Let $k$ be a field of characteristic $0$.

Let $H$ be a cocommutative connected filtered bialgebra over $k$. ("Connected" means that the counit, restricted to the $0$-th part of the filtration, is an isomorphism. Since $H$ is a connected filtered bialgebra, it is automatically a Hopf algebra.)

The convolution algebra $\mathcal L\left(H,H\right)$ of all $k$-linear maps $H\to H$ is local; its maximal ideal is the ideal $\mathfrak g\left(H,H\right)$ of all $k$-linear maps $H\to H$ sending $1$ to $0$. Denoting by $e$ the unity of the algebra $\mathcal L\left(H,H\right)$ (this $e$ is the map which sends every $x\in H$ to $\varepsilon\left(x\right)\cdot 1_H$), we can define an exponential mapping $\exp^{\ast}: \mathfrak g\left(H,H\right)\to e+\mathfrak g\left(H,H\right)$ and a logarithm mapping $\log^{\ast}: e+\mathfrak g\left(H,H\right)\to \mathfrak g\left(H,H\right)$ by the formulae

$\exp^{\ast}\left(f\right)=\sum\limits_{i\geq 0}\dfrac{f^{\ast i}}{i!}$

and

$\log^{\ast}\left(e+f\right)=\sum\limits_{i\geq 1}\dfrac{\left(-1\right)^{i-1}}{i}f^{\ast i}$,

where $f^{\ast i}$ denotes the $i$-th power of $f$ in the convolution algebra $\mathcal L\left(H,H\right)$. Both of these power series converge pointwise because $H$ is filtered. It is easy to see (and annoying to write up...) that $\exp^{\ast}\circ\log^{\ast}=\mathrm{id}$ and $\log^{\ast}\circ\exp^{\ast}=\mathrm{id}$.

So far we have been doing general stuff, not using the cocommutativity of $H$, and not using the fact that $H=H$ either (i. e., we could just as well have done all the above in $\mathcal L\left(C,A\right)$ with $C$ being a connected filtered $k$-coalgebra, and $A$ a $k$-algebra). But in the case of cocommutative $H$, we have something nice:

**Theorem (Patras, Reutenauer, Garsia?).** The map $\log^{\ast}\mathrm{id}$ is a projection of $H$ onto the subspace consisting of all primitive elements of $H$.

I can prove this using lots and lots of computations and an umbral-calculus trick (I can send the proof to anyone interested, but it's boredom guaranteed). The paper where I know this theorem from gives three references I cannot get any quick use of ([R2] [GR] [P1] as referenced in the Example on page 4; two of these are Solomon descent algebra literature, and the third is a book by Reutenauer that I'll one day read from one end to the other, but not now...). There is a shortcut through finite-dimensionality and dualization, but I don't like this kind of approaches as they use the fact of $k$ being a field (whereas the Theorem holds for $k$ being any commutative ring with $1$).

While I would be glad to hear a nicer proof of the Theorem, here is the

**Actual question:** For every $s\in\mathbb N$, we can define a mapping $\exp_s^{\ast}: \mathfrak g\left(H,H\right)\to e+\mathfrak g\left(H,H\right)$ by the formula

$\exp_s^{\ast}\left(f\right)=\sum\limits_{i=0}^s \dfrac{f^{\ast i}}{i!}$.

This is just the exponential series, stopped at $i=s$ and applied to $f$.

Now, I conjecture that $\log^{\ast}\left(\exp_s^{\ast}\left(\mathrm{id}-e\right)\right)$ is a projection of $H$ onto the subspace $C^{i+}$, where $\left(C^{\ell}\right)_{\ell\geq 0}$ is the coradical filtration of the coalgebra $H$ (note that $C^0 = H^0 = k\cdot 1$, and that $C^1 = k\cdot 1 + \left(\text{subspace of primitive elements of }H\right)$), and $S^+$ denotes $S\cap \mathrm{Ker}\varepsilon$ for every subspace $S$ of $H$.

The only evidence I have for this is that it's true for $s=0$ (obviously), for $s=1$ (this is the above Theorem) and for $s=\infty$ (not exactly a natural number, but a reasonable value to plug in the conjecture). Don't these data cry for a deeper meaning?...