The geometry of Brauer graph algebras and cluster mutations

TL;DR

This paper explores the relationship between ribbon graphs, Brauer graph algebras, and surface triangulations, demonstrating how geometric transformations correspond to algebraic mutations and constructing explicit tilting complexes.

Contribution

It establishes a novel connection between surface geometry and Brauer graph algebras, including a dual construction and explicit tilting complexes for geometric moves.

Findings

A surface with marked points determines a unique Brauer graph algebra up to derived equivalence.

Diagonal rotations in m-angulations correspond to Whitehead moves in dual graphs.

Explicit tilting complexes reflect geometric mutations in the algebraic setting.

Abstract

In this paper we establish a connection between ribbon graphs and Brauer graphs. As a result, we show that a compact oriented surface with marked points gives rise to a unique Brauer graph algebra up to derived equivalence. In the case of a disc with marked points we show that a dual construction in terms of dual graphs exists. The rotation of a diagonal in an m-angulation gives rise to a Whitehead move in the dual graph, and we explicitly construct a tilting complex on the related Brauer graph algebras reflecting this geometrical move.