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.

The affine Lie algebra as a semidirect product

Example

The Lie algebra of affine transformations of $V$ is $g l (V) ⋉ V$ , with bracket

$[(A, u), (B, v)] = ([A, B], A v - B u) .$

Facts & Assumptions

Given: A vector space $V$ , with $g l (V)$ acting on the abelian Lie algebra $V$ by evaluation.

[L1]

The semidirect bracket is that of Semidirect products of Lie algebras.

Verification

technique · direct block-matrix computation

1.1

Represent $(A, u)$ on $V \oplus k$ by $M (A, u) = (\begin{array}{cc} A & u \\ 0 & 0 \end{array})$ , where $u : k \to V$ sends $1$ to $u$ . Multiplication gives $M (A, u) M (B, v) = (\begin{array}{cc} A B & A v \\ 0 & 0 \end{array})$ .

constructalgebra

2.1

Subtracting the reversed product yields $[M (A, u), M (B, v)] = M ([A, B], A v - B u)$ , exactly the bracket in [L1] because $V$ is abelian.

step 1.1L1algebra

3.1

Thus the block realization identifies the affine Lie algebra with the stated semidirect product.

step 2.1∎

Depends on

Semidirect products of Lie algebras

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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 (standard reference, not scraped)