DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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.

Normalizer of a Lie subalgebra

Definition

Let $g$ be a Lie algebra and let $h \subseteq g$ be a Lie subalgebra (Lie subalgebras, ideals, and center). The normalizer of $h$ in $g$ is

$N_{g} (h) = {x \in g : [x, h] \subseteq h} .$

It is a Lie subalgebra containing $h$ : for $x, y \in N_{g} (h)$ and $h \in h$ , Jacobi gives $[[x, y], h] = [x, [y, h]] - [y, [x, h]]$ , a difference of two elements of $h$ ; and $h \subseteq N_{g} (h)$ because $h$ is a subalgebra. Moreover $h$ is an ideal of $N_{g} (h)$ by the defining condition.

Depends on

Lie subalgebras, ideals, and center

Used by

A maximal abelian subalgebra that is not a Cartan subalgebra Counterexample
Cartan subalgebra Definition
Cartan subalgebra and roots of sl₂ Example
Cartan subalgebras of a direct sum Example
Diagonal Cartan subalgebra and roots of slₙ Example
Root systems B₂ and C₂ from matrix Lie algebras Example
A Cartan subalgebra of an arbitrary Lie algebra means a maximal abelian subalgebra False statement
The central reduction of U(sl2) is simple away from the finite-dimensional central characters Lemma
Split Cartan subalgebras of classical matrix Lie algebras Proposition
The normalizer acts on Lie algebra cohomology Proposition
Cartan subalgebras are exactly maximal toral subalgebras Theorem
Cayley transforms connect theta-stable Cartans in the classification Theorem
Centralizer of a regular semisimple element is Cartan Theorem
Conjugacy of Cartan subalgebras Theorem
Every real Cartan subalgebra is conjugate to a theta-stable one Theorem
Existence of Cartan subalgebras Theorem
Root-space decomposition Theorem

Dependency tree · two levels

3 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)