Folded Gentle Algebras

TL;DR

This paper introduces folded gentle algebras, a new class generalizing certain tilted algebras, and demonstrates they share many properties of gentle algebras, including module classification and derived equivalence closure.

Contribution

It defines folded gentle algebras using folding techniques and proves they retain key properties of gentle algebras, including module classification and derived invariance.

Findings

Classified indecomposable modules via string and band modules.

Described Auslander-Reiten sequences and irreducible morphisms.

Proved folded gentle algebras are closed under derived equivalence.

Abstract

We use folding techniques to define a new class of gentle-like algebras that generalise the iterated tilted algebras of type $C$ and $\widetilde{C}$, which we call folded gentle algebras. We then show that folded gentle algebras satisfy many of the same remarkable properties of gentle algebras, and that the proof of these properties follows directly from folding arguments. In particular, we classify the indecomposable modules of folded gentle algebras in terms symmetric and asymmetric string and band modules. We classify the Auslander-Reiten sequences over these algebras, showing that irreducible morphisms between string modules are given by adding/deleting hooks and cohooks to/from strings. Finally, we show that the class of folded gentle algebras are closed under derived equivalence.