Graded Brauer Tree Algebras

TL;DR

This paper constructs non-negative gradings on Brauer tree algebras for any tree shape by transferring gradings via derived equivalences, and analyzes automorphism groups to establish grading uniqueness.

Contribution

It introduces a method to transfer gradings from star-shaped Brauer tree algebras to arbitrary trees using derived equivalences and tilting complexes, and proves grading uniqueness up to Morita equivalence.

Findings

Successfully transferred gradings to all Brauer tree algebras.

Computed the group of outer automorphisms fixing simple modules.

Proved the uniqueness of grading up to Morita equivalence.

Abstract

In this paper we construct non-negative gradings on a basic Brauer tree algebra $A_{\Gamma}$ corresponding to an arbitrary Brauer tree $\Gamma$ of type $(m,e)$. We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra $A_S$, whose tree is a star with the exceptional vertex in the middle, to $A_{\Gamma}$. The grading on $A_S$ comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green's walk around $\Gamma$ (cf. [\ref{Zak}]). By computing endomorphism rings of these tilting complexes we get graded algebras. We also compute ${\rm Out}^K(A_{\Gamma})$, the group of outer automorphisms that fix isomorphism classes of simple $A_{\Gamma}$-modules, where $\Gamma$ is an arbitrary Brauer tree, and we prove that there is unique grading on $A_{\Gamma}$ up to graded Morita equivalence and rescaling.