2011/02/25 by Seok‐Jin Kang, Seok-Jin Kang, Se-jin Oh +5
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.1102.5165
We corrected typos (and a few small errors) and changed the definition of KLR algebras to more general version
openalex publication_date 2011/02/25 · arxiv created 2012/08/20 · arxiv updated 2012/08/21 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras R and their cyclotomic quotients Rλ 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=(aij)i,j ∈ I and let K0(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) → K0(R) and that Φ is an isomorphism if aii≠ 0 for all i∈ I. Let B(∞) and B(λ) be the crystals of Uq-(\g) and V(λ), respectively, where V(λ) is the irreducible highest weight Uq(\g)-module. We denote by \mathfrakB(∞) and \mathfrakB(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 \mathfrakB(∞) and \mathfrakB(λ), and show that there exist crystal isomorphisms \mathfrakB(∞) ≃ B(∞) and \mathfrakB(λ) ≃ B(λ). One of the key ingredients of our approach is the perfect basis theory for generalized Kac-Moody algebras.