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...ociated cluster, a derived equivalence of the Brauer graph algebra whose graph is the triangulation, a derived equivalence of the Brauer tree algebra of the dual graph of the triangulation and, using =-=[23, 28]-=- (see also [5, 20]), a derived equivalence of endomorphism algebras in Frobenius categories in the cluster context. We observe that in the case of Brauer graph algebras the Brauer graph algebra associ...

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...k of Franco [11, §6.1] and Buan-Iyama-Reiten-Smith [4, Defn. 1.1]. In the latter case, a slightly different approach is used, whereby any arrow joining two boundary (or frozen) vertices is considered to be frozen and hence does not contribute an F-term relation, while in our case, we may have internal arrows with both end-points being boundary vertices (see Figures 23 and 28). Buan-Iyama-Reiten-Smith [4] give a description [4, Thm. 6.6] of the endomorphism algebras of some cluster-tilting objects over preprojective algebras as frozen Jacobian algebras in the sense of [4, Defn 1.1]. DemonetLuo [8] give a 2-Calabi-Yau categorification C of the Grassmannian Gr(2, n) using frozen Jacobian algebras in the sense of [4, Defn. 1.1]; these algebras are the endomorphism algebras of cluster-tilting objects in C (see [8, Thm. 1.3]). Definition 3.7. We write AD for the dimer algebra AQ(D) associated to the dimer model Q(D). It follows from the defining relations that, for any vertex I ∈ Q0(D), the product in AD of the arrows in any cycle that starts at I and bounds a face is the same. We denote this element by uI , and write u = ∑ I∈Q0(D) uI . (3.2) It similarly follows from the relations that u c...

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...gories to an equivalence between the stable category of graded Cohen-Macaulay Λ-modules and the bounded derived category of modules over a path algebra of type Dn. 1. Introduction In a previous paper =-=[10]-=-, we constructed ice quivers with potential arising from triangulations of polygons and we proved that the frozen parts of their frozen Jacobian algebras are orders. We proved that the categories of C...