This is an expository work primarily on the relation of indecomposable representation of quivers and the positive root systems of the underlying Dynkin diagrams. This work with the background of Gabriel's result on the relation of indecomposable representation of ADE Dynkin quivers and the positive roots system of underlying Dynkin diagram explore the possible relation of exact sequences of modules of quivers and the generators of the associated Lie algebra. Considerable portion of this work started with prof. Igusa's observation on the orientation of simply-laced quivers (especially for the simplest ones, we will see the case of A type and their connection to the generators of the Lie algebra associated by the positive roots of the underlying Dynkin diagrams.
Yuichi Miyake (Wed,) studied this question.