# Dynkin Diagram

In the mathematical field of Lie theory, a Dynkin diagram is a type of graph with some edges doubled or tripled (drawn as a double or triple line), and with any multiple edges directed, satisfying certain constraints. They are of interest firstly because they classify semisimple Lie algebras over algebraically closed fields, and give rise to Weyl groups, which are many (but not all) of the finite reflection groups. They also arise in other contexts. They are named for Eugene Dynkin; see history, below.

There is ambiguity in the terminology: in some cases Dynkin diagrams are assumed directed, in which case they correspond to root systems and semisimple Lie algebras, while in other cases they are assumed undirected, in which case they correspond to Weyl groups; the and directed diagrams yield the same undirected diagram, correspondingly named In this article, "Dynkin diagram" means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named.

### Other articles related to "dynkin diagram, diagram":

Triality
... Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because the group ... all simple Lie groups, Spin(8) has the most symmetrical Dynkin diagram, D4 ... The diagram has four nodes with one node located at the center, and the other three attached symmetrically ...
Outer Automorphism Groups of Complex Lie Groups
... may permute them, while preserving the structure of the associated Dynkin diagram ... In this way one may identify the automorphism group of the Dynkin diagram of G with a subgroup of Out(G) ... D4 has a very symmetric Dynkin diagram, which yields a large outer automorphism group of Spin(8), namely Out(Spin(8)) = S3 this is called triality ...
Satake Diagram - Definition
... A Satake diagram is obtained from a Dynkin diagram by blackening some vertices, and connecting other vertices in pairs by arrows, according to certain rules ... Then G(K) has a Dynkin diagram with respect to some choice of positive roots of T ... This Dynkin diagram has a natural action of the Galois group of K/k ...

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