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...HomC(KQ)(Y, Z), the composition is defined by (g ◦ f)i = ∑ i1+i2=i gi1 ◦ F i1(fi2) for all i ∈ Z. In [22], Happel proves that Db(KQ) has Auslander-Reiten triangles. For a Dynkin quiver Q, he shows in =-=[21]-=- that the Auslander-Reiten quiver of Db(KQ) is Z∆ where ∆ is the underlying Dynkin diagram of Q. Then the Auslander-Reiten quiver of C(KQ) is Z∆/ϕ, where ϕ is the graph automorphism induced by τ−1[1]....

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...Introduction In 2001, Fomin and Zelevinsky introduced a new class of algebras with a rich combinatorial structure, called cluster algebras [17], [18] motivated by canonical bases and total positivity =-=[29]-=-, [34], [35]. Since their emergence, cluster algebras have generated a lot of interest, coming in particular from their links with many other subjects: combinatorics, Poisson geometry, integrable syst...

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...uction In 2001, Fomin and Zelevinsky introduced a new class of algebras with a rich combinatorial structure, called cluster algebras [17], [18] motivated by canonical bases and total positivity [29], =-=[34]-=-, [35]. Since their emergence, cluster algebras have generated a lot of interest, coming in particular from their links with many other subjects: combinatorics, Poisson geometry, integrable systems, T...

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...h potential and certain Frobenius categories given by Cohen-Macaulay modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to =-=[4, 13, 40, 41]-=- for a general background on this subject and also to [2, 3, 14, 25, 27, 28] for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representatio...

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...nd thick(T ) = C, where thick(T ) is the smallest full triangulated subcategory of C containing T and closed under isomorphisms and direct summands. Theorem 4.5 ([30, Theorem 4.3], [25, Theorem 2.2], =-=[7]-=-). Let C be an algebraic triangulated KrullSchmidt category. If C has a tilting object T , then there exists a triangle-equivalence C → Kb(proj EndC(T )). Now we get the following theorem which is ana...

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...iated an important algebra, P(Q,W ), called the Jacobian algebra. For each triangulation of a bordered surface with marked points, a quiver was introduced by Caldero, Chapoton and Schiffler in type A =-=[11]-=- and Fomin, Shapiro and Thurston in general cases [16]. They showed that the combinatorics of triangulations of the surface correspond to that of the cluster algebra defined by the quiver. Later in [3...

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...luster categories, were widely generalized in several directions. One of these generalizations, was given by Amiot [1] by using quivers with potential, introduced by Derksen, Weyman and Zelevinsky in =-=[15]-=-. A quiver with potential is a quiver Q together with a linear combination W of cyclic paths in Q. To each quiver with potential is associated an important algebra, P(Q,W ), called the Jacobian algebr...

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... spaces, and, last but not least, representations of finite-dimensional algebras. In seminal articles, Marsh, Reineke, Zelevinsky [36], Buan, Marsh, Reineke, Reiten, Todorov [9] and Caldero, Chapoton =-=[10]-=- have shown that the important class of acyclic cluster algebras could be modeled with categories constructed from representations of quivers. Under these categorifications, clusters are represented b...

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...an algebra. For each triangulation of a bordered surface with marked points, a quiver was introduced by Caldero, Chapoton and Schiffler in type A [11] and Fomin, Shapiro and Thurston in general cases =-=[16]-=-. They showed that the combinatorics of triangulations of the surface correspond to that of the cluster algebra defined by the quiver. Later in [32], Labardini-Fragoso associated to each triangulation...

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...h potential and certain Frobenius categories given by Cohen-Macaulay modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to =-=[4, 13, 40, 41]-=- for a general background on this subject and also to [2, 3, 14, 25, 27, 28] for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representatio...

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...able systems, Teichmüller spaces, and, last but not least, representations of finite-dimensional algebras. In seminal articles, Marsh, Reineke, Zelevinsky [36], Buan, Marsh, Reineke, Reiten, Todorov =-=[9]-=- and Caldero, Chapoton [10] have shown that the important class of acyclic cluster algebras could be modeled with categories constructed from representations of quivers. Under these categorifications,...

