Editing Dynkin diagram (section)

== Classification of semisimple Lie algebras ==
{{Further|Semisimple Lie algebra#Classification}}

The fundamental interest in Dynkin diagrams is that they classify [[semisimple Lie algebra]]s over [[algebraically closed field]]s. 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:
* <math>A_n</math>: <math>\mathfrak {sl}_{n+1}</math>, the [[special linear Lie algebra]].
* <math>B_n</math>: <math>\mathfrak{so}_{2n+1}</math>, the odd-dimensional [[special orthogonal Lie algebra]].
* <math>C_n</math>: <math>\mathfrak {sp}_{2n}</math>, the [[symplectic Lie algebra]].
* <math>D_n</math>: <math>\mathfrak{so}_{2n}</math>, the even-dimensional [[special orthogonal Lie algebra]] (<math>n>1</math>).
For the exceptional groups, the names for the Lie algebra and the associated Dynkin diagram coincide.