JournalsprimsVol. 52, No. 2pp. 141–205

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

  • Laurent Demonet

    Nagoya University, Japan
  • Xueyu Luo

    Nagoya University, Japan
Ice Quivers with Potential Arising from Once-punctured Polygons and Cohen–Macaulay Modules cover
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Abstract

Given a tagged triangulation of a once-punctured polygon PP^* with nn 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 K[X]K[X]-order Λ\Lambda. Then we show that the stable category of the category of Cohen–Macaulay Λ\Lambda-modules is equivalent to the cluster category C\mathcal C of type DnD_n. It gives a natural interpretation of the usual indexation of cluster tilting objects of C\mathcal C by tagged triangulations of PP^*. Moreover, it extends naturally the triangulated categorification by C\mathcal C of the cluster algebra of type DnD_n to an exact categorification by adding coefficients corresponding to the sides of PP. Finally, we lift the previous equivalence of categories to an equivalence between the stable category of graded Cohen–Macaulay Λ\Lambda-modules and the bounded derived category of modules over a path algebra of type DnD_n.

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