Abelian Hall categories

TL;DR

This paper constructs a finite length monoidal abelian category from a quiver that categorifies the preprojective K-theoretic Hall algebra, providing canonical bases and structures related to quantum toroidal algebras.

Contribution

It introduces a new abelian category associated with quivers that categorifies the Hall algebra and offers canonical bases for quantum groups.

Findings

Provides a categorification of the Hall algebra for quivers.

Establishes a basis for the positive half of quantum toroidal algebras.

Shows the abelian category has natural renormalized r-matrices.

Abstract

To a quiver we associate a finite length monoidal abelian category which categorifies the corresponding preprojective K-theoretic Hall algebra of Varagnolo-Vasserot. The simples in this category provide a (dual) canonical basis of the Hall algebra. In particular, if the quiver is affine, this provides a basis for the positive half of the corresponding quantum toroidal algebra. We also show that this abelian category is naturally endowed with renormalized r-matrices.