Cluster Algebras, Quantum Groups, and Decorated Character Varieties

这项研究项目处于代数和几何的十字路口。一个特别的焦点是簇代数和模空间的理论。簇代数是由Fomin和Zlevinsky于2001年发现的一类与整数反对称矩阵相关的代数。自诞生以来，团簇理论的快速发展已经与数学和物理的许多不同领域产生了巨大的相互作用，包括表象理论、纽结理论和高能物理。模空间是将某些固定形状的对象或物理中特定系统的解分类的几何空间。这个项目将探索模空间和簇代数之间的联系，以加深他们的理解，并在不同领域之间建立相似之处。此外，PI将指导本科生研究项目，培训研究生，并支持一次研究研讨会。本项目将在簇代数的框架下研究模空间的量子几何。它还探索了装饰角色种类、量子群和传奇结之间的丰富联系，在各个方向都找到了新的结果。更详细地说，该项目将涉及四个主题。(1)提供了一个实现量子群的刚性团簇模型，从团簇代数的角度对量子群的许多性质有了新的解释。(2)研究量化装饰字品种的自然基础，包括具体的网库结构图。(3)探索Legendrian纽结的精确拉格朗日填充与其增强变种的簇种之间的内在对应关系。作为应用，它将解决正辫子以外的传奇结的无限填充问题。(4)它将使用Legendrian纽结理论的工具在广义Richardson品种上引入集群结构，证实Leclerc关于开放Richardson品种集群性质的猜想。这一奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The research project lies at the crossroads of algebra and geometry. A particular focus is on the theory of cluster algebras and moduli spaces. Cluster algebras, discovered by Fomin and Zelevinsky in 2001, are a class of algebras associated with integer skew-symmetric matrices. Since its inception, the rapid developments of cluster theory have found tremendous interactions with many different areas of mathematics and physics, including representation theory, knot theory, and high energy physics. Moduli spaces are geometric spaces that classify objects of some fixed shapes or solutions to specific systems in physics. This project will explore the connections between moduli spaces and cluster algebras to further their understanding and build parallels between the different areas. In addition, the PI will mentor undergraduate research projects, train graduate students, and support a research seminar. This project will investigate the quantum geometry of moduli spaces under the framework of cluster algebras. It also explores the fruitful connections between decorated character varieties, quantum groups, and Legendrian knots, finding new results in all directions. In more detail, the project will touch on four topics. (1) It will provide a rigid cluster model realizing quantum groups, obtaining new interpretations of many properties of quantum groups from the perspective of cluster algebras. (2) It will study the natural bases of the quantized decorated character varieties, including a concrete diagrammatic construction of web bases. (3) It will explore an intrinsic correspondence between the exact Lagrangian fillings of Legendrian knots and the cluster seeds of their augmentation varieties. As an application, it will solve the infinite-filling problem for Legendrian knots beyond positive braids. (4) It will use tools from Legendrian knot theory to introduce a cluster structure on generalized Richardson varieties, confirming a conjecture of Leclerc on the cluster nature of open Richardson varieties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.