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.