Proposition 11

Let G be a Lie group. (i) The cogebra $\mathscr{T}^{(\infty)}(G)$, with convolution, is a bigebra (Algebra, Chapter III, § 11, no. 4). (ii) Let c be the coproduct on $\mathscr{T}^{(\infty)}(G)$. Let $t \in \mathscr{T}^{(\infty)}(G)$ and write

Then $c(t^\vee) = \sum_{i=1}^n t_i^\vee \otimes t_{i'}^\vee$.

We prove (i). In the definition of bigebra referred to, condition (1) follows from Propositions 2 and 3 and condition (2) follows from Differentiable and Analytic Manifolds, R, 13.5.1. Let d be the mapping $g \mapsto (g, g)$ of G into $G \times G$. Then $c = d_*$ and hence c is an algebra morphism (Propositions 6 and 9), which is condition (3). Let $t \in T_g^{(\infty)}(G)$, $t' \in T_{g'}^{(\infty)}(G)$ have no constant term and $\lambda, \lambda'$ be elements of K; then $\varepsilon_g \otimes t', t \otimes \varepsilon_{g'}, t \otimes t'$ are without constant term (Differentiable and Analytic Manifolds, R, 13.4.1) and hence the constant term of $(\lambda \varepsilon_g + t) * (\lambda' \varepsilon_{g'} + t')$ is $\lambda \lambda'$; hence condition (4) holds.

We prove (ii). By Propositions 8 and 9,