DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Derivations of Lie algebras

Definition

A derivation of a Lie algebra $g$ is a linear map $D : g \to g$ satisfying

$D [x, y] = [D x, y] + [x, D y]$

for all $x, y$ . The vector space of derivations is denoted $Der (g)$ . For $x \in g$ , the linear map

${ad}_{x} : g \to g, {ad}_{x} (y) = [x, y],$

is called an inner derivation once its derivation law is established by Jacobi in the following proposition.

Depends on

Used by

Regular element and rank Definition
Root and root space Definition
Semidirect products of Lie algebras Definition
Toral and maximal toral subalgebras Definition
The root-space decomposition classifies real semisimple Lie algebras with no extra data False statement
Generalized weight spaces of a nilpotent subalgebra Lemma
The adjoint action preserves the associated graded of a two-sided ideal Lemma
The Killing length of a root is nonzero Lemma
Brackets of root spaces Proposition
Derivations form a Lie algebra and inner derivations an ideal Proposition
Derivations preserve the nilradical in characteristic zero Proposition
First cohomology is derivations modulo inner derivations Proposition
Cartan subalgebras are exactly maximal toral subalgebras Theorem
Centralizer of a regular semisimple element is Cartan Theorem
Existence of Cartan subalgebras Theorem
The Lie bracket from infinitesimals and the adjoint action Theorem

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

Etingof, MIT 18.745 notes, derivations in §3.2 (standard reference, not scraped)