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.

表示理论是通过线性变换对向量空间的作用进行对称的研究。它遵循了长期的数学传统，即通过进行线性近似来研究困难问题。这一想法的自然性意味着，代表理论是与许多数学分支的联系，特别是代数的几何形状，代数拓扑和组合学。三角调查类别的概念可以追溯到英国数学家在1960年代曼彻斯特大学和曼彻斯特大学的弗兰克·阿达姆斯（Frank Adams）的工作，由1950年代和1960年代的paroth paroth paroth and paroth。如今，代表理论通常是使用三角形类别制定的，该类别允许使用强大的同源代数方法，并通过几何，拓扑甚至数学物理学提供进一步的交叉。表示理论的一个基本思想是研究某些发电机，或“构建块”，从中可以构建所有表示。投影物体及其对莫里塔理论和倾斜理论的概括起源于经典的同源代数，在过去的40年中，它们具有爆炸性的发展，与谎言理论，量子代数，组合，代数几何学，代数几何学和数学物理学和数学物理学。 Scur的引理说，简单的表示“彼此垂直”，而Jordan-Hölder定理说，所有表示都可以由简单的表示形式构建，是本科代数探索的核心组成部分。简单的收藏（SMC）和简单的系统（SMS）的笔记是三角形类别中的集合，满足了Schur的引理和Jordan-Hölder定理，并为简单对象提供了同源框架。缺乏对简单对象的倾斜理论来阻止许多强有力的机器和组合问题的倾斜理论，以构成许多强大的家具问题。拟议的研究将通过开发理论来纠正这一问题，从而通过利用该提案及其合作者的最新观点来传达从莫里塔理论和倾斜理论到简单对象的理论，即简单对象是一种“负面群集倾斜对象”。拟议的研究将提供用于构建旧（突变）新简单对象集的方法，该方法将为某些长期存在的开放问题（例如Auslander-Reiten猜想）提供新的观点； - - 投影对象和简单对象之间的字典，这将为模块化表示理论提供新的方法；以及 - 研究由同源代数（例如稳定条件的空间）引起的几何空间的离散框架。