Cluster algebra structures on module categories over quantum affine algebras

Abstract

We study monoidal categorifications of certain monoidal subcategories ${\mathcal{C}}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_{\infty }$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category ${\mathcal{C}}_{\mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.

66 pages