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 nonsplit two-dimensional representation

Example

Let the one-dimensional abelian Lie algebra $k t$ act on $V = k e_{1} \oplus k e_{2}$ by

$t e_{1} = 0, t e_{2} = e_{1} .$

Then $k e_{1}$ is invariant but has no invariant complement.

Facts & Assumptions

Given: The displayed nilpotent action on a two-dimensional vector space.

[L1]

Stable subspaces and complete reducibility are those of Irreducible, completely reducible, and faithful representations.

Verification

technique · direct

1.1givenalgebra

The action is a representation because the acting Lie algebra is generated by one element and its chosen operator commutes with itself. Also $t (k e_{1}) = 0$ , so $k e_{1}$ is stable.

2.1step 1.1algebra

Every line complementary to $k e_{1}$ is spanned by $e_{2} + a e_{1}$ for some $a \in k$ , but $t (e_{2} + a e_{1}) = e_{1}$ does not belong to that line. Hence no complementary line is stable.

3.1step 1.1step 2.1L1∎

The invariant short filtration $0 \subset k e_{1} \subset V$ therefore does not split into subrepresentations, providing the claimed nonsplit example and, by [L1], a failure of complete reducibility.

Depends on

Irreducible, completely reducible, and faithful representations

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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, irreducibility in §4.3, printed pp. 52–53 (standard reference, not scraped)