Exercise 5, § 6

Denote by H(G) the graded $\mathbf{R}$-algebra $H(\Omega (G))$ (cf. Exerc. 4).

a) The space $H^p(G)$ is zero for $p > n$ and of dimension one for $p=n$ and $p= 0$. The space $H^1(G)$ can be identified canonically with $\mathfrak{c}^*$, where $\mathfrak{c}= L(C(G))$.

b) Assume from now on that G is semi-simple. Show that $H^1(G) = H^2(G) = 0$ (cf. Chap. I, §6, Exerc. 1).

c) Denote by $B(\mathfrak{g})$ the space of G-invariant symmetric bilinear forms on $\mathfrak{g}$; for $b\in B(\mathfrak{g})$ and $x, y, z$ in $\mathfrak{g}$, put $\widetilde{b}(x, y, z) =b([x, y], z)$. Show that the map $b \rightarrow \widetilde{b}$ defines an isomorphism from $B(\mathfrak{g})$ to $H^3(G)$ (let $\omega \in H^3(G)$; prove that, for all $x\in \mathfrak{g}$, there exists a unique linear form $f(x)$ on $\mathfrak{g}$ such that $df(x) =i(x)\omega$, and consider the form $(x, y) \rightarrow -\langle y, f(x)\rangle$ ).

d) Show that the dimension of the $\mathbf{R}$-vector space $H^3(G)$ is equal to the number of simple ideals of $\mathfrak{g}$.