Simple-mindedness in triangulated categories

表示理论是通过线性变换在向量空间上的作用来研究对称性的一门学科；它遵循了一个长期存在的数学传统，即通过线性近似来研究难题。这种想法的自然性意味着表示理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20世纪50年代英国数学家弗兰克·亚当斯在曼彻斯特大学的代数拓扑学工作，并在20世纪60年代由巴黎的格罗滕狄克学派发展起来。如今，表征理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。表示理论的一个基本思想是研究某些生成器，或所有表示都可以构建的“构建块”。射影对象起源于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年里实现了爆炸性的发展，与李论、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，有一种更古老的发生器：简单物体，自19世纪80年代舒尔以来一直在研究。舒尔引理说简单的表示是“彼此垂直的”，Jordan-Hölder定理说所有的表示都可以由简单的表示构建，这是全世界本科代数课程的核心组成部分。简单集合（SMC）和简单系统（SMS）的概念是满足舒尔引理和Jordan-Hölder定理的三角化范畴中对象的集合，并为简单对象提供了同调框架。简单物体倾斜理论的森田理论的缺失，阻碍了表征理论中许多强大的同调和组合方法在基本问题上的应用。本研究将利用作者及其合作者最近提出的简单物体是一种“负簇倾斜物体”的观点，通过发展从森田理论和倾斜理论到简单物体理论的成熟技术来纠正这一问题。提出的研究将提供从旧（突变）构造新简单对象集的新方法，这将为一些长期存在的开放性问题（如Auslander-Reiten猜想）提供新的视角；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；并且，-一个用于研究由同调代数产生的几何空间的离散框架，例如稳定性条件的空间。

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.