# Shuffle Hopf algebra of Multiple zeta values

Wenchuan Hu  School of Mathematics, Sichuan University, Chengdu, 610064, P. R.
China<EMAIL_ADDRESS>,  Hongyu Xiang  School of Mathematics, Sichuan
University, Chengdu, 610064, P. R. China<EMAIL_ADDRESS>and  Bin Zhang
School of Mathematics, Sichuan University, Chengdu, 610064, P. R. China
<EMAIL_ADDRESS>
###### Abstract.

The shuffle relation among multiple zeta values is algebraically expressed as
the shuffle algebra. In this paper, the shuffle algebra structure for multiple
zeta values is extended to a Hopf algebra structure, for which the key idea is
the lifting of the shuffle multiplication to Chen fractions as the function
multiplication. The linear span of Chen fractions can be equipped with a
locality Hopf algebra structure, and the pushforward of the coproduct gives us
the desired construction on the shuffle algebra.

Keywords: multiple zeta values; shuffle algebras; Chen fractions; locality
algebras; locality Hopf algebras

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 The shuffle algebra for MZVs
    2. 2.2 Cones and fractions
    3. 2.3 Locality Hopf algebras
  3. 3 The Locality Hopf algebra of Chen fractions
    1. 3.1 Locality algebra of Chen fractions
    2. 3.2 The lifting of shuffle product
    3. 3.3 Locality Hopf algebra of Chen fractions
  4. 4 The shuffle Hopf algebra
    1. 4.1 The shuffle Hopf algebra
    2. 4.2 Operators on the shuffle algebra
    3. 4.3 Characterization of the coproduct on the shuffle algebra

##  1\. Introduction

In this paper, if there is no specific explanation, we denote

  * •

$\mathbb{Q}$: the rational number field,

  * •

$\mathbb{R}$: the real number field,

  * •

$K$: any field,

  * •

$\mathbb{Z}_{\geq 1}$: the set of all positive integers,

  * •

$\mathbb{R}_{\geq 0}$: the set of all non-negative real numbers.

Multiple zeta values (MZVs) are the values of the multiple zeta series

(1)
$\zeta(s_{1},\cdots,s_{k})=\sum_{n_{1}>\cdots>n_{k}>0}\frac{1}{n_{1}^{s_{1}}\cdots
n_{k}^{s_{k}}}$

at convergent points with positive integer arguments, that is, $s_{1}>1$,
$s_{i}\in\mathbb{Z}_{\geq 1},i=2,\cdots,k$. MZVs and their generalizations
have been studied extensively from different points of view since 1990s with
connections to number theory, algebraic geometry, mathematical physics,
quantum groups and knot theory [BBBL, BK, Car, Gon, GM, H3, Kre, Ter, GZ1].

There are rich relations among MZVs over $\mathbb{Q}$, the structure of these
relations is one of the main interesting topics in the study of MZVs. There
are two intuitive families of relations:

  * •

stuffle relation, based on the series expression (1) of MZVs;

  * •

shuffle relation, based on the iterated integral expression of MZVs:

(2) $\zeta(s_{1},\cdots
s_{k})=\underbrace{\int_{0}^{1}\frac{dt}{t}\int_{0}^{t}\frac{dt}{t}\cdots\int_{0}^{t}\frac{dt}{t}}_{s_{1}-1}\int_{0}^{t}\frac{dt}{1-t}\cdots\underbrace{\int_{0}^{t}\frac{dt}{t}\cdots\int_{0}^{t}\frac{dt}{t}}_{s_{k}-1}\int_{0}^{t}\frac{dt}{1-t}$

These relations can be promoted as stuffle multiplication $*$ and shuffle
multiplication X on the vector space

(3) ${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}=\mathbb{Q}{\bf
1}\bigoplus_{k\in\mathbb{Z}_{>0},\vec{s}\in\mathbb{Z}_{\geq
1}^{k}}\mathbb{Q}[\vec{s}],$

which is called the stuffle algebra and the shuffle algebra for MZVs. The
subspace space

(4) ${\mathcal{H}}^{0}=\mathbb{Q}{\bf
1}\oplus\bigoplus_{k\in\mathbb{Z}_{>0},\vec{s}\in\mathbb{Z}_{\geq
1}^{k},s_{1}>1}\mathbb{Q}[\vec{s}].$

is a subalgebra with respect to both multiplications, and MZVs can be viewed
as an algebra homomorphism

(5) $\zeta:{\mathcal{H}}^{0}\to\mathbb{R}$

for both multiplications. Therefore

$\\{\zeta(\vec{s}*\vec{t})-\zeta(\vec{s}{\mbox{\cyrs X}}\vec{t})=0\ |\
\vec{s}\in\mathbb{Z}_{\geq 1}^{k},s_{1}>1,\vec{t}\in\mathbb{Z}_{\geq
1}^{\ell},t_{1}>1,k,\ell\in\mathbb{Z}_{>0}\\}$

is a family of $\mathbb{Q}$-linear relations among MZVs, which is called
double shuffle relation.

Ihara-Kaneko-Zagier [IKZ] extended the algebra homomorphisms (5) to algebra
homomorphisms

$\zeta_{*},\zeta_{\mbox{\cyrs X}}:{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\to\mathbb{R}[T],$

and extended the double shuffle relation to regularized double shuffle
relation.

Zagier’s conjecture [Zag]: Regularized double shuffle relation gives all
$\mathbb{Q}$-linear relations among MZVs.

This conjecture looks so beautiful, but a complete solution seems quite
distant. Recent research focus on the relatively easier part: regularized
double shuffle relation gives all known $\mathbb{Q}$-linear relations among
MZVs, through it is still very challenging.

One possible spot of difficulty is, though the algebraic formulation provides
a beautiful expression of the double shuffle relations, but at the same time
it may also hide deeper structures. Our main result in this paper supports
this idea: the shuffle algebra for MZVs carries a natural coproduct, it is in
fact a Hopf algebra.

We use Chen fractions as our model for MZVs to explore the coproduct
structure. As showed in [GPZ2, GX], the model for MZVs by Chen fractions gives
the shuffle relation for MZVs, the shuffle multiplication for MZVs is lifted
to the function multiplication for Chen fractions. A key observation is that
Chen fractions, as functions, carry more operations like partial derivatives.

This model or lifting behaves well in the locality setting [CGPZ]. A locality
relation in a set is a symmetric binary relation. It is a useful tool to
explore singularity and a tool to reveal hidden structures. The linear span of
Chen fractions is not an algebra under the function multiplication, but it is
a locality polynomial algebra for this multiplication [GPZ1].

The locality polynomial algebra of Chen fractions carries partial derivatives,
which can be used to define a locality Hopf algebra structure for Chen
fractions. The pushforward of this coproduct gives a coproduct compatible with
the shuffle multiplication. Thus the shuffle algebra for MZVs is indeed a Hopf
algebra.

This paper is structured as following. In Section 2, we recall the backgrounds
for our study, shuffle algebra, cones and fractions, and locality Hopf
algebra. Section 3 is devoted to explore the locality Hopf algebra structure
of Chen fractions, and in Section 4, we show that the induced coproduct on
shuffle algebra of MZVs gives a Hopf algebra structure on
${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$.

This constructed Hopf algebra for MZVs on will be used in our paper in
preparation on the duality of shuffle Hopf algebra. It is also a part of our
project to extend double shuffle relation by renormalization.

##  2\. Preliminaries

Let us first recall some necessary backgrounds on shuffle algebra for MZVs,
geometry of fractions, and locality algebras.

