From Quantum Loop Group to Coherent Satake Category via Fusion Product

This lecture by Ilya Dumanski from MIT explores the connection between quantum loop groups and coherent Satake categories through fusion products. Discover partial progress toward proving the Cautis-Williams conjecture for simply-laced types, which states that the category of perverse coherent sheaves on the affine Grassmanian has a cluster structure. Learn how the coherent Satake category relates to finite-dimensional modules over the affine quantum group, with the Feigin-Loktev fusion product for current algebra modules serving as the critical bridge between these mathematical structures. Understand how this approach helps construct cluster short exact sequences of perverse coherent sheaves using Q-systems, which are exact sequences of modules over the quantum affine group.