Categorification of cluster algebras

箭图是一类非常广泛的对象，最初的目的是研究给定向量空间的子空间。在对半单李代数进行分类的过程中，人们发现某些图在李理论(即所谓的动态图)中起着非常重要的作用。多亏了Gabriel和Ringel的工作，有限域上的箭图表示理论可以归类为同类型量子包络代数的正部分。在他令人难以置信的论文中，Lusztig设法构造了量子包络代数的一个新的基，该基也给出了所有最高权不可约表示的基。受与正则基相关的思想的启发，Fomin和Zlevinsky从一个有限集合开始，然后“突变”，从而构建了一个代数。由突变生成的代数是簇代数。这一突变过程可能与颤动的突变有关。在Buan，Marsh，Reineke，Reiten和Todorov的一篇文章中，证明了相关箭图的路径代数上的有界导出模范畴的某个商，给出了非循环型情况下的簇代数的范畴结构。这个项目的目的将是寻找一种不同情况下的簇代数的分类(几何情况来自草本)。这类工作有望在代数、几何、组合学甚至数学物理中得到应用。

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.