Triangulated monoidal categorifications of finite type cluster algebras

Abstract

We propose a framework of monoidal categorification of finite type cluster algebras involving triangulated monoidal categories. Namely, given a Dynkin quiver $Q$, we consider the bounded homotopy category $\mathcal{K}_Q^{(1)}$ of a symmetric monoidal category $\mathcal{H}_Q^{(1)}$ that we define in terms of the Auslander-Reiten theory of $Q$. Using some iterated mapping cone procedure, we construct a distinguished family $\{ C_{\bullet}[\beta] \}_{\beta \in \Delta_+}$ of chain complexes in $\mathcal{K}_Q^{(1)}$ characterized (up to isomorphism) by homological conditions similar to those of higher exact sequences appearing in the context of higher homological algebra. We then prove that the distinguished triangle in $\mathcal{K}_Q^{(1)}$ given by each mapping cone categorifies an exchange relation in the finite type cluster algebra $\mathcal{A}_Q$ with initial exchange quiver $Q$ (for a suitable choice of frozen variables). As a consequence, we obtain that for each positive root $\beta$, the Euler characteristic of $C_{\bullet}[\beta]$ coincides with the truncated $q$-character of the simple module $L[\beta]$ in the HL category $\mathcal{C}_{\xi}^{(1)}$ categorifying the cluster variable $x[\beta]$ of $\mathcal{A}_Q$ via Hernandez-Leclerc's monoidal categorification. Along the way, we establish a uniform formula for the dominant monomial of $L[\beta]$ in all types $A_n$ and $D_n$ for arbitrary orientations (agreeing with Brito-Chari's results in type $A_n$).