtext></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math>.</div></li><li>(3).<div>We show that if R is a local ring of dimension 3 with <math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mo>!</mo></mrow></mfrac><mo>∈</mo><mi>R</mi></math>. Then <math><msub><mrow><mtext>Um</mtext></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>[</mo><mi>X</mi><mo>]</mo><mo>)</mo><mo>=</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mtext>Sp</mtext></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>[</mo><mi>X</mi><mo>]</mo><mo>)</mo></math>.</div></li><li>(4).<div>We also show that if <math><mi>R</mi><mo>=</mo><msub><mrow><mo>⊕</mo></mrow><mrow><mi>i<

在本文中，我们证明了以下结果：(1)。设R是一个3维的光滑仿射代数，在一个3维的代数闭域K上！∈K，则我们证明Um4(R)=e1Sp4(R)和Um4(R[X])=e1Sp4（R[X]）。我们也证明了如果R是一个4维的光滑仿射代数在一个4维的代数闭域K上！∈K，假设WE(R)可整除，则Um3(R)=e1SL3(R)。结果表明，如果R是一个4维的光滑仿射代数，在一个代数闭域K上，具有4！∈K，假设WE(R)可整除，则Um4(R)=e1Sp4(R)。我们证明，如果R是一个3维的局部环，其中13!∈R。然后Um4 (R [X]) = e1Sp4 (R [X]),(4)。我们还证明了如果R=⊕i≥0Ri是3维局部环上的一个分级环，其中13!∈R。然后Um4 (R) = e1Sp4 (R)。

引用次数: 0

Brauer graphs of solvable groups with a vertex of degree one 顶点为1度的可解群的布劳尔图

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-11-10 DOI: 10.1016/j.jalgebra.2025.10.035

James P. Cossey

Let G be a finite group, let p be a fixed prime, and let B be a p-block of G. The (ordinary) Brauer graph

Γ (B)

is defined to be the graph whose vertices are labeled by

Irr (B)

, and χ and ψ are adjacent if χ and ψ have a Brauer character

φ \in {IBr}_{p} (B)

in common. Recent results suggest that, unlike for instance symmetric groups, the Brauer graphs of blocks of solvable groups should be ‘‘very connected” in some sense. In this paper, we continue that theme, and show that a certain ‘‘local condition” on the graph guarantees global information. In particular, we show that if

Γ (B)

has a vertex of degree one, then the defect group D of B is elementary abelian, the graph has diameter at most three, each Brauer character has a unique lift, and the vertices corresponding to the non-lifts in

Irr (B)

form a complete subgraph.

设G是一个有限群，p是一个固定素数，B是G的一个p块。定义（普通）Brauer图Γ(B)为顶点用Irr(B)标记的图，如果χ和ψ有一个共同的Brauer字符φ∈IBrp(B)，则χ和ψ相邻。最近的结果表明，与对称群不同，可解群块的Brauer图在某种意义上应该是“非常连接”的。在本文中，我们继续这一主题，并证明了图上的某个“局部条件”保证了全局信息。特别地，我们证明了如果Γ(B)有一个度为1的顶点，则B的缺陷群D是初等阿贝尔，图的直径最多为3，每个Brauer字符有一个唯一的提升，并且Irr(B)中的非提升对应的顶点形成一个完整的子图。

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引用次数: 0

Normality conditions in the Sylow p-subgroup of Sym(pn) and its associated Lie algebra Sym（pn）的Sylow p子群及其相关李代数的正态性条件

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-10-31 DOI: 10.1016/j.jalgebra.2025.10.033

Riccardo Aragona, Norberto Gavioli, Giuseppe Nozzi

In this work, we give a description of the structure of the normal subgroups of a Sylow p-subgroup

W_{n}

Sym (p^{n})

, showing that they contain a term from the lower central series with bounded index. To this end, we explicitly determine the terms of the upper and the lower central series of

W_{n}

. We provide a similar description of these series in the Lie algebra associated to

W_{n}

, giving a new proof of the equality of their terms in both the group and algebra contexts. Finally, we calculate the growth of the normalizer chain starting from an elementary abelian regular subgroup of

W_{n}

本文给出了Sym（pn）的Sylow p-子群Wn的正规子群的结构，证明了它们包含有界索引下中心序列的一项。为此，我们明确地确定了n的上、下中心级数的项。我们在与n相关的李代数中对这些级数进行了类似的描述，给出了它们的项在群和代数上下文中相等的一个新的证明。最后，我们从n的一个初等阿贝尔正则子群出发，计算了正则链的增长。

引用次数: 0

Morita theory for quantales 森田量子理论

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-10-31 DOI: 10.1016/j.jalgebra.2025.10.034

Bachuki Mesablishvili

Morita theory for quantales is developed. The main result of the paper is a characterization of those quantaloids (categories enriched in the symmetric monoidal closed category of sup-lattices) that are equivalent to module categories over quantales. Based on this characterization, necessary and sufficient conditions are derived for two quantales to be Morita-equivalent, i.e. have equivalent module categories. As an application, it is shown that the category of internal sup-lattices in a Grothendieck topos is equivalent to the module category over a suitable chosen ordinary quantale.

