Mathematics.

lie theory

Dynkin Diagrams

Representation Theory60 minDifficulty8 out of 10

You should know: root systems

Overview

A Dynkin diagram is a graph that encodes the complete combinatorial data of a root system — and hence of a semisimple Lie algebra. Nodes represent simple roots, edges (and their multiplicities and arrow directions) record the angles between them. The classification of connected Dynkin diagrams is precisely the classification of simple Lie algebras: An, Bn, Cn, Dn for n ≥ 1, and the exceptionals G₂, F₄, E₆, E₇, E₈.

Intuition

Think of a Dynkin diagram as a blueprint for a root system. Each dot is a simple root (a 'building block' direction). A single edge between two dots means they meet at 120°; a double edge means 135°; a triple edge means 150°. The arrow on a multi-edge points from the longer root to the shorter root, recording the length ratio.

Formal Definition

Definition

Given a root system Φ with simple roots Δ = {α₁, …, αₙ}, the Dynkin diagram is a graph on n nodes where the edge between nodes i and j is determined by the product of Cartan integers.

aij=⟨αi,αj∨⟩=2⟨αi,αj⟩⟨αj,αj⟩a_{ij} = \langle \alpha_i, \alpha_j^\vee \rangle = \frac{2\langle \alpha_i, \alpha_j \rangle}{\langle \alpha_j, \alpha_j \rangle}
Cartan matrix entries
Number of edges between i and j:  nij=aijaji∈{0,1,2,3}\text{Number of edges between } i \text{ and } j: \; n_{ij} = a_{ij} a_{ji} \in \{0,1,2,3\}
Edge multiplicity
Arrow: i→j  ⟺  aij<aji  ⟺  ∣αi∣>∣αj∣\text{Arrow: } i \to j \iff a_{ij} < a_{ji} \iff |\alpha_i| > |\alpha_j|
Arrow convention (longer to shorter)

Notation

NotationMeaning
AnA_nChain of n nodes with single edges; corresponds to sl_{n+1}(ℂ)
BnB_nChain ending in a double edge with arrow toward shorter root; so(2n+1)
CnC_nChain ending in a double edge with arrow toward longer root; sp(2n)
DnD_nChain with a fork at the end; so(2n)
E6,E7,E8E_6, E_7, E_8Exceptional diagrams with branch

Properties

Connectedness and simplicity

A Dynkin diagram is connected if and only if the corresponding semisimple Lie algebra is simple\text{A Dynkin diagram is connected if and only if the corresponding semisimple Lie algebra is simple}

Symmetry of Cartan matrix

aii=2 and aij∈{0,−1,−2,−3} for i≠ja_{ii} = 2 \text{ and } a_{ij} \in \{0,-1,-2,-3\} \text{ for } i \neq j

No cycles in simply-laced diagrams

The diagrams An,Dn,E6,E7,E8 are trees (cycle-free graphs) with all single edges\text{The diagrams } A_n, D_n, E_6, E_7, E_8 \text{ are trees (cycle-free graphs) with all single edges}

Worked Examples

  1. 1

    A₃ has 3 simple roots α₁, α₂, α₃ all of equal length.

    Δ={α1,α2,α3}\Delta = \{\alpha_1, \alpha_2, \alpha_3\}
  2. 2

    Adjacent roots meet at 120° so a₁₂ = a₂₁ = -1, a₂₃ = a₃₂ = -1, a₁₃ = 0.

    A=(2−10−12−10−12)A = \begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}
  3. 3

    The diagram is a straight chain of 3 nodes with single edges: ○-○-○.

    ∘ ⁣− ⁣∘ ⁣− ⁣∘\circ \!-\!\circ \!-\!\circ

✓ Answer

A₃ is a chain of 3 nodes with single edges, corresponding to sl₄(ℂ).

Practice Problems

Mediumfree response

Write out the Cartan matrix for G₂ and verify it gives 3 edges between the two nodes.

Common Mistakes

Common Mistake

Confusing the arrow direction in Dynkin diagrams

The arrow points from the longer root to the shorter root. In Bn the long root is on the left and the short root on the right, so the arrow points right.

Common Mistake

Thinking Bn and Cn have the same Dynkin diagram

Both have a double edge, but the arrow direction is reversed: Bn has ⇒ (long to short) while Cn has ⇐ (short to long). They are different Lie algebras.

Quiz

Which Dynkin diagram corresponds to the Lie algebra sp(2n,ℂ)?
How many nodes does the E₈ Dynkin diagram have?

Historical Background

Eugene Dynkin introduced these diagrams in 1947 as part of his streamlined approach to classifying semisimple Lie algebras. Prior to Dynkin's work, the classification — due to Killing and Cartan — required heavy analytic machinery. Dynkin's combinatorial encoding made the classification accessible and launched a tradition of using decorated graphs in Lie theory.

  1. 1947

    Dynkin introduces the diagrams in his classification of simple Lie algebras

    Eugene Dynkin

  2. 1950s

    Coxeter and others connect Dynkin diagrams to reflection groups and Coxeter systems

    Harold Coxeter

Summary

  • A Dynkin diagram encodes a root system: nodes are simple roots, edges record angles and length ratios.
  • Single edge: 120°, double edge: 135°, triple edge: 150°; arrows point from longer to shorter root.
  • Connected diagrams classify simple Lie algebras: An, Bn, Cn, Dn, G₂, F₄, E₆, E₇, E₈.
  • The Cartan matrix (aᵢⱼ = ⟨αᵢ, αⱼ∨⟩) is completely determined by the Dynkin diagram.

References

  1. BookHumphreys, J.E. — Introduction to Lie Algebras and Representation Theory (1972), Chapter IV
  2. WebsiteWikipedia — Dynkin diagram
  3. WebsiteMathWorld — Dynkin Diagram