DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Universal enveloping algebra

Definition

Let $g$ be a Lie algebra over $k$ . In $T (g)$ , regard $[x, y] \in g$ as an element of tensor degree one, and let $I$ be the two-sided ideal generated by

$x \otimes y - y \otimes x - [x, y] (x, y \in g) .$

The universal enveloping algebra is

$U (g) = T (g) / I .$

Write

$ι_{g} : g ⟶ U (g)$

for the composite of the degree-one inclusion with the quotient map. This definition makes $U (g)$ a unital associative algebra and $ι_{g}$ a linear map. It does not assert that $ι_{g}$ is injective; that conclusion will come from PBW.

Universal enveloping algebra

Definition

Depends on

Dependency tree · 2 levels

Sources