ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

A semidirect-product Lie algebra from a linear action

Example

If $ρ : g \to g l (V)$ is a representation and $V$ is regarded as an abelian Lie algebra, then

$[(x, u), (y, v)] = ([x, y], ρ (x) v - ρ (y) u)$

makes $g \oplus V$ a Lie algebra $g ⋉_{ρ} V$ , with $V$ an abelian ideal.

Facts & Assumptions

Given: A Lie-algebra representation $ρ$ on a vector space $V$ .

[L1]

The semidirect construction uses a Lie map into derivations (Semidirect products of Lie algebras), and its bracket satisfies Jacobi (The semidirect-product bracket satisfies Jacobi).

Verification

technique · direct

1.1

The zero bracket makes $V$ abelian, and every endomorphism of $V$ is then a derivation because both sides of the derivation identity are zero. Thus $ρ$ has the target required by [L1], and substituting the zero bracket on $V$ gives the displayed formula.

givenL1algebra

2.1

For $u, v \in V$ , $[(0, u), (0, v)] = (0, 0)$ , while $[(x, w), (0, v)] = (0, ρ (x) v)$ lies in $0 \oplus V$ . Hence $V$ is an abelian ideal; it is also the kernel of the projection to $g$ .

step 1.1algebra

3.1

The Jacobi lemma in [L1] and steps 1.1–2.1 verify the claimed semidirect Lie algebra and its ideal.

step 1.1step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, semidirect products in §3.3 and representations in §4.1 (standard reference, not scraped)