Coherent categorification of quantum loop algebras: The SL(2) case

Abstract

We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type A 1 ( 1 ) {A_{1}^{(1)}} to the category of equivariant perverse coherent sheaves on the nilpotent cone of type A. We prove that this equivalence is weakly monoidal. This gives a representation-theoretic categorification of the preprojective K-theoretic Hall algebra considered by Schiffmann and Vasserot. Using this categorification, we compare the monoidal categorification of the quantum open unipotent cells of type A 1 ( 1 ) {A_{1}^{(1)}} given by Kang, Kashiwara, Kim, Oh and Park in terms of quiver-Hecke algebras with the one given by Cautis and Williams in terms of equivariant perverse coherent sheaves on the affine Grassmannians.