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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.

表示论是通过线性变换对向量空间的作用来研究对称性的理论;它遵循了通过线性近似来研究困难问题的长期数学传统。这个概念的自然性意味着表示论与数学的许多分支，特别是代数几何、代数拓扑和组合学有着密切的联系。三角范畴的概念可以追溯到英国数学家弗兰克·亚当斯在20世纪50年代曼彻斯特大学的代数拓扑工作，并在20世纪60年代由巴黎的格罗滕迪克学校发展。如今，表示论通常使用三角范畴来表述，这允许使用同调代数的强大方法，并提供与几何，拓扑甚至数学物理的进一步交叉。表示论的一个基本思想是研究某些生成元，或者说是“积木”，所有的表示都可以从这些生成元中构建出来。投射对象起源于60多年前的经典同调代数，并将其推广到森田理论和倾斜理论，在过去的40年里取得了爆炸性的发展，与李群理论、量子代数、组合数学、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19世纪80年代舒尔以来一直在研究。舒尔引理认为简单的表示是“相互垂直的”，而乔丹-赫尔德定理认为所有的表示都可以从简单的表示中建立出来，这是世界各地本科代数课程的核心组成部分。简单集合（SMC）和简单系统（SMS）的概念是满足Schur引理和Jordan-Hölder定理的三角范畴中的对象的集合，并为简单对象提供了同调框架。简单对象的倾斜理论的森田理论的缺乏阻止了许多强大的同调和组合方法在表示论的基本问题中的应用。拟议的研究将纠正这一问题，通过发展理论，运输发达的技术从森田理论和倾斜理论的理论，利用最近的观点，简单的对象是一种“负集群倾斜对象”的提议者和他的合作者开发的简单对象。这项研究将提供从旧的简单对象构造新的简单对象集的方法。（mutation），这将为Auslander-Reiten猜想等一些长期存在的问题提供新的视角;-一个射影对象和简单对象之间的字典，这将为模表示理论提供新的方法;和，-一个离散的框架，用于研究几何空间所产生的同调代数，如空间的稳定条件。