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.

Standard representations of classical matrix Lie algebras

Example

The matrix Lie algebras ${g l}_{n}$ , ${s l}_{n}$ , ${s o}_{n}$ , and ${s p}_{2 n}$ act on their defining vector spaces by matrix multiplication.

Facts & Assumptions

Given: One of the displayed bracket-closed matrix Lie algebras over its defining field, and its defining vector space $V$ .

[L1]

A representation is equivalently a bilinear action satisfying $[X, Y] v = X (Y v) - Y (X v)$ (Representations of Lie algebras).

Verification

technique · direct

1.1

For all endomorphisms $X, Y$ and $v \in V$ , the commutator definition gives $[X, Y] v = (X Y - Y X) v = X (Y v) - Y (X v)$ .

givenalgebra

2.1

Matrix multiplication is bilinear in the matrix and vector variables. Together with step 1.1, this verifies both conditions in the equivalence [L1], so restriction to each named bracket-closed matrix Lie algebra is a representation.

step 1.1L1algebra∎

Depends on

Representations of Lie algebras

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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, Examples 11.1 and classical groups in §6, printed pp. 38–44 and 62 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §4.1, printed pp. 49–50 (standard reference, not scraped)