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

TL;DR

This paper develops a graph theoretic framework for analyzing the derived categories of gentle algebras, focusing on indecomposables, homological forms, and invariants, with applications to related algebraic structures.

Contribution

It introduces simple graph-based tools to study derived categories of gentle algebras, connecting homological forms, Coxeter transformations, and classical invariants.

Findings

Homological quadratic form of gentle algebras is non-negative and of Dynkin type.

Classifies indecomposable perfect complexes via roots of the quadratic form.

Relates Coxeter polynomial to Avella-Alaminos Geiss invariant and derives consequences for Brauer graph algebras.

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.