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.

Quotient Lie algebras

Definition

For an ideal $i ⊴ g$ , the quotient vector space $g / i$ equipped with

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

is the quotient Lie algebra. The bracket is representative-independent and satisfies all Lie identities by The quotient Lie-algebra bracket is well-defined. The canonical projection $π : g \to g / i$ is linear and satisfies $π ([x, y]) = [π (x), π (y)]$ .

Depends on

The quotient Lie-algebra bracket is well-defined

Used by

Kernels, images, and the first isomorphism theorem for Lie algebras 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

Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.3, before Lemma 5.17 (standard reference, not scraped)