Categorification of highest weight modules over quantum generalized Kac–Moody algebras

Let $U_q(\mathfrak g)$ be a quantum generalized Kac–Moody algebra and let $V(\Lambda)$ be the integrable highest weight $U_q(\mathfrak g)$-module with highest weight $\Lambda$. We prove that the cyclotomic Khovanov–Lauda–Rouquier algebra $R^\Lambda$ provides a categorification of $V(\Lambda)$.