DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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The PBW filtration by tensor degree on the enveloping algebra

Definition

Let $g$ be a complex Lie algebra and let $U (g)$ be its universal enveloping algebra from The universal enveloping algebra as a tensor quotient. For $n \geq 0$ , let $F_{n} U (g)$ be the image in $U (g)$ of the truncated tensor algebra

$⨁_{k = 0}^{n} g^{\otimes k} \subseteq T (g) .$

The increasing sequence

$F_{0} U (g) \subseteq F_{1} U (g) \subseteq F_{2} U (g) \subseteq \dots$

is the PBW filtration by tensor degree. Its associated graded algebra is

$gr U (g) : = ⨁_{n \geq 0} F_{n} U (g) / F_{n - 1} U (g),$

with the convention $F_{- 1} U (g) = 0$ .

Depends on

The universal enveloping algebra as a tensor quotient

Used by

Dependency tree · one level

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Sources

Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)