Categories over quantum affine algebras and monoidal categorification

Abstract

Let $U_{q}'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine $\mathit{ADE}$ type, and $\mathcal{C}_{\mathfrak{g}}^{0}$ the Hernandez-Leclerc category of finite-dimensional $U_{q}'(\mathfrak{g})$-modules. For a suitable infinite sequence $\widehat{w}_{0}= \cdots s_{i_{-1}}s_{i_{0}}s_{i_{1}} \cdots$ of simple reflections, we introduce subcategories $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$ of $\mathcal{C}_{\mathfrak{g}}^{0}$ for all $a \leqslant b \in \mathbf{Z} \sqcup\{\pm \infty \}$. Associated with a certain chain $\mathfrak{C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(\mathfrak{C})$ in $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$, which consists of Kirillov-Reshetikhin modules. The category $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$ provides a monoidal categorification of the cluster algebra $K(\mathcal{C}_{\mathfrak{g}}^{[a,b]})$, whose set of initial cluster variables is $[M(\mathfrak{C})]$. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on $\mathcal{C}_{\mathfrak{g}}^{-}$ by Hernandez-Leclerc since it is $\mathcal{C}_{\mathfrak{g}}^{[-\infty,0]}$, and is also applicable to $\mathcal{C}_{\mathfrak{g}}^{0}$ since it is $\mathcal{C}_{\mathfrak{g}}^{[-\infty,\infty]}$.