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

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.

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The universal enveloping algebra as a tensor quotient

Definition

Let $g$ be a complex Lie algebra, and let $T (g) = ⨁_{n \geq 0} g^{\otimes n}$ be its tensor algebra. Let $I \subseteq T (g)$ be the two-sided ideal generated by all elements

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

The universal enveloping algebra of $g$ is the quotient algebra

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

The image of $x \in g$ in $U (g)$ is again denoted by $x$ . The defining relation is therefore

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

This is the quotient algebra used throughout the page for PBW, the Casimir element, and the Harish-Chandra projection.

Used by

Central character of a Lie algebra module Definition
The Harish-Chandra projection Definition
The PBW filtration by tensor degree on the enveloping algebra Definition
The quadratic Casimir element Definition
Lie algebra actions extend to unital actions of the enveloping algebra Proposition
PBW gives an ordered monomial basis for the enveloping algebra Theorem

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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