###  2.1. The shuffle algebra for MZVs

The expression (2) of MZVs can be put in a simple form

$x_{0}^{s_{1}-1}x_{1}\cdots x_{0}^{s_{k}-1}x_{1},$

thus a MZV can be viewed as an element in $\mathbb{Q}\langle
x_{0},x_{1}\rangle$–the noncommutative polynomial algebra over $\mathbb{Q}$ in
variables $x_{0},x_{1}$. Of course, an MZV is in the subspace
$x_{0}\mathbb{Q}\langle x_{0},x_{1}\rangle x_{1}$–the subspace spanned by
monomials started with $x_{0}$ and ended with $x_{1}$. Meanwhile, we will use
$\mathbb{Q}\langle x_{0},x_{1}\rangle x_{1}$ to denote the linear spanned of
monomials ended with $x_{1}$.

For any alphabet ${\mathcal{A}}$, let $W$ be the set of words (including the
empty word ${\bf 1}$) in letters in ${\mathcal{A}}$. Then the linear space
$\mathbb{Q}W$ with a basis $W$ carries a shuffle multiplication X [H3], which
is inductively defined as follows:

$w{\mbox{\cyrs X}}{\bf 1}={\bf 1}{\mbox{\cyrs X}}w=w,$ $a_{1}{\mbox{\cyrs
X}}b_{1}\cdots b_{\ell}=a_{1}b_{1}\cdots b_{\ell}+b_{1}(a_{1}{\mbox{\cyrs
X}}b_{2}\cdots b_{\ell}).$ $a_{1}\cdots a_{k}{\mbox{\cyrs
X}}b_{1}=a_{1}(a_{2}\cdots a_{k}{\mbox{\cyrs X}}b_{1})+b_{1}a_{1}\cdots
a_{k},$ $a_{1}\cdots a_{k}{\mbox{\cyrs X}}b_{1}\cdots
b_{\ell}=a_{1}(a_{2}\cdots a_{k}{\mbox{\cyrs X}}b_{1}\cdots
b_{\ell})+b_{1}(a_{1}\cdots a_{k}{\mbox{\cyrs X}}b_{2}\cdots b_{\ell}).$

for $w\in\mathbb{Q}W$, $a_{i},b_{j}\in{\mathcal{A}}$, $i=1,\cdots,k$,
$j=1,\cdots,\ell$.

In the case of ${\mathcal{A}}=\\{x_{0},x_{1}\\}$, $\mathbb{Q}W$ is the
noncommutative polynomial algebra $\mathbb{Q}\langle x_{0},x_{1}\rangle$, the
subspaces $\mathbb{Q}\langle x_{0},x_{1}\rangle x_{1}$ and
$x_{0}\mathbb{Q}\langle x_{0},x_{1}\rangle x_{1}$ are subalgebras under the
shuffle multiplication. The subalgebra $\mathbb{Q}\langle x_{0},x_{1}\rangle
x_{1}$ is called the shuffle algebra for MZVs.

Let us introduce a notation. For a set $G$, let ${\mathcal{H}}_{G}$ be the
vector space over $\mathbb{Q}$ with a basis:

$\\{{\bf 1}\\}\cup\bigcup_{k\in\mathbb{Z}_{>0}}G^{k},$

that is

${\mathcal{H}}_{G}=\mathbb{Q}{\bf
1}\oplus\bigoplus_{k\in\mathbb{Z}_{>0},\vec{s}\in G^{k}}\mathbb{Q}[\vec{s}].$

We are mostly interested in the cases $G=\mathbb{Z}_{\geq 0},\mathbb{Z}_{\geq
1}$. For $\vec{s}=(s_{1},\cdots,s_{k})$ $\in\mathbb{Z}_{\geq 1}^{k}$, $k$ is
called the depth of $[\vec{s}]$ and $|\vec{s}|=s_{1}+\cdots+s_{k}$ is called
the weight of $[\vec{s}]$.

In this notation, the way to present the expression (2) gives a linear
isomorphism

(6) $\rho:{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}\to\mathbb{Q}\langle
x_{0},x_{1}\rangle x_{1},\ [s_{1},\cdots,s_{k}]\to x_{0}^{s_{1}-1}x_{1}\cdots
x_{0}^{s_{k}-1}x_{1},$

the shuffle product is pull back to a multiplication on
${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$, which we still denote by X. With this
multiplication, ${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$ is also called the
shuffle algebra for MZVs. Then

$\rho^{-1}(x_{0}\mathbb{Q}\langle x_{0},x_{1}\rangle
x_{1})=\bigoplus_{k\in\mathbb{Z}_{>0},\vec{s}\in\mathbb{Z}_{\geq
1}^{k},s_{1}\geq 2}\mathbb{Q}[\vec{s}]$

is a subalgebra.

Inside the space ${{\mathcal{H}}_{\mathbb{Z}_{\geq 0}}}$, let

$I:\bigoplus_{k\in\mathbb{Z}_{>0},\vec{s}\in\mathbb{Z}_{\geq
0}^{k}}\mathbb{Q}[\vec{s}]\longrightarrow\bigoplus_{k\in\mathbb{Z}_{>0},\vec{s}\in\mathbb{Z}_{\geq
0}^{k}}\mathbb{Q}[\vec{s}]$

be the linear operator defined on the basis as

$I([s_{1},s_{2},\cdots,s_{k}])=[s_{1}+1,s_{2},\cdots,s_{k}].$

Then the shuffle product on ${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$ is
characterized as in the following proposition.

######  Proposition 2.1.

[GZ3] The shuffle product on ${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$ is the
restriction of the unique commutative multiplication X on
${{\mathcal{H}}_{\mathbb{Z}_{\geq 0}}}$ which makes $I$ a Rota-Baxter operator
of weight $0$ on $\bigoplus_{k\in\mathbb{Z}_{>0},\vec{s}\in\mathbb{Z}_{\geq
0}^{k}}\mathbb{Q}[\vec{s}]$ and

${\bf 1}{\mbox{\cyrs X}}[\vec{s}]=[\vec{s}]{\mbox{\cyrs X}}{\bf 1}=[\vec{s}],$
$[0]{\mbox{\cyrs X}}[\vec{s}]=[0,\vec{s}].$

######  Remark 2.1.

$($[Guo]$)$ A Rota-Baxter operator $I$ of weight $0$ in an algebra $A$ is a
linear operator such that

$I(a)I(b)=I(aI(b))+I(I(a)b).$

###  2.2. Cones and fractions

In the linear space $\mathbb{R}^{k}$, a (polyhedral convex) cone is the
intersection of finitely many half spaces, or equivalently it is generated by
finitely many rays, $\sum_{i=1}^{n}\mathbb{R}_{\geq 0}v_{i}$. If the
generating set $\\{v_{i}\\}$ is linearly independent, then it is called
simplicial. The dimension of a cone is the dimension of the subspace spanned
by vectors in the cone. A face of a cone $C$ is a subset of the form $\\{x\
|u(x)=0\\}\cap C$, where $u\in(\mathbb{R}^{k})^{*}$, and $u(x)\geq 0$ for any
$x\in C$. A face of a cone is also a cone.

A subdivision of a cone $C$ is a set $\\{C_{1},\cdots,C_{r}\\}$ of cones such
that

  1. (i)

$C=\cup_{i=1}^{r}C_{i}$,

  2. (ii)

$C_{1},\cdots,C_{r}$ have the same dimension as $C$, and

  3. (iii)

$C_{1},\cdots,C_{r}$ intersect along their faces, i.e., $C_{i}\cap C_{j}$ is a
face of both $C_{i}$ and $C_{j}$ for $1\leq i,j\leq r$.

If all $C_{i},i=1,\cdots r$ are simplicial, then it is called a simplicial
subdivision. Let $\mathbb{R}{\mathcal{C}}_{k}$ be the linear space generated
by simplicial cones in $\mathbb{R}^{k}$ modulo simplicial subdivision and
cones containing non-trivial subspace, and $\mathbb{R}{\mathcal{C}}$ be the
direct limit.

