A graph theoretic model for the derived categories of gentle algebras and their homological bilinear forms

Abstract

We customize the existing models for the bounded derived category of gentle algebras to obtain simple graph theoretic tools to analyze indecomposable objects, Auslander-Reiten triangles, and their interaction with the associated homological bilinear forms and the Coxeter transformation. We apply these tools to explore related new and classical derived invariants. We exhibit the non-negativity and Dynkin type of the homological quadratic form of a gentle algebra, classify indecomposable perfect complexes by means of its roots, describe the Coxeter polynomial and relate it with the Avella-Alaminos Geiss invariant. We also derive some consequences for Brauer graph algebras.

Corrected misprints; few reformulated comments and added auxiliary minor diagrams to improve readability; simplified and/or removed a part of elementary claims and arguments in old Proposition 3.14 (which is now presented as Lemma 3.14)