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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The centralizer of a Cartan element from its vanishing roots

Statement

Let $g$ be a complex semisimple Lie algebra and let $g = h \oplus ⨁_{α \in Φ} g_{α}$ be its root-space decomposition relative to a Cartan subalgebra $h$ . For $h \in h$ ,

$C_{g} (h) = h \oplus ⨁_{α (h) = 0} g_{α} .$

Facts & Assumptions

Given: A complex semisimple Lie algebra $g$ , an element $h \in h$ in a Cartan subalgebra, and the root-space decomposition of $g$ relative to $h$ .

Proof

technique · direct

1.1givenalgebra

Write $x = x_{0} + \sum_{α} x_{α}$ using the root-space decomposition supplied in the Statement, with $x_{0} \in h$ and $x_{α} \in g_{α}$ . Then $[h, x] = \sum_{α} α (h) x_{α}$ .

2.1step 1.1∎

The directness of the root-space decomposition implies that $[h, x] = 0$ holds exactly when $x_{α} = 0$ for every root with $α (h) \neq 0$ . Thus the centralizer consists precisely of $h$ together with the root spaces on which $h$ vanishes.

Depends on

Root-space decomposition relative to a Cartan subalgebra

Used by

Regular semisimple elements form a dense open subset Lemma

Dependency tree · two levels

4 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)