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

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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 canonical map to U(g) is a Lie homomorphism

Statement

For every $x, y \in g$ , the canonical map satisfies

$ι_{g} ([x, y]) = ι_{g} (x) ι_{g} (y) - ι_{g} (y) ι_{g} (x) .$

Consequently $ι_{g} : g \to U (g)_{L i e}$ is a Lie-algebra homomorphism into the commutator Lie algebra.

Facts & Assumptions

Given: The quotient presentation of $U (g)$ and canonical linear map $ι_{g}$ from Universal enveloping algebra.

[L1]

Every generator $x \otimes y - y \otimes x - [x, y]$ of the defining ideal has zero image in the quotient.

Proof

technique · direct

1.1

Applying the quotient map to the relator in [L1] gives $ι_{g} (x) ι_{g} (y) - ι_{g} (y) ι_{g} (x) - ι_{g} ([x, y]) = 0$ , which is the displayed identity.

givenL1algebra

2.1

Since $ι_{g}$ is linear by construction and step 1.1 is bracket preservation, it is a Lie-algebra homomorphism. No injectivity has been used.

step 1.1∎

Depends on

Universal enveloping algebra

Used by

An enveloping algebra need not be commutative False statement
Functoriality of the enveloping algebra Proposition
The PBW filtration is multiplicative and has commutative associated graded Proposition
Universal property of the enveloping algebra Theorem

Dependency tree · two levels

4 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)