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 $\mathrm{\Pi }$ (or equivalently, a set of positive roots $R^{+}$). The Dynkin 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. 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.)

Title	Dynkin diagram
Canonical name	DynkinDiagram
Date of creation	2013-03-22 13:28:05
Last modified on	2013-03-22 13:28:05
Owner	bwebste (988)
Last modified by	bwebste (988)
Numerical id	5
Author	bwebste (988)
Entry type	Definition
Classification	msc 17B20