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

TY - JOUR

T1 - Green groupoids of 2-Calabi–Yau categories, derived Picard actions, and hyperplane arrangements

AU - Jørgensen, Peter

AU - Yakimov, Milen

N1 - Publisher Copyright: © 2022 American Mathematical Society.

PY - 2022/11

Y1 - 2022/11

N2 - 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.

AB - 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.

KW - Cluster category

KW - cluster tilting object

KW - Deligne groupoid

KW - Gorenstein singularity

KW - maximal rigid object

KW - silting theory

KW - tilting theory

UR - http://www.scopus.com/inward/record.url?scp=85139570092&partnerID=8YFLogxK

U2 - 10.1090/tran/8770

DO - 10.1090/tran/8770

M3 - Journal article

AN - SCOPUS:85139570092

SN - 0002-9947

VL - 375

SP - 7981

EP - 8031

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 11

ER -