Exercise 8, § 6

Let $k$ be a field of characteristic zero and E a graded left bigebra over $k$ (Algebra, Chap. III, §11, no. 4, Def. 3). Assume that E is anti-commutative and anti-cocommutative, and that $E_m= 0$ for $m$ sufficiently large. Denote by P the set of primitive elements of E (cf. Chap. II, §1).

a) Show that every homogeneous element of P is of odd degree (expand $c(x^m)$ for $m$ large). Deduce that there is a canonical morphism $\varphi :\wedge \wedge (P)\rightarrow E$ of graded left bigebras (the graded left bigebra structure of $\wedge \wedge (P)$ being that defined in Algebra, Chap. III, §11, Exerc. 6).

b) Show that $\varphi$ is an isomorphism (adapt the proof of Th. 1 of Chap. II, §1, no. 6).