Categorification of cluster algebras

簇代数的分类

• 批准号: 2697138
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2697138
• 关键词: Categorification; cluster; algebras

Quivers are a very broad class of objects that were originally intended to study subspaces of a given vector space. On the route to classifying all the semisimple Lie algebras it was discovered that certain diagrams play a very important role in Lie theory (the so-called Dynkin diagrams). If we add an orientation to such a diagram we obtain a Dynkin quiver.Thanks to work by Gabriel and Ringel it turns out that quiver representation theory over a finite field can categorify the positive part of the quantum envelopping algebra of the same type. In his incredible paper Lusztig managed to construct a new basis of the quantum envelopping algebra which also gives bases of all of the highest weight irreducible representations. Motivated by ideas related to canonical bases Fomin and Zelevinsky constructed an algebra by starting from a finite set and then "mutating". The algebras generated by the mutations are cluster algebras. This mutation process can be related to a mutation of quivers. In a paper by Buan, Marsh, Reineke, Reiten and Todorov a certain quotient of the bound derived category of modules over the path algebra of a related quiver was shown to give us the categorical structure of a cluster algebra in the case of acyclic type. The aim of this project would be to search for a categorification of cluster algebras in a different case (the geometric case coming from grassmannians). This sort of work could hope to find applications in algebra, geometry, combinatorics and even mathematical physics.

箭图是一类非常广泛的对象，最初用于研究给定向量空间的子空间。在对所有半单李代数进行分类的过程中，人们发现某些图在李理论中起着非常重要的作用（所谓的Dynkin图）。如果我们在这样的图上加上一个方向，我们就得到了一个Dynkin表示。由于Gabriel和Ringel的工作，有限域上的表示理论可以对同一类型的量子表示代数的正部分进行分类。在他的令人难以置信的文件Lusztig设法建立一个新的基础上的量子代数也使基地的所有最高权重的不可约表示。受规范基相关思想的启发，Fomin和Zelevinsky从有限集开始构建了一个代数，然后“突变”。由突变生成的代数是簇代数。这个突变过程可能与箭袋的突变有关。在一份文件中Buan，马什，Reineke，Reiten和托多罗夫一定商的束缚导范畴的模块的道路代数的一个相关的dogs被证明给我们的范畴结构的集群代数的情况下，非循环型。这个项目的目的是在不同的情况下（来自格拉斯曼的几何情况）寻找簇代数的分类。这类工作有望在代数、几何、组合数学甚至数学物理中找到应用。