Exercise 11, § 2

Lie Groups and Lie Algebras, II

Bourbaki marks this one as harder than the rest.

Let Y and Z be two disjoint sets (Y being endowed with a total order structure), let $X = Y \cup Z$ and let $l = L(X)$ . Let $\alpha_Y$ be the submodule of $l$ generated by Z and $[l, l]$ ; it is an ideal of the Lie algebra $l$ . We shall make explicit a basic family of $\alpha_Y$ .

Let M be the set of positive integer-valued functions on Y, with finite support. If $m \in M$ , we denote by $\theta^m$ the endomorphism of $l$ given by

\theta^m = \prod_{y \in Y} (\operatorname{ad} y)^{m(y)},

the product being relative to the order relation given on Y. We denote by

S

the subset of

Y \times M

formed by the pairs

(u, m)

such that there exists

y \in Y

for which

y > u

and

m(y) \geq 1

. Prove that the elements

\theta^m(u) \text{ for } (u, m) \in S \quad \text{ and } \quad \theta^m(z) \text{ for } (z, m) \in Z \times M

form a basic family of the Lie algebra

\alpha_Y

(Reduce to the case where Y is finite, and reason by induction on Card(Y). Use the cor. of prop. 10, applied to the greatest element $y$ of Y, and apply the induction hypothesis to $Y - \{y\}$ .)

Solution

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