Proposition 7

There exists on the algebra $U$ one and only one coproduct $c$ which makes $U$ into a bigebra such that the elements of $\sigma(g)$ are primitive. The bigebra $(U, c)$ is cocommutative; its counit is the linear form $\varepsilon$ such that the constant term (Chapter I, § 2, no. 1) of every element $x$ of $U$ is $\varepsilon(x) . 1$. The canonical filtration $(U_n)_{n \geq 0}$ of $U$ is compatible with this bigebra structure.

(a) Let $x \in g$; we write $c_0(x) = \sigma(x) \otimes 1 + 1 \otimes \sigma(x) \in U \otimes U$. If $x, y$ are in $g$, then

whence By the universal property of $U$ (Chapter I, § 2, no. 1, Proposition 1), there exists one and only one unital algebra homomorphism such that $c(\sigma(x)) = \sigma(x) \otimes 1 + 1 \otimes \sigma(x)$ for $x \in g$. This proves the uniqueness assertion of Proposition 7.

(b) We show that $c$ is coassociative. The linear mappings $c'$ and $c''$ of $U$ into $U \otimes U \otimes U$ defined by

are unital algebra homomorphisms which coincide on $\sigma(g)$ since, for $a \in \sigma(g)$, whence the result.

(c) We show that $c$ is cocommutative. Let $\tau$ be the automorphism of $U \otimes U$ such that $\tau(a \otimes b) = b \otimes a$ for $a, b$ in $U$. The mappings $\tau \circ c$ and $c$ of $U$ into $U \otimes U$ are unital algebra homomorphisms which coincide on $\sigma(g)$, whence the result.

(d) We show that $\varepsilon$ is a counit for $c$. The mappings $(\mathrm{Id}_U \otimes \varepsilon) \circ c$ and (ε ⊗ Id_U) ∘ c, of U into U are unital algebra homomorphisms which coincide with Id_U on σ(g).

(e) We know that U_0 = K . 1, U_n ⊂ U_{n+1}, U = ⋃_{n ≥ 0} U_n and U_n · U_m ⊂ U_{n+m} (Chapter I, § 2, no. 6). Let a_1, ..., a_n be in σ(g). Then

where I(i) denotes the set of permutations of {1, n} which are increasing in each of the intervals {1, i} and {i + 1, n}. As U_n is the K-module generated by the products of at most n elements of σ(g), formula (8) implies that the filtration (U_n) is compatible with the bigebra structure of (U, c).