Combinatorial aspects of derived representation theory

派生表示论的组合方面

• 批准号: 2436975
• 金额: --
• 依托单位: Lancaster University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2436975
• 关键词: Combinatorial; aspects; derived; representation; theory

Calabi-Yau (CY) triangulated categories provide a homological context for ideas that first originated in mathematical physics. In recent years they have acquired independent interest in pure mathematics, particularly in relation to cluster-tilting theory and Bridgeland stability conditions. A triangulated category is Calabi-Yau if it satisfies a particularly nice form of Serre duality: for an integer n, the nth shift functor [n] is naturally isomorphic to the Serre functor S; such a category is called n-CY. When n >1, much is known about the homological and combinatorial structure of such categories, e.g. (higher) cluster-tilting theory and higher homological algebra. The picture when n < 1 is much less well developed despite there being mainstream examples, e.g. stable module categories of symmetric algebras are (-1)-CY and their perfect derived categories are 0-CY. The aim of this project is to develop the structure theory for negative CY categories, for example by classifying (co)torsion pairs in discrete CY triangulated categories, and describing simple-minded systems for stable module categories.

Calabi-Yau （CY）三角分类为最初起源于数学物理的思想提供了一个同源背景。近年来，他们对纯数学产生了独立的兴趣，特别是在簇倾斜理论和布里奇兰稳定性条件方面。三角化范畴是Calabi-Yau，如果它满足一个特别好的Serre对偶形式：对于整数n，第n移函子[n]自然同构于Serre函子S；这样的类别被称为n-CY。当n >1时，对于这些范畴的同调和组合结构，例如（高）簇倾斜理论和高同调代数，我们已经知道了很多。尽管有主流的例子，但当n < 1时的图像发展得不太好，例如对称代数的稳定模范畴是(-1)-CY，它们的完美派生范畴是0-CY。本计划的目的是发展负CY范畴的结构理论，例如，通过在离散CY三角范畴中分类（co）扭转对，并描述稳定模范畴的简单系统。