Tilting Modules Over Path Algebras Of Dynkin Type

Representation theory of algebras is a new branch of algebra which began at the seventies of the last century. Tilting theory is an important tool to study representation theory of algebras. Tilting theory originated from the work of Bornstein,Gelfand and Ponomarev. For the sake of proving famous Gabriel theorem, they introduced reflection functor and Coxter functor. In the recent years, due to there being intrinsic relationships between tilting theory and quantum group, Lie algebra and other algebraic branch, tilting theory has been one of the international hot topics. In 1998, I. Reiten was an invited speaker whose topics is "Tilting theory and quasitilted algebras" at the International Congress of Mathematicians in Berlin. The main purpose of this paper is to study structural characteristics of the tilting module over path algebra of type Dynkin in its AR-quiver. It takes advantages of AR-quiver analysis and APR- tilting translation. We proved: if T_A is a tilting A-module over a path algebra of type A_n or a characteristic tilting A-module over a path algebra of type D_n,E₆,E₇, E₈,then there are points corresponding indecomposable direct summands of T_A in theÏ„- orbit of each edge.