Brauer graph algebras

TL;DR

This paper provides a comprehensive overview of Brauer graph algebras, exploring their definitions, interpretations, connections to surface triangulations, and detailed descriptions of their representation theory.

Contribution

It offers a detailed exposition of Brauer graph algebras, including their relation to special biserial and gentle algebras, and links to surface triangulations and derived equivalences.

Findings

Connection between Brauer graph algebras and decorated ribbon graphs

Derived equivalences via flips of diagonals in triangulations

Complete description of the Auslander Reiten quiver

Abstract

These lecture notes on Brauer graph algebras are the result of a series of four lectures given at the CIMPA research school in Mar del Plata, Argentina, in March 2016. After motivating the study of Brauer graph algebras by relating them to special biserial algebras, the definition of Brauer graph algebras is given in great detail with many examples to illustrate the concepts. This is followed by a short section on the interpretation of Brauer graphs as decorated ribbon graphs. A section on gentle algebras and their graphs, trivial extensions of gentle algebras, admissible cuts of Brauer graph algebras and a first connection of Brauer graph algebras with Jacobian algebras associated to triangulations of marked oriented surfaces follows. The interpretation of flips of diagonals in triangulations of marked oriented surfaces as derived equivalences of Brauer graph algebras and the comparison of derived equivalences of Brauer graph algebras with derived equivalences of frozen Jacobian algebras is the topic of the next section. In the last section, after defining Green's walk around the Brauer graph, a complete description of the Auslander Reiten quiver of a Brauer graph algebra is given.