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-lambda 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 is an element of I and let K-0(R) be the Grothendieck group of finitely generated projective graded R-modules. We prove that there exists an injective algebra homomorphism Phi : U-A(-) (g) -> K-0(R) and that Phi is an isomorphism if a(ii) not equal 0 for all i is an element of I. Let B(infinity) and B(lambda) be the crystals of U-q(-) (g) and V (lambda), respectively, where V (lambda) is the irreducible highest weight U-q(g)-module. We denote by B(infinity) and B(lambda) the isomorphism classes of irreducible graded modules over R and R-lambda, respectively. If a(ii) not equal 0 for all i is an element of I, we define the U-q(g)-crystal structures on B(infinity) and B(lambda), and show that there exist crystal isomorphisms B(infinity) similar or equal to B(infinity) and B(lambda) similar or equal to B(lambda). One of the key ingredients of our approach is the perfect basis theory for generalized Kac-Moody algebras.