Mathematics.

lie theory

Root Systems

Representation Theory90 minDifficulty9 out of 10

You should know: hilbert spaces, group mathematics

Overview

A root system is a finite set of nonzero vectors in a Euclidean space satisfying certain symmetry conditions with respect to reflections. Root systems encode the symmetry structure of semisimple Lie algebras and provide the combinatorial data needed to classify them completely. The celebrated classification — types A, B, C, D, and exceptional types G₂, F₄, E₆, E₇, E₈ — is one of the deepest results in mathematics.

Intuition

Imagine a highly symmetric arrangement of arrows (vectors) in a plane or higher-dimensional space where reflecting the whole arrangement through any arrow's perpendicular hyperplane maps the set to itself, and the only lengths that appear are at most two distinct values. This rigid combinatorial structure — the root system — completely determines which semisimple Lie algebras can exist.

Formal Definition

Definition

Let V be a finite-dimensional real inner-product space. A root system in V is a finite set Φ ⊂ V \ {0} satisfying four axioms.

(R1) span⁡RΦ=V\text{(R1) } \operatorname{span}_{\mathbb{R}} \Phi = V
Φ spans V
(R2) α∈Φ  ⟹  Rα∩Φ={α,−α}\text{(R2) } \alpha \in \Phi \implies \mathbb{R}\alpha \cap \Phi = \{\alpha, -\alpha\}
Only scalar multiples ±α appear
(R3) α∈Φ  ⟹  sα(Φ)=Φ,sα(β)=β−⟨β,α∨⟩ α\text{(R3) } \alpha \in \Phi \implies s_\alpha(\Phi) = \Phi, \quad s_\alpha(\beta) = \beta - \langle \beta, \alpha^\vee \rangle\,\alpha
Reflection-invariance
(R4) ∀ α,β∈Φ,⟨β,α∨⟩=2⟨β,α⟩⟨α,α⟩∈Z\text{(R4) } \forall\, \alpha, \beta \in \Phi,\quad \langle \beta, \alpha^\vee \rangle = \frac{2\langle \beta, \alpha \rangle}{\langle \alpha, \alpha \rangle} \in \mathbb{Z}
Cartan integers are integers

Notation

NotationMeaning
Φ\PhiRoot system
α∨=2α⟨α,α⟩\alpha^\vee = \frac{2\alpha}{\langle \alpha,\alpha\rangle}Coroot of α
⟨β,α∨⟩\langle \beta, \alpha^\vee \rangleCartan integer
WWWeyl group generated by reflections sα
Φ+,  Φ−\Phi^+,\; \Phi^-Positive and negative roots

Properties

Finite Weyl group

The Weyl group W=⟨sα:α∈Φ⟩ is a finite group acting faithfully on V\text{The Weyl group } W = \langle s_\alpha : \alpha \in \Phi \rangle \text{ is a finite group acting faithfully on } V

Root length ratio constraint

⟨α,β∨⟩⟨β,α∨⟩∈{0,1,2,3}for α≠±β\langle \alpha, \beta^\vee \rangle \langle \beta, \alpha^\vee \rangle \in \{0,1,2,3\} \quad \text{for } \alpha \neq \pm\beta

Simple roots form a basis

A set of simple roots Δ⊂Φ+ forms a basis of V and every root is a Z-linear combination of Δ\text{A set of simple roots } \Delta \subset \Phi^+ \text{ forms a basis of } V \text{ and every root is a } \mathbb{Z}\text{-linear combination of } \Delta

Worked Examples

  1. 1

    A₂ consists of 6 roots arranged at 60° intervals.

    Φ={±e1,±e2,±(e1−e2)} (in a suitable 2D embedding)\Phi = \{\pm e_1, \pm e_2, \pm(e_1-e_2)\} \text{ (in a suitable 2D embedding)}
  2. 2

    All roots have equal length. The Weyl group is the dihedral group of order 6 (symmetries of an equilateral triangle).

    W≅S3,∣Φ∣=6W \cong S_3,\quad |\Phi| = 6
  3. 3

    Simple roots are Δ = {α₁, α₂} with Cartan matrix entries ⟨α₁, α₂∨⟩ = ⟨α₂, α₁∨⟩ = -1.

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

✓ Answer

A₂ is a rank-2 root system with 6 roots, Weyl group S₃, and corresponds to the Lie algebra sl₃(ℂ).

Practice Problems

Hardfree response

Show that if α, β ∈ Φ with α ≠ ±β, then ⟨α, β∨⟩⟨β, α∨⟩ ∈ {0, 1, 2, 3}.

Hardproof writing

Prove that the Weyl group of a root system Φ acts simply transitively on the set of Weyl chambers.

Common Mistakes

Common Mistake

Assuming all roots in a root system have equal length

Only simply-laced root systems (A, D, E types) have all roots the same length. Types B, C, F₄, G₂ have two root lengths.

Common Mistake

Confusing the root system with the root lattice

The root system Φ is a finite set of vectors; the root lattice Q = ℤ-span(Φ) is an infinite lattice. They are different objects.

Quiz

Which condition ensures Cartan integers ⟨β, α∨⟩ are integers?
How many roots does the root system G₂ contain?

Historical Background

Root systems emerged from Wilhelm Killing's 1888–1890 classification of simple Lie algebras over ℂ, later made rigorous by Élie Cartan in his 1894 thesis. The purely combinatorial reformulation in terms of root systems and Dynkin diagrams was developed by Eugene Dynkin in the 1940s, giving an elegant and elementary approach to the classification.

  1. 1888

    Killing classifies simple Lie algebras, discovering exceptional types

    Wilhelm Killing

  2. 1894

    Cartan rigorises the classification in his doctoral thesis

    Élie Cartan

  3. 1947

    Dynkin introduces Dynkin diagrams as a clean encoding of root systems

    Eugene Dynkin

Summary

  • A root system is a finite set of nonzero vectors in a Euclidean space, closed under reflections, with integer Cartan integers.
  • Simple roots form a basis; every root is a non-negative or non-positive integer combination of simple roots.
  • The Weyl group encodes the symmetries of the root system.
  • Root systems classify semisimple Lie algebras: types An, Bn, Cn, Dn, and exceptionals G₂, F₄, E₆, E₇, E₈.

References

  1. BookHumphreys, J.E. — Introduction to Lie Algebras and Representation Theory (1972), Chapter III
  2. WebsiteWikipedia — Root system
  3. WebsiteMathWorld — Root System