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

TL;DR

This paper constructs a categorical equivalence linking representations of quiver-Hecke algebras to equivariant perverse coherent sheaves, providing a new categorification of quantum loop algebras in the SL(2) case.

Contribution

It introduces a novel equivalence of categories that categorifies the quantum loop algebra for SL(2) using geometric and algebraic methods.

Findings

Establishes a weakly monoidal equivalence of categories.

Provides a categorification of the preprojective K-theoretic Hall algebra.

Compares different monoidal categorifications of quantum unipotent cells.

Abstract

We construct an equivalence of graded Abelian categories from a category of representations of the quiver-Hecke algebra of type $A_1^{(1)}$ 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-Vasserot. Using this categorification, we compare the monoidal categorification of the quantum open unipotent cells of type $A_1^{(1)}$ given by Kang-Kashiwara-Kim-Oh-Park in terms of quiver-Hecke algebras with the one given by Cautis-Williams in terms of equivariant perverse coherent sheaves on the affine Grassmannians.