Derived equivalence classification of Brauer graph algebras

TL;DR

This paper classifies Brauer graph algebras up to derived equivalence using a complete set of invariants, connecting algebraic, geometric, and topological perspectives to understand their structure.

Contribution

It proves the completeness of Antipov's derived invariants for Brauer graph algebras and introduces a geometric interpretation via orbit invariants of line fields.

Findings

Complete classification of Brauer graph algebras up to derived equivalence.

Connection between derived invariants and orbit invariants of line fields.

Extension of Brauer graph algebras through $A_{ abla}$-trivial extensions of Fukaya categories.

Abstract

We classify Brauer graph algebras up to derived equivalence by showing that the set of derived invariants introduced by Antipov is complete. These algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. Motivated by the connection between Brauer graph algebras and gentle algebras we consider $A_{\infty}$-trivial extensions of partially wrapped Fukaya categories associated to surfaces with boundary. This construction naturally enlarges the class of Brauer graph algebras and provides a way to construct derived equivalences between Brauer graph algebras with the same derived invariants. As part of the proof we provide an interpretation of derived invariants of Brauer graph algebras as orbit invariants of line fields under the action of the mapping class group.