## Abstract

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 K_{0}(R) be the Grothendieck group of finitely generated projective graded R-modules. We prove that there exists an injective algebra homomorphism $\Phi: U-{{\mathbb A}}-(\mathfrak{g}) \rightarrow K-0(R)$ 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.

Original language | English |
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Article number | 1250116 |

Journal | International Journal of Mathematics |

Volume | 23 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2012 |

### Bibliographical note

Funding Information:The first and third authors were supported by KRF Grant # 2007-341-C00001 and NRF Grant # 2010-0010753. The second author was supported by NRF Grant # 2010-0019516 and BK21 Mathematical Sciences Division.

## Keywords

- Categorification
- crystals
- Khovanov-Lauda-Rouquier algebras
- perfect bases
- quantum generalized Kac-Moody algebras