Higher Dimensional Homological Algebra

高维同调代数

• 批准号: EP/P016014/1
• 负责人: Peter Jorgensen
• 金额: $ 30.73万
• 依托单位: Newcastle University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FP016014%2F1
• 关键词: Higher; Dimensional; Homological; Algebra

Classic homological algebra had its origin some 60 years ago in algebraic topology. Since then, it has grown to a theory with applications to many areas of mathematics, including combinatorics, geometry, and representation theory. Classic homological algebra is often phrased as the theory of abelian and triangulated categories.Higher dimensional homological algebra is a new development of the last decade. It is the theory of n-abelian categories and (n+2)-angulated categories, where n is a positive integer. In these categories, the role previously played by 1-extensions is taken over by n-extensions. Note that the case n=1 gives ordinary abelian and triangulated categories, hence classic homological algebra. We refer to (n+2)-angulated categories because the case n=1 gives triangulated categories. Higher dimensional homological algebra is currently very active. It has applications to algebraic geometry, combinatorics, and the representation theory of finite dimensional algebras. There are substantial contributions from a number of strong mathematicians (Iyama, Keller, Reiten). Some of the combinatorial structures which appear, like higher dimensional cyclic polytopes, are novel to homological algebra and representation theory.This project will provide three key items currently missing in higher dimensional homological algebra: Tilting objects, higher dimensional derived categories, and higher dimensional model categories. We will define tilting objects in higher dimensional homological algebra and show a version of the Ingalls-Thomas bijections between support tilting objects, torsion classes, intermediate t-structures, and non-crossing partitions. This will enhance and illuminate the links to combinatorics. We will define higher dimensional derived categories and higher dimensional model categories. This will provide the right context for higher dimensional tilting theory, and give a comprehensive framework for higher dimensional homological algebra.

经典同调代数起源于大约 60 年前的代数拓扑。从那时起，它已经发展成为一种应用于许多数学领域的理论，包括组合学、几何和表示论。经典的同调代数通常被表述为阿贝尔范畴和三角范畴的理论。高维同调代数是近十年来的新发展。它是n-阿贝尔范畴和(n+2)-角范畴的理论，其中n是正整数。在这些类别中，以前由 1-扩展所扮演的角色被 n-扩展所取代。请注意，n=1 的情况给出了普通的阿贝尔和三角范畴，因此是经典的同调代数。我们指的是 (n+2) 角度类别，因为 n=1 的情况给出了三角类别。高维同调代数目前非常活跃。它适用于代数几何、组合学和有限维代数的表示论。许多强大的数学家（Iyama、Keller、Reiten）做出了重大贡献。出现的一些组合结构，如高维循环多胞体，对于同调代数和表示论来说是新颖的。该项目将提供目前在高维同调代数中缺失的三个关键项目：倾斜对象、高维派生类别和高维模型类别。我们将在高维同调代数中定义倾斜物体，并展示支撑倾斜物体、扭转类、中间 t 结构和非交叉分区之间的 Ingalls-Thomas 双射的版本。这将增强并阐明与组合学的联系。我们将定义更高维度的派生类别和更高维度的模型类别。这将为高维倾斜理论提供正确的背景，并为高维同调代数提供全面的框架。

项目成果

Infinite friezes and triangulations of annuli

无限的饰带和环带的三角剖分

• DOI: http://dx.10.48550/arxiv.2007.09411
• 发表时间: 2020
• 作者: Baur K

ON HIGHER TORSION CLASSES

关于更高的扭转等级

• DOI: http://dx.10.1017/nmj.2022.8
• 发表时间: 2022
• 期刊: Nagoya Mathematical Journal
• 影响因子: 0.8
• 作者: ASADOLLAHI J

Infinite friezes and triangulations of annuli

无限的饰带和环带的三角剖分

• DOI: http://dx.10.1142/s0219498824502074
• 发表时间: 2023
• 期刊: Journal of Algebra and Its Applications
• 影响因子: 0.8
• 作者: Baur K

Lattice bijections for string modules, snake graphs and the weak Bruhat order

字符串模块、蛇图和弱 Bruhat 阶的格双射

• DOI: http://dx.10.48550/arxiv.1811.06064
• 发表时间: 2018
• 作者: Canakci I

Friezes, weak friezes, and T-paths

饰带、弱饰带和 T 形路径

• DOI: http://dx.10.48550/arxiv.2005.06230
• 发表时间: 2020
• 作者: Canakci I