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.

The enveloping algebra of a one-dimensional abelian Lie algebra

Example

If $g = k x$ is one-dimensional and abelian, then $U (g) ≅ k [t]$ , with $ι_{g} (x)$ corresponding to $t$ .

Facts & Assumptions

Given: The abelian Lie algebra $g = k x$ .

[L1]

The enveloping algebra of an abelian Lie algebra is its symmetric algebra (The enveloping algebra of an abelian Lie algebra is symmetric).

Verification

technique · direct

1.1

The symmetric algebra $S (k x)$ has one basis monomial $x^{n}$ in every degree $n \geq 0$ , and multiplication satisfies $x^{m} x^{n} = x^{m + n}$ .

givenalgebra

2.1

Sending $t^{n} \mapsto x^{n}$ therefore defines a bijective unital algebra map $k [t] \to S (k x)$ . Composing with [L1] gives $k [t] ≅ U (g)$ and sends $t$ to $ι_{g} (x)$ .

step 1.1L1algebra∎

Depends on

The enveloping algebra of an abelian Lie algebra is symmetric

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, Example 12.2, printed p. 69 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, Example 5.1, printed p. 71 (standard reference, not scraped)