A simplical fraction in $\mathbb{R}^{k}$ is a fraction of the form

$\frac{1}{L_{1}\cdots L_{n}},$

where $L_{1},L_{2},\cdots,L_{n}$ are linearly independent linear functions.
Let $\mathbb{R}{\mathcal{S}}_{k}$ be the linear span of simplicial fractions
in $\mathbb{R}^{k}$, and $\mathbb{R}{\mathcal{S}}$ be the direct limit.

Notice that for a simplicial cone $C$ in $\mathbb{R}^{k}$ with linearly
independent generators $v_{1},\cdots v_{k}$,
$v_{i}=\sum\limits_{j=1}^{k}a_{ji}e_{j}$, let
$L_{i}=L_{v_{i}}=\sum\limits_{j=1}^{k}a_{ji}z_{j}$ and $A_{C}=[a_{ij}]$, then

$\frac{|\det(A)|}{L_{1}\cdots
L_{k}}=(-1)^{k}\int\cdots\int_{C}e^{x_{1}z_{1}+\cdots+x_{k}z_{k}}dx_{1}\cdots
dx_{k}$

for any given $z\in\check{C}^{-}$, here $\check{C}^{-}=\\{y\ |\
(y,c)<0,\forall c\in C\\}$. So

$\Phi(C)=\frac{|\det(A)|}{L_{1}\cdots L_{k}}$

defines an element in $\mathbb{Q}{\mathcal{S}}_{k}$ for a simplicial cone $C$
of dimension $k$ in $\mathbb{R}^{k}$ which is compatible with simplicial
subdivisions. By changes of variables we can define a linear map

$\Phi:\mathbb{R}{\mathcal{C}}_{k}\to\mathbb{R}{\mathcal{S}}_{k},$

and by taking direct limit, we have

$\Phi:\mathbb{R}{\mathcal{C}}\to\mathbb{R}{\mathcal{S}}.$

######  Proposition 2.2.

[GPZ2] The map $\Phi$ is a linear isomorphism.

By using cones, we have a nice geometric criterion for linear independence of
simplicial fractions. Recall that a family of cones is properly positioned if
they intersect along faces, and the union contains no non-trivial linear
subspace. The supporting cone of a simplicial fraction $\frac{1}{L_{1}\cdots
L_{n}}$ is the cone generated by linearly independent linear functions
$L_{1},\cdots,L_{n}$.

######  Proposition 2.3.

For a finite set of simplicial fractions, if the denominators are not
pairwisely proportional, and their supporting cones are properly positioned,
then this set is linearly independent.

This geometric criterion can be extended to more general case. A simplex
fraction if a fraction of the form

$\frac{1}{L_{1}^{s_{1}}\cdots L_{n}^{s_{n}}}$

with linearly independent linear functions $L_{1},\cdots,L_{n}$. The cone
generated by $L_{1},\cdots,L_{n}$ is called the supporting cone of this
simplex fraction.

######  Proposition 2.4.

For a finite set of simplex fractions, if the denominators are not pairwisely
proportional, and their supporting cones can be chosen to be properly
positioned, then this set is linearly independent.

###  2.3. Locality Hopf algebras

We recall some backgrounds on locality structures from [CGPZ] in this part.

  * •

A locality set is a couple $(X,\top)$ where $X$ is a set and

$\top:=X\times_{\top}X\subseteq X\times X$

is a binary symmetric relation, which is called a locality relation on $X$.
For $x_{1},x_{2}\in X$, denote $x_{1}\top x_{2}$ if $(x_{1},x_{2})\in\top$.
For a subset $U\subset X$, the polar subset of $U$ is

$U^{\top}:=\\{x\in X\,|\,(x,U)\subseteq\top\\}.$

  * •

For locality sets $(X,\top_{X})$ and $(Y,\top_{Y})$, a map $f:X\to Y$ is
called a locality map if

$x_{1}\top_{X}x_{2}\Longrightarrow f(x_{1})\top_{Y}f(x_{2}),\quad\forall
x_{1},x_{2}\in X.$

Algebraic structures can be generalised to the locality setting.

  * •

A locality vector space is a vector space $V$ equipped with a locality
relation $\top$ which is compatible with the linear structure on $V$ in the
sense that, for any subset $X$ of $V$, $X^{\top}$ is a linear subspace of $V$.

  * •

A (nonunitary) locality algebra over $K$ is a locality vector space $(A,\top)$
over a field $K$ together with a bilinear map

$m_{A}:A\times_{\top}A\to A,(u,v)\mapsto u\cdot v=m_{A}(x,y)$

satisfying the following associativity and distributivity:

    1. (a)

For $u,v,w\in A$ with $u\top v,u\top w,v\top w$,

$(u\cdot v)\top w,\quad u\top(v\cdot w),\quad(u\cdot v)\cdot w=u\cdot(v\cdot
w).$

    2. (b)

For $u,v,w\in A$ with $u\top w,v\top w$,

$(u+v)\cdot w=u\cdot w+v\cdot w,\quad w\cdot(u+v)=w\cdot u+w\cdot v,$
$(ku)\cdot w=k(u\cdot w),\quad u\cdot(kw)=k(u\cdot w),\ k\in K.$

  * •

A unitary locality algebra is a locality algebra $(A,\top,m_{A})$ with a unit
$1_{A}$ such that $1_{A}\top u$ for each $u\in A$, and

$1_{A}\cdot u=u\cdot 1_{A}=u.$

  * •

Let $(A,\top_{A})$ be a locality algebra. A subspace $B$ of $A$ is called a
(resp. unitary) locality subalgebra of $A$ if, for

$\top_{B}:=\top_{A}\cap(B\times B)\vspace{-.2cm}$

the pair $(B,\top_{B})$ is a (resp. unitary) locality algebra.

  * •

A locality algebra $A$ with a grading $A=\oplus_{n\geq 0}A_{n}$ is called a
locality graded algebra if $m_{A}((A_{m}\times A_{n})\cap\top_{A})\subseteq
A_{m+n}$ for all $m,n\in\mathbb{Z}_{\geq 0}$. The locality graded algebra is
called connected if $A_{0}=K$.

  * •

Given two locality algebras $(A_{i},\top_{i}),i=1,2$, a (resp. unitary)
locality algebra homomorphism is a linear map $\varphi:A_{1}\longrightarrow
A_{2}$ such that $a\top_{1}b$ implies $\varphi(a)\top_{2}\varphi(b)$ and
$\varphi(a\cdot b)=\varphi(a)\cdot\varphi(b)$ (resp. and
$\varphi(1_{A_{1}})=1_{A_{2}}$).

Coalgebra structure can also be generalized to locality setting. For a
locality set $(X,\top)$ and an integer $k\geq 2$, denote

$X^{{}_{\top}k}:=X\times_{\top}\cdots\times_{\top}X=\\{(x_{1},\cdots,x_{k})\in
X^{k}\ |\ x_{i}\top x_{j},1\leq i\not=j\leq k\\}$

and for a locality vector space $(C,\top)$ over $\mathbb{Q}$, let

$C^{\otimes_{\top}n}=\underbrace{C\otimes_{\top}\cdots\otimes_{\top}C}_{n\text{
factors }}$

denote the image of the composition map
$\mathbb{Q}(C^{{}_{\top}n})\rightarrow\mathbb{Q}(C^{n})\rightarrow C^{\otimes
n}$, where the first map is the inclusion and the second map is the quotient
map modulo multilinearity.

