Exercise 1, § 6

Let $g$ be a semi-simple Lie algebra and $\rho$ a finite-dimensional representation of $g$ on $M$.

(a) If $\rho$ is simple and non-zero, $H^p(g, M) = \{0\}$ for all $p$. (Use Proposition 1 and Exercise 12 (j) of § 3.)

(b) For all $\rho$, $H^1(g, M) = \{0\}$. (If $\rho$ is simple and non-zero, apply (a). If $\rho$ is zero, use the fact that $g = \mathscr{D}g$. In the general case, argue by induction on the dimension of $\rho$: if $N$ is a sub-$g$-module of $M$ distinct from $\{0\}$ and $M$, use the exact sequence:

established in Exercise 12 (f) of § 3.) Hence recover Remark 2 of no. 2.

(c) Deduce from (b) a proof of Theorem 2. (Use Exercise 12 (g) of § 3.) Deduce also from (b) that every derivation of $g$ is inner. (Use Exercise 12 (h) of § 3.)

(d) For all $\rho$, $H^2(g, M) = \{0\}$. (Arguing as for (b), it suffices to consider the case where $\rho = 0$. Let $c \in H^2(g, M)$. Consider, following Exercise 12 (i) of § 3, the central extension $h$ of $g$ by $M$ defined by $c$. The adjoint representation of $h$ defines a representation of $g$ on $h$. By Theorem 2, $M$ admits in $h$ a supplement which is stable under $g$ and hence the extension is trivial. Hence $c = 0$ by Exercise 12 (i) of § 3.)

(e) Deduce from (b) and (d) a proof of Theorem 5. (As in the text, it can be reduced to the case where the radical is commutative.)