Abelian hereditary fractionally calabi-yau categories

As a generalization of a Calabi-Yau category, we will say a k-linear Hom-finite triangulated category is fractionally Calabi-Yau if it admits a Serre functor double struck S sign and there is an n>0 with double struck S sign≅ ⁿ[m]. An abelian category will be called fractionally Calabi-Yau if its bounded derived category is. We provide a classification up to derived equivalence of abelian hereditary fractionally Calabi-Yau categories (for algebraically closed k). They are:(1) the category of finite-dimensional representations of a Dynkin quiver;(2) the category of finite-dimensional nilpotent representations of a cycle;(3) the category of coherent sheaves on an elliptic curve or a weighted projective line of tubular type.To obtain this classification, we introduce generalized 1-spherical objects and use them to obtain results about tubes in hereditary categories (which are not necessarily Calabi-Yau).