Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras

Let $C$ be the category of finite-dimensional modules over a simply-laced quantum affine algebra $U_q\left(\hat{g}\right)$. For any height function ξ and $\ell \in Z_{\ge 1}$, we introduce certain subcategories $C_{\ell }^{\le \xi }$ of $C$, and prove that the quantum Grothendieck ring $K_t\left(C_{\ell }^{\le \xi }\right)$ of $C_{\ell }^{\le \xi }$ admits a quantum cluster algebra structure. We classify the real prime simple modules (call them the Hernandez–Leclerc modules) in $C_1^{\le \xi }$ in terms of their highest l-weight monomials by relating this subcategory $C_1^{\le \xi }$ to the cluster category $C_1^{\le \xi }$ of the Dynkin quiver given by ξ. For any ℓ, we formulate the modified Hernandez-Leclerc conjectures, and prove them for the subcategories $C_{\ell }^{\le \xi }$ whose Grothendieck rings are cluster algebras of finite type.