Ice quivers with potential associated with triangulations and Cohen-Macaulay modules over orders

Abstract:

Given a triangulation of a polygon $P$ with $n$ vertices, we associate an ice quiver with potential such that the frozen part of the associated Jacobian algebra has the structure of a Gorenstein tiled $K[x]$-order $\Lambda$. Then we show that the stable category of the category of Cohen-Macaulay $\Lambda$-modules is equivalent to the cluster category $\mathcal {C}$ of type $A_{n-3}$. It gives a natural interpretation of the usual indexation of cluster tilting objects of $\mathcal {C}$ by triangulations of $P$. Moreover, it extends naturally the triangulated categorification by $\mathcal {C}$ of the cluster algebra of type $A_{n-3}$ to an exact categorification by adding coefficients corresponding to the sides of $P$. Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen-Macaulay $\Lambda$-modules and the bounded derived category of modules over a path algebra of type $A_{n-3}$.