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 [15] . This leads to the categorification of the crystal B ( ∞ ) of the quantum Borcherds algebra U q ( g ) and its irreducible highest weight crystal B ( λ ) for arbitrary Borcherds-Cartan data. Additionally, we study the cyclotomic categorification of irreducible highest weight U q ( g ) -modules.