Equivalences between cluster categories

Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object T in a hereditary abelian category H, we verify that the tilting functor HomH(T,−) induces a triangle equivalence from the cluster category C(H) to the cluster category C(A), where A is the quasi-tilted algebra EndHT. Under the condition that one of derived categories of hereditary abelian categories H, H is triangle equivalent to the derived category of a hereditary algebra, we prove that the cluster categories C(H) and C(H) are triangle equivalent to each other if and only if H and H 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 [FZ2, FZ5], by taking H to be the representation category of a valued Dynkin quiver and T a BGP-tilting (or APR-tilting, in other words).