Categorification of Quantum Generalized Kac-Moody Algebras and Crystal Bases

Abstract

We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^\lambda$ which give a categrification 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 $A=(a_{ij})_{i,j \in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $\Phi: U_\A^-(\g) \to K_0(R)$ and that $\Phi$ is an isomorphism if $a_{ii}\ne 0$ for all $i\in I$. Let $B(\infty)$ 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 $\mathfrak{B}(\infty)$ and $\mathfrak{B}(\lambda)$ the isomorphism classes of irreducible graded modules over $R$ and $R^\lambda$, respectively. If $a_{ii}\ne 0$ for all $i\in I$, we define the $U_q(\g)$-crystal structures on $\mathfrak{B}(\infty)$ and $\mathfrak{B}(\lambda)$, and show that there exist crystal isomorphisms $\mathfrak{B}(\infty) \simeq B(\infty)$ and $\mathfrak{B}(\lambda) \simeq B(\lambda)$. One of the key ingredients of our approach is the perfect basis theory for generalized Kac-Moody algebras.