Quiver Hecke algebras for Borcherds-Cartan datum II

We give the crystal structure of the Grothendieck group $G_0(R)$ of irreducible modules over the quiver Hecke algebra $R$ constructed in \cite{TW2023}. This leads to the categorification of the crystal $B(\infty)$ of the quantum Borcherds algebra $U_q(\mathscr g)$ and its irreducible highest weight crystal $B(\lambda)$ for arbitrary Borcherds-Cartan data. Additionally, we study the cyclotomic categorification of irreducible highest weight $U_q(\mathscr g)$-modules.