The associated graded algebras of Brauer graph algebras I: finite representation type

Abstract

It is well known that Brauer graph algebras coincide with symmetric special biserial algebras and there has been a lot of work on Brauer graph algebras and their representation theory. Given a Brauer graph algebra A associated with a Brauer graph G, we denote by gr(A) the graded algebra associated with the radical filtration of A. We give a criterion for gr(A) to be representation-finite in terms of the graded degrees of vertices in G. Moreover, when gr(A) is representation-finite, we give the precise relationship between the Auslander–Reiten quiver of A and the Auslander–Reiten quiver of gr(A).