## Abstract

We present a construction of (faithful) group actions via derived equivalences in the general categorical setting of algebraic 2-Calabi–Yau triangulated categories. To each algebraic 2-Calabi–Yau category C satisfying standard mild assumptions, we associate a groupoid G_{C}, named the green groupoid of C, defined in an intrinsic homological way. Its objects are given by a set of representatives mrig C of the equivalence classes of basic maximal rigid objects of C, arrows are given by mutation, and relations are given by equating monotone (green) paths in the silting order. In this generality we construct a homomorphsim from the green groupoid G_{C} to the derived Picard groupoid of the collection of endomorphism rings of representatives of mrig C in a Frobenius model of C; the latter canonically acts by triangle equivalences between the derived categories of the rings. We prove that the constructed representation of the green groupoid G_{C} is faithful if the index chamber decompositions of the split Grothendieck groups of basic maximal rigid objects of C come from hyperplane arrangements. If Σ^{2 ∼}= id and C has finitely many equivalence classes of basic maximal rigid objects, we prove that G_{C} is isomorphic to a Deligne groupoid of a hyperplane arrangement and that the representation of this groupoid is faithful.

Original language | English |
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Journal | Transactions of the American Mathematical Society |

Volume | 375 |

Issue | 11 |

Pages (from-to) | 7981-8031 |

Number of pages | 51 |

ISSN | 0002-9947 |

DOIs | |

Publication status | Published - Nov 2022 |

## Keywords

- Cluster category
- cluster tilting object
- Deligne groupoid
- Gorenstein singularity
- maximal rigid object
- silting theory
- tilting theory