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.

Tensor, dual, and Hom representation formulas

Example

For representations $V, W$ , the induced actions are

$x (v \otimes w) = x v \otimes w + v \otimes x w,$

$(x λ) (v) = - λ (x v), x T = ρ_{W} (x) T - T ρ_{V} (x) .$

Facts & Assumptions

Given: Representations $V, W$ of the same Lie algebra.

[L1]

These constructions are asserted in Direct-sum, dual, Hom, and tensor representations.

Verification

technique · direct expansion illustrating the signs

1.1givenalgebra

Applying two tensor operators to $v \otimes w$ produces the two unmixed commutator terms $[x, y] v \otimes w$ and $v \otimes [x, y] w$ ; the mixed terms $x v \otimes y w$ and $y v \otimes x w$ occur with opposite signs and cancel.

1.2givenalgebra

On the dual, two applications give $(x (y λ) - y (x λ)) (v) = λ (y x v - x y v) = - λ ([x, y] v)$ , which is exactly the displayed dual action of $[x, y]$ .

1.3givenalgebra

On Hom, expanding the commutator of $T \mapsto ρ_{W} (x) T - T ρ_{V} (x)$ and its $y$ -analogue cancels the mixed composites and leaves $ρ_{W} ([x, y]) T - T ρ_{V} ([x, y])$ .

2.1step 1.1step 1.2step 1.3L1∎

These computations verify the representation identity for all three formulas in [L1] and show why the dual minus sign and Hom subtraction are necessary.

Depends on

Direct-sum, dual, Hom, and tensor representations

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

Etingof, MIT 18.745 notes, §11.2, printed pp. 62–63 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §4.2, printed pp. 50–52 (standard reference, not scraped)