On visual representations of Lie algebras: the type Al between the old and the new

Abstract

This paper is about some graph isomorphisms between the Auslander-Reiten quivers of the path algebras of quivers with underlying Dynkin diagrams of type Al, and the weight diagrams or the weight graphs relative to some basic (adjoint) representations of a semi-simple complex Lie algebra associated to the same Dynkin diagrams. The crucial point is that, in the attempt to establish the above relationships, we constructed the oriented weight diagrams and the oriented weight graphs with respect to particular orientations on the Dynkin diagrams of type A1. Our main goal is to furnish another applications of weight theory and visual representations.