发展了量子的森田理论。本文的主要结果是描述了那些等价于量子上的模范畴的量子类（丰富于超晶格的对称单轴闭范畴的范畴）。在此基础上，导出了两个量子是森田等价的充要条件，即具有等价的模范畴。作为一个应用，证明了格罗滕迪克拓扑中内部超格的范畴等价于一个适当选择的普通量子上的模范畴。

引用次数: 0

Explicit construction of the maximal subgroups of the Monster 怪物的极大子群的显式构造

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-10-30 DOI: 10.1016/j.jalgebra.2025.10.016

Heiko Dietrich, Melissa Lee, Anthony Pisani, Tomasz Popiel

Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group

M

, the Monster. The aim of this work is to present an mmgroup database of maximal subgroups of

M

: for each conjugacy class

C

of maximal subgroups in

M

, we construct explicit group elements in mmgroup and prove that these elements generate a group in

C

. Our generators and the computations verifying correctness are available in accompanying code. The maximal subgroups of

M

have been classified in a number of papers spanning several decades; our work constitutes an independent verification of the constructions in these papers. We also correct the claim that

M

has a maximal subgroup

{PSL}_{2} (59)

, and hence identify a new maximal subgroup

59 : 29

Seysen的Python包mmgroup提供了在零星的简单组M （Monster）内进行快速计算的功能。本文的目的是建立M的极大子群的mmgroup数据库：对于M的极大子群的共轭类C，我们在mmgroup中构造了显式群元素，并证明了这些元素在C中生成了一个群。我们的生成器和验证正确性的计算在附代码中提供。几十年来，许多论文对M的极大子群进行了分类；我们的工作构成了对这些论文结构的独立验证。我们还修正了M有一个极大子群PSL2（59）的说法，从而确定了一个新的极大子群59:29。

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引用次数: 0

Is the projective cover of the trivial module in characteristic 11 for the sporadic simple Janko group J4 a permutation module? 散发性简单Janko群J4的特征11平凡模的投影盖是置换模吗？

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-10-30 DOI: 10.1016/j.jalgebra.2025.10.018

Jürgen Müller

We determine the ordinary character of the projective cover of the trivial module in characteristic 11 for the sporadic simple Janko group

J_{4}

, and answer the question posed in the title.

我们确定了散乱的简单Janko群J4的特征11平凡模的投影盖的一般性质，并回答了题目中的问题。

引用次数: 0

Constructing 2-dimensional Lubin-Tate formal groups over Zp (I) 构造Zp (I)上的二维Lubin-Tate形式群

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-10-30 DOI: 10.1016/j.jalgebra.2025.10.027

Ramla Abdellatif , Mabud Ali Sarkar

In this paper, we construct a class of 2-dimensional formal groups over

Z_{p}

that provide a higher-dimensional analogue of the usual 1-dimensional Lubin-Tate formal groups, then we initiate the study of the extensions generated by their

p^{n}

-torsion points. For instance, we prove that the coordinates of the

p^{\infty}

-torsion points of such a formal group generate an abelian extension over a certain unramified extension of

Q_{p}

, and we study some ramification properties of these abelian extensions. In particular, we prove that the extension generated by the coordinates of the p-torsion points is in general totally ramified.

本文在Zp上构造了一类二维形式群，它们提供了一维Lubin-Tate形式群的高维模拟，然后我们开始研究它们的pn-扭转点所产生的扩展。例如，我们证明了这种形式群的p∞-扭转点的坐标在Qp的某个未分形扩展上生成了一个阿贝尔扩展，并研究了这些阿贝尔扩展的一些分形性质。特别地，我们证明了由p-扭转点的坐标所产生的扩展一般是完全分枝的。

引用次数: 0

The fields of values of the Isaacs' head characters 以撒头字符值的字段

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-10-30 DOI: 10.1016/j.jalgebra.2025.10.032

Gabriel Navarro

We determine the fields of values of the Isaacs' head characters of a finite solvable group.

我们确定了有限可解群的艾萨克头字符值的域。

引用次数: 0

Characteristic polynomials and some combinatorics for finite Coxeter groups 有限Coxeter群的特征多项式和一些组合

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-10-30 DOI: 10.1016/j.jalgebra.2025.10.026

Shoumin Liu, Yuxiang Wang

Let W be a finite Coxeter group with Coxeter generating set

S = {s_{1}, \dots, s_{n}}

, and ρ be a complex finite dimensional representation of W. The characteristic polynomial of ρ is defined as

d (S, ρ) = \det [x_{0} I + x_{1} ρ (s_{1}) + \dots + x_{n} ρ (s_{n})],

where I is the identity operator. In this paper, we show the existence of a combinatorics structure within W, and thereby prove that for any two complex finite dimensional representations

ρ_{1}

and

ρ_{2}

of W,

d (S, ρ_{1}) = d (S, ρ_{2})

if and only if

ρ_{1} ≅ ρ_{2}

设W是一个有限Coxeter群，其Coxeter生成集S={s1，…，sn}， ρ是W的一个复有限维表示。ρ的特征多项式定义为d(S,ρ)=det (x0I+x1ρ(s1)+⋯+xnρ(sn)]，其中I是单位算子。本文证明了W中一个组合结构的存在性，从而证明了对于W的任意两个有限维复表示ρ1和ρ2， d(S,ρ1)=d（S,ρ2）当且仅当ρ1 = ρ2。

引用次数: 0

The splitting lemma in any characteristic 任何特征的分裂引理

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Algebra

Pub Date : 2025-10-29 DOI: 10.1016/j.jalgebra.2025.10.023

Gert-Martin Greuel, Gerhard Pfister

We give a simple proof of the splitting lemma in singularity theory, also known as generalized Morse lemma, for formal power series over arbitrary fields. Our proof for the uniqueness of the residual part in any characteristic is new and was previously unknown in characteristic two. Beyond the formal case, we give proofs for algebraic power series and for convergent real and complex analytic power series, which are new for non-isolated singularities.

对任意域上的形式幂级数给出了奇异理论中分裂引理的一个简单证明，即广义莫尔斯引理。我们对任意特征中残差部分唯一性的证明是新的，在特征二中是未知的。在形式上的证明之外，我们给出了代数幂级数和收敛实幂级数和复解析幂级数的证明，这是关于非孤立奇点的新证明。

引用次数: 0

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