Ice Quivers with Potential Arising from Once-punctured Polygons and Cohen–Macaulay Modules

  • Laurent Demonet

    Nagoya University, Japan
  • Xueyu Luo

    Nagoya University, Japan

Abstract

Given a tagged triangulation of a once-punctured polygon with vertices, we associate an ice quiver with potential such that the frozen part of the associated frozen Jacobian algebra has the structure of a Gorenstein -order . Then we show that the stable category of the category of Cohen–Macaulay -modules is equivalent to the cluster category of type . It gives a natural interpretation of the usual indexation of cluster tilting objects of by tagged triangulations of . Moreover, it extends naturally the triangulated categorification by of the cluster algebra of type to an exact categorification by adding coefficients corresponding to the sides of . Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen–Macaulay -modules and the bounded derived category of modules over a path algebra of type .

Cite this article

Laurent Demonet, Xueyu Luo, Ice Quivers with Potential Arising from Once-punctured Polygons and Cohen–Macaulay Modules. Publ. Res. Inst. Math. Sci. 52 (2016), no. 2, pp. 141–205

DOI 10.4171/PRIMS/177