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

1.1constructalgebra

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 $M (A, u) M (B, v) = (\begin{array}{cc} A B & A v \\ 0 & 0 \end{array})$ .

2.1step 1.1L1algebra

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.

3.1step 2.1∎

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

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)