Cluster Algebras, Quantum Groups, and Decorated Character Varieties

Michigan State University, East Lansing MI

Investigators

Abstract

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.