TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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Root-space decomposition relative to a Cartan subalgebra

Statement

Assume the Axiom of Choice. Let $g$ be a finite-dimensional complex semisimple Lie algebra and $h \subseteq g$ a Cartan subalgebra. For $α \in h^{*}$ , set

$g_{α} : = {x \in g : [h, x] = α (h) x for every h \in h} .$

Then there is a finite set $Φ \subseteq h^{*} ∖ {0}$ such that

$g = h \oplus ⨁_{α \in Φ} g_{α},$

and the nonzero summands are exactly the root spaces.

Facts & Assumptions

Given: The Axiom of Choice, a finite-dimensional complex semisimple Lie algebra $g$ , and a Cartan subalgebra $h \subseteq g$ in the nilpotent self-normalizing sense.

[A1]

The Axiom of Choice is The Axiom of Choice.

[F1]

Under [A1], a Cartan subalgebra in this sense is maximal toral (Cartan subalgebras are exactly maximal toral subalgebras).

[F2]

Every maximal toral subalgebra of a finite-dimensional complex semisimple Lie algebra has a finite simultaneous root-space decomposition, with zero weight space exactly the subalgebra (Finite semisimple Cartan, root and string structure).

Proof

technique · direct

1.1A1F1F2

By [A1, F1], $h$ is maximal toral. Its adjoint operators therefore commute and are diagonalizable; [F2] supplies their simultaneous finite weight-space decomposition.

2.1F2step 1.1

The zero weight space is the centralizer $C_{g} (h)$ , which equals $h$ by [F2]. The other nonzero weight spaces are exactly the displayed $g_{α}$ by their simultaneous-eigenvector definition.

3.1step 1.1step 2.1∎

Collecting the finitely many nonzero weights gives the finite set $Φ$ , and the simultaneous eigenspace decomposition from step 1.1 becomes the stated direct sum decomposition.

Depends on

Used by

The regular root-hyperplane arrangement in a Cartan subalgebra Definition
The Chevalley–Eilenberg Laplacian is scalar on weight components Lemma
The extremal weight cochain of a Weyl element is closed and unique Lemma
Brackets of root spaces add their roots Proposition
Opposite root spaces bracket to the Killing-dual line Proposition
The centralizer of a Cartan element from its vanishing roots Proposition
Under Choice, the Killing form pairs only opposite root spaces Proposition
Triangular decomposition from a chosen positive root system Theorem

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)