Folded Gentle Algebras

Abstract

We use folding techniques to define a new class of gentle-like algebras that generalise the iterated tilted algebras of type $C$ and $\tilde{C}$, which we call folded gentle algebras. We show that folded gentle algebras satisfy many of the same properties of gentle algebras, and that the proof of these properties follows directly from folding arguments. As a subclass of clannish algebras (with irreducible quadratic relations), we show that the classification of indecomposable modules (in terms of symmetric and asymmetric strings and bands) can be recovered from folding techniques, and that this proof technique provides further explicit detail on the classification of band modules. In particular, our paper includes a classification of indecomposable modules over the algebra $K⟨x,y⟩/⟨p(x),q(y)⟩$, where $p$ and $q$ are monic, irreducible, quadratic polynomials over $K$. In addition, we classify the Auslander-Reiten sequences of folded gentle 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 is closed under derived equivalence.

69 pages. Changes from v2: Minor corrections throughout. Included a reference to the [Bennett-Tennenhaus, Crawley-Boevey] paper on semilinear clannish algebras and a commentary on its implications to the paper's results. Previous results that were only implicitly mentioned in v2 have been made more explicit. Included an additional appendix with a comprehensive worked example