Derived invariants for surface algebras

Abstract

In this paper we study the derived equivalences between surface algebras, introduced by David-Roesler and Schiffler [11]. 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 ϵ of generators of the fundamental group of the surface π1(S), we associate to any cut d a weight wϵ(d)∈Z2g+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 wϵ(d) up to homeomorphism of the surface. Surface algebras are gentle and of global dimension ≤2, and any surface algebras coming from the same surface (S,M) are cluster equivalent, in the sense of [2]. To prove that the weight is a derived invariant we strongly use results about cluster equivalent algebras from [2].Furthermore we also show that for surface algebras the invariant defined for gentle algebras by Avella-Alaminos and Geiss in [6] is determined by the weight.