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

Peter Jørgensen, Milen Yakimov

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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 GC, 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 GC 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 GC is faithful if the index chamber decompositions of the split Grothendieck groups of basic maximal rigid objects of C come from hyperplane arrangements. If Σ2 ∼= id and C has finitely many equivalence classes of basic maximal rigid objects, we prove that GC is isomorphic to a Deligne groupoid of a hyperplane arrangement and that the representation of this groupoid is faithful.

Original languageEnglish
JournalTransactions of the American Mathematical Society
Volume375
Issue11
Pages (from-to)7981-8031
Number of pages51
ISSN0002-9947
DOIs
Publication statusPublished - Nov 2022

Keywords

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

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