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 Casimir element in U(sl_2)

Example

Over a characteristic-zero field, the expression

$Ω = e f + f e + \frac{1}{2} h^{2}$

defines an element of $U ({s l}_{2})$ . This example does not assert or use its centrality.

Facts & Assumptions

Given: The standard basis $e, f, h$ of ${s l}_{2}$ and its images in $U ({s l}_{2})$ over a characteristic-zero field.

[L1]

The enveloping algebra is a unital associative quotient in which such finite sums and products are defined (Universal enveloping algebra).

[L2]

For the PBW order $f < h < e$ , one has $e f = f e + h$ (PBW reordering in sl_2).

Verification

technique · direct

1.1

Characteristic zero makes $2$ invertible, and [L1] therefore makes the displayed finite polynomial in $e, f, h$ a well-defined enveloping-algebra element.

givenL1algebra

2.1

Using [L2], it has the PBW-normal expression $Ω = 2 f e + h + \frac{1}{2} h^{2}$ . This is an equality of elements, not a centrality computation.

step 1.1L2algebra

3.1

Thus the stated Casimir expression and its normal form are justified; centrality is deliberately deferred to the later Casimir and central-character treatment.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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 13.9, printed p. 75 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, Example 5.6, printed p. 73 (standard reference, not scraped)