Graded Brauer tree algebras

Abstract

a b s t r a c tIn 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.

The singularity and cosingularity categories of $C^*BG$ for groups with cyclic Sylow $p$-subgroups

“…In more detail, the projective indecomposables are uniserial, and the radical filtration is isomorphic to its associated graded. Some readers may find it helpful to refer to the paper Bogdanic [12] which recalls details of Brauer tree algebras and shows that the stable grading comes from a grading of the Brauer tree algebras themselves.…”

The Differential Graded Stable Category of a Self-Injective Algebra

“…Theorem 5.1. (Bogdanic, [2], Theorem 4.3 and Lemma 4.9) Let S be the star with n vertices and multiplicity m. Let A S be graded so that soc(A S ) is in degree nm. Let Γ be any Brauer tree with n vertices and multiplicity m. Then A Γ admits a non-negative grading such that soc(A Γ ) is in degree nm, and there is an equivalence D b (A S -grmod) → D b (A Γ -grmod).…”

Automorphismes, graduations et catégories triangulées

“…Une algèbre de Brauer associéeà une ligne avec multiplicité 1 est isomorphè a une extension triviale de l'algèbre d'un carquois de type A où aucun sommet n'est la source (resp. le but) de deux flèches distinctes [Bog,Proposition 8.1]. Le théorème 6.18 montre alors qu'une telle algèbre de Brauer (et donc toute algèbre de Brauerà multiplicité triviale) satisfait la propriété de relèvement.…”

“…Pour chaque i, on fixe V i un sous-kE-module de J(kP i ) tel que J(kP Le cas des blocsà défaut cyclique aétéétudié en détail par Bogdanic [3]. La conjecture de Broué [4] sur les blocsà défaut abélien prédit que les conclusions du théorème 6.10 sont vraies si D est abélien (la conjecture prédit plus précisément que les blocs seront dérivé-équivalents).…”