TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Root reflections preserve the root set

Statement

Assume the Axiom of Choice. Let $h$ be a Cartan subalgebra of a finite-dimensional complex semisimple Lie algebra $g$ with root set $Φ$ (Root and root space). For roots $α, β \in Φ$ , the reflected functional $s_{α} (β)$ (Root reflection defined by a coroot) is again a root.

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ and roots $α, β$ .

[A1]

The Axiom of Choice is The Axiom of Choice; it licenses the root-string, coroot, and reflection facts in [L1] and [L2].

[L1]

The set ${k \in Z : g_{β + k α} \neq 0}$ of indices $k$ with $β + k α$ a root or zero is a nonempty interval ${- p, \dots, q}$ of consecutive integers with $p - q = β (h_{α})$ (The root-string property).

[L2]

The reflection is $s_{α} (λ) = λ - λ (h_{α}) α$ , and $β (h_{α}) \in Z$ (Root reflection defined by a coroot, Cartan integers are integers, Coroot of a Lie-algebra root).

Proof

technique · direct

1.1

By [L1] applied to the pair $(α, β)$ there are integers $p, q \geq 0$ with $p - q = β (h_{α})$ and such that $β + k α \in Φ \cup {0}$ for every $k$ with $- p \leq k \leq q$ .

A1L1L2

2.1

By [L2] we may rewrite $s_{α} (β) = β - β (h_{α}) α = β + (q - p) α$ , and the index $k = q - p$ satisfies $- p \leq q - p \leq q$ because $p, q \geq 0$ . Hence $s_{α} (β) = β + k α$ with $- p \leq k \leq q$ , so $s_{α} (β) \in Φ \cup {0}$ by step 1.1.

L1L2step 1.1algebra

3.1

Finally $s_{α} (β) \neq 0$ : if $β + (q - p) α = 0$ then $β$ is a scalar multiple of $α$ , so $s_{α} (β) = 0$ would mean $β = β (h_{α}) α$ ; but then $β$ and $α$ are proportional roots and the reflection of a nonzero functional is nonzero because $s_{α}$ is an involutive linear automorphism of $h^{*}$ (Root reflection defined by a coroot) with $s_{α} (α) = - α \neq 0$ . Hence $s_{α} (β) \in Φ$ , as claimed.

step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

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Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)