  * •

Let $(C,\top)$ be a locality vector space over a field $K$ and let
$\Delta:C\to C\otimes C$ be a linear map. $(C,\top,\Delta)$ is a
(noncounitary) locality coalgebra if it satisfies the following two conditions

    1. (1)

for any $U\subset C$

(7) $\Delta(U^{\top})\subset U^{\top}\otimes_{\top}U^{\top},$

which is equivalent to for any $c\in C$,

$\Delta(c^{\top})\subseteq c^{\top}\otimes_{\top}c^{\top}.$

    2. (2)

the coassociativity holds:

$({\rm Id}_{C}\otimes\Delta)\,\Delta=(\Delta\otimes{\rm Id}_{C})\,\Delta.$

    * –

If in addition, there is a counit, namely a linear map $\varepsilon:C\to K$
such that $({{\rm Id}}_{C}\otimes\varepsilon)\,\Delta=(\varepsilon\otimes{{\rm
Id}}_{C})\,\Delta={{\rm Id}}_{C}$, then $(C,\top,\Delta,\varepsilon)$ is
called a counitary locality coalgebra.

    * –

A graded locality coalgebra is a locality coalgebra
$(C,\top,\Delta)$ with a grading $C=\oplus_{n\geq 0}C_{n}$ such that, for any
$U\subseteq C$,

(8) $\Delta(C_{n}\cap U^{\top})\subseteq\bigoplus_{p+q=n}(C_{p}\cap
U^{\top})\otimes_{\top}(C_{q}\cap U^{\top}).$

Moreover, the graded locality coalgebra is connected if

$\bigoplus_{n\geq 1}C_{n}=\ker\varepsilon,C_{0}=K\,J,$

where $J$ is the unique element of $C_{0}$ with $\varepsilon(J)=1_{K}$.

  * •

A locality bialgebra is a sextuple $(B,\top,m,u,\Delta,\varepsilon)$
consisting of a unitary locality algebra $(B,\top,m,u)$ and a counitary
locality coalgebra $\left(B,\top,\Delta,\varepsilon\right)$ that are locality
compatible in the sense that $\Delta$ and $\varepsilon$ are locality algebra
homomorphisms.

A locality bialgebra $B$ is called graded if there is a $\mathbb{Z}_{\geq
0}$-grading $B=\oplus_{n\geq 0}B_{n}$ with respect to which $B$ is both a
graded locality algebra and a graded locality coalgebra. In this case, $B$ is
called connected if $J=1_{B}$.

  * •

A locality Hopf algebra is a septuple $(B,\top,m,\Delta,u,\varepsilon,S)$ such
that $(B,\top,m,\Delta,u,\varepsilon)$ is a locality bialgebra, $S:B\to B$ is
called an antipode which is a linear map mutually independent with ${\rm
Id}_{B}$ and

$S\star{\rm Id}_{B}={\rm Id}_{B}\star S=u\circ\,\varepsilon.$

A connected graded bialgebra is in fact a hopf algebra.

######  Proposition 2.5.

$($ Exercise 3.3 of [Man]$)$ Any connected filtered bialgebra
$(A,m,u,\Delta,\varepsilon)$ is a Hopf algebra.

Similarly, we have the locality version.

######  Proposition 2.6.

$($Proposition 5.6 of [CGPZ]$)$ Let
$\left(B,\top,m,u,\Delta,\varepsilon\right)$ be a connected locality
bialgebra, then it is a locality Hopf algebra, the antipode is given by

$S=\sum_{k=0}^{\infty}(u\circ\varepsilon-{\rm Id})^{\star k}.$

##  3\. The Locality Hopf algebra of Chen fractions

In this part, we study the structure of the space of spanned by Chen
fractions.

###  3.1. Locality algebra of Chen fractions

For the variables set
$\\{x_{i}\\}_{i\in\mathbb{Z}_{\geq 1}}$, and a finite nonempty subset
$I=\\{i_{1},\cdots,i_{k}\\}\subset\mathbb{Z}_{\geq 1}$, recall that a Chen
fraction with variables in $\\{x_{i}\ |i\in I\\}$ is a fraction of the form

$\frac{1}{(x_{\sigma(i_{1})}+x_{\sigma(i_{2})}+\cdots+x_{\sigma(i_{k})})^{s_{1}}(x_{\sigma(i_{2})}+\cdots+x_{\sigma(i_{k})})^{s_{2}}\cdots
x_{\sigma(i_{k})}^{s_{k}}},$

where $s_{1},\cdots,s_{k}\in\mathbb{Z}_{\geq 1}$, and $\sigma\in S_{k}$–the
symmetric group of $\\{1,\cdots,k\\}$. We also use the notation

${\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\sigma(i_{1})},\cdots,x_{\sigma(i_{k})}\end{array}]}}$

to denote it.

Let ${F^{ch}}$ be the set of all Chen fractions in any finite nonempty subset
of $\\{x_{i}\\}_{i\in\mathbb{Z}_{\geq 1}}$ and the constant function $1$, and
$\mathbb{Q}{F^{ch}}$ be the subspace of functions in variables
$\\{x_{i}\\}_{i\in\mathbb{Z}_{\geq 1}}$ spanned by ${F^{ch}}$ over
$\mathbb{Q}$.

######  Proposition 3.1.

The set ${F^{ch}}$ is a basis of $\mathbb{Q}{F^{ch}}$.

###### Proof.

The supporting cone of a Chen fraction
${\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\sigma(i_{1})},\cdots,x_{\sigma(i_{k})}\end{array}]}}$ can be chosen as the
cone generated by

$x_{\sigma(i_{1})}+x_{\sigma(i_{2})}+\cdots+x_{\sigma(i_{k})},x_{\sigma(i_{2})}+\cdots+x_{\sigma(i_{k})},\cdots,x_{\sigma(i_{k})}.$

Any set of this kind of cones are properly positioned, so by Lemma 2.4, the
set ${F^{ch}}$ is linearly independent. ∎

In general, $\mathbb{Q}F^{ch}$ is not an algebra under the multiplication of
functions, which we will sometimes denote by $``\cdot"$ to make it visible.

######  Example 3.1.

By [GPZ2], the product of Chen fractions

$\frac{1}{(x_{1}+x_{2})x_{1}}\cdot\frac{1}{(x_{1}+x_{3})x_{1}}\not\in\mathbb{Q}{F^{ch}}.$

The situation is much better in the locality setting. In [GPZ1], a locality
relation $\perp^{Q}$ on ${F^{ch}}$ is introduced by using an inner product
$Q$. Under an orthonormal basis of $Q$, let $\\{x_{i}\\}$ be the coordinates
with respect to this basis, then for Chen fractions in variables
$\\{x_{i}\\}$, the locality relation is

$f_{1}\perp^{Q}f_{2},\,\text{if they are Chen fractions in different
variables},$

and $1\perp^{Q}f$ for any Chen fraction $f$. This locality relation is
extended to $\mathbb{Q}{F^{ch}}$, which we denote by
$\top_{\mathbb{Q}{F^{ch}}}$, or $\top$ if there is no ambiguity .

######  Proposition 3.2.

The restriction of function multiplication makes
$(\mathbb{Q}{F^{ch}},\top_{\mathbb{Q}{F^{ch}}},1)$ into a unitary commutative
locality algebra.

###  3.2. The lifting of shuffle product

The locality function multiplication of Chen fractions is closely related to
the shuffle product of MZVs.

