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

TL;DR

This paper constructs a link between triangulations of polygons, ice quivers with potentials, and Cohen-Macaulay modules over orders, providing a categorification of cluster algebras of type A(n-3).

Contribution

It introduces a new association between triangulations and ice quivers with potentials, extending cluster categorification to include coefficients via Cohen-Macaulay modules.

Findings

The Jacobian algebra forms a Gorenstein tiled K[x]-order.

The stable category of Cohen-Macaulay modules is equivalent to the cluster category of type A(n-3).

The categorification extends to an exact one with coefficients and lifts to derived categories.

Abstract

Given a triangulation of a polygon P with n vertices, we associate an ice quiver with potential such that the associated Jacobian algebra has the structure of a Gorenstein tiled K[x]-order L. Then we show that the stable category of the category of Cohen-Macaulay L-modules is equivalent to the cluster category C of Dynkin type A(n-3). It gives a natural interpretation of the usual indexation of cluster tilting objects of C by triangulations of P. Moreover, it extends naturally the triangulated categorification by 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 L-modules and the bounded derived category of modules over a quiver of type A(n-3).