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 , with an inner-product space, choose a base or simple roots (or equivalently, a set of positive roots ). The Dynkin diagram associated to is a graph whose vertices are . If and are distinct elements of the root system, we add 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), . 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.)
|Date of creation||2013-03-22 13:28:05|
|Last modified on||2013-03-22 13:28:05|
|Last modified by||bwebste (988)|