Geometric aspects of representations and cohomology of finite dimensional algebras

有限维代数表示和上同调的几何方面

• 批准号: 0500946
• 负责人: Julia Pevtsova
• 金额: --
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500946&HistoricalAwards=false
• 关键词: Geometric; aspects; representations; cohomology; finite

Pevtsova proposes to investigate aspects of the modular representation theory of various finite dimensional algebras over a field of positive characteristic, and of the representation theory of related algebras over fields of characteristic zero. Built upon earlier work of Pevtsova and others which employs local methods in modular representation theory, the proposed research aims tocreate new invariants and to achieve a better understanding of existing ones. For example, Pevtsova proposes in joint work with Friedlander to produce finer invariants than support varieties by considering restrictions of representations to subalgebras isomorphic to group algebras of a cyclic group. In another project, joint with Witherspoon, Pevtsova will seek a representation-theoretic description ofsupport varieties for representations of a restricted quantum Lie algebra at roots of unity. In a baby sisterof this project, Pevtsova and Witherspoon propose to develop a new description of the rank variety for a class of local algebras over fields of characteristic zero, and define rank varieties for certain non-cocommutative Hopf algebras, identifying the new varieties with the existing geometric constructions which use Hochschild cohomology. In a third project, Pevtsova proposes to establish a geometric correspondence between certain triangulated module categories associated to reduced enveloping algebras of a Lie algebra.The proposed research begins with the philosophy of understanding a difficult, complicated algebraic structure by studying a family of simple structures embedded in the original. The simple objects are well understood so that the challenge is to combine this understanding to gain some information about the original, difficult structure. This idea is applied to the study of formal algebraic objects arising as symmetries of familiar structures. In the process, the interplay of two different mathematical fields, geometry and algebra, is used extensively, revealing beautiful connections and enabling applications to both areas. A second aspect of this proposal includes the mentoring of female undergraduate students, participation in a popular mathematical program for advanced high school students, and the editing of a special volume of amathematical journal.

Pevtsova 提议研究正特征域上的各种有限维代数的模表示论以及特征零域上的相关代数的表示论的各个方面。 这项研究以 Pevtsova 等人的早期工作为基础，采用模块化表示理论中的局部方法，旨在创建新的不变量并更好地理解现有的不变量。例如，Pevtsova 提议与 Friedlander 合作，通过考虑与循环群的群代数同构的子代数表示的限制，产生比支持簇更精细的不变量。在与 Witherspoon 合作的另一个项目中，Pevtsova 将寻求对单位根处受限量子李代数表示的支持簇的表示理论描述。在这个项目的一个小姐妹中，Pevtsova 和 Witherspoon 提议为特征零域上的一类局部代数的秩簇开发一种新的描述，并为某些非共交换 Hopf 代数定义秩簇，用使用 Hochschild 上同调的现有几何构造来识别新簇。在第三个项目中，Pevtsova 提议在与李代数的约简包络代数相关的某些三角模块类别之间建立几何对应关系。所提出的研究始于通过研究嵌入在原始代数中的一系列简单结构来理解困难、复杂的代数结构的哲学。简单的物体已经被很好地理解了，因此挑战是结合这种理解来获得一些关于原始的、困难的结构的信息。这个想法应用于对作为熟悉结构的对称性而产生的形式代数对象的研究。在此过程中，几何和代数这两个不同数学领域的相互作用被广泛使用，揭示了美丽的联系并实现了这两个领域的应用。该提案的第二个方面包括指导女本科生、参与针对高级高中生的流行数学项目以及编辑数学期刊特刊。