Graded Brauer Tree Algebras

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.