Mutation classes of certain quivers with potentials as derived equivalence classes

TL;DR

This paper characterizes certain surfaces whose associated quivers with potentials have Jacobian algebras all derived equivalent, identifying specific classes and providing explicit quivers, thus advancing understanding of derived equivalence in cluster theory.

Contribution

It classifies surfaces with derived equivalent Jacobian algebras from triangulations and analyzes exceptional mutation classes, offering explicit quivers and new insights into derived equivalence classes.

Findings

Surfaces of genus g with boundary and one marked point per boundary have all associated Jacobian algebras derived equivalent.

All quivers in these classes have the same number of arrows.

Explicit quivers for each derived equivalence class are provided.

Abstract

We characterize the marked bordered unpunctured oriented surfaces with the property that all the Jacobian algebras of the quivers with potentials arising from their triangulations are derived equivalent. These are either surfaces of genus g with b boundary components and one marked point on each component, or the disc with 4 or 5 points on its boundary. We show that for each such marked surface, all the quivers in the mutation class have the same number of arrows, and the corresponding Jacobian algebras constitute a complete derived equivalence class of finite-dimensional algebras whose members are connected by sequences of Brenner-Butler tilts. In addition, we provide explicit quivers for each of these classes. We consider also 10 of the 11 exceptional finite mutation classes of quivers not arising from triangulations of marked surfaces excluding the one of the quiver X_7, and show that all the finite-dimensional Jacobian algebras in such class (for suitable choice of potentials) are derived equivalent only for the classes of the quivers E_6^(1,1) and X_6.