Let ${\mathcal{A}}=\\{x_{i}\\}_{i\in\mathbb{Z}_{\geq 0}}$ be an alphabet, as
we already see, the linear space $\mathbb{Q}W$ with a basis $W$ of words in
${\mathcal{A}}$ carries the shuffle product X. Let $W_{1}$ be the set of words
of the form

$x_{0}^{s_{1}-1}x_{i_{1}}\cdots x_{0}^{s_{k}-1}x_{i_{k}}$

with $s_{1},\cdots,s_{k}\in\mathbb{Z}_{\geq 1}$, and $i_{1},\cdots,i_{k}$ are
distinct, and the empty word ${\bf 1}$. The set $W_{1}$ has a natural locality
relation $\top_{W_{1}}$

$(x_{0}^{s_{1}-1}x_{i_{1}}\cdots
x_{0}^{s_{k}-1}x_{i_{k}})\top_{W_{1}}(x_{0}^{t_{1}-1}x_{j_{1}}\cdots
x_{0}^{t_{\ell}-1}x_{j_{\ell}})$

if

$\\{i_{1},\cdots,i_{k}\\}\cap\\{j_{1},\cdots,j_{\ell}\\}=\emptyset,$

and

${\bf 1}\top_{W_{1}}w$

for any $w\in W_{1}$. This locality relation extends to a locality relation
$\top_{\mathbb{Q}W_{1}}$. It is easy to see that the restriction of the
shuffle product in $\mathbb{Q}W$ equips $\mathbb{Q}W_{1}$ with a unitary
commutative locality algebra structure.

Obviously,

######  Proposition 3.3.

The the linear map $\psi$

$\psi:\mathbb{Q}W_{1}\to\mathbb{Q}\langle x_{0},x_{1}\rangle x_{1},$

which on $W_{1}$ is defined as

${\bf 1}\longmapsto{\bf 1},$ $x_{0}^{s_{1}-1}x_{i_{1}}\cdots
x_{0}^{s_{k}-1}x_{i_{k}}\mapsto x_{0}^{s_{1}-1}x_{1}\cdots
x_{0}^{s_{k}-1}x_{1}$

is a unitary locality algebra homomorphism, where on $\mathbb{Q}\langle
x_{0},x_{1}\rangle x_{1}$, the locality relation $\top_{\mathbb{Q}\langle
x_{0},x_{1}\rangle x_{1}}=\mathbb{Q}\langle x_{0},x_{1}\rangle
x_{1}\times\mathbb{Q}\langle x_{0},x_{1}\rangle x_{1}$ is trivial.

######  Proposition 3.4.

[GPZ1] The the linear map $\phi$

$\phi:\mathbb{Q}{F^{ch}}\to\mathbb{Q}W_{1},$

which on ${F^{ch}}$ is defined as

$1\longmapsto{\bf 1},$ ${\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{i_{1}},\cdots,x_{i_{k}}\end{array}]}}\mapsto x_{0}^{s_{1}-1}x_{i_{1}}\cdots
x_{0}^{s_{k}-1}x_{i_{k}}$

is a unitary locality algebra isomorphism from
$(\mathbb{Q}F^{ch},\top_{\mathbb{Q}{F^{ch}}},\cdot)$ to
$(\mathbb{Q}W_{1},\top_{\mathbb{Q}W_{1}},{\mbox{\cyrs X}})$, where $\cdot$ is
the restriction of function multiplication.

######  Remark 3.1.

This Proposition shows the advantage to study in locality setting, this linear
map $\phi$ is not an algebra homomorphism without locality:

$\phi({\scriptsize{[\begin{array}[]{c}1\\\
x_{1}\end{array}]}}{\scriptsize{[\begin{array}[]{c}1\\\
x_{1}\end{array}]}})=\phi(\frac{1}{x_{1}^{2}})=\phi({\scriptsize{[\begin{array}[]{c}2\\\
x_{1}\end{array}]}})=x_{0}x_{1}$

but

$\phi({\scriptsize{[\begin{array}[]{c}1\\\ x_{1}\end{array}]}}){\mbox{\cyrs
X}}\phi({\scriptsize{[\begin{array}[]{c}1\\\
x_{1}\end{array}]}})=x_{1}{\mbox{\cyrs X}}x_{1}=2x_{1}x_{1}.$

By Proposition 3.3, Proposition 3.4, and Eq. (6),

######  Corollary 3.1.

The composition

$\pi=\rho^{-1}\circ\psi\circ\phi:\mathbb{Q}{F^{ch}}\to{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}$

lifts the shuffle product X on ${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$ to
locality function multiplication of Chen fractions.

###  3.3. Locality Hopf algebra of Chen fractions

As a space of functions, the space $\mathbb{Q}{F^{ch}}$ carries a family of
partial derivatives $\frac{\partial}{\partial x_{i}}$, $i\in\mathbb{Z}_{\geq
1}$, they are essential for our construction. Let

$\partial_{i}=-\frac{\partial}{\partial x_{i}},$
$d_{i,j}={\partial}_{i}-{\partial}_{j},$

where we take ${\partial}_{0}=0$, then we have [GPZ2]

(13) $\displaystyle{\scriptsize{[\begin{array}[]{c}s_{1},s_{2},\cdots,s_{k}\\\
x_{i_{1}},x_{i_{2}},\cdots,x_{i_{k}}\end{array}]}}=\frac{d_{i_{1},i_{0}}^{s_{1}-1}\circ
d_{i_{2},i_{1}}^{s_{2}-1}\cdots\circ
d_{i_{k},i_{k-1}}^{s_{k}-1}}{(s_{1}-1)!(s_{2}-1)!\cdots(s_{k}-1)!}{\scriptsize{[\begin{array}[]{c}1,1,\cdots,1\\\
x_{i_{1}},x_{i_{2}},\cdots,x_{i_{k}}\end{array}]}},$

here we take $i_{0}=0$ for convenience.

We now construct a coproduct
$\Delta:\mathbb{Q}{F^{ch}}\longrightarrow\mathbb{Q}{F^{ch}}\otimes\mathbb{Q}{F^{ch}}$
on the basis ${F^{ch}}$ by requiring that it commutes with partial derivatives
in the following sense, for any $i\in\mathbb{Z}_{\geq 1}$,

(14) $\Delta\circ{\partial}_{i}=(1\otimes{\partial}_{i}+{\partial}_{i}\otimes
1)\circ\Delta.$

It is easy to check the following lemma.

######  Lemma 3.1.

If a linear operator
$\Delta:\mathbb{Q}{F^{ch}}\longrightarrow\mathbb{Q}{F^{ch}}\otimes\mathbb{Q}{F^{ch}}$
satisfying Eq. (14), then

$(1\otimes\Delta)\circ(1\otimes{\partial}_{i}+{\partial}_{i}\otimes 1)$
$=(1\otimes 1\otimes{\partial}_{i}+1\otimes{\partial}_{i}\otimes
1+{\partial}_{i}\otimes 1\otimes 1)\circ(1\otimes\Delta)$

and

$(\Delta\otimes 1)\circ(1\otimes{\partial}_{i}+{\partial}_{i}\otimes 1)$
$=(1\otimes 1\otimes{\partial}_{i}+1\otimes{\partial}_{i}\otimes
1+{\partial}_{i}\otimes 1\otimes 1)\circ(\Delta\otimes 1).$

So if we define

$\Delta:\mathbb{Q}{F^{ch}}\longrightarrow\mathbb{Q}{F^{ch}}\otimes\mathbb{Q}{F^{ch}}$

on the basis ${F^{ch}}$ by

  1. (1)

For the constant function $1$,

$\Delta(1)=1\otimes 1.$

  2. (2)

For $s_{1}=\cdots=s_{k}=1$,

(15) $\Delta({\scriptsize{[\begin{array}[]{c}1,1,\cdots,1\\\
x_{i_{1}},x_{i_{2}},\cdots,x_{i_{k}}\end{array}]}}):=\sum_{j=0}^{k}{\scriptsize{[\begin{array}[]{c}1,\cdots,1\\\
x_{i_{1}},\cdots,x_{i_{j}}\end{array}]}}\otimes{\scriptsize{[\begin{array}[]{c}1,\cdots,1\\\
x_{i_{j+1}},\cdots,x_{i_{k}}\end{array}]}}.$

  3. (3)

Inductively,

$\displaystyle\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{j}+1,\cdots,s_{k}\\\
x_{i_{1}},\cdots,x_{i_{j}},\cdots,x_{i_{k}}\end{array}]}}):=\frac{1\otimes
d_{i_{j},i_{j-1}}+d_{i_{j},i_{j-1}}\otimes
1}{s_{j}}\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{j},\cdots,s_{k}\\\
x_{i_{1}},\cdots,x_{i_{j}},\cdots,x_{i_{k}}\end{array}]}}).$

Then we have

######  Proposition 3.5.

The linear operator $\Delta$ is coassociative.

