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: 0629156
• 负责人: Julia Pevtsova
• 金额: $ 6.72万
• 依托单位: University of Washington
• 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: 2005-12-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0629156&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建议研究在正性特征的领域上各种有限维代数的模块化表示理论的各个方面，以及特征零字段上相关代数的表示理论。 拟议的研究旨在基于Pevtsova和其他采用局部方法的局部方法，其旨在创造新的不变性，并对现有的研究更好地理解。例如，Pevtsova在与Friedlander的联合合作中提出，通过考虑对循环基团的组代数的代表构成同构代数来产生比支持品种的不变性。在另一个项目（与Witherspoon的联合）中，Pevtsova将寻求代表理论描述，以供应品种，以表示统一根源有限的量子谎言代数。在这个项目的一个小姐妹中，Pevtsova和Witherspoon提议在特征零领域中开发一类本地代数的等级的新描述，并为某些非协商Hopf代数定义了等级品种，从而确定了使用Hochschscheld Coohomology的现有几何结构的新品种。在第三个项目中，Pevtsova提议在某些三角模块类别之间建立几何对应关系，与减少谎言代数的包围代数相关的包围代数相关。拟议的研究始于理解一种困难，复杂的代数结构的哲学，通过研究原始原始中的简单结构的家族。简单的对象是充分理解的，因此挑战是将此理解结合起来，以获取有关原始，困难的结构的一些信息。这个想法应用于对熟悉结构对称的形式代数对象的研究。在此过程中，广泛使用了两个不同数学字段的几何和代数的相互作用，从而揭示了美丽的连接并启用了两个领域的应用程序。该提案的第二个方面包括指导女性本科生，参加高级高中生的流行数学计划以及特殊卷的杂志。