Dynkin diagram

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

Given a reduced root system $R\subset E$, with $E$ an inner-product space, choose a baseor simple roots $\mathrm{\Pi }$ (or equivalently, a set of positive roots $R^{+}$). TheDynkin diagram associated to $R$ is a graph whose vertices are $\mathrm{\Pi }$. 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. Thisnumber is obivously positive, and an integer since it is the product of 2quantities that the axioms of a root system require to be integers. By theCauchy-Schwartz inequality, and the fact that simple roots are neveranti-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 aretrees, and have at most one multiple edge. In fact, all Dynkin diagramsfall into 4 infinite families, and 5 exceptional cases, in exact parallel to theclassification of simple Lie algebras.

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