# 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 |