Categories over quantum affine algebras and monoidal categorification

Abstract

Let $U_q^{\prime }(\mathfrak{g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and ${\mathcal{C}}_{\mathfrak{g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q^{\prime }(\mathfrak{g})$-modules. For a suitable infinite sequence ${\hat{w}}_0=\cdots s_{i_{-1}}{s}_{i_0}{s}_{i_1}\cdots$ of simple reflections, we introduce subcategories ${\mathcal{C}}_{\mathfrak{g}}^{[a,b]}$ of ${\mathcal{C}}_{\mathfrak{g}}^0$ for all $a\le b\in \mathbb{Z}\bigsqcup \{\pm \infty \}$. 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 ${\mathcal{C}}_{\mathfrak{g}}^{[-\infty ,\infty ]}$.

10 pages. This paper is an announcement whose details will appear elsewhere