Math 2.0

Nonassociative rings and algebras Definition

Root system

Builds on Euclidean spaceInner productReflection

Definition

Let EE be a finite-dimensional real vector space with an inner product (⋅,⋅)(\cdot, \cdot), a Euclidean space. For a nonzero α∈E\alpha \in E, let sα(x)=x−2(x,α)(α,α)αs_\alpha(x) = x - \frac{2 (x, \alpha)}{(\alpha, \alpha)} \alpha be the reflection in the hyperplane orthogonal to α\alpha. A root system in EE is a finite subset Φ⊆E\Phi \subseteq E such that:

  1. Φ\Phi spans EE and 0∉Φ0 \notin \Phi;
  2. for each α∈Φ\alpha \in \Phi, the only multiples of α\alpha in Φ\Phi are α\alpha and −α-\alpha;
  3. for each α∈Φ\alpha \in \Phi, the reflection sαs_\alpha maps Φ\Phi onto Φ\Phi;
  4. for all α,β∈Φ\alpha, \beta \in \Phi, the number ⟨β,α⟩=2(β,α)(α,α)\langle \beta, \alpha \rangle = \frac{2 (\beta, \alpha)}{(\alpha, \alpha)} is an integer.

The elements of Φ\Phi are the roots, and dim⁡E\dim E is the rank of Φ\Phi.

Remarks

Example: let EE be the plane x1+x2+x3=0x_1 + x_2 + x_3 = 0 in ℝ3\R^3, with the standard basis ε1,ε2,ε3\varepsilon_1, \varepsilon_2, \varepsilon_3 of ℝ3\R^3. The six vectors εi−εj\varepsilon_i - \varepsilon_j with i≠ji \ne j form the root system A2A_2, whose roots point to the corners of a regular hexagon. They span EE; each sαs_\alpha swaps two coordinates, so it permutes Φ\Phi; and all numbers ⟨β,α⟩\langle \beta, \alpha \rangle lie in {0,±1,±2}\set{0, \pm 1, \pm 2}. Non-example: with e1,e2e_1, e_2 orthonormal, the set {±e1,±e2,±(e1+e2)}\set{\pm e_1, \pm e_2, \pm (e_1 + e_2)} fails clause 3, since se1(e1+e2)=−e1+e2s_{e_1}(e_1 + e_2) = -e_1 + e_2 is not in the set. Some authors drop clause 2 and call the systems that satisfy it reduced; then {±e1,±2e1}\set{\pm e_1, \pm 2 e_1} in ℝ\R is the non-reduced system BC1BC_1. Some authors also drop the spanning condition in clause 1.

Reading path

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Euclidean space, object, no evidence levelEuclidean spaceobjectInner product, concept, no evidence levelInner productconceptReflection, concept, no evidence levelReflectionconceptRoot system, definition, no evidence levelRoot systemdefinition
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  1. Euclidean space object
  2. Inner product concept
  3. Reflection concept
  4. Root system definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Root_system, Definition
About this page
TagY32EWAATPG
URIm2://Y32EWAATPG
Kinddefinition
WritingW1 stated
MSC17B22, 20F55
DefinesRoot system
UsesEuclidean space, Inner product, Reflection
Changed3 November 2026

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