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

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 $V(\Lambda)$.