Categorification of quantum generalized kac-moody algebras and crystal bases

Seok Jin Kang, Se Jin Oh, Euiyong Park

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras R and their cyclotomic quotients Rλ which give a categorification of quantum generalized Kac-Moody algebras. Let UA(g) be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix A = (aij)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 aii ≠ 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 Uq(g)-module. We denote by B(∞) and B(λ) the isomorphism classes of irreducible graded modules over R and Rλ, respectively. If aii ≠ 0 for all i ⋯ I, we define the Uq(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 languageEnglish
Article number1250116
JournalInternational Journal of Mathematics
Issue number11
StatePublished - 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.


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


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