Corollary 4

Tag 09VC, permanent. Cite it as [Bourbaki, Tag 09VC].

Lie Groups and Lie Algebras, chapter I, § 2. Enveloping algebra of a Lie algebra

Let ${S'}^n \subset T^n$ be the set of homogeneous symmetric tensors of order n. Suppose that K is a field of characteristic 0. Then the composite mapping of the canonical mappings

S^n \to {S'}^n \to U_n

is an isomorphism of the vector space $S^n$ onto a supplement of $U_{n-1}$ in $U_n$ .

This follows from Corollary 1 taking $W = {S'}^n$ .

Suppose henceforth that K is a field of characteristic 0. Let $\eta_n$ be the mapping of $S^n$ into $U_n$ just defined. Let $U^n = \eta_n(S^n)$ . The vector space $U$ is the direct sum of the $U^n$ . The $\eta_n$ define an isomorphism $\eta$ of the vector space $S = \sum_n S^n$ onto the vector space $U = \sum_n U^n$ , called the canonical isomorphism of S onto U; this is not an algebra isomorphism. We have the commutative diagram:

\begin{array}{ccccc} & & U^n & & \\ & \nearrow \psi_n & \uparrow \eta_n & \searrow \theta_n & \\ {S'}^n & & & & G^n \\ & \searrow \tau_n & & \nearrow \omega_n & \\ & & S^n & & \end{array}

where each arrow represents a vector space isomorphism. If $x_1, x_2, \ldots, x_n$ are in $g$ , $\eta_n$ maps the product $x_1 x_2 \ldots x_n$ , calculated in $S$ , to the element

\frac{1}{n!} \sum_{\sigma \in S_n} x_{\sigma(1)} x_{\sigma(2)} \cdots x_{\sigma(n)}

calculated in $U$ .

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

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Cites

Corollary 1