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.

Adjoint and trivial representations

Example

Every Lie algebra $g$ acts on itself by $x \cdot y = [x, y]$ , the adjoint representation. It also acts on any vector space $V$ by $x \cdot v = 0$ , the trivial representation.

Facts & Assumptions

Given: A Lie algebra $g$ over $k$ and an arbitrary vector space $V$ over $k$ .

[L1]

A representation requires $[ρ (x), ρ (y)] = ρ ([x, y])$ (Representations of Lie algebras).

[L2]

The adjoint map is a Lie-algebra homomorphism (Derivations form a Lie algebra and inner derivations an ideal).

Verification

technique · direct

1.1

For the adjoint action, [L2] gives $[{ad}_{x}, {ad}_{y}] = {ad}_{[x, y]}$ , exactly the identity in [L1].

L1L2

1.2

For the trivial action, both $[0, 0]$ and the operator assigned to $[x, y]$ are zero, so [L1] holds.

L1algebra

2.1

Hence both formulas define representations; the adjoint action is trivial precisely when $g$ is abelian.

step 1.1step 1.2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, Example 11.1, printed p. 62 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, Examples 4.2–4.3, printed p. 49 (standard reference, not scraped)