Folded Gentle Algebras
Drew Damien Duffield
TL;DR
The paper develops folded gentle algebras by folding gentle-type quivers with crease loops, yielding a non-algebraically-closed analogue of the gentle framework for type C. It establishes a robust folding/unfolding correspondence to connect folded algebras with unfolded gentle algebras, enabling a complete combinatorial and categorical treatment via strings and bands, including symmetric and asymmetric variants. A full classification of indecomposable modules (strings and bands) is obtained, together with Auslander-Reiten sequences, and the module category is described through explicit unfolding functors. Finally, the authors prove closure under derived equivalence using repetitive algebras and tilting theory, mirroring known results for gentle algebras and extending the reach of tame representation theory to a broader Dynkin/Euclidean family.
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.