Q:

Hopf algebra structure on the ring of quasisymmetric functions

I'm looking for a particular description of the Hopf algebra structure on the ring of quasisymmetric functions.
Let me illustrate by giving this kind of description for the Hopf algebra of symmetric functions.
Fix a ground field $k$.
Edit: I'm happy to assume $k=\mathbb{Q}$. As darijgrinberg pointed out in the comments, $\mathrm{QSym}$ only has the Lyndon monomial basis when $\mathrm{char}(k)=0$.
Let $\Lambda$ be the ring of symmetric functions over $k$. Putting a Hopf algebra structure on a $k$-algebra $R$ is the same as putting a group structure on $\mathrm{Hom}_{k-alg}(R,S)$ for each $k$-algebras $S$ such that postcomposition is a group homomorphism. Since $\Lambda$ is a polynomial algebra in the elementary symmetric functions $e_1,e_2,\ldots$, we can identify $\mathrm{Hom}_{k-alg}(\Lambda,S)$ with the set of sequences $(s_1,s_2,\ldots)$ of elements of $S$ (where $f:\Lambda\to S$ is identified with $(f(e_2),f(e_2),\ldots)$). If we further identify the sequence $(s_1,s_2,\ldots)$ with the power series $1+s_1T+s_2T^2+\ldots\in 1+TS[[T]]$, then the group structure (coming from the Hopf algebra structure on $\Lambda$) is multiplication of power series.
I am hoping for something similar for the ring of quasisymmetric functions. Specifically, I would like, for each $k$-algebra $S$, a group $G(S)$ (which won't generally be abelian) and a bijection $\mathrm{Hom}_{k-alg}(\mathrm{QSym}_k,S)\leftrightarrow G(S)$, such that the group multiplication $G(S)\times G(S) \to G(S)$ is induced by the comultiplication on $\mathrm{QSym}$. I'm hoping the description of $G(S)$ is as explicit as possible (like the group $1+TS[[T]]$ in the case of symmetric functions).
It is known that $\mathrm{QSym}$ is a polynomial algebra. One possible free generating set is the monomial quasi-symmetric functions
$$
M_{(s_1,\dots,s_k)}:=\sum_{i_1<\ldots<i_k}x_{i_1}^{s_1}\ldots x_{i_k}^{s_k},
$$
where $(s_1,\ldots,s_k)$ runs through the Lyndon words (a Lyndon word is one that is lexicographically smaller than all of its proper right subwords). This means we can identify $\mathrm{Hom}_{k-alg}(\mathrm{QSym},S)$ with the set of set maps $\{\text{Lyndon words}\}\to S$. However, under this identification it looks like a huge mess to describe the group structure, since (as far as I know) it is not easy to read off the comultiplication of $M_{(s_1,\ldots,s_k)}$ as a sum of simple tensors of other monomial quasisymmetric functions indexed by Lyndon words.

A:

Not sure  that it helps but if I'm not mistaken, the $k$-Hopf algebra $Qsym$ you are looking for is the topological dual completed Hopf algebra $k \langle\!\langle y_i ,  i \geq 1 \rangle\! \rangle$ of power series in an infinity of non commutative variables with coproduct $\Delta_\star (y_n) = \sum_{p+q = n} y_p \otimes y_q$ (with $y_0 = 1$). 
As a result, $Hom_{k\textrm{-alg}}(QSym,A)$ is identified with diagonal power series $\Phi_\star = \sum a_s y^s \in A\langle\!\langle y_i ,  i \geq 1 \rangle\! \rangle$. Diagonal meaning $\Delta_\star \Phi_\star  = \Phi_\star \widehat{\otimes} \Phi_\star$ and $\varepsilon(\Phi_\star) = 1$.  
Here's a detailled article on the subject: http://www1.mat.uniroma1.it/people/malvenuto/Duality.pdf . Hope it clarifies things for you.

