Dynkin diagram

Dynkin diagrams are a combinatorial way of representing the information in a root system. Their primary advantage is that they are easier to write down, remember, and analyze than explicit representations of a root system. They are an important tool in the classification of simple Lie algebras.

Given a reduced root system $R\subset E$ , with $E$ an inner-product space, choose a base or simple roots $\Pi$ (or equivalently, a set of positive roots $R^+$ ). The Dynkin diagram associated to $R$ is a graph whose vertices are $\Pi$ . If $\pi_i$ and $\pi_j$ are distinct elements of the root system, we add $m_{ij}=\frac{-4(\pi_i,\pi_j)^2}{(\pi_i,\pi_i)(\pi_j,\pi_j)}$ lines between them. This number is obivously positive, and an integer since it is the product of 2 quantities that the axioms of a root system require to be integers. By the Cauchy-Schwartz inequality, and the fact that simple roots are never anti-parallel (they are all strictly contained in some half space), $m_{ij}\in\{0,1,2,3\}$ . Thus Dynkin diagrams are finite graphs, with single, double or triple edges. Fact, the criteria are much stronger than this: if the multiple edges are counted as single edges, all Dynkin diagrams are trees, and have at most one multiple edge. In fact, all Dynkin diagrams fall into 4 infinite families, and 5 exceptional cases, in exact parallel to the classification of simple Lie algebras.

(Does anyone have good Dynkin diagram pictures? I'd love to put some up, but am decidedly lacking.)