Exercise 7, § 6

Let K be a connected compact Lie group.

a) Let $f: K\rightarrow G$ be a surjective homomorphism with finite kernel. Show that the homomorphism $H(f^*) : H(G)\rightarrow H(K)$ is an isomorphism.

b) Show that the algebra $H(G\times K)$ can be identified canonically with the skew tensor product $H(G)^g\otimes H(K)$.

c) Deduce from a) and b) that H(G) is isomorphic to $H(C(G)_0)^g\otimes H(D(G))$ as an algebra. Show that the algebra $H(C(G)_0)$ is isomorphic to $\wedge \wedge (\mathfrak{c}^*)$, with $\mathfrak{c}= L(C(G))$.