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Again, there is a general and a concrete question:

Concrete question. Let $H$ be a connected graded bialgebra over a commutative ring $k$ with unity (where graded means $\mathbb N$-graded). Then, it is known that $H$ is a Hopf algebra, thus has an antipode $S$. Let $E:H\to H$ be the $k$-linear map which sends every homogeneous element $x\in H_n$ to $nx$ for every $n\in\mathbb N$. (Note that $E$ is both a derivation and a coderivation, i. e., it satisfies $E\left(ab\right)=aE\left(b\right)+E\left(a\right)b$ for all $a$ and $b$, and $\Delta\circ E = \left(E\otimes \mathrm{id} + \mathrm{id}\otimes E\right)\circ\Delta$.) Let $*$ denote convolution of $k$-linear maps $H\to H$.

It is known that $\left(E\ast S\right)\circ \left(E\ast S\right) = E\circ\left(E\ast S\right)$ whenever $H$ is commutative or cocommutative. (The usual way to state this in characteristic $0$ is by saying that the map which sends every homogeneous $x\in H_n$ to $\frac{1}{n}\left(E\ast S\right)\left(x\right)$ (for all $n$) is a projection - the so-called Dynkin idempotent of $H$. For $H$ cocommutative, this projection is the projection on the primitive part of $H$; for its kernel, see Patras, Reutenauer, On Dynkin and Klyachko idempotents in graded bialgebras, Corollary 7. For commutative $H$ the dual assertions hold.)

I have also checked (in the $3$rd degree of the Malvenuto-Reutenauer Hopf algebra) that $\left(E\ast S\right)\circ \left(E\ast S\right) = E\circ\left(E\ast S\right)$ does not hold for general $H$.

The question is whether $\left(E\ast S\right)\circ \left(E\ast S\right) = E\circ\left(E\ast S\right)$ holds for involutive $H$ ("involutive" means that $S^2=\mathrm{id}$). If not generally, at least in characteristic $0$ ?

General question. What are good examples of involutive (this means $S^2=\mathrm{id}$) connected graded Hopf algebras which are neither commutative nor cocommutative? I know that for every Hopf algebra $H$, the ideal $H\cdot \left(\left(S^2-\mathrm{id}\right)\left(H\right)\right)\cdot H$ is a Hopf ideal of $H$ (unless I am mistaken, it is a nice easy exercise), and the quotient Hopf algebra of $H$ by this ideal is the largest involutive quotient Hopf algebra of $H$. Unfortunately it does not seem to have been studied anywhere. I know of only two connected graded Hopf algebras which are neither commutative nor cocommutative: Loday-Ronco and Malvenuto-Reutenauer. The latter is definitely non-involutive, but I can reasonably compute by hand only up to degree $3$, and the involutive quotient in degree $3$ is not a counterexample when $2$ is invertible in $k$. As for the former, I don't have a grasp on it yet; I don't even know whether it is involutive.

EDIT: So it seems that the answer to the Concrete Question is No for the $4$-th graded component of a certain sub-Hopf algebra of the Hopf algebra of ordered (rooted) trees ($\mathbf H_o$ in Loic Foissy's "Ordered forests and parking functions" (arXiv:1007.1547v3). Why am I sure about this? I am not. I have spent hours with the computations, did them a second time writing them up, corrected everything that gave different results on these two tries. But who knows how often I made the same mistake twice? (No, I am not wanting anyone to check this stuff by hand.)

Also, after a total of 2 days wasted on this problem, I am surely not going to do the same for the next natural question: is $\log\operatorname*{id}$ (the logarithm being taken in the convolution algebra) still an idempotent in an involutive connected graded Hopf algebra? (This would be the "Eulerian idempotent" rather than then Dynkin one. These two idempotents seem to be strangely related in some sense, but I have not figured out how exactly. It's as if the Dynkin one tries to be a kind of "logarithmic derivative" of the Eulerian one, but the non-commutativity of the convolution algebra puts a spoke in its wheel.)

Now that the General Question is sufficiently answered with Loic Foissy's website (which is not saying that there can ever be enough combinatorial Hopf algebras), let me ask the Real question: What is the right software to solve such problems in 2011? Sorry for spamming MO with yet another software question, but I cannot believe there hasn't been anything written for such cases yet.

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Great question! –  Theo Johnson-Freyd Dec 27 '11 at 2:17
    
PS. Taking things like a tensor product of a commutative with a cocommutative graded connected Hopf algebra doesn't produce any counterexample. In fact, if the statement works for two connected graded Hopf algebras $H$, then it also works for their tensor product (nice exercise). –  darij grinberg Dec 27 '11 at 12:27
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The answers to both the Concrete Question and the analogous question for $\log\operatorname*{id}$ are "No". By that I mean that now I am sufficiently convinced of the correctness of my Sage 5.0 code. I am still surprised that I had to go all the way up to a $5$-th graded component to disprove the $\log\operatorname*{id}$ conjecture (and up to a $4$-th one for Dynkin), but there is no guarantee that my counterexample is minimal.

Correspondence with Marcelo Aguiar gave me a clue towards a different direction: What is the largest involutive quotient of the Malvenuto-Reutenauer Hopf algebra? It clearly surjects onto the abelianization of the Malvenuto-Reutenauer Hopf algebra. In degree $\leq 5$ (at least), this surjection is an isomorphism. What about higher degrees? (This, of course, would explain why I could not find counterexamples in Malvenuto-Reutenauer.) What about the Hopf algebra of ordered forests? (Note that my counterexamples don't work in the Hopf algebra of ordered forests itself; they work in a proper Hopf subalgebra of it.) Both of these Hopf algebras are free and cofree (I think this was proven by Loic Foissy); is there a theorem that the largest involutive quotient of a free-and-cofree connected graded Hopf algebra must equal to its abelianization?

EDIT: The largest involutive quotient of the Malvenuto-Reutenauer Hopf algebra is not equal to its abelianization, and $\log\operatorname*{id}$ is not idempotent on this quotient. The same holds for the quotient of the Malvenuto-Reutenauer Hopf algebra modulo dual equivalence (this is one of the two mutually dual Hopf algebras introduced by Poirier and Reutenauer). I just checked this on Sage, which found a counterexample in degree $7$.

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