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

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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
Split Cartan subalgebras of classical matrix Lie algebras 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)