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

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Universal enveloping algebra

Definition

Let $g$ be a Lie algebra over $k$ . In $T (g)$ , regard $[x, y] \in g$ as an element of tensor degree one, and let $I$ be the two-sided ideal generated by

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

The universal enveloping algebra is

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

Write

$ι_{g} : g ⟶ U (g)$

for the composite of the degree-one inclusion with the quotient map. This definition makes $U (g)$ a unital associative algebra and $ι_{g}$ a linear map. It does not assert that $ι_{g}$ is injective; that conclusion will come from PBW.

Depends on

Used by

PBW filtration on the enveloping algebra Definition
The Casimir element in U(sl₂) Example
Injectivity is not part of the enveloping quotient definition False statement
The canonical map to U(g) is a Lie homomorphism Lemma
The enveloping algebra of an abelian Lie algebra is symmetric Proposition
Lie representations are U(g)-modules Theorem
Universal property of the enveloping algebra Theorem

Dependency tree · two levels

12 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, §12.1, printed pp. 69–70 (standard reference, not scraped)
Kirillov, An Introduction to Lie Groups and Lie Algebras, §5.1, printed pp. 71–72 (standard reference, not scraped)