Representation-Finite Algebras whose Auslander-Reiten Quiver is Planar

Abstract

Throughout the paper A will denote a finite-dimensional connected and basic /c-algebra of finite representation type. We assume the ground field k to be algebraically closed. By FA we denote the Auslander-Reiten quiver of A, which has as vertices the isomorphism classes of indecomposable left A-modules and in which we put an arrow from the class of M to the class of N whenever there exists an irreducible map from M to N. Algebras whose associated Auslander-Reiten quivers have no oriented cycles have played an important role in representation theory: hereditary algebras, tree algebras, simply connected algebras, etc. In this paper we study some conditions which enable us, in some cases, to decide, in terms of the quiver with relations of A, whether FA has oriented cycles or not. Our conditions give, a constructive characterization of all finite representation type algebras A whose Auslander-Reiten quiver FA is planar, that is, FA has no oriented cycles and every Auslander-Reiten sequence in FA has at most two indecomposable summands in its middle term. We thank Prof. F. Larrion for some suggestions about the presentation of this paper.