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

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.