丛代数是近十多年来的一个热门研究领域，而它的基也一直处于研究重点。作为丛代数诞生的初始动机和核心问题，Fomin-Zelevinsky猜想某些丛代数的生成元单项是量子群对偶典范基的元素。作为新的研究思路，Hernandez-Leclerc猜想另一些丛代数的生成元单项是量子仿射代数的不可约表示。通过研究丛代数的generic基，Geiss-Leclerc-Schroer证明了一个Fomin-Zelevinsky猜想的类比：生成元单项是包络代数的对偶半典范基的元素。最近，申请人定义了丛代数的共同三角基，并利用它验证了Fomin-Zelevinsky猜想和Hernandez-Leclerc猜想。..上述猜想中的那些丛代数是源于李理论的两类丛代数的特例。本课题拟结合表示论、范畴化和几何方法，研究这两类丛代数上的generic基和共同典范基的存在性与构造，进一步发掘丛代数和李理论之间的联系。

Cluster algebras have been a hot topic over the past decade, and their bases have always been important subjects. As the motivating and core question of cluster algebras, Fomin-Zelevinsky conjectured that the cluster monomials of some cluster algebras belong to the dual canonical bases of quantum groups. As a new approach, Hernandez-Leclerc conjectured that the cluster monomials of some other cluster algebras are irreducible representations of quantum affine algebras. By studying the generic bases of cluster algebras, Geiss-Leclerc-Schroer proved an analogue of the Fomin-Zelevinsky conjecture: the cluster monomials belong to the dual semi-canonical bases of enveloping algebras. Recently, the applicant defined the common triangular bases of cluster algebras, and thus verified the Fomin-Zelevinsky conjecture and Hernandez-Leclerc conjecture...Those cluster algebras appearing in the conjectures above are special cases of two types of cluster algebras arising from Lie theory. The present project expects, by combining representation theory, categorification, and geometric methods, to study the existence and construction of the generic bases and the common triangular bases of these two types of cluster algebras, and to further explore the relation between cluster algebras and Lie theory.