Dynkin diagram

Dynkin diagram
Jgmoxness · CC BY-SA 3.0 · source
Name	Dynkin diagram
Caption	Simple Dynkin diagram examples: A_n, D_n, E_n series
Type	Graphical classification tool
Introduced	1940s
Introduced by	Eugenio Calabi, Raymond Paley, Nathan Jacobson
Area	Sophus Lie theory, Claude Chevalley theory, Hermann Weyl theory
Related	Cartan matrix, Weyl group, Coxeter graph

Dynkin diagram is a finite graph encoding the angles and lengths between simple roots of a root system and classifying simple complex Lie algebras, finite reflection groups, and related algebraic structures. Developed in the mid‑20th century, these diagrams serve as combinatorial invariants that link representation theory, algebraic groups, and singularity theory. They appear in the study of Élie Cartan's classification, the structure of Chevalley groups, and in connections to mathematical physics such as the ADE classification.

Definition and basic properties

A Dynkin diagram is a finite graph whose vertices correspond to simple roots of a root system and whose edges encode the relative inner products determined by a Cartan matrix. Each vertex often carries a mark indicating root length; multiple edges and arrows record nonsymmetric pairings as in the diagrams for B_n, C_n, and G_2. The classification links directly to simple finite-dimensional Lie algebras over the complex numbers, with each connected diagram uniquely determining a complex simple Lie algebra such as sl(n,C), so(n,C), or exceptional algebras like E_8. Diagrams obey constraints coming from integrality and positivity conditions derived from Élie Cartan's work and later formalized by Claude Chevalley and Hermann Weyl.

Classification of Dynkin diagrams

The connected Dynkin diagrams fall into the classical infinite families A_n, B_n, C_n, and D_n and the exceptional types E_6, E_7, E_8, F_4, and G_2. This ADE list parallels classifications found in the study of simple singularities by Vladimir Arnolʹd and in the McKay correspondence linking finite subgroups of SU(2) to ADE classification. Cartan matrices associated to each diagram are integral, symmetrizable matrices that satisfy positivity properties exploited by Jean-Pierre Serre and Nathan Jacobson in structural results for Lie algebras. The classification also appears in the theory of finite Coxeter groups studied by H.S.M. Coxeter and in the geometry of regular polytopes examined by Ludwig Schläfli.

Construction and Coxeter–Dynkin relations

A Dynkin diagram can be constructed from a symmetric bilinear form on a Euclidean space by choosing a system of simple roots; vertices represent these roots and edges encode angles via labels that are derived from the entries of the Cartan matrix. The Coxeter–Dynkin relations define reflections s_i with relations (s_i s_j)^{m_{ij}} = 1 where integers m_{ij} are determined by the diagram; these relations underpin the structure of the associated Weyl group and related Coxeter groups. The pioneering computation of reflection relations was developed in contexts involving Wilhelm Killing and Élie Cartan, and later synthesized by H.S.M. Coxeter in his study of reflection groups and polytopes.

Applications in Lie algebras and algebraic groups

Dynkin diagrams classify isomorphism classes of complex simple Lie algebras and connected reductive algebraic groups, guiding the construction of root decompositions, highest‑weight modules, and representation rings studied by Harish-Chandra and George Lusztig. They determine conjugacy classes of maximal tori in groups such as SL_n, SO_n, and Sp_{2n} and inform the classification of parabolic subalgebras used in the theory of Borel subgroups and Tits buildings. In number theory and arithmetic geometry, diagrams appear in the structure theory of Chevalley groups over finite fields and in the classification of reductive group schemes considered by Alexander Grothendieck.

Extensions and related diagrams (affine, extended, twisted)

Affinization of a Dynkin diagram yields affine Dynkin diagrams (also called extended diagrams) that classify affine Kac–Moody algebras introduced by Victor Kac and Robert Moody. Twisted affine diagrams correspond to automorphisms of finite diagrams and lead to different classes of infinite-dimensional algebras relevant to integrable models studied by Ludwig Faddeev and Mikhail Semenov-Tian-Shansky. The extended and twisted diagrams play roles in the classification of loop algebras, in the theory of generalized Cartan matrices, and in applications to conformal field theory investigated by Belavin and Alexander Zamolodchikov.

Representations, root systems, and Weyl groups

Dynkin diagrams determine the combinatorics of weight lattices, dominant weights, and highest‑weight representations central to the work of Élie Cartan, Harish-Chandra, and Weyl. The associated root system yields reflection symmetries forming the Weyl group, whose representation theory connects with Hecke algebras analyzed by Iwahori and Hecke. Diagram symmetries correspond to outer automorphisms of Lie algebras and to diagram folding techniques used to derive non‑simply laced types from simply laced ones; these methods appear in the study of modular invariants in the work of Pasquier and Ocneanu. In geometry and topology, Dynkin types index singularity strata in moduli spaces considered by Michael Atiyah and Raoul Bott and appear in the classification of certain four‑manifold intersection forms as studied by Simon Donaldson.

Category:Dynkin diagrams