Monoidal categorification and quantum affine algebras III

Let $U_q^{\mathrm{\prime }}(\mathfrak{g})$ be an arbitrary quantum affine algebra of either untwisted or twisted type, and let ${\mathcal{C}}_{\mathfrak{g}}^0$ be its Hernandez-Leclerc category. We denote by $\mathsf{B}$ the braid group determined by the simply-laced finite type Lie algebra $\mathsf{g}$ associated with $U_q^{\mathrm{\prime }}(\mathfrak{g})$. For any complete duality datum $\mathbb{D}$ and any sequence of simple roots of $\mathsf{g}$, we construct the corresponding affine cuspidal modules and affine determinantial modules and study their key properties including T-systems. Then, for any element $b$ of the positive braid monoid ${\mathsf{B}}^{+}$, we introduce a distinguished subcategory ${\mathcal{C}}_{\mathfrak{g}}^{\mathbb{D}}(b)$ of ${\mathcal{C}}_{\mathfrak{g}}^0$ categorifying the specialization of the bosonic extension $\widehat{\mathcal{A}}(b)$ at $q^{1\mathrm{/}2}=1$ and investigate its properties including the categorical PBW structure. We finally prove that the subcategory ${\mathcal{C}}_{\mathfrak{g}}^{\mathbb{D}}(b)$ provides a monoidal categorification of the (quantum) cluster algebra $\widehat{\mathcal{A}}(b)$, which significantly generalizes the earlier monoidal categorification developed by the authors.