## Abstract

Let A be a finite-dimensional algebra over a field k. The derived Picard group DPic_{k}(A) is the group of triangle auto-equivalences of D^{b}(modA) induced by two-sided tilting complexes. We study the group DPic_{k}(A) when A is hereditary and k is algebraically closed. We obtain general results on the structure of DPic_{k}(A), as well as explicit calculations for many cases, including all finite and tame representation types. Our method is to construct a representation of DPic_{k}(A) on a certain infinite quiver Γ^{irr}. This representation is faithful when the quiver A of A is a tree, and then DPic_{k}(A) is discrete. Otherwise a connected linear algebraic group can occur as a factor of DPic_{k}(A). When A is hereditary, DPic_{k}(A) coincides with the full group of k-linear triangle auto-equivalences of D^{b}(modA). Hence, we can calculate the group of such auto-equivalences for any triangulated category D equivalent to D^{b}(modA). These include the derived categories of piecewise hereditary algebras, and of certain noncommutative spaces introduced by Kontsevich and Rosenberg.

Original language | English |
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Pages (from-to) | 341-368 |

Number of pages | 28 |

Journal | Compositio Mathematica |

Volume | 129 |

Issue number | 3 |

DOIs | |

State | Published - 1 Dec 2001 |

## Keywords

- Derived category
- Finite dimensional algebra
- Picard group
- Quiver