Green groupoids of 2-Calabi–Yau categories, derived Picard actions, and hyperplane arrangements

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 \mathscr {C} satisfying standard mild assumptions, we associate a groupoid G C \mathscr {G}_{ \mathscr {C} } , named the green groupoid of C \mathscr {C} , defined in an intrinsic homological way. Its objects are given by a set of representatives m r i g C mrig\mathscr {C} of the equivalence classes of basic maximal rigid objects of C \mathscr {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 \mathscr {G}_{ \mathscr {C} } to the derived Picard groupoid of the collection of endomorphism rings of representatives of m r i g C mrig\mathscr {C} in a Frobenius model of C \mathscr {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 \mathscr {G}_{ \mathscr {C} } is faithful if the index chamber decompositions of the split Grothendieck groups of basic maximal rigid objects of C \mathscr {C} come from hyperplane arrangements. If Σ 2 ≅ i d \Sigma ^2 \cong id and C \mathscr {C} has finitely many equivalence classes of basic maximal rigid objects, we prove that G C \mathscr {G}_{ \mathscr {C} } is isomorphic to a Deligne groupoid of a hyperplane arrangement and that the representation of this groupoid is faithful.