Another Description Of Dimension Vector Of Simple Representation Of ∏～λ(Q)

Quiver algebra plays an important role in many areas of mathematics, for example,representation theory, algebraic geometry, dierential geometry, Kac-Moody algebrasand quantum groups. Dimension vector of representation of quiver algebra gives an de-scription of dimension of vector spaces in the representation, it determines the repre-sentation to some extent. This thesis gives another description of the dimension vectorof a simple representation of the deformed preprojective algebraλ(Q). It has threechapters. The first chapter gives some basic definitions and notations about quivers andasserts the classification of quivers, and gives the main result of this thesis. In the secondchapter, we remain that an KQmodule can be recognized as anλ(Q)-module only insome specific case, and sketch the Kac's theorem, the key to the theory of preprojectivealgebra. Chapter three is the main body of this thesis. In this chapter, we completethe proof of our main theorem, i.e, another description of dimension vector of simplerepresentation of the deformed preprojective algebraλ(Q).