Simple-mindedness in triangulated categories

三角范畴中的头脑简单

基本信息

批准号：
EP/V050524/1
负责人：
David Pauksztello
金额：
$ 43.84万
依托单位：
Lancaster University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2022
资助国家：
英国
起止时间：
2022 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV050524%2F1
关键词：
Simple mindedness triangulated categories

项目摘要

Representation theory is a the study of symmetry via the action of linear transformations on vector spaces; it follows a long-standing mathematical tradition of studying difficult problems by taking linear approximations. The naturalness of this idea means that representation theory sits at a nexus with many branches of mathematics, particularly, algebraic geometry, algebraic topology and combinatorics.The concept of a triangulated category goes back to the work of British mathematician Frank Adams in algebraic topology at the University of Manchester in the 1950s and was developed by the Grothendieck school in Paris in the 1960s. Nowadays, representation theory is often formulated using triangulated categories, which permits the use of powerful methods of homological algebra and provides further crossovers with geometry, topology and even mathematical physics. A basic idea in representation theory is to study certain generators, or "building blocks" out of which all representations can be built. Originating in classic homological algebra over 60 years ago, projective objects, and their generalisations into Morita theory and tilting theory have enabled explosive development over the past 40 years with deep connections to Lie theory, quantum algebra, combinatorics, algebraic geometry and mathematical physics.However, there is a much older kind of generator: simple objects, which have been studied since Schur in the 1880s. Schur's lemma, which says that simple representations are "perpendicular to each other", and the Jordan-Hölder theorem, which says that all representations can be built out of simple representations, are core components of undergraduate algebra curricula all over the world. The notions of simple-minded collection (SMC) and simple-minded system (SMS) are collections of objects in triangulated categories satisfying both Schur's lemma and the Jordan-Hölder theorem and provide the homological framework for simple objects.The absence of a Morita theory of tilting theory for simple objects prevents the application of many powerful homological and combinatorial methods to basic problems in representation theory. The proposed research will rectify this problem by developing the theory to transport well-developed techniques from Morita theory and tilting theory to the theory of simple objects by exploiting a recent perspective developed by the proposer and his collaborators that simple objects are a kind of "negative cluster-tilting object". The proposed research will provide- methods for constructing new sets of simple objects from old (mutation), which will provide new perspectives to some long-standing open problems such as the Auslander-Reiten Conjecture;- a dictionary between projective objects and simple objects, which will provide new methods for modular representation theory; and,- a discrete framework for studying geometric spaces arising out of homological algebra such as spaces of stability conditions.

表示论是通过向量空间上的线性变换来研究对称性；它遵循通过采用线性近似来研究难题的长期数学传统。这种想法的自然性意味着表示论与许多事物都有联系。数学的分支，特别是代数几何、代数拓扑和组合学。三角范畴的概念可以追溯到英国数学家 Frank Adams 在英国大学的代数拓扑中的工作。曼彻斯特于 20 世纪 50 年代诞生，并由巴黎的格洛腾迪克学派于 20 世纪 60 年代发展。如今，表示论通常使用三角范畴来表述，这允许使用强大的同调代数方法，并提供与几何、拓扑甚至数学物理的进一步交叉。表示论的基本思想是研究某些生成器，或可以构建所有表示的“构建块”，起源于 60 多年前的经典同调代数。射影物体及其对森田理论和倾斜理论的推广在过去 40 年中实现了爆炸性发展，与李理论、量子代数、组合学、代数几何和数学物理有着深刻的联系。然而，还有一种更古老的生成器：简单对象，自 1880 年代 Schur 引理以来一直在研究，该引理说简单表示“彼此垂直”，以及 Jordan-Hölder。该定理指出，所有表示都可以由简单表示构建出来，该定理是全世界本科代数课程的核心组成部分。简单集合 (SMC) 和简单思维系统 (SMS) 的概念是对象的集合。满足 Schur 引理和 Jordan-Hölder 定理的三角范畴，并为简单对象提供同调框架。简单对象倾斜理论的 Morita 理论的缺乏阻碍了许多强大的应用所提出的研究将通过利用提出者开发的最新观点，将成熟的技术从森田理论和倾斜理论转移到简单物体的理论，从而纠正这个问题。他的合作者认为简单对象是一种“负集群倾斜对象”，所提出的研究将提供从旧的（变异）简单对象中构造新的简单对象集的方法，这将为一些长期存在的开放性问题提供新的视角。作为Auslander-Reiten 猜想；- 射影对象和简单对象之间的字典，它将为模表示理论提供新方法；以及，- 用于研究由同调代数产生的几何空间（例如稳定条件空间）的离散框架。