Proposition 5

Tag 0A1X, permanent. Cite it as [Bourbaki, Tag 0A1X].

Lie Groups and Lie Algebras, chapter II, § 1. Enveloping bigebra of a Lie algebra

Let $f : E \to E'$ be a bigebra morphism. If $x$ is a primitive element of $E$ , then $f(x)$ is a primitive element of $E'$ and the restriction of $f$ to $P(E)$ is a Lie algebra homomorphism $P(f) : P(E) \to P(E')$ .

Let $c$ (resp. $c'$ ) be the coproduct of $E$ (resp. $E'$ ). Since $f$ is a cogebras morphism $c' \circ f = (f \otimes f) \circ c$ , whence

\begin{align*} c'(f(x)) &= (f \otimes f)(c(x)) = (f \otimes f)(x \otimes 1 + 1 \otimes x) \\ &= f(x) \otimes 1 + 1 \otimes f(x), \end{align*}

for

x

primitive. Hence

f

maps

P(E)

into

P(E')

and

f([x, y]) = [f(x), f(y)]

since

f

is an algebra homomorphism.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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Cites

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