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...property, called cluster tilting objects. Since that time, these categories, called cluster categories, were widely generalized in several directions. One of these generalizations, was given by Amiot =-=[1]-=- by using quivers with potential, introduced by Derksen, Weyman and Zelevinsky in [15]. A quiver with potential is a quiver Q together with a linear combination W of cyclic paths in Q. To each quiver ...

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...cts: combinatorics, Poisson geometry, integrable systems, Teichmüller spaces, and, last but not least, representations of finite-dimensional algebras. In seminal articles, Marsh, Reineke, Zelevinsky =-=[36]-=-, Buan, Marsh, Reineke, Reiten, Todorov [9] and Caldero, Chapoton [10] have shown that the important class of acyclic cluster algebras could be modeled with categories constructed from representations...

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...1] and Fomin, Shapiro and Thurston in general cases [16]. They showed that the combinatorics of triangulations of the surface correspond to that of the cluster algebra defined by the quiver. Later in =-=[32]-=-, Labardini-Fragoso associated to each triangulation σ of a bordered surface with marked points a potential W (σ) on the corresponding quiver Q(σ). He proved that flips of triangulations are compatibl...

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... 0 and 0→ N → E′ →M → 0 are non-split short exact sequences then X ′MX ′ N = X ′ E +X ′ E′ . Therefore, the map X ′ is called a cluster character. Remark 3.27. The only difference with the setting of =-=[19]-=- is that CM(Λ) does not have finite dimensional morphism spaces over K. However, the considered morphism spaces are finitely generated as Γσ-modules and CM(Λ) has finite dimensional morphism spaces ov...

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...ically equivalent terms. For each arrow a ∈ Q1, the cyclic derivative ∂a is the K-linear map⊕ i≥1KQi,cyc → KQ defined on cycles by ∂a(a1 · · · ad) = ∑ ai=a ai+1 · · · ada1 · · · ai−1. Definition 2.1. =-=[8]-=- An ice quiver with potential is a triple (Q,W,F ), where (Q,W ) is a quiver with potential and F is a subset of Q0. Vertices in F are called frozen vertices. The frozen Jacobian algebra is defined by...

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...ficient per side of the polygon and the situation is invariant by rotation of the polygon. More precisely, using the cluster character X ′ defined in [19, section 3] on Frobenius categories (see also =-=[37]-=-), we get the following theorem. Theorem 1.4 (Theorem 3.29). If K is algebraically closed, the category CM(Λ) categorifies through the cluster character X ′ a cluster algebra structure on the homogene...

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...h potential and certain Frobenius categories given by Cohen-Macaulay modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to =-=[4, 13, 40, 41]-=- for a general background on this subject and also to [2, 3, 14, 25, 27, 28] for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representatio...

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... ) = ⊕ i∈Z HomDb(KQ)(F iX,Y ), where X and Y are objects in Db(KQ). For f ∈ HomC(KQ)(X,Y ) and g ∈ HomC(KQ)(Y, Z), the composition is defined by (g ◦ f)i = ∑ i1+i2=i gi1 ◦ F i1(fi2) for all i ∈ Z. In =-=[22]-=-, Happel proves that Db(KQ) has Auslander-Reiten triangles. For a Dynkin quiver Q, he shows in [21] that the Auslander-Reiten quiver of Db(KQ) is Z∆ where ∆ is the underlying Dynkin diagram of Q. Then...

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...lso to [2, 3, 14, 25, 27, 28] for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representation theory and cluster categories has been found =-=[2, 14, 25, 31]-=-. We will enlarge this connection by studying the frozen Jacobian algebras associated with triangulations of surfaces from the viewpoint of Cohen-Macaulay representation theory. Through this paper, le...

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...tegory of modules over a path algebra of type An−3. 1. Introduction In 2001, Fomin and Zelevinsky introduced a new class of algebras with a rich combinatorial structure, called cluster algebras [17], =-=[18]-=- motivated by canonical bases and total positivity [29], [34], [35]. Since their emergence, cluster algebras have generated a lot of interest, coming in particular from their links with many other sub...

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...modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to [4, 13, 40, 41] for a general background on this subject and also to =-=[2, 3, 14, 25, 27, 28]-=- for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representation theory and cluster categories has been found [2, 14, 25, 31]. We will enla...

