Dynkin Diagram

A Dynkin diagram is a graph that encodes the geometric structure of a root system, and thereby the structure of a corresponding simple Lie algebra or simple Lie group. Named after the Soviet mathematician Eugene Dynkin, who systematically classified them in the 1940s, Dynkin diagrams are the key combinatorial objects in the classification of simple Lie groups and the construction of Chevalley groups.

Each Dynkin diagram consists of nodes representing simple roots, connected by edges that encode the angles between them. A single edge indicates a 120° angle, a double edge indicates 135°, and a triple edge indicates 150°. The diagrams are classified into four infinite families — A_n, B_n, C_n, and D_n — corresponding to the classical simple Lie groups, and five exceptional cases — E6, E7, E8, F4, and G2 — corresponding to the exceptional groups. The same diagrams govern the classification of finite-dimensional simple Lie algebras, affine Kac-Moody algebras, and the finite simple groups of Lie type.

The Dynkin diagram is the ultimate structural invariant: it reduces the continuous infinity of a Lie group to a finite graph, and yet it preserves every essential property. The fact that the same diagram governs both continuous and finite symmetries is not a coincidence but a proof that the combinatorial structure of roots is more fundamental than the topology of the group itself.