View source for Leibniz-Hopf algebra

Let $\mathbf{Z}\langle Z \rangle$ be the [[free associative algebra]] on $Z = \{Z_1,Z_2,\ldots\}$ over the integers. Give $\mathbf{Z}\langle Z \rangle$ a [[Hopf algebra]] structure by means of the following co-multiplication, augmentation, and antipode:
$$
\mu(Z_n) = \sum_{i+j=n\,;\,\,i,j \ge 0} Z_i \otimes Z_j
$$
where
$$
Z_0 = 1
$$
$$
\epsilon(Z_n) = 0,\ \ n=1,2,\ldots
$$
$$
\iota(Z_n) = \sum_{i_1+\cdots+i_k = n} (-1)^k Z_{i_1} \cdots Z_{i_k} \ ,
$$
where the sum is over all strings $i_1,\ldots,i_k$, $i_j \ge 1$, such that $i_1+\cdots+i_k = n$. This makes $\mathbf{Z}\langle Z \rangle$ a Hopf algebra, called the Leibniz–Hopf algebra. This Hopf algebra is important, e.g., in the theory of curves of non-commutative formal groups (see [[Formal group]]) [[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]]. Its commutative quotient $\mathbf{Z}[Z]$, with the same co-multiplication, is the underlying Hopf algebra of the (big) Witt vector functor $R \mapsto W(R)$ (see [[Witt vector]]) and it plays an important role in the classification theory of unipotent commutative algebraic groups and in the theory of commutative formal groups (amongst other things) [[#References|[a3]]].

The Leibniz–Hopf algebra $\mathbf{Z}\langle Z \rangle$ is free as a $\mathbf{Z}$-module and graded. Its graded dual is also a Hopf algebra, whose underlying algebra is the overlapping shuffle algebra $\mathrm{OSh}(\mathbf{N})$. As a $\mathbf Z$-module, $\mathrm{OSh}(\mathbf{N})$ is free with basis $\mathbf{N}^*$, the free monoid (see [[Free semi-group]]) of all words in the alphabet $\mathbf{N}$ with the duality pairing $\mathbf{Z}\langle Z \rangle \times \mathrm{OSh}(\mathbf{N}) \rightarrow \mathbf{Z}$ given by
$$
\left({ w , Z_{i_1}\cdots Z_{i_r} }\right) = \begin{cases}1,& w = i_1\cdots i_r ,\\ 0 &\text{otherwise}.\end{cases}
$$

The overlapping shuffle product of two such words $u=(a_1,\ldots,a_s)$, $v = (b_1,\ldots,b_t)$ is equal to
$$
a \times_{\mathrm{OSh}} b = \sum_{f,g} (c_1,\ldots,c_r)
$$
where the sum is over all $r \in \mathbf{N}$ and pairs of order-preserving injective mappings $f : \{1,\ldots,s\} \rightarrow \{1,\ldots,r\}$, $g : \{1,\ldots,t\} \rightarrow \{1,\ldots,r\}$ such that $\mathrm{im}(f) \cup \mathrm{im}(g) = \{1,\ldots,r\}$, and where
$$
c_i = a_{f^{-1}(i)} + b_{g^{-1}(i)}\ ,\ \ \ i = 1,\ldots,r
$$
with $a_{f^{-1}(i)} = 0 $ if $f^{-1}(i) = \emptyset$ , and similarly for $b_{g^{-1}(i)}$.

For example,
$$
(a)(b_1,b_2) = (a,b_1,b_2) + (b_1,a,b_2) + (b_1,b_2,a) + (a+b_1,b_2) + (b_1,a+b_2) \ .
$$
The terms of maximal length of the overlapping shuffle product form the shuffle product, see [[Shuffle algebra]].

A word $w \in \mathbf{N}^*$, $w = (a_1,\ldots,a_s)$, is elementary if the greatest common divisor of $a_1,\ldots,a_s$ is $1$. With this terminology, the Ditters–Scholtens theorem [[#References|[a4]]], [[#References|[a5]]] says that, as an algebra over $\mathbf{Z}$, the overlapping shuffle algebra $\mathrm{Osh}(\mathbf{Z})$ is the free commutative polynomial algebra with as generators the elementary concatenation powers of elementary Lyndon words (see [[Lyndon word]]). (E.g., the third concatenation power of $(a_1,a_2)$ is $(a_1,a_2,a_1,a_2,a_1,a_2)$.) In contrast with the case of the [[shuffle algebra]], this theorem already holds over $\mathbf{Z}$ (not just over $\mathbf{Q}$).

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> E.J. Ditters, "Curves and formal (co)groups" ''Invent. Math.'' , '''17''' (1972) pp. 1–20</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> E.J. Ditters, "Groupes formels" , ''Lecture Notes'' , Univ. Paris XI: Orsay (1974)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) {{MR|0506881}} {{MR|0463184}} {{ZBL|0454.14020}} </TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Hazewinkel, "The Leibniz Hopf algebra and Lyndon words" ''Preprint AM CWI'' , '''9612''' (1996)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top"> A.C.J. Scholtens, "$S$-Typical curves in non-commutative Hopf algebras" , Free Univ. Amsterdam (1996) (Thesis)</TD></TR>
</table>

====References====
<table>
<TR><TD valign="top">[b1]</TD> <TD valign="top"> Michiel Hazewinkel, "The Algebra of Quasi-Symmetric Functions is Free over the Integers", ''Advances in Mathematics'' '''164''' (2001) 283–300 {{DOI|10.1006/aima.2001.2017}}</TD></TR>
</table>

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