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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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
Generalized casimir on restricted kac moody modules Definition
Kac moody verma module Definition
Smooth and K-finite vectors for SL2(R), and the (g,K)-module Definition
The annihilator of a module over an enveloping algebra Definition
The central reduction of the enveloping algebra at a central character Definition
The Harish-Chandra projection Definition
The PBW filtration by tensor degree on the enveloping algebra Definition
The quadratic Casimir element Definition
Universal affine vacuum module Definition
Verma modules Definition
Central actions on nilradical cohomology factor through the Harish–Chandra projection Lemma
Dixmier's lemma: endomorphisms of a simple module over a countable-dimensional algebra Lemma
Enveloping quotient kernels and augmentation intersections Lemma
Free Lie construction for finite Kac Moody generators Lemma
Highest- and lowest-weight submodules at the exceptional parameters Lemma
PBW basis of the universal affine vacuum module Lemma
PBW for countably presented Kac Moody Lie algebras Lemma
The quadratic Casimir element is independent of the choice of dual bases Lemma
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)