A graph theoretic model for the derived categories of gentle algebras and their homological bilinear forms
Jesús Arturo Jiménez González, Andrzej Mróz
TL;DR
This work develops a discrete graph-theoretic framework to study the bounded derived category of gentle algebras by encoding bound quivers as marked ribbon graphs. It links combinatorial walks on these graphs to indecomposable objects (perfect and non-perfect), AR-triangles, and homological invariants such as the Euler form and Coxeter transformation, notably expressing key invariants through incidence matrices. The main results establish that the Cartan data and Euler form are governed by incidence graphs, yield non-negativity and Dynkin-type classification of q_A, and relate the Coxeter polynomial to the Avella-Alaminos-Geiss invariant. These tools produce derived-invariant graph-theoretic invariants and provide a natural bridge to Brauer graph algebras, while yielding a root-system perspective on indecomposable objects in the positive-case setting.
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.