DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-22

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Root reflection defined by a coroot

Definition

Assume the Axiom of Choice. Let $α$ be a root of a finite-dimensional complex semisimple Lie algebra $g$ with respect to a Cartan subalgebra $h$ (Root and root space) and let $h_{α} \in h$ be its coroot (Coroot of a Lie-algebra root). The root reflection defined by $α$ is the linear map

$s_{α} : h^{*} ⟶ h^{*}, s_{α} (λ) = λ - λ (h_{α}) α .$

It is linear and involutive: $s_{α} (α) = α - 2 α = - α$ because $α (h_{α}) = 2$ , and $s_{α}$ fixes every $λ$ with $λ (h_{α}) = 0$ . In particular $s_{α}$ is an automorphism of the vector space $h^{*}$ with $s_{α}^{2} = id$ , since $s_{α} (λ) (h_{α}) = - λ (h_{α})$ .

Depends on

Used by

The Weyl reflection in sl₂ Example
Simple reflections preserve weight multiplicities Lemma
Root reflections are induced by inner automorphisms Proposition
The roots form a reduced crystallographic Euclidean root system Proposition
Root reflections preserve the root set Theorem
Roots of a complex semisimple Lie algebra form a reduced crystallographic root system Theorem

Dependency tree · two levels

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Sources

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I, Lectures 19–24 (standard reference, not scraped)