Derived invariants for surface algebras

Abstract

In this paper we study the derived equivalences between surface algebras, introduced by David-Roesler and Schiffler. Each surface algebra arises from a cut of an ideal triangulation of an unpunctured marked Riemann surface with boundary. A cut can be regarded as a grading on the Jacobian algebra of the quiver with potential (Q,W) associated with the triangulation. Fixing a set $\epsilon$ of generators of the fundamental group of the surface, we associate to any cut $d$ a weight $w^\epsilon(d)\in\mathbb Z^{2g+b}$, where $g$ is the genus of $S$ and $b$ the number of boundary components. The main result of the paper asserts that the derived equivalence class of the surface algebra is determined by the corresponding weight $w^\epsilon(d)$ up to homeomorphism of the surface. Surface algebras are gentle and of global dimension $\leq 2$, and any surface algebras coming from the same surface $(S,M)$ are cluster equivalent, in the sense of Amiot and Oppermann. To prove that the weight is a derived invariant we strongly use Amiot Oppermann's results on cluster equivalent algebras. Furthermore we also show that for surface algebras the invariant defined for gentle algebras by Avella-Alaminos and Geiss, is determined by the weight.