Exact Structures in Representation Theory

表示论中的精确结构

基本信息

批准号：
RGPIN-2019-04465
负责人：
Brüstle, Thomas
金额：
$ 1.89万
依托单位：
Bishop's University
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2021
资助国家：
加拿大
起止时间：
2021-01-01 至 2022-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=737301
关键词：
Exact Structures Representation Theory

项目摘要

The aim of this project is to develop a theory of stability conditions in a general setting and thus advance the knowledge in this field. The objectives of this proposal are to: 1) study stability conditions in the new context of exact categories, and 2) interpret the matrix reduction technique as a change of exact structure. The approach is, in both cases, to work with exact structures that have been defined by Quillen in his fundamental work on K-theory. They provide a framework that allows the use of methods from homological algebra, but is strictly more general than the setting of abelian categories. Stability conditions have been studied systematically in geometry and theoretical physics. The motivation stems from a phenomenon called mirror symmetry, relating two seemingly unrelated geometrical objects relevant to theoretical physics. The matrix reduction technique is an elementary tool that has been used by the Kiev school to obtain fundamental results in representation theory of algebras. The formal setup for matrix reduction has been given using a certain setup called called BOCSes. The disadvantage of this approach is that iterated reduction of matrices, per se not a complicated technique, produces more and more complicated descriptions of objects in the categories to be studied. We propose here a fundamentally novel approach: instead of changing the objects, we suggest to keep the same category, but to change its exact structure. We plan to apply this technique of reductions of exact structures to solve questions related to representation theory and mirror symmetry.

该项目的目的是发展一般环境下的稳定性条件理论，从而推进该领域的知识。该提案的目标是：1）研究精确类别的新背景下的稳定性条件，2）。在这两种情况下，将矩阵约简技术解释为精确结构的改变，这些方法都是使用 Quillen 在其 K 理论基础工作中定义的精确结构，它们提供了一个允许使用方法的框架。来自同调代数，但严格来说更稳定性条件在几何和理论物理学中得到了系统的研究，其动机源于一种称为镜像对称的现象，它将两个看似不相关的与理论物理学相关的几何对象联系起来。基辅学派已将其用于代数表示论的基本结果。使用称为 BOCS 的某种设置给出了矩阵简化的形式。这种方法的缺点是矩阵的迭代简化，本身不是一个矩阵。复杂的技术，对要研究的类别中的对象产生越来越复杂的描述，我们在这里提出了一种根本上新颖的方法：我们建议保持相同的类别，而不是改变其确切的结构。应用这种精确结构简化技术来解决与表示论和镜像对称相关的问题。