Graded Brauer tree algebras

In this paper we construct non-negative gradings on a basic Brauer tree algebra AΓ corresponding to an arbitrary Brauer tree Γ of type (m,e). We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra AS, whose tree is a star with the exceptional vertex in the middle, to AΓ. The grading on AS 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 Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras.We also compute , the group of outer automorphisms that fix the isomorphism classes of simple AΓ-modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on AΓ up to graded Morita equivalence and rescaling.