DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.

Depends on

Used by

No hidden linear relations in degree one Corollary
PBW spanning by ordered monomials Lemma
The PBW filtration is multiplicative and has commutative associated graded Proposition

Dependency tree · two levels

7 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, §§12.2 and 13.1, printed pp. 70 and 74–75 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.2, printed pp. 72–74 (standard reference, not scraped)