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) [a1], [a2], [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) [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 [a4], [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

[a1]	E.J. Ditters, "Curves and formal (co)groups" Invent. Math. , 17 (1972) pp. 1–20
[a2]	E.J. Ditters, "Groupes formels" , Lecture Notes , Univ. Paris XI: Orsay (1974)
[a3]	M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) MR0506881 MR0463184 Zbl 0454.14020
[a4]	M. Hazewinkel, "The Leibniz Hopf algebra and Lyndon words" Preprint AM CWI , 9612 (1996)
[a5]	A.C.J. Scholtens, "$S$-Typical curves in non-commutative Hopf algebras" , Free Univ. Amsterdam (1996) (Thesis)

References

[b1]	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