###### Proof.

By Equation (15), $\Delta$ is the image of the deconcatenation coproduct under
the isomorphism $\phi$ in Proposition 3.4, so it is coassociative in this case
[H2]. In general, $\Delta$ is coassociative by Lemma 3.1. ∎

######  Lemma 3.2.

For $f,g\in\mathbb{Q}{F^{ch}}$, if $f\top g$, then

$\Delta(f\cdot g)=\Delta(f)\cdot\Delta(g).$

###### Proof.

We only need to prove this for $f,g\in{F^{ch}}$. The conclusion is obvious for
$f=1$ or $g=1$.

Let $f={\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}}$,
$g={\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}}$, since $f\top g$, we know that
$x_{i_{1}},\cdots,x_{i_{m}}$, $x_{j_{1}},\cdots,x_{j_{n}}$ are distinct.

For $s_{1}=\cdots=s_{m}=t_{1}=\cdots=t_{n}=1$, this is compatibility between X
and the deconcatenation coproduct, which is known [H2].

In general, we do induction on $w=s_{1}+\cdots+s_{m}+t_{1}+\cdots+t_{n}$,
which is true for $w=m+n$. Now assume it is true for $w=k\geq m+n$, and for
$w=k+1$, so there is $s_{l}\geq 2$ for some $1\leq l\leq m$, (or $t_{j}\geq 2$
for some $1\leq j\leq n$, the proof is similar), then

$\displaystyle\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}}\cdot{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}})$ $\displaystyle=$
$\displaystyle\Delta(\frac{d_{i_{l},i_{l-1}}}{s_{l}-1}({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{1}-1,\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{l}},\cdots,x_{i_{m}}\end{array}]}})\cdot{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}})$ $\displaystyle\overset{1}{=}$
$\displaystyle\Delta(\frac{d_{i_{l},i_{l-1}}}{s_{l}-1}({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{1}-1,\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{l}},\cdots,x_{i_{m}}\end{array}]}}\cdot{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}}))$ $\displaystyle=$
$\displaystyle\frac{1\otimes d_{i_{l},i_{l-1}}+d_{i_{l},i_{l-1}}\otimes
1}{s_{l}-1}(\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{1}-1,\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{l}},\cdots,x_{i_{m}}\end{array}]}}\cdot{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}}))$ $\displaystyle=$
$\displaystyle\frac{1\otimes d_{i_{l},i_{l-1}}+d_{i_{l},i_{l-1}}\otimes
1}{s_{l}-1}(\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{l}-1,\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{l}},\cdots,x_{i_{m}}\end{array}]}})\cdot\Delta({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}}))$ $\displaystyle\overset{2}{=}$
$\displaystyle\frac{1\otimes d_{i_{l},i_{l-1}}+d_{i_{l},i_{l-1}}\otimes
1}{s_{l}-1}(\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{l}-1,\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{l}},\cdots,x_{i_{m}}\end{array}]}}))\cdot\Delta({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}})$ $\displaystyle=$
$\displaystyle\Delta(\frac{d_{i_{l},i_{l-1}}}{s_{l}-1}({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{1}-1,\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{l}},\cdots,x_{i_{m}}\end{array}]}}))\cdot\Delta({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}})$ $\displaystyle=$
$\displaystyle\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{l},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{l}},\cdots,x_{i_{m}}\end{array}]}})\cdot\Delta({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}}),$

as required.

The equalities marked with number $1$ and $2$ follow from the fact that the
variables in the one part are different from the variables in the other part,
which is the locality condition. ∎

Define a linear map $\varepsilon:\mathbb{Q}{F^{ch}}\longrightarrow\mathbb{Q}$
on the basis set ${F^{ch}}$ as

$\varepsilon(f)=\left\\{\begin{array}[]{lc}1,&{\rm if}\,f=1\\\ 0,&{\rm
otherwise}\end{array}\right.,$

then $\mathbb{Q}{F^{ch}}=\mathbb{Q}{1}\oplus Ker\varepsilon$.

######  Lemma 3.3.

The quadruple $(\mathbb{Q}{F^{ch}},\top,\Delta,\varepsilon)$ is a counitary
locality coalgebra.

###### Proof.

We first prove that

$\Delta(\\{f\\}^{\top})\subseteq\\{f\\}^{\top}\otimes_{\top}\\{f\\}^{\top}$

for any $f\in{F^{ch}}$. We know $1\in\\{f\\}^{\top}$, and

$\Delta(1)=1\otimes 1\in\\{f\\}^{\top}\otimes_{\top}\\{f\\}^{\top}.$

Now for $g={\scriptsize{[\begin{array}[]{c}s_{1},s_{2},\cdots,s_{k}\\\
x_{i_{1}},x_{i_{2}},\cdots,x_{i_{k}}\end{array}]}}\in\\{f\\}^{\top}$, by the
definition of $\Delta$, we know

$\Delta({\scriptsize{[\begin{array}[]{c}s_{1},s_{2},\cdots,s_{k}\\\
x_{i_{1}},x_{i_{2}},\cdots,x_{i_{k}}\end{array}]}})\in\\{f\\}^{\top}\otimes_{\top}\\{f\\}^{\top},$

so we have the conclusion.

Next we show $\varepsilon$ is a counit. For any $h\in\mathbb{Q}{F^{ch}}$,

$(\varepsilon\otimes{{\rm Id}})\Delta(h)=\varepsilon(1)h=h,$ $({{\rm
Id}}\otimes\varepsilon)\Delta(h)=\varepsilon(1)h=h.$

By Proposition 3.5, $\Delta$ is coassociative, hence
$(\mathbb{Q}{F^{ch}},\top,\Delta,\varepsilon)$ is a counitary locality
coalgebra. ∎

######  Proposition 3.6.

The sextuple $(\mathbb{Q}{F^{ch}},\top,\cdot,\Delta,1,\varepsilon)$ is a
locality Hopf algebra.

###### Proof.

By Proposition 3.2, $(\mathbb{Q}{F^{ch}},\top,\cdot,1)$ is a unitary locality
algebra, by Lemma 3.3, $(\mathbb{Q}{F^{ch}},\top,\Delta,\varepsilon)$ is a
counitary locality coalgebra, and by Lemma 3.2, $\Delta$ is a locality algebra
homomorphism with respect to the multiplication $``\cdot"$. So we only need
check that $\varepsilon$ is a locality algebra homomorphism:

  * •

$\varepsilon(1)=1=\varepsilon(1)\cdot\varepsilon(1)$,

  * •

for $f,g\in{F^{ch}}$ such that $f\top\ g$ and $f\neq 1$ or $g\neq 1$,

$\varepsilon(f\cdot g)=0=\varepsilon(f)\cdot\varepsilon(g).$

So $(\mathbb{Q}{F^{ch}},\top,\cdot,\Delta,1,\varepsilon)$ is a locality
bialgebra.

