REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: The Geometric Satake equivalence is an equivalence of categories (really, of 2-categories) between the representations of a complex semisimple lie algebra and some geometrically defined category (of perverse sheaves).  These are both difficult categories to understand monoidally, and the usual proofs of this equivalence are far from explicit.  Thankfully, in type A, both these categories have nice subcategories which admit combinatorial and algebraic descriptions: Bott-Samelson sheaves and tensor products of fundamental representations.  We describe the morphism algebras in both categories by generators and relations, giving a straightforward proof that they are equivalent. For time reasons, we probably only address sl_2.
Moreover, the morphism algebras above admit q-deformations, leading to a Quantum Algebraic Satake Equivalence (with no corresponding geometry at the moment). If time permits, we will reveal what is Satake-equivalent to U_q(sl_2) at a root of unity.