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

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Cartan integers are integers

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 Cartan integer $⟨ β, α^{\lor} ⟩ : = β (h_{α}) = \frac{2 B (H_{α}, H_{β})}{B (H_{α}, H_{α})}$ is an integer.

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 and coroot facts in [L1] and [L2].

[L1]

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

[L2]

The coroot $h_{α} = 2 H_{α} / α (H_{α})$ and the Killing-dual vector are as in Coroot of a Lie-algebra root (Root and root space supplies the root set).

Proof

technique · direct

1.1

By [L1] applied to the roots $α, β$ there are nonnegative integers $p, q$ with $p - q = β (h_{α})$ .

A1L1algebra

2.1

Since $p$ and $q$ are integers, their difference $β (h_{α})$ is an integer; this is the claimed integrality. The displayed formula for the Cartan integer is the definition of $H_{α}$ and $h_{α}$ from [L2].

L1L2step 1.1algebra∎

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)