Proposition 9

Tag 0A27, permanent. Cite it as [Bourbaki, Tag 0A27].

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

For every integer $n \geq 0$ , let $U^n$ be the vector subspace of $U$ generated by the $\sigma(x)^n$ for $x \in \mathfrak{g}$ .

(a) The sequence $(U^n)_{n \geq 0}$ is a graduation of the vector space $U$ compatible with its cogebra structure.

Let $U$ be given the graduation $(U^n)$ .

(b) The canonical mapping $\eta : S(\mathfrak{g}) \to U$ is an isomorphism of graded cogebras.

Let $x \in \mathfrak{g}$ and $n \in \mathbf{N}$ . Then

c_S(x^n) = c_S(x)^n = (x \otimes 1 + 1 \otimes x)^n = \sum_{i=0}^n \binom{n}{i} x^i \otimes x^{n-i}

since $c_S$ is an algebra homomorphism. Similarly, by (10),

c(\eta(x^n)) = c(\sigma(x)^n) = c(\sigma(x))^n = (\sigma(x) \otimes 1 + 1 \otimes \sigma(x))^n

= \sum_{i=0}^n \binom{n}{i} \sigma(x)^i \otimes \sigma(x)^{n-i} = \sum_{i=0}^n \binom{n}{i} \eta(x^i) \otimes \eta(x^{n-i}),

whence

(\eta \otimes \eta)(c_S(x^n)) = c(\eta(x^n)).

As the $x^n$ , for $x \in \mathfrak{g}$ and $n \in \mathbf{N}$ , generate the vector space $S(\mathfrak{g})$ , $(\eta \otimes \eta) \circ c_S = c \circ \eta$ and $\eta$ is a cogebra isomorphism.

On the other hand, formula (10) shows that $\eta(S^n(\mathfrak{g})) = U^n$ , which completes the proof of (a) and (b) taking account of the fact that the graduation of $S(\mathfrak{g})$ is compatible with its cogebra structure.

The graduation $(U^n)_{n \geq 0}$ of $U$ is called the canonical graduation.

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

Cited by

Nothing.

Cites

Nothing.