Extended Admissible Dissections of Marked Surfaces and Piano Algebras

TL;DR

This paper introduces extended admissible dissections of marked surfaces and constructs associated piano algebras, establishing their derived equivalences and connections to cluster categories, advancing the understanding of surface-related algebraic structures.

Contribution

It defines extended admissible dissections and constructs piano algebras, linking them to cluster categories and proving derived equivalences for algebras from homeomorphic surfaces.

Findings

Piano algebras are quasi-isomorphic to endomorphism rings in certain cluster categories.

An additive equivalence exists between the cluster category and the perfect derived category of a piano algebra.

Piano algebras from homeomorphic discs are proven to be derived equivalent.

Abstract

We introduce the notion of extended admissible dissections of a marked surface, building upon the notion of an admissible dissection of a marked surface by Amiot--Plamondon--Schroll. For each extended admissible dissection we construct a differential graded algebra, called a piano algebra, which may be viewed in some sense as a differential graded analogue of a gentle algebra. We show that for a marked disc without punctures, a piano algebra is quasi-isomorphic to the graded endomorphism ring of a classical generator of the Paquette--Y\i ld\i r\i m completion of the discrete cluster category of Dynkin type $A_{\infty}$, labelled $\overline{\mathcal{C}}_n$. We use previous results of the authors to show that there exists an additive equivalence between $\overline{\mathcal{C}}_n$ and the perfect derived category of a specific piano algebra, that sends triangles with two indecomposable terms to triangles with two indecomposable terms. We use this equivalence to prove that any two piano algebras coming from homeomorphic marked discs are derived equivalent.