Categorification of Quantum Kac-Moody Algebras via Quiver Hecke Algebras and Cyclotomic Quotients

This monograph presents a rigorous examination of the categorification of quantum Kac-Moody algebras Uq(g) through the lens of Quiver Hecke algebras, also known as Khovanov-Lauda-Rouquier (KLR) algebras. We focus specifically on the cyclotomic quotients R() of these algebras and their role in categorifying irreducible highest weight modules V(). We detail the construction of the KLR algebra R() associated with a symmetrizable Cartan datum and define the cyclotomic ideal I. The central result discussed is the isomorphism between the Grothendieck ring of the category of finitely generated projective graded R-modules and the integral form of the highest weight module V(). Furthermore, we explore the Brundan-Kleshchev isomorphism, which connects cyclotomic KLR algebras of type A to classical cyclotomic Hecke algebras, thereby unifying the geometric and algebraic approaches to categorification. The structural properties of crystal bases within this categorical framework are also addressed.