Now we show that $(\mathbb{Q}{F^{ch}},\top,\cdot,\Delta,1,\varepsilon)$ is a
connected graded locality bialgebra. Let

$F_{0}:=\\{1\\},$
$F_{k}:=\\{{\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}}\,|s_{1}+\cdots+s_{m}=k\,\\},$

then

${F^{ch}}=\bigcup_{k=0}^{\infty}F_{k},$

and $\mathbb{Q}{F^{ch}}$ is graded as

$\mathbb{Q}{F^{ch}}=\bigoplus_{k=0}^{\infty}\mathbb{Q}F_{k}.$

It is graded by degree of fractions, so $(\mathbb{Q}{F^{ch}},\top,\cdot)$ is a
connected locality algebra.

To show that $(\mathbb{Q}{F^{ch}},\top,\Delta)$ is a connected counitary
locality coalgebra under the grading above, for any
$U\subseteq\mathbb{Q}{F^{ch}}$, $1\in U^{T}$, we have

$\Delta(1)=1\otimes 1\in(\mathbb{Q}F_{0}\cap
U^{\top})\otimes_{\top}(\mathbb{Q}F_{0}\cap U^{\top}).$

Now we prove

$\Delta(\mathbb{Q}F_{k}\cap
U^{\top})\subseteq\bigoplus_{p+q=k}(\mathbb{Q}F_{p}\cap
U^{\top})\otimes_{\top}(\mathbb{Q}F_{q}\cap U^{\top})$

by induction on $k$. If $\sum_{i}a_{i}h_{i}\in U^{\top}$ with
$h_{i}\in{F^{ch}}$, by the definition of locality relation in
$\mathbb{Q}{F^{ch}}$, we know $h_{i}\in U^{\top}$, so we only need to prove it
for basis elements. Let
$h={\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}}\in F_{k}\cap U^{\top}$, which implies
${\scriptsize{[\begin{array}[]{c}1,\cdots,1\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}}\in U^{\top}$.

  1. (1)

If $s_{1}=\cdots=s_{m}=1$( $m=k$),

$\Delta({\scriptsize{[\begin{array}[]{c}1,\cdots,1\\\
x_{i_{1}},\cdots,x_{i_{k}}\end{array}]}})=\sum_{j=0}^{k}{\scriptsize{[\begin{array}[]{c}1,\cdots,1\\\
x_{i_{1}},\cdots,x_{i_{j}}\end{array}]}}\otimes{\scriptsize{[\begin{array}[]{c}1,\cdots,1\\\
x_{i_{j+1}},\cdots,x_{i_{k}}\end{array}]}},$

which belongs to $\bigoplus_{p+q=k}(\mathbb{Q}F_{p}\cap
U^{\top})\otimes_{\top}(\mathbb{Q}F_{q}\cap U^{\top})$.

  2. (2)

If $s_{j}\geq 2$ for some $1\leq j\leq m$, then
${\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{j}-1,\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{j}},\cdots,x_{i_{m}}\end{array}]}}\in U^{\top}$,

$\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{j},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{j}},\cdots,x_{i_{m}}\end{array}]}}):=\frac{1\otimes
d_{i_{j},i_{j-1}}+d_{i_{j},i_{j-1}}\otimes
1}{s_{j}-1}\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{j}-1,\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{j}},\cdots,x_{i_{m}}\end{array}]}}),$

by inductive hypothesis,

$\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{j}-1,\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{j}},\cdots,x_{i_{m}}\end{array}]}})\in\bigoplus_{p+q=k-1}(\mathbb{Q}F_{p}\cap
U^{\top})\otimes_{\top}(\mathbb{Q}F_{q}\cap U^{\top}),$

so

$\Delta(h)\in\bigoplus_{p+q=k}(\mathbb{Q}F_{p}\cap
U^{\top})\otimes_{\top}(\mathbb{Q}F_{q}\cap U^{\top}).$

It is obvious that

$\bigoplus_{n\geq
1}\mathbb{Q}F_{n}=\ker\varepsilon,\mathbb{Q}F_{0}=\mathbb{Q}\,1.$

Hence $(\mathbb{Q}{F^{ch}},\top,\Delta,\varepsilon)$ is a connected locality
graded coalgebra. Thus $(\mathbb{Q}{F^{ch}},\top,\cdot,\Delta,1,\varepsilon)$
is a connected locality bialgebra. By Proposition 2.6,
$(\mathbb{Q}{F^{ch}},\top,\cdot,\Delta,1,\varepsilon)$ is a locality Hopf
algebra. ∎

##  4\. The shuffle Hopf algebra

Now we use the locality algebra homomorphism

$\pi:\mathbb{Q}{F^{ch}}\longrightarrow{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$

in Corollary 3.1 to construct a coproduct on the shuffle algebra
$({{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}},{\mbox{\cyrs X}})$. By the definition
of $\pi$,

$\pi({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{i_{1}},\cdots,x_{i_{k}}\end{array}]}})=[s_{1},\cdots,s_{k}].$

###  4.1. The shuffle Hopf algebra

For $[s_{1},\cdots,s_{k}]\in{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$, take
subsets $\\{i_{1},\cdots,i_{k}\\}$ and $\\{j_{1},\cdots,j_{k}\\}$ of $k$
elements in $\mathbb{Z}_{\geq 1}$, by the definition of
$\Delta:\mathbb{Q}{F^{ch}}\to\mathbb{Q}{F^{ch}}\otimes\mathbb{Q}{F^{ch}}$, we
know $\Delta$ commutes with changes of variables:

$\big{(}\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{i_{1}},\cdots,x_{i_{k}}\end{array}]}})\big{)}|_{x_{i_{1}}=x_{j_{1}},\cdots,x_{i_{k}}=x_{j_{k}}}=\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{j_{1}},\cdots,x_{j_{k}}\end{array}]}}),$

thuse have

######  Lemma 4.1.

The element
$(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{i_{1}},\cdots,x_{i_{k}}\end{array}]}})$ does not depend on the choice of
$i_{1},\cdots,i_{k}$ if they are distinct.

Therefore the linear map

${\Delta_{\geq 1}}:{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\longrightarrow{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\otimes{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}},$

which on basis elements is given by

${\Delta_{\geq 1}}({\bf 1}):={\bf 1}\otimes{\bf 1},$ ${\Delta_{\geq
1}}([s_{1},\cdots,s_{k}]):=(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{i_{1}},\cdots,x_{i_{k}}\end{array}]}}),$

for any subset $\\{i_{1},\cdots,i_{k}\\}$ of $k$ elements of $\mathbb{Z}_{\geq
1}$ is well-defined.

######  Proposition 4.1.

The linear map

${\Delta_{\geq 1}}:{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\longrightarrow{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\otimes{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$

is coassociative.

