Varieties of modules and representations of Frobenius kernels of reductive groups

还原群 Frobenius 核的各种模和表示

• 批准号: EP/K022997/1
• 负责人: Paul Levy
• 金额: $ 12.21万
• 依托单位: Lancaster University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FK022997%2F1
• 关键词: Varieties; modules; representations; Frobenius; kernels

An algebra A is a type of mathematical object with a structure satisfying certain properties; a representation for A is a space on which A acts in a way which is compatible with its structure. Varieties of modules arise when we consider the set of *all* possible representations (of a given dimension) for A. Beginning with elementary examples, one obtains surprisingly rich and complex geometric structures parametrizing the n-dimensional modules.One natural question to ask is the following: given an algebra A, can we identify the irreducible components of the variety of n-dimensional A-modules? In general, this turns out to be a hard question. The existing methods for tackling it mostly depend on fairly restrictive properties of the algebra A. In the research proposed here, we will investigate varieties of modules for a particular class of algebras: group algebras of elementary abelian p-groups of rank 2. The problem of describing these varieties of modules has an alternative interpretation in relation to cohomology of the second Frobenius kernel of the group of invertible n x n matrices, due to work of Suslin, Friedlander and Bendel. In order to tackle this specific problem, we will have to develop some new methods for studying varieties of modules, adapting earlier results of Crawley-Boevey and Schroer.

代数A是一类数学对象，其结构满足某些性质; A的表示是一个空间，A以与其结构相容的方式作用于该空间。当我们考虑A的（给定维数的）* 所有 * 可能表示的集合时，出现了各种各样的模。从基本的例子开始，我们可以得到参数化n维模的丰富而复杂的几何结构。一个自然的问题是：给定一个代数A，我们能确定n维A-模簇的不可约分支吗？总的来说，这是一个很难回答的问题。现有的解决方法主要依赖于代数A的相当严格的性质。在这里提出的研究中，我们将研究一类特殊代数的模的变种：秩为2的初等交换p-群的群代数。问题描述这些品种的模块有一个替代的解释关系到上同调的第二弗罗贝纽斯内核组的可逆n × n矩阵，由于工作的Suslin，Friedlander和本德尔。为了解决这个具体问题，我们将不得不发展一些新的方法来研究各种各样的模，适应克劳利-博维和施罗德的早期结果。

项目成果

Generic singularities of nilpotent orbit closures

幂零轨道闭合的一般奇点

• DOI: 10.1016/j.aim.2016.09.010
• 发表时间: 2015-02
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: 付保华;Daniel Juteau;Paul Levy;Eric Sommers
• 通讯作者: Eric Sommers

On nilpotent commuting varieties and cohomology of Frobenius kernels

关于 Frobenius 核的幂零交换簇和上同调

• DOI: 10.1016/j.jalgebra.2014.11.014
• 发表时间: 2015
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Ngo N

On varieties of commuting nilpotent matrices

• DOI: 10.1016/j.laa.2014.03.032
• 发表时间: 2013-08
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: N. Ngo;Klemen vSivic

Commuting varieties and cohomological complexity theory

• DOI: 10.1112/jlms.12650
• 发表时间: 2022-07
• 期刊: Journal of the London Mathematical Society
• 作者: Paul D. Levy;N. Ngo;Klemen Šivic

Mixed nilpotent commuting varieties of Lie algebras of exceptional type

特殊类型李代数的混合幂零交换簇

• 作者: Levy, P.D.