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...h potential and certain Frobenius categories given by Cohen-Macaulay modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to =-=[4, 13, 40, 41]-=- for a general background on this subject and also to [2, 3, 14, 25, 27, 28] for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representatio...

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...modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to [4, 13, 40, 41] for a general background on this subject and also to =-=[2, 3, 14, 25, 27, 28]-=- for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representation theory and cluster categories has been found [2, 14, 25, 31]. We will enla...

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...a3, b2c2, c2a2 − a3c1, b3c3 − c1b2, c3a3〉. 4 L. DEMONET AND X. LUO 12 3 4 5 6 7 a bc d e f g h i αβ γ Figure 2. Note that this ice quiver with potential can be constructed from preprojective algebras =-=[8, 20]-=-. 2.2. Ice quivers with potential arising from triangulations. We recall the definition of triangulations of polygons and introduce our definition of ice quivers with potential arising from triangulat...

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...modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to [4, 13, 40, 41] for a general background on this subject and also to =-=[2, 3, 14, 25, 27, 28]-=- for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representation theory and cluster categories has been found [2, 14, 25, 31]. We will enla...

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...rder A, if K(x)⊗RA is a semisimple K(x)-algebra, then we call A an isolated singularity. By using the notions above, we have the following well-known results in Auslander-Reiten theory. Theorem 3.18 (=-=[4, 38, 39]-=-). Let A be an R-order. If A is an isolated singularity, then (1) [4, Chapter I, Proposition 8.3] The construction τ gives an equivalence CM(A)→ CM(A), where CM(A) is the quotient of CM(A) by the idea...

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...modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to [4, 13, 40, 41] for a general background on this subject and also to =-=[2, 3, 14, 25, 27, 28]-=- for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representation theory and cluster categories has been found [2, 14, 25, 31]. We will enla...

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...ved that flips of triangulations are compatible with mutations of quivers with potential. This is partially generalized to the case of tagged triangulations by Labardini-Fragoso and Cerulli Irelli in =-=[12]-=-, and completely by Labardini-Fragoso in [33]. The aim of this paper is to enhance some of these known results by considering ice quivers with potential and certain Frobenius categories given by Cohen...

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... In 2001, Fomin and Zelevinsky introduced a new class of algebras with a rich combinatorial structure, called cluster algebras [17], [18] motivated by canonical bases and total positivity [29], [34], =-=[35]-=-. Since their emergence, cluster algebras have generated a lot of interest, coming in particular from their links with many other subjects: combinatorics, Poisson geometry, integrable systems, Teichmu...

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...modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to [4, 13, 40, 41] for a general background on this subject and also to =-=[2, 3, 14, 25, 27, 28]-=- for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representation theory and cluster categories has been found [2, 14, 25, 31]. We will enla...

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...modules. The study of CohenMacaulay modules (or lattices) over orders is a classical subject in representation theory. We refer to [4, 13, 40, 41] for a general background on this subject and also to =-=[2, 3, 14, 25, 27, 28]-=- for recent results about connections with tilting theory. Recently a strong connection between Cohen-Macaulay representation theory and cluster categories has been found [2, 14, 25, 31]. We will enla...

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...ed in [17] from Plücker coordinates. Similar results are obtained in the categorifications of the cluster structures of coordinate rings of (general) Grassmannians independently by Baur, King, Marsh =-=[6]-=- (see also Jensen, King, Su [26, Theorem 4.5]). Usually, the cluster category C(KQ) is constructed as an orbit category of the bounded derived category Db(KQ). We can reinterpret this result in this c...

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...(x2) (x) R · · · R R ... ... ... . . . ... ... (x2) (x2) (x2) · · · R R (x2) (x2) (x2) · · · (x) R n×n. (1.1) Remark 1.2. The order Λ is an almost Bass order of type (IVa), see for example =-=[23]-=-. Theorem 1.3 (Theorems 2.9, 2.25, 3.8, 3.13 and 3.16). (1) For any triangulation σ of P , we can map each edge i of σ to the indecomposable Cohen-Macaulay Λ-module eFΓσei where ei is the idempotent o...