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 $Φ: U_\A^-(\g) \to K_0(R)$ 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.

arXiv commentWe corrected typos (and a few small errors) and changed the definition of KLR algebras to more general version