Cluster algebras and tilting theory

本项目主要研究簇代数与有限维代数的表示理论之间的关系。有限维代数通常被描述为具有关系的箭图的路代数。于是代数的模对应于代数的表示。簇代数是具有特殊组合结构的交换代数。簇代数理论是一个快速发展的领域，它涉及到数学的许多领域。它的一个最活跃的分支是它的连接表示理论的有限维代数已建立在PI的联合工作与卡尔德罗和Chapoton，并独立地由布安，马什，Reineke，Reiten和托多罗夫。他们对某些三角范畴（簇范畴）的构造为倾斜理论提供了意想不到的新见解，并产生了一类全新的有限维代数--簇倾斜代数。当簇代数在2002年由Fomin和Zelevinsky引入时，它们的最初动机来自表示论，它是现代代数的一个分支，主要研究科学模型的对称性。研究模型的对称性往往比直接研究模型更有成效，而且表示理论在物理、化学以及其他数学领域中有着广泛的应用。簇代数为整个表示论中出现的基本模式提供了一个数学框架。令人惊讶的是，这些模式也在其他科学分支中观察到，先验地，与表征理论无关。这激发了进一步发展理论的集群代数，这一项目将作出贡献。

The project focuses on the relation between cluster algebras and the representation theory of finite dimensional algebras.Finite dimensional algebras are often described as path algebras of quivers with relations. The modules of the algebra then correspond to the representations of the quiver. Cluster algebras are commutative algebras with a special combinatorial structure. The theory of cluster algebras is a fast developing field which is related to many areas of mathematics. One of its most active branches is its connection to the representation theory of finite dimensional algebras which has been established in the PI's joint work with Caldero and Chapoton and, independently, by Buan, Marsh, Reineke, Reiten and Todorov. Their construction of certain triangulated categories, the cluster categories, has provided unexpected new insights in tilting theory and has given rise to a whole new class of finite dimensional algebras, the cluster-tilted algebras.When cluster algebras were introduced by Fomin and Zelevinsky in 2002,their original motivation came from representation theory, which is abranch of modern algebra that is concerned with the study ofsymmetries of scientific models. Studying the symmetries of a model isoften more fruitful than studying the model directly, andrepresentation theory has found many applications in physics andchemistry, as well as in other mathematical fields. The cluster algebras provide a mathematical framework for fundamental patterns which occur throughout representation theory. Surprisingly, these patterns are also observed in various other branches of science which, a priori, are not related to representation theory. This motivates a further development ofthe theory of cluster algebras to which this project will contribute.