Dynkindiagrammen

Dynkindiagrammen, commonly written as Dynkin diagrams in English, are a class of graphs used in the study of Lie algebras and their representations. They were introduced by the Russian-born mathematician Eugene Dynkin in 1947 as a tool to classify semisimple Lie algebras over complex numbers.

Each diagram consists of nodes (vertices) connected by edges. Nodes correspond to simple roots in a root system; edges, which can be single, double, or triple, encode the angles between roots and their relative lengths. In traditional diagrams, arrows point toward the node representing the shorter root when multiple edges are present.

Connected Dynkin diagrams classify finite-dimensional complex semisimple Lie algebras, with types A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4 and G_2, where n is a positive integer and the exceptional types E, F, and G occur only for specific small ranks. There are also affine (extended) Dynkin diagrams associated with affine Lie algebras, obtained by adding a node to the finite diagram.

Dynkin diagrams serve as a combinatorial shorthand for the Cartan matrix of a Lie algebra and are closely related to root systems and Weyl groups. They also play a role in representation theory, mathematical physics, and geometry, where the diagrams help describe symmetries and possible decompositions of representations.

Beyond classification, diagram automorphisms of Dynkin diagrams correspond to outer automorphisms of the associated Lie algebras and can lead to symmetries between representations. The concept remains a foundational tool in the study of Lie theory.