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.

PBW basis for the Heisenberg Lie algebra

Example

Let $h$ have basis $x, y, z$ with $[x, y] = z$ and $z$ central. For the order $x < y < z$ , the elements

$x^{a} y^{b} z^{c} (a, b, c \geq 0)$

form a basis of $U (h)$ .

Facts & Assumptions

Given: The Heisenberg Lie algebra with the displayed supplied ordered basis.

[L1]

PBW gives a basis of weakly increasing monomials for any supplied ordered basis (Poincaré–Birkhoff–Witt theorem).

Verification

technique · direct PBW specialization

1.1

A weakly increasing word in the order $x < y < z$ consists uniquely of $a$ copies of $x$ , then $b$ copies of $y$ , then $c$ copies of $z$ , and is therefore $x^{a} y^{b} z^{c}$ .

givenalgebra

1.2

The enveloping relation is $y x = x y - z$ , while centrality gives $z x = x z$ and $z y = y z$ . These formulas concretely move every inversion toward the ordered form.

givenalgebra

2.1

By [L1], the ordered forms identified in step 1.1 are linearly independent as well as spanning, so they are a basis; step 1.2 is the corresponding reordering rule.

step 1.1step 1.2L1∎

Depends on

Poincaré–Birkhoff–Witt theorem

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, PBW examples in §13.1, printed pp. 74–75 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.2, printed pp. 72–75 (standard reference, not scraped)