Séminaires : Séminaire d'Algèbre

A dimer model is a quiver with faces embedded in a surface. Dimer models on the disk and torus are particularly well-studied, though these theories have remained largely separate. Various “consistency conditions” may be imposed on dimer models on the disk or torus with implications relating to 3-Calabi-Yau properties and categorification. We extend many of these definitions and results to the setting of general surfaces with boundary. We show that the completed dimer algebra of a “strongly consistent” dimer model is bimodule internally 3-Calabi-Yau with respect to its boundary idempotent. As a consequence, the Gorenstein-projective module category of the completed boundary algebra of a suitable dimer model categorifies the cluster algebra given by its underlying ice quiver. We give a class of examples of annulus models satisfying the requisite conditions. 

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