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

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.

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

Semidirect products of Lie algebras Definition
Derivations form a Lie algebra and inner derivations an ideal Proposition

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)