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 $\mathcal{C}$ satisfying standard mild assumptions, we associate a groupoid ${\mathcal{G}}_{\mathcal{C}}$, named the green groupoid of $\mathcal{C}$, defined in an intrinsic homological way. Its objects are given by a set of representatives $\mathrm{mrig}\mathcal{C}$ of the equivalence classes of basic maximal rigid objects of $\mathcal{C}$, 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 ${\mathcal{G}}_{\mathcal{C}}$ to the derived Picard groupoid of the collection of endomorphism rings of representatives of $\mathrm{mrig}\mathcal{C}$ in a Frobenius model of $\mathcal{C}$; the latter canonically acts by triangle equivalences between the derived categories of the rings. We prove that the constructed representation of the green groupoid ${\mathcal{G}}_{\mathcal{C}}$ is faithful if the index chamber decompositions of the split Grothendieck groups of basic maximal rigid objects of $\mathcal{C}$ come from hyperplane arrangements. If ${\Sigma }^2\cong \mathrm{id}$ and $\mathcal{C}$ has finitely many equivalence classes of basic maximal rigid objects, we prove that ${\mathcal{G}}_{\mathcal{C}}$ is isomorphic to a Deligne groupoid of a hyperplane arrangement and that the representation of this groupoid is faithful.

Minor corrections after helpful comments. 48 pages

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