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Concept
Dynkin diagram
Résumé
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra.
The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means directed Dynkin diagram, and undirected Dynkin diagrams will be explicitly so named.
The fundamental interest in Dynkin diagrams is that they classify semisimple Lie algebras over algebraically closed fields. One classifies such Lie algebras via their root system, which can be represented by a Dynkin diagram. One then classifies Dynkin diagrams according to the constraints they must satisfy, as described below.
Dropping the direction on the graph edges corresponds to replacing a root system by the finite reflection group it generates, the so-called Weyl group, and thus undirected Dynkin diagrams classify Weyl groups.
They have the following correspondence for the Lie algebras associated to classical groups over the complex numbers:
, the special linear Lie algebra.
, the odd-dimensional special orthogonal Lie algebra.
, the symplectic Lie algebra.
, the even-dimensional special orthogonal Lie algebra ().
For the exceptional groups, the names for the Lie algebra and the associated Dynkin diagram coincide.
Dynkin diagrams can be interpreted as classifying many distinct, related objects, and the notation "An, Bn, ...
Source officielle
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