Categories over quantum affine algebras and monoidal categorification

Abstract

Let U-q'(g) be a quantum affine algebra of untwisted affine ADE type, and C-g(0) the Hernandez-Leclerc category of finite-dimensional U-q'(g)-modules. For a suitable infinite sequence (w) over cap (0) = ... s(i-1)s(i0)s(i1) ... of simple reflections, we introduce subcategories C-g([a,b]) of C-g(0) for all a <= b is an element of Z (sic) {+/-infinity}. Associated with a certain chain C of intervals in [a, b] we construct a real simple commuting family M(C) in C-g([a,b]), which consists of Kirillov-Reshetikhin modules. The category C-g([a,b]) provides a monoidal categorification of the cluster algebra K(C-g([a,b])), whose set of initial cluster variables is [M(C)]. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on C-g(-) by Hernandez-Leclerc since it is C-g([-infinity,0]), and is also applicable to C-g(0) since it is C-g([-infinity,infinity]).