## Abstract

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).

Original language | English |
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Pages (from-to) | 2708-2750 |

Number of pages | 43 |

Journal | International Mathematics Research Notices |

Volume | 2012 |

Issue number | 12 |

DOIs | |

Publication status | Published - Jun 2012 |

Externally published | Yes |