Quantum groups of Borcherds-Cartan type and Khovanov-Lauda-Rouquier algebras

TL;DR

This paper constructs Khovanov-Lauda-Rouquier algebras for quantum groups associated with quivers, including those with loops, and demonstrates their role in categorifying the negative part and highest weight modules of these quantum groups.

Contribution

It introduces a categorification framework for quantum groups of Borcherds-Cartan type using KLR algebras, extending previous categorification results to quivers with loops.

Findings

Indecomposable projective modules realize the canonical basis.

Cyclotomic KLR algebras categorify irreducible highest weight modules.

The approach applies to quantum groups associated with quivers with loops.

Abstract

We categorify a class of quantum groups associated with quivers, possibly with loops, by constructing the corresponding Khovanov-Lauda-Rouquier algebras (KLR) algebras $R$. We prove that the indecomposable projective $R$-modules realize the canonical basis of the negative part $U^-$ of the quantum group. Moreover, for $\Lambda \in P^+$, the cyclotomic KLR algebra $R^\Lambda$ provide a categorification of the irreducible highest weight $U$-module $V(\Lambda)$.