Stability conditions on Brauer tree algebras

TL;DR

This paper investigates the structure of the space of stability conditions for Brauer tree algebras, revealing topological properties and differences from geometric cases, with a focus on the algebra $A_3$.

Contribution

It demonstrates that the connected component of the stability space for $A_3$ is simply connected and explores the properties of the Bridgeland homomorphism in this context.

Findings

The space of stability conditions for $A_3$ is simply connected.

The Bridgeland homomorphism is not a covering map in this case.

The study links algebraic stability conditions to topological properties.

Abstract

We study the space of stability conditions attached to the derived category of $A_{n}$-mod for $A_{n}$ the Brauer tree algebra of the line with $n$ edges. These algebras arise in the study of cyclic defect blocks of group algebras, and they are also related to the zig-zag algebras introduced by Huerfano and Khovanov. We show that for the Brauer tree algebra $A_{3}$, the connected component of the natural heart of the space of stability conditions is simply connected. However, unlike certains examples arising in geometry, the Bridgeland homomorphism is not a covering map.