Exercise 12, § 1

Suppose that K is a field of characteristic $p > 0$. (a) Let $g$ be a Lie $p$-algebra (Chapter I, § 1, Exercise 20) and let $\tilde{U}$ be its

† The exercise in question assumes that K is a field, but nothing is lost if this hypothesis is suppressed.

restricted enveloping bigebra (Chapter I, § 2, Exercise 6). Show that there exists on $\tilde{U}$ one and only one bigebra structure which is compatible with its algebra structure and for which the elements of $g$ are primitive. Show that $\tilde{U}$ is cocommutative and that $P(\tilde{U}) = g$.

(b) If $n$ is an integer $\geqslant 0$, let $\tilde{U}_n$ denote the vector subspace of $\tilde{U}$ generated by the products $x_1 \ldots x_n$ where $x_i \in g$ for all $i$. Show that $(\tilde{U}_n)_{n \geqslant 0}$ is a filtration of $\tilde{U}$ compatible with its bigebra structure.

(c) Let $E$ be a bigebra and let $g = P(E)$ be the Lie algebra of its primitive elements. The mapping $x \mapsto x^p$ leaves $g$ stable and gives $g$ a Lie $p$-algebra structure. Show that the canonical injection $g \to E$ can be extended uniquely to a bigebra morphism $\tilde{U} \to E$ and that this morphism is injective (use the fact that, if $(x_i)_{i \in I}$ is a basis of $g$ with a total ordering, the monomials $\prod x_i^{n_i}$ $(0 \leqslant n_i < p)$ form a basis of $U$ (loc. cit.) and argue as in the proof of Lemma 2).