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.



Hopf algebra structure on the ring of quasisymmetric functions
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.


