Let $k$ be a field of characteristic $0$ and let $\mathfrak{g}$ be a finite dimensional Lie algebra over $k$. $\mathfrak{g}$ corresponds to a formal group scheme $\mathcal{G} = \text{Spf} (U(\mathfrak{g})^{\*})$. In fact, there is an equivalence of categories between finite-dimensional Lie algebras and infinitesimal formal group schemes over $k$ with finite-dimensional tangent space. This is elaborated on [here](https://amathew.wordpress.com/2012/06/23/formal-lie-theory-in-characteristic-zero/) by Akhil Mathew.


There should be an exponential map $\text{exp} : \mathfrak{g} \rightarrow \mathcal{G}$. This is mentioned at the end of the blog post above, but I can't figure out how to construct it explicitly.


Any help on this would be much appreciated.



Exponential map of a Formal Group Scheme
$\newcommand{\g}{\mathfrak{g}}$
In a way this is tautological, in the sense that $\mathcal G$ can be seen as a formal exponentation of $\g$, although I don't think there is an actual exponential map from one to the other in general. Rather, there is a map (in fact an isomorphism of formal schemes) from the formal completion of $\g$ at the origin to $\mathcal G$.


Namely, the dual $U(\g)^\*$ can be identified as a (topological) algebra with the algebra $\widehat S(\g^\*)$ of formal power series on $\g^\*$, ie with the formal completion of $\mathcal O(\g)$ w.r.t to the ideal of function vanishing at the origin.


This identification, should indeed be thought as pulling back functions through the exponential map: indeed, the induced coproduct on $\widehat S(\g^\*)$ is expressed through the Baker-Campbell-Hausdorff formula $$\forall x,y \in \g, \Delta(f)(x\otimes y)=f(BCH(x,y))$$ where BCH is the Lie formal power series defined on formal variables $a,b$ by 
$$BCH(a,b):=\log(e^ae^b).$$


