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

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Centralizer dimension from vanishing roots

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 $H \in h$ , $g^{H} = \ker ({ad}_{H}) = h \oplus ⨁_{\begin{array}{c} α \in Φ \\ α (H) = 0 \end{array}} g_{α},$ and consequently $\dim g^{H} = \dim h + # {α \in Φ : α (H) = 0} .$ In particular $g^{H} = h$ exactly for the regular elements $H \in h_{r e g}$ of Regular root hyperplanes.

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ and an element $H \in h$ .

[A1]

The Axiom of Choice is The Axiom of Choice; it licenses [L1] and the regular-set definition [L2].

[L1]

$g = h \oplus ⨁_{α \in Φ} g_{α}$ is a direct sum, each $g_{α}$ is the eigenspace of ${ad}_{h}$ with eigenvalue $α$ , and $g_{α}$ is one-dimensional (Root-space decomposition, Root and root space, Root spaces of a complex semisimple Lie algebra are one-dimensional).

[L2]

$h_{r e g} = {H \in h : α (H) \neq 0 for all α \in Φ}$ (Regular root hyperplanes).

Proof

technique · direct

1.1

Write $x = H_{0} + \sum_{α \in Φ} x_{α}$ with $H_{0} \in h$ and $x_{α} \in g_{α}$ , using the direct sum [L1]. Then ${ad}_{H} (x) = \sum_{α} α (H) x_{α}$ because $h$ is abelian, and this vanishes exactly when $α (H) x_{α} = 0$ for every root.

A1L1algebra

2.1

Hence $\ker ({ad}_{H}) = h \oplus ⨁_{α (H) = 0} g_{α}$ and its dimension is $\dim h$ plus the number of roots vanishing at $H$ , by the direct sum of [L1]. By [L2] that number is zero exactly when $H \in h_{r e g}$ , in which case $g^{H} = h$ .

L1L2step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

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