Equivalences between cluster categories

Bin Zhu

Published 2005-11-15, updated 2006-06-19Version 2

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 $\mathcal{H}$, we verify that the tilting functor Hom$_\mathcal{H}(T,-)$ induces a triangle equivalence from the cluster category $\mathcal{C(H)}$ to the cluster category $\mathcal{C}(A)$, where $A$ is the quasi-tilted algebra End$_{\mathcal{H}}T.$ Under the condition that one of derived categories of hereditary abelian categories $\mathcal{H},$ $\mathcal{H}'$ is triangle equivalent to the derived category of a hereditary algebra, we prove that the cluster categories $\mathcal{C(H)}$ and $\mathcal{C(H')}$ are triangle equivalent to each other if and only if $\mathcal{H}$ and $\mathcal{H}'$ 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 $\mathcal{H}$ to be the representation category of a valued Dynkin quiver and $T$ a BGP-tilting (or APR-tilting, in other words).