Cluster algebra structures on module categories over quantum affine algebras

Abstract

We study monoidal categorifications of certain monoidal subcategories C J $\mathcal {C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures on their Grothendieck rings K ( C J ) $K(\mathcal {C}_J)$ are closely related to the category of finite-dimensional modules over quiver Hecke algebra of type A ∞ $A_\infty$ via the generalized quantum Schur–Weyl duality functors. In particular, when the quantum affine algebra is of type A $A$ or B $B$ , the subcategory coincides with the monoidal category C g 0 $\mathcal {C}_{\mathfrak {g}}^0$ introduced by Hernandez–Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.