Exceptional Cycles in the Bounded Derived Categories of Quivers

An exceptional n-cycle in a Hom-finite triangulated category with Serre functor has been recently introduced by Broomhead,Pauksztello and Ploog.When n=1,it is a spherical object.We explicitly determine all the exceptional cycles in the bounded derived category Db（kQ） of a finite quiver Q without oriented cycles.In particular,if Q is an Euclidean quiver,then the length type of exceptional cycles in Db（kQ） is exactly the tubular type of Q;if Q is a Dynkin quiver of type Em（m=6,7,8）,or Q is a wild quiver,then there are no exceptional cycles in Db（kQ）;and if Q is a Dynkin quiver of type An or D_n,then the length of an exceptional cycle in Db（kQ） is either h or h/2,where h is the Coxeter number of Q.