Exercise 11, § 1

Let E be a bigebra and let I be a finite set. We are given a basis $(e_\alpha)$ of E indexed by the elements $\alpha$ of $\mathbf{N}^I$ and such that: (i) $e_0 = 1$; (ii) $c(e_\gamma) = \sum_{\alpha + \beta = \gamma} e_\alpha \otimes e_\beta$ for all $\gamma \in \mathbf{N}^I$.

Condition (ii) implies that E is isomorphic as a cogaebra to the cogaebra $TS(K^I)$ of symmetric tensors of $K^I$ (cf. Algebra, Chapter IV, § 5, no. 7).

(a) Let $E' = \mathrm{Hom}(E, K)$ be the dual algebra of E and let $\Lambda = K[[((X_i)_{i \in I})]]$ be the algebra of formal power series in indeterminates $X_i$ indexed by I. For all $u \in E'$ let $\phi_u$ be the formal power series

where $x_\alpha = \prod_{i \in I} X_i^{\alpha(i)}$.

Show that $u \mapsto \phi_u$ is an isomorphism of $E'$ onto $\Lambda$ and that this isomorphism maps the topology of simple convergence on $E'$ to the product topology on $\Lambda$.

(b) Let $c_{\alpha \beta \gamma}$ be the constants of structure of the algebra E relative to the basis $(e_\alpha)$. Then

We write $x$ instead of $(X_i)_{i \in I}$ and $y$ instead of $(Y_i)_{i \in I}$, where the $Y_i$ are new indeterminates. Show that there exists a family $f(x, y) = (f_i(x, y))_{i \in I}$ of formal power series in the variables $x, y$ such that

where we write, as above, $x^\alpha = \prod_i X_i^{\alpha(i)}$ and similarly for $y^\beta$ and $f(x, y)^\gamma$.

Show that $f(x, y)$ is a formal group law over K of dimension $n = \mathrm{Card}(I)$, in the sense of Chapter I, § 1, Exercise 24†; define a Lie algebra isomorphism of this formal group law onto the Lie algebra $P(E)$, which has basis the $e_\alpha$ with $|\alpha| = 1$.

(c) Conversely, show that every formal group law over K in the $x$ and $y$ can be obtained by the above procedure, which is unique up to isomorphism.

(d) When K is a $\mathbf{Q}$-algebra, apply the above to the enveloping bigebra of a Lie algebra $g$ with a finite basis over K. Deduce the existence (and uniqueness, up to isomorphism) of a formal group law with $g$ as Lie algebra.

(e) Give explicitly the bigebra corresponding to the formal group law in one parameter $f(x, y) = x + y$ (resp. $f(x, y) = x + y + xy$).