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.

PBW filtration on the enveloping algebra

Definition

Let $q : T (g) ↠ U (g)$ be the defining quotient map. For $n \geq 0$ , define

$F_{n} U (g) = q (⨁_{0 \leq r \leq n} g^{\otimes r}),$

and put $F_{- 1} U (g) = 0$ . This is the PBW filtration. It is increasing and exhaustive: every tensor-algebra element has finite degree support, hence every enveloping-algebra element belongs to some $F_{n}$ .

Equivalently, $F_{n}$ is spanned by products of at most $n$ elements from the image $ι_{g} (g)$ , with the empty product included. This definition does not assume that $ι_{g}$ is injective.

PBW filtration on the enveloping algebra

Definition

Depends on

Dependency tree · 2 levels

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