TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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.

Universal property of the free Lie algebra

Statement

Let $V$ be a complex vector space and let $g$ be a complex Lie algebra (Lie algebras over a field). Every linear map $f : V \to g$ extends uniquely to a homomorphism of Lie algebras $L (V) \to g$ . Assume the Axiom of Choice for the basis used below.

Facts & Assumptions

Given: A complex vector space $V$ , a complex Lie algebra $g$ , and a linear map $f : V \to g$ .

[L1]

$L (V)$ is the Lie subalgebra of the tensor algebra $T (V)$ generated by $V$ (Free Lie algebra on a vector space).

[L2]

Every linear map $V \to A$ into a unital associative algebra $A$ extends uniquely to a unital algebra homomorphism $T (V) \to A$ (Universal property of the tensor algebra).

[L3]

The canonical map $ι_{g} : g \to U (g)$ satisfies $ι_{g} ([x, y]) = ι_{g} (x) ι_{g} (y) - ι_{g} (y) ι_{g} (x)$ (The canonical map to U(g) is a Lie homomorphism).

[L4]

Under the Axiom of Choice, $g$ has a basis; after ordering it, the degree-one PBW corollary makes $ι_{g}$ injective (Every vector space has a basis, No hidden linear relations in degree one).

Proof

technique · direct

1.1

By [L2] applied to the composition of $f$ with the injective canonical map $ι_{g} : g ↪ U (g)$ , there is a unique unital algebra homomorphism $\hat{f} : T (V) \to U (g)$ extending $ι_{g} \circ f$ .

L2L4algebra

2.1

The restriction of $\hat{f}$ to $L (V)$ takes values in the image of $g$ and is a Lie-algebra homomorphism: for $x, y \in L (V)$ one has $\hat{f} ([x, y]) = \hat{f} (x) \hat{f} (y) - \hat{f} (y) \hat{f} (x)$ , and by induction on the generation of $L (V)$ each $\hat{f} (x)$ lies in the image of $g$ , where the bracket of two images is the image of the bracket by [L3]; hence the composite $g : L (V) \to g$ obtained by restricting $\hat{f}$ and inverting the injective canonical map from [L4] is a Lie homomorphism $L (V) \to g$ extending $f$ .

L1L3L4step 1.1algebra

3.1

Uniqueness: if $g_{1}, g_{2} : L (V) \to g$ are Lie homomorphisms agreeing on $V$ , then the set of $x \in L (V)$ with $g_{1} (x) = g_{2} (x)$ is a Lie subalgebra containing $V$ ; since $L (V)$ is generated as a Lie algebra by $V$ , it is all of $L (V)$ .

L1algebra∎

Depends on

Used by

Lie algebra presented by generators and relations Definition
Serre presentation theorem Theorem

Dependency tree · two levels

23 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

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)