Geometric aspects of representations and cohomology of finite dimensional algebras

Pevtsova建议研究各种有限维代数在正特征域上的模表示理论的各个方面，以及相关代数在特征为零的域上的表示理论。基于Pevtsova和其他人在模块化表示理论中使用局部方法的早期工作，提出的研究旨在创建新的不变量，并更好地理解现有的不变量。例如，Pevtsova在与Friedlander的联合工作中提出，通过考虑与循环群的群代数同构的子代数的表示的限制，可以产生比支持变量更精细的不变量。在与Witherspoon合作的另一个项目中，Pevtsova将寻求在单位根处受限量子李代数表示的支持变体的表示理论描述。Pevtsova和Witherspoon在本课题的子论文中提出了对特征为零的域上的一类局部代数的秩变化的一种新的描述，并定义了某些非协易Hopf代数的秩变化，并将这些新变化与已有的使用Hochschild上同的几何构造进行了识别。在第三个项目中，Pevtsova提出在与李代数的约简包络代数相关的某些三角化模类别之间建立几何对应关系。提出的研究从理解一个困难的，复杂的代数结构的哲学开始，通过研究一个家族的简单结构嵌入原来。简单的对象已经被很好地理解了，所以挑战是结合这些理解来获得一些关于原始的、困难的结构的信息。这一思想被应用于研究由熟悉结构的对称而产生的形式代数对象。在这个过程中，两个不同的数学领域——几何和代数——的相互作用被广泛使用，揭示了美丽的联系，并使这两个领域的应用成为可能。该建议的第二个方面包括对女本科生进行指导，参与一个受欢迎的高级高中学生数学计划，以及编辑一本数学期刊专刊。

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.