TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Root-space decomposition relative to a Cartan subalgebra

Statement

Let $g$ be a 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: A complex semisimple Lie algebra $g$ and a Cartan subalgebra $h \subseteq g$ .

Proof

technique · direct

1.1

Because $h$ is a Cartan subalgebra of a complex semisimple Lie algebra, the commuting operators $ad (h)$ for $h \in h$ are simultaneously diagonalizable, so $g$ decomposes into common eigenspaces for the adjoint action of $h$ .

given

2.1

The common eigenspace for the zero functional is the centralizer of $h$ , which equals $h$ because $h$ is self-normalizing. Every other common eigenfunctional is nonzero and contributes a subspace of the displayed form $g_{α}$ .

step 1.1

3.1

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.

step 1.1step 2.1∎

Depends on

The Killing form is invariant and nondegenerate on a complex semisimple Lie algebra

Used by

The regular root-hyperplane arrangement in a Cartan subalgebra Definition
Central elements lie in the zero-weight subspace of U(g) 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
The Killing form pairs only opposite root spaces Proposition
The quadratic Casimir eigenvalue on a highest-weight module is (λ,λ+2ρ) Proposition
The root set is a reduced crystallographic root system Theorem
Triangular decomposition from a chosen positive root system Theorem

Dependency tree · two levels

2 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)