Equivalences between cluster categories

Tilting theory in cluster categories of hereditary algebras has been developed in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, preprint, arXiv: math.RT/0402075, 2004, Adv. Math., in press; A. Buan, R. Marsh, I. Reiten, Cluster-tilted algebras, preprint, arXiv: math.RT/0402054, 2004; Trans. Amer. Math. Soc., in press]. Some of them are already proved for hereditary abelian categories there. In the present paper, all basic results about tilting theory are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object T in a hereditary abelian category , we verify that the tilting functor induces a triangle equivalence from the cluster category to the cluster category , where A is the quasi-tilted algebra . Under the condition that one of derived categories of hereditary abelian categories , is triangle equivalent to the derived category of a hereditary algebra, we prove that the cluster categories and are triangle equivalent to each other if and only if and are derived equivalent, by using the precise relation between cluster-tilted algebras (by definition, the endomorphism algebras of tilting objects in cluster categories) and the corresponding quasi-tilted algebras proved previously. As an application, we give a realization of “truncated simple reflections” defined by Fomin–Zelevinsky on the set of almost positive roots of the corresponding type [S. Fomin, A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. 154 (1) (2003) 63–121; S. Fomin, A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. 158 (3) (2003) 977–1018], by taking to be the representation category of a valued Dynkin quiver and T a BGP-tilting object (or APR-tilting, in other words).