Given a coalgebra $U$ with a marked element $1\in U$, define its lie algebra to be the set of primitive elements ($u \in U|\Delta(u) = u \otimes 1 + 1\otimes u$, where $\Delta: U \rightarrow U\otimes U$ is the comultiplication map). Define $Cospec(U) = \{u \in U| \Delta(u) = u \otimes u\}$
Let $S$ be a scheme, and $M$ be a quasi-coherent $\mathcal{O}_S$ module. Messing (Crystals associated to Barsotti-Tate groups) defines a PD algebra $\Gamma(M)$, and gives it the structure of a co-algebra. This done, it makes sense to talk its lie algebra, and because $\Gamma(M)$ has divided powers, it's possible to define the exponential map from the lie algebra to $Cospec(\Gamma(M)$.
On the other hand, given $G = \varinjlim G_i$ a filtered direct limit of finite locally free $S$-schemes (for example $G$ a p-divisible group, or $G = \mathbb{G}_a$), one can look at $U = \varinjlim U_i$, the corresponding co-algebra. Now, if $S_o \rightarrow S$ is an immersion defined by an ideal $I$ with nilpotent divided powers, and if $U$ was a flat bi-algebra (which would certainly be the case if $U$ arose from a p-divisible group $G$), then Messing defines an exponential map from $ILie(U)$ to ker($\Gamma(S,Cospec(U)) \rightarrow \Gamma(S_0,Cospec(U))$) where $\Gamma$ denotes global sections.
I don't see how these two definitions are related, as in the first one, the PD structure on $\Gamma(M)$ is intrinsic to $M$. How does one construct the exponential map for an extension of a p-divisible group by a vector group? Finally, even if one does construct the exponential for a vector extension, how does one see that the exponential map is functorial, given that the construction for vector groups and for p-divisible groups are so different? (eg why does the exponential commute with the canonical homomorphism from a vector extension of G to G?)
