Exercise 9, § 6

Denote by $m: G\times G\rightarrow G$ the map such that $m(g, h) =gh$ for $g, h$ in G. Identify $H(G\times G)$ with $H(G)^g\otimes H(G)$ (Exerc. $7b$)$)$, so that $m^*$ defines a homomorphism of algebras $c: H(G)\rightarrow H(G)^g\otimes H(G)$.

a) Show that $(H(G), c)$ is an anti-commutative and anti-cocommutative graded left bigebra (observe that the map $g \rightarrow g^{-1}$ induces on $H^p(G)$ the multiplication by $(-1)^p$). b) Let P(G) be the graded subspace of H(G) consisting of the primitive elements; deduce from Exerc. 8 that there is an isomorphism of graded bigebras $\wedge \wedge (P(G))\rightarrow H(G)$.

c) Show that dim$_{\mathbf{R}}P(G) =r$ (use Exerc. $6b$)$)$.

d) Deduce that the polynomial $\sum_{i\geq 0}b_i(G)X^i$ is of the form

where $c$ is the dimension of $C(G),l$ the rank of D(G), and the $k_i$ are integers $\geq 1$; we have $k_1+\cdots +k_l=\frac{1}{2}$Card $R(G,T)^{12}$.