Proposition 8

Let E be a bigebra with coproduct denoted by cE and let h be a Lie algebra homomorphism of g into P(E)* (no. 2, Proposition 4). The unital algebra homomorphism f : U → E such that f(σ(x)) = h(x) for all x ∈ g is a bigebra morphism.

We show that (f ⊗ f) ∘ c = c_E ∘ f. These are two unital algebra homomorphisms of U into E ⊗ E and, for a ∈ σ(g),

since f(a) ∈ P(E). Similarly if ε_E is the counit of E, ε_E ∘ f is a unital algebra homomorphism U → K which is zero on σ(g) (no. 1, Proposition 1) and therefore coincides with ε.

It follows from Propositions 5 and 8 that the mapping f ↦ f ∘ σ defines a one-to-one correspondence between bigebra homomorphisms U(g) → E and Lie algebra homomorphisms g → P(E).