Real modules, cluster algebras, and Kac-Moody root systems

The proposed project plans to work on quantum affine algebras and cluster algebras in mathemtaics. Quantum affine algebras are a family of quantum groups. Quantum groups were introduced by Drinfeld and Jimbo indepently aroud 1985. Cluster algebras were introduced by Fomin and Zelevinsky around 2000. Cluster algebras are found to have connections with many different areas of mathematics and physics.In the proposal, we will work on the following problems. 1. study real modules over quantum affine algebras and cluster monomials in Grassmannian cluster algebras, 2. study rigid indecomposable modules in Grassmannian cluster categories, 3. give a combinatorial description of real and imaginary roots in the Kac-Moody root system associated to Gr(k,n), 4. apply Grassmannian cluster algebras to study questions from physics: amplituhedrons and cluster algebraic functions. The applicant plans to collaborate with Karin Baur on Grassmannian cluster categories and real roots in Kac-Moody root systems in parts 2 and 3 of the proposal, with Ran Tessler on amplituhedrons in part 4, and with Marcus Spradlin on cluster algebraic functions in part 4 of the proposal.

This project investigates the representation theory of quantum affine algebras and its connections to combinatorics, cluster algebras, and mathematical physics. Building on the PI's previous work, the research explores structures such as real prime modules of quantum affine algebras, dual canonical bases of cluster algebras, braid group symmetries, Kac-Moody root systems. The project has produced explicit formulas and classifications that have been applied in theoretical physics, for instance in the study of scattering amplitudes and N=4 super Yang-Mills theory. It also advances understanding of tropical geometry, Grassmannian cluster categories, and the relations between different categorifications of cluster algebras. Through this work, the project has strengthened international collaborations, contributed to the development of mathematical tools for representation theory, and connected abstract algebraic structures to applications in physics.