Unipotent completion of free group

Question

Whilst I am reading articles on unipotent completion to understand its basic construction, I found something confusing. Let $F$ be a free group of rank 2 whose generating letters are $x$ and $y$ and let $K$ be a field of characteristic 0. Then the unipotent completion of $F$ over $K$ would be the set of group-like elements in the completed Hopf algebra $K[F]^\wedge=\varprojlim K[F]/I^n$ where $I=\ker(K[F]\to K)$ is the kernel of the augmentation map. Note that $K[F]^\wedge$ is isomorphic to the ring of non-commutative formal power series in two variables $K\langle\!\langle X,Y\rangle\!\rangle$, i.e. $K[F]^\wedge\simeq K\langle\!\langle X,Y\rangle\!\rangle$. Now, here is the point I am confusing: why is the natural embedding $F\to K[F]^\wedge\simeq K\langle\!\langle X,Y\rangle\!\rangle$ given explicitly by $x\mapsto e^X$ and $y\mapsto e^Y$? Is this a chosen one?

More precisely, I guess that the natural embedding $F\to K[F]^\wedge$ would be $x\mapsto x$ and $y\mapsto y$ (up to the abuse of notations). So, I guess that the explicit form is given by the isomorphism $K[F]^\wedge\simeq K\langle\!\langle X,Y\rangle\!\rangle$. But I think that the map $x\mapsto 1+X$, $y\mapsto 1+Y$ also gives an isomorphism, is it wrong?

Answer 1

You're absolutely right. The last map you mention is called the Magnus expansion. One of the reasons it's interesting is that this formula is valid over $\mathbb{Z}$ and induces an injective map $$F\rightarrow \mathbb{Z}\langle \langle X,Y\rangle\rangle$$ which you can use to prove that $F$ is residually torsion-free-nilpotent (I believe this was Magnus motivation).

On the other hand, the map $x\mapsto e^X, y\mapsto e^Y$ is an Hopf algebra morphism, hence it identifies the unipotent completion of $F$, with the exponential group of the degree completion of the free Lie algebra on $X,Y$. This is also very useful, and implies for example that $F$ is 1-formal which is an important property.

This is a somewhat distinguished choice for such an isomorphism, at least once you've chosen generators, but by no means is it canonical and there are in fact many of those. For example, another important one is obtained from the monodromy of the flat regular connection on $P^1(\mathbb{C})-\{0,1,\infty\}$ given by $$d- (\frac xz + \frac y{1-z}).$$ This one is defined only for $K=\mathbb{C}$ but have some extra nice properties. It has a version for any field of characteristic 0 but those are much less explicit (they are constructed using so-called Drinfeld associators).