Proposition 12

Let G, H be two Lie groups and $\phi$ a morphism of G into H. Then $\phi_*$ is a bigebra morphism of $\mathscr{T}^{(\infty)}(G)$ into $\mathscr{T}^{(\infty)}(H)$.

This follows from Proposition 6 and Differentiable and Analytic Manifolds, R, 13.5.1.

Let G be a Lie group. The restrictions of the convolution and the coproduct to U(G) define a bigebra structure on U(G). We have U(G)^{\vee} = U(G). If $\phi : G \to H$ is a Lie group morphism, we denote by U($\phi$) the mapping $t \mapsto \phi_*(t)$ of U(G) into U(H); this is a bigebra morphism. If $\psi : H \to L$ is another Lie group morphism, then U($\psi \circ \phi$) = U($\psi$) \circ U($\phi$). If $\phi$ is an immersion (resp. a submersion), U($\phi$) is injective (resp. surjective) by Differentiable and Analytic Manifolds, R, 13.2.3. In particular, if H is a Lie subgroup of G, U(H) is identified with a subalgebra of U(G), the coproduct on U(H) being the restriction of the coproduct on U(G). If H is open in G, then U(H) = U(G). If G_1, G_2 are Lie groups, U(G_1 \times G_2) is identified with U(G_1) \times U(G_2). The primitive elements of U(G) are those of T_e(G) (Differentiable and Analytic Manifolds, R, 13.5.3).

Again let $\phi : G \to H$ be a Lie group morphism. If gr U(G) is identified TS(T_e(G)) and gr U(H) with TS(T_e(H)), then gr U($\phi$) is identified with TS(T_e($\phi$)) (Differentiable and Analytic Manifolds, R, 13.3.5). We apply this to the isomorphism $g \mapsto g^{-1}$ of G onto G^{\vee}; then T_e($\phi$) = -1 and hence