Categorification and supercategorification of highest weight modules over quantum generalized Kac-Moody algebras

다국어 초록 (Multilingual Abstract)

Let $U_q(\g)$ be the quantum generalized Kac-Moody algebras and let $V(\Lambda)$ be the
integrable highest weight $U_q(\g)$-module. We prove that $V(\Lambda)$
can be categorified via the cyclotomic quiver Hecke algebra $\KLR^\Lambda$ and
supercategorified via the cyclotomic quiver Hecke superalgebras $\R^\Lambda$, simultaneously.
Moreover, since $U^-_q(\g)$ is the projective limit of
$V(\Lambda)$, $U^-_q(\g)$ can be also categorified via the quiver Hecke algebra $\KLR$ and
supercategorified via the quiver Hecke superalgebras $\R$, simultaneously.

목차 (Table of Contents)

• Abstract
• 1 Introduction
• 2 Quantum generalized Kac-Moody algebras
• 2.1 Generalized Kac-Moody algebras
• 2.2 Quantum deformation
• 2.3 Lower crystal bases
• 2.4 Upper crystal bases
• 2.5 Upper global bases
• 2.6 Perfect bases
• 3 Categorification
• 3.1 The quiver Hecke algebra R
• 3.2 The algebra R(n \alpha_i$)
• 3.3 The Poincar'e-Birkhoff-Witt-type basis
• 3.4 Quantum Serre relation
• 3.5 Crystal structure and strong perfect bases
• 3.6 The functors $E_i$, $F_i$ and $\overline{F}_i$
• 3.7 Cyclotomic quotient
• 3.8 Categorification
• 4 Supercategorification
• 4.1 Supercategories and superbimodules
• 4.2 The quiver Hecke superalgebra R
• 4.3 Strong perfect basis of [Rep(R(\beta))]
• 4.4 The superfunctors $E_i$, $F_i$ and $\overline{F}_i$
• 4.5 Cyclotomic quotient
• 4.6 The superfunctors $E_i^\Lambda$ and $F_i^\Lambda$
• 4.7 Supercategorification
• Abstract (in Korean) 130
• Acknowledgement (in Korean) 131