Graded Brauer tree algebras

Abstract

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