Abstract. In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph. As in the previous article, the proofs rely on the Calabi-Yau property of the cluster category. 1.

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of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin–Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.

Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau 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

Abstract. We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.

Abstract. We show that for cluster algebras associated with finite quivers without oriented cycles (with no coefficients), a seed is determined by its cluster, as conjectured by Fomin and Zelevinsky. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

(joint work with Idun Reiten) Let k be a field and Q a finite quiver without oriented cycles. Let kQ be the path algebra of Q and mod kQ the category of k-finite-dimensional right kQ-modules. The cluster category CQ was introduced in [1] (for general Q) and, independently, in [4] (for Q of type An). It is defined as the orbit category of the bounded derived category D b (mod kQ) under the action of the automorphism Σ −1 ◦ S 2, where S is the suspension (=shift) functor of the derived category and Σ its Serre functor, characterized by the Serre duality formula D Hom(X, Y) = Hom(Y, ΣX), where D is the duality functor Homk(?, k). The motivation behind this definition was to find a ‘categorification ’ of the cluster algebras introduced by Fomin-Zelevinsky in [6]. This program has been quite successful, cf. e.g. [3] [5] and the references given there. The cluster category has the following properties (explained below in more detail): a) CQ is a triangulated category. In fact, it is even an algebraic triangulated

We give a geometric realization of cluster categories of type Dn using a polygon with n vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices. 0

Abstract. We show that a certain orbit category considered by Keller encodes the combinatorics of the m-clusters 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 Auslander-Reiten translate τ.) The cluster category C(H) is a triangulated Krull-Schmidt 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

We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a lattice. We also prove some new results within the theory of quiver representations. We show that the finitely generated, exact abelian, and extension-closed subcategories of the representations of a quiver Q without oriented cycles are in natural bijection with the cluster tilting objects in the associated cluster category. We also show these subcategories are exactly the finitely generated categories that can be obtained as the semistable objects with respect to some stability condition. 1.