Categorification of quantum generalized kac-moody algebras and crystal bases

We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras R and their cyclotomic quotients R^λ which give a categorification of quantum generalized Kac-Moody algebras. Let U_A(g) be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix A = (a_ij)_{i, j ⋯ I} and let K₀(R) be the Grothendieck group of finitely generated projective graded R-modules. We prove that there exists an injective algebra homomorphism $\Phi: U-{{\mathbb A}}-(\mathfrak{g}) \rightarrow K-0(R)$ and that Φ is an isomorphism if a_ii ≠ 0 for all i ⋯ I. Let B(∞) and B(λ) be the crystals of $U-q-(\mathfrak{g})$ and V(λ), respectively, where V(λ) is the irreducible highest weight U_q(g)-module. We denote by B(∞) and B(λ) the isomorphism classes of irreducible graded modules over R and R^λ, respectively. If a_ii ≠ 0 for all i ⋯ I, we define the U_q(g)-crystal structures on B(∞) and B(λ), and show that there exist crystal isomorphisms B(∞) ≃ B(∞) and B(λ) ≃ B(λ). One of the key ingredients of our approach is the perfect basis theory for generalized Kac-Moody algebras.