Diagrams and Roots

Abstract

Our aim is to determine when a quiver without oriented cycles has finite representation type. As we will see in the next chapter, this only depends on the underlying graph of the quiver, that is, on the graph which is obtained by forgetting the orientation of the arrows. In this chapter, we describe the relevant graphs. They are known as Dynkin diagrams and Euclidean diagrams, these occur in many parts of mathematics. We introduce roots, and we discuss further tools needed later, such as the Coxeter transformations. The content of this chapter is mainly basic combinatorics and linear algebra.