On auto-equivalences and complete derived invariants of gentle algebras

Abstract

We study triangulated categories which can be modeled by an oriented marked surface $\mathcal{S}$ and a line field $\eta$ on $\mathcal{S}$. This includes bounded derived categories of gentle algebras and -- conjecturally -- all partially wrapped Fukaya categories introduced by Haiden-Katzarkov-Kontsevich. We show that triangle equivalences between such categories induce diffeomorphisms of the associated surfaces preserving orientation, marked points and line fields up to homotopy. This shows that the pair $(\mathcal{S},\eta )$ is a triangle invariant of such categories and prove that it is a complete derived invariant for gentle algebras of arbitrary global dimension. We deduce that the group of auto-equivalences of a gentle algebra is an extension of the stabilizer subgroup of $\eta$ in the mapping class group and a group, which we describe explicitely in case of triangular gentle algebras. We show further that diffeomorphisms associated to spherical twists are Dehn twists.

References in corpus (1)

• Combinatorial rigidity of arc complexes

Cited by in corpus (3)

• Derived equivalence classification of Brauer graph algebras
• A geometric model for the module category of a skew-gentle algebra
• A geometric realization of silting theory for gentle algebras