Cluster algebras and tilting theory II

该项目的重点是集群代数及其关系的表示理论的有限维代数。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，涉及数学的许多领域。PI将继续研究与黎曼曲面相关的特殊类型的簇代数;他将引入广义簇变量并使用它们来构建这些簇代数的规范基，他将开发簇代数内部显式计算的组合模型。PI还将研究簇倾斜代数，这是某些有限维代数，其模对应于簇代数的元素。簇代数是Fomin和Zelevinsky于2002年引入的，其最初的动机来自于现代代数的一个分支表示论，它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果，表示论在物理和化学以及其他数学领域中有许多应用。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。这促使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 cluster algebras and their relation to the representation theory of finite dimensional algebras. 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. The PI will continue his study of the special type of cluster algebras associated to Riemann surfaces; he will introduce generalized cluster variables and use them to construct canonical bases for these cluster algebras, and he will develop combinatorial models for explicit computations inside the cluster algebra. The PI will also investigate cluster-tilted algebras, which are certain finite dimensional algebras, whose modules correspond to elements of the cluster algebra. Cluster-tilted algebras provide a new point of view on tilting theory over hereditary algebras, and their module categories carry interesting information about the cluster algebra.When cluster algebras were introduced by Fomin and Zelevinsky in 2002, their original motivation came from representation theory, which is a branch of modern algebra that is concerned with the study of symmetries of scientific models. Studying the symmetries of a model is often more fruitful than studying the model directly, and representation theory has found many applications in physics and chemistry 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 of the theory of cluster algebras to which this project will contribute.