PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Functoriality of the enveloping algebra

Statement

A Lie-algebra homomorphism $f : g \to h$ induces a unique unital algebra homomorphism

$U (f) : U (g) ⟶ U (h)$

such that $U (f) ι_{g} = ι_{h} f$ . Moreover $U (id) = id$ and $U (g f) = U (g) U (f)$ .

Facts & Assumptions

Given: Lie-algebra homomorphisms between Lie algebras over $k$ .

[L1]

Each canonical map $ι_{h}$ is a Lie map into the commutator algebra (The canonical map to U(g) is a Lie homomorphism).

[L2]

Such Lie maps extend uniquely from $g$ to $U (g)$ (Universal property of the enveloping algebra).

Proof

technique · direct

1.1

The composite $ι_{h} f : g \to U (h)_{L i e}$ is a Lie map by [L1], so [L2] supplies the unique unital algebra map $U (f)$ with the stated generator equation.

L1L2

1.2

Both $U ({id}_{g})$ and ${id}_{U (g)}$ compose with $ι_{g}$ to $ι_{g}$ , so uniqueness in [L2] makes them equal.

L2algebra

2.1

For $g \overset{f}{\to} h \overset{g}{\to} l$ , both $U (g f)$ and $U (g) U (f)$ send $ι_{g}$ to $ι_{l} g f$ . Uniqueness in [L2] therefore gives $U (g f) = U (g) U (f)$ .

step 1.1L2algebra

3.1

The construction preserves identities and composition and is consequently functorial.

step 1.1step 1.2step 2.1∎

Depends on

Used by

Enveloping algebra of a direct sum Proposition

Dependency tree · two levels

5 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)