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54
Cluster Categories for Algebras of Global Dimension 2 and . . .
, 2008
"... Let k be a field and A a finitedimensional kalgebra of global dimension ≤ 2. We construct a triangulated category CA associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When CA is Homfinite, we prove that it is 2CY and endowed with a canonical cluster ..."
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Cited by 46 (7 self)
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Let k be a field and A a finitedimensional kalgebra of global dimension ≤ 2. We construct a triangulated category CA associated to A which, if A is hereditary, is triangle equivalent to the cluster category of A. When CA is Homfinite, we prove that it is 2CY and endowed with a canonical clustertilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by GeissLeclercSchröer and by BuanIyamaReitenScott. Our results rely on quivers with potential. Namely, we introduce a cluster category C (Q,W) associated to a quiver with potential (Q, W). When it is Jacobifinite we prove that it is endowed with a clustertilting object whose endomorphism algebra is isomorphic
Cluster structures for 2CalabiYau categories and unipotent groups
"... Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This c ..."
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Cited by 46 (11 self)
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Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This class of 2CalabiYau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2CalabiYau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related
Mutation of clustertilting objects and potentials
 Amer. Journal Math. (2008
"... Abstract. We prove that mutation of clustertilting objects in triangulated 2CalabiYau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2CYtilted algebras and Jacobian algebras associated with quivers with potentials. We show that cl ..."
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Cited by 21 (4 self)
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Abstract. We prove that mutation of clustertilting objects in triangulated 2CalabiYau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2CYtilted algebras and Jacobian algebras associated with quivers with potentials. We show that clustertilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3CY algebras. The nearly Morita equivalence for 2CYtilted algebras is shown to hold for the finite length modules over Jacobian algebras.
Cluster tilting for higher Auslander algebras
"... Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the ca ..."
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Cited by 19 (6 self)
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Abstract. The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representationfinite algebras and Auslander algebras. The nAuslanderReiten translation functor τn plays an important role in the study of ncluster tilting subcategories. We study the category Mn of preinjectivelike modules obtained by applying τn to injective modules repeatedly. We call a finite dimensional algebra Λ ncomplete if Mn = add M for an ncluster tilting object M. Our main result asserts that the endomorphism algebra EndΛ(M) is (n + 1)complete. This gives an inductive construction of ncomplete algebras. For example, any representationfinite hereditary algebra Λ (1) is 1complete. Hence the Auslander algebra Λ (2) of Λ (1) is 2complete. Moreover, for any n ≥ 1, we have an ncomplete algebra Λ (n) which has an ncluster tilting object M (n) such that Λ (n+1) = End Λ (n)(M (n)). We give the presentation of Λ (n) by a quiver with relations. We apply our results to construct ncluster tilting subcategories of derived categories of ncomplete algebras. Contents 1. Our results 3 1.1. ncluster tilting in module categories 4
Defining an m−cluster category
, 2005
"... Abstract. We show that a certain orbit category considered by Keller encodes the combinatorics of the mclusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. T ..."
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Cited by 17 (2 self)
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Abstract. We show that a certain orbit category considered by Keller encodes the combinatorics of the mclusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. This allows us to give typeuniform proofs of certain results of Fomin and Reading in the simply laced cases. For Φ any root system, Fomin and Zelevinsky [FZ] define a cluster complex ∆(Φ), a simplicial complex on Φ≥−1, the almost positive roots of Φ. Its facets (maximal faces) are called clusters. In [BM+], starting in the more general context of a finite dimensional hereditary algebra H over a field K, Buan et al. define a cluster category C(H) = D b (H)/τ −1 [1]. (D b (H) is the bounded derived category of representations of H; more will be said below about it, its shift functor [1], and its AuslanderReiten translate τ.) The cluster category C(H) is a triangulated KrullSchmidt category. We will be mainly interested in the case where H is a path algebra associated to the simply laced root system Φ, in which case we write C(Φ) for C(H). There is a bijection V taking Φ≥−1 to the indecomposables of C(Φ). A (cluster)tilting set
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 14 (9 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
Compact corigid objects in triangulated categories and cotstructures
"... Abstract. In the work of Hoshino, Kato and Miyachi, [11], the authors look at tstructures induced by a compact object, C, of a triangulated category, T, which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a nondegenerate tstructure on ..."
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Cited by 13 (6 self)
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Abstract. In the work of Hoshino, Kato and Miyachi, [11], the authors look at tstructures induced by a compact object, C, of a triangulated category, T, which is rigid in the sense of Iyama and Yoshino, [12]. Hoshino, Kato and Miyachi show that such an object yields a nondegenerate tstructure on T whose heart is equivalent to Mod(End(C) op). Rigid objects in a triangulated category can the thought of as behaving like chain differential graded algebras (DGAs). Analogously, looking at objects which behave like cochain DGAs naturally gives the dual notion of a corigid object. Here, we see that a compact corigid object, S, of a triangulated category, T, induces a structure similar to a tstructure which we shall call a cotstructure. We also show that the coheart of this nondegenerate cotstructure is equivalent to Mod(End(S) op), and hence an abelian subcategory of T. Suppose T is a triangulated category with set indexed coproducts and let
Generalized cluster complexes via quiver representations
 DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NEW
, 2007
"... We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using d−cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d−compa ..."
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Cited by 13 (0 self)
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We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using d−cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d−compatibility degree (−−) on any pair of “colored ” almost positive real Schur roots which generalizes previous definitions on the noncolored case, and call two such roots compatible provided the d−compatibility degree of them is zero. Associated to the root system Φ corresponding to the valued quiver, by using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices and d−compatible subsets as simplicies. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is the generalized cluster complex defined by Fomin and Reading.
On the Structure of CalabiYau Categories with a Cluster Tilting Subcategory
 DOCUMENTA MATH.
, 2007
"... We prove that for d ≥ 2, an algebraic dCalabiYau triangulated category endowed with a dcluster tilting subcategory is ..."
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Cited by 9 (0 self)
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We prove that for d ≥ 2, an algebraic dCalabiYau triangulated category endowed with a dcluster tilting subcategory is
Cluster categories and selfinjective algebras: type A
, 2006
"... We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually ucluster categories. ..."
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Cited by 8 (3 self)
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We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually ucluster categories.