Grassmannian cluster categories and Braid groups

The proposed project is about two active fields of research in pure mathematics: representation theory and cluster algebras. Representation theory is an area of mathematics which studies modules over algebras. It has many deep connections with mathematical physics, number theory, and geometry. Cluster algebras were introduced by Fomin and Zelevinsky in the beginning of this century. Cluster algebras are found and play an important role in different contexts of mathematics and physics: Poisson gemeotry, representation theory, Teichmuller theory, BPS states, scattering amplitudes. More precisely, we will work on the following problems. 1. Study categorifications of cluster algebra structures on the homogeneous coordinate rings of Grassmannians. 2. Try to construct some new braidings. The new braidings are useful in studying flat deformations of symmetric algebras and studying Nichols algebras. We shall use various programming tools, algorithms and theoretical results to work our way through this ambitious project. We plan to pursue this research in close collaboration with Professor K. Baur, as well as Professors A. Berenstein, J. Greenstein.

Cluster algebras were introduced by Fomin and Zelevinsky around 2000. Cluster algebras play an important role in different areas of mathematics and physics. One of the topic of the project is to study Grassmannian cluster categories. We classified rigid indecomposable modules in the Grassmannian cluster category. We also studied relation between representations of quantum affine algebras and Grassmannians. Another topic of the project is braid group. Using the Yang-Baxter equation and the ansatz for the Baxterization of the models, we introduced 4-CB algebras which are analogue of BMW algebras. We also studied presentations of Boolean reflection monoids using quiver mutations.