Quiver Hecke algebras for Borcherds-Cartan datum II

We give the crystal structure of the Grothendieck group $G_0\left(R\right)$ of irreducible modules over the quiver Hecke algebra R constructed in [15]. This leads to the categorification of the crystal $B(\infty )$ of the quantum Borcherds algebra $U_q\left(g\right)$ and its irreducible highest weight crystal $B\left(\lambda \right)$ for arbitrary Borcherds-Cartan data. Additionally, we study the cyclotomic categorification of irreducible highest weight $U_q\left(g\right)$-modules.