###### Proof.

Notice that

${\Delta_{\geq 1}}\circ\pi=(\pi\otimes\pi)\Delta,$

by the definition and the Lemma 4.1, then

$\displaystyle(1\otimes{\Delta_{\geq 1}}){\Delta_{\geq 1}}$
$\displaystyle=(1\otimes{\Delta_{\geq
1}})(\pi\otimes\pi)\Delta=(\pi\otimes{\Delta_{\geq 1}}\pi)\Delta$
$\displaystyle=(\pi\otimes\pi\otimes\pi)(1\otimes\Delta)\Delta,$

and

$\displaystyle({\Delta_{\geq 1}}\otimes 1){\Delta_{\geq 1}}$
$\displaystyle=({\Delta_{\geq 1}}\otimes
1)(\pi\otimes\pi)\Delta=({\Delta_{\geq 1}}\pi\otimes\pi)\Delta$
$\displaystyle=(\pi\otimes\pi\otimes\pi)(\Delta\otimes 1)\Delta,$

by Proposition 3.5, we get the conclusion. ∎

######  Lemma 4.2.

For any set with $m+n$ distinct positive integers

$\\{i_{1},\cdots,i_{m},i_{m+1},\cdots,i_{m+n}\\},$

and $(s_{1},\cdots,s_{m})\in\mathbb{Z}_{\geq
1}^{m},(t_{1},\cdots,t_{n})\in\mathbb{Z}_{\geq 1}^{n}$, we have

$(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}}\cdot{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}})={\Delta_{\geq
1}}([s_{1},\cdots,s_{m}]{\mbox{\cyrs X}}[t_{1},\cdots,t_{n}]).$

###### Proof.

Let

(24) $\displaystyle{\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}}\cdot{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{i_{m+1}},\cdots,x_{i_{m+n}}\end{array}]}}=\sum
C_{{\scriptsize{[\begin{array}[]{c}v_{1,j},\cdots,v_{m+n,j}\\\
l_{1,j},\cdots,l_{m+n,j}\end{array}]}}}{\scriptsize{[\begin{array}[]{c}v_{1,j},\cdots,v_{m+n,j}\\\
l_{1,j},\cdots,l_{m+n,j}\end{array}]}},$

where $C_{{\scriptsize{[\begin{array}[]{c}v_{1,j},\cdots,v_{m+n,j}\\\
l_{1,j},\cdots,l_{m+n,j}\end{array}]}}}\in\mathbb{Q}$,
$[l_{1,j},\cdots,l_{m+n,j}]$ is a permutation of

$[x_{i_{1}},\cdots,x_{i_{m}},x_{i_{m+1}},\cdots,x_{i_{m+n}}].$

Apply $\pi$ on Equation (24),

$[s_{1},\cdots,s_{m}]{\mbox{\cyrs X}}[t_{1},\cdots,t_{n}]=\sum
C_{{\scriptsize{[\begin{array}[]{c}v_{1,j},\cdots,v_{m+n,j}\\\
l_{1,j},\cdots,l_{m+n,j}\end{array}]}}}[v_{1,j},\cdots,v_{m+n,j}].$

Hence

$\displaystyle{\Delta_{\geq 1}}([s_{1},\cdots,s_{m}]{\mbox{\cyrs
X}}[t_{1},\cdots,t_{n}])$ $\displaystyle=$ $\displaystyle{\Delta_{\geq
1}}\big{(}\sum C_{{\scriptsize{[\begin{array}[]{c}v_{1,j},\cdots,v_{m+n,j}\\\
l_{1,j},\cdots,l_{m+n,j}\end{array}]}}}[v_{1,j},\cdots,v_{m+n,j}]\big{)}$
$\displaystyle=$ $\displaystyle(\pi\otimes\pi)\Delta\big{(}\sum
C_{{\scriptsize{[\begin{array}[]{c}v_{1,j},\cdots,v_{m+n,j}\\\
l_{1,j},\cdots,l_{m+n,j}\end{array}]}}}{\scriptsize{[\begin{array}[]{c}v_{1,j},\cdots,v_{m+n,j}\\\
l_{1,j},\cdots,l_{m+n,j}\end{array}]}}\big{)}$ $\displaystyle=$
$\displaystyle(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}}\cdot{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{i_{m+1}},\cdots,x_{i_{m+n}}\end{array}]}}),$

as required. ∎

Define a linear map $\varepsilon_{\geq 1}:{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\longrightarrow\mathbb{Q}$ on basis elements as $\varepsilon_{\geq 1}({\bf
1})=1$ and $\varepsilon_{\geq 1}([\vec{s}])=0$ for $\vec{s}\in\mathbb{Z}_{\geq
1}^{k}\,(k\geq 1)$.

######  Proposition 4.2.

The quintuple $({{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}},{\mbox{\cyrs
X}},{\Delta_{\geq 1}},{\bf 1},\varepsilon_{\geq 1})$ is a bialgebra.

###### Proof.

We need to check $\Delta$ and $\varepsilon_{\geq 1}$ are algebra homomorphisms
on basis elements. For $\vec{s}\in\mathbb{Z}_{\geq 1}^{k}$,
$\vec{t}\in\mathbb{Z}_{\geq 1}^{l}$, $k,l\geq 1$,

$\varepsilon_{\geq 1}({\bf 1}{\mbox{\cyrs X}}{\bf 1})=1=\varepsilon_{\geq
1}({\bf 1})\varepsilon_{\geq 1}({\bf 1}),$ $\varepsilon_{\geq 1}({\bf
1}{\mbox{\cyrs X}}[\vec{s}])=\varepsilon_{\geq 1}([\vec{s}]{\mbox{\cyrs
X}}{\bf 1})=\varepsilon_{\geq 1}([\vec{s}])=0=\varepsilon_{\geq
1}([\vec{s}])\varepsilon_{\geq 1}({\bf 1}),$ $\varepsilon_{\geq
1}([\vec{s}]{\mbox{\cyrs X}}[\vec{t}])=0=\varepsilon_{\geq
1}([\vec{s}])\varepsilon_{\geq 1}([\vec{t}]).$

The coproduct $\Delta_{\geq 1}$ is an algebra homomorphism if one of the
elements is ${\bf 1}$, for $x\in{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$,

$\Delta_{\geq 1}(x{\mbox{\cyrs X}}{\bf 1})=\Delta_{\geq 1}({\bf 1}{\mbox{\cyrs
X}}x)=\Delta_{\geq 1}(x)=\Delta_{\geq 1}({\bf 1}){\mbox{\cyrs X}}\Delta_{\geq
1}(x)=\Delta_{\geq 1}(x){\mbox{\cyrs X}}\Delta_{\geq 1}({\bf 1}).$

For basis elements $[s_{1},\cdots,s_{m}],[t_{1},\cdots,t_{n}]$, let

$\\{i_{1},\cdots,i_{m},j_{1},\cdots,j_{n}\\}$

be any $m+n$ distinct elements in $\mathbb{Z}_{\geq 1}$, then by Lemma 4.2,
Lemma 3.2 and Corollary 3.1,

$\displaystyle{\Delta_{\geq 1}}([s_{1},\cdots,s_{m}]{\mbox{\cyrs
X}}[t_{1},\cdots,t_{n}])=(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}}\cdot{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}})$ $\displaystyle=$
$\displaystyle(\pi\otimes\pi)\big{(}\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}})\cdot\Delta({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}})\big{)}$ $\displaystyle=$
$\displaystyle\big{(}(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{m}\\\
x_{i_{1}},\cdots,x_{i_{m}}\end{array}]}})\big{)}{\mbox{\cyrs
X}}\big{(}(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{n}\\\
x_{j_{1}},\cdots,x_{j_{n}}\end{array}]}})\big{)}$ $\displaystyle=$
$\displaystyle{\Delta_{\geq 1}}([s_{1},\cdots,s_{m}]){\mbox{\cyrs
X}}{\Delta_{\geq 1}}([t_{1},\cdots,t_{n}]),$

we have the conclusion. ∎

######  Theorem 4.1.

The bialgebra $({{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}},{\mbox{\cyrs X}},{\bf
1},{\Delta_{\geq 1}},\varepsilon_{\geq 1})$ is a Hopf algebra.

###### Proof.

By Proposition 2.5, we only need show $({{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}},{\mbox{\cyrs X}},{\bf 1},{\Delta_{\geq 1}},\varepsilon_{\geq 1})$ is a
connected graded bialgebra.

The space ${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$ has the grading by weight,
that is

$H_{0}:={\bf 1},$ $H_{k}:=\\{[s_{1},\cdots,s_{m}]\in\mathbb{Z}_{\geq
1}^{m}\,|\,s_{1}+\cdots+s_{m}=k\\},$ ${{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}=\bigoplus_{k=0}^{\infty}\mathbb{Q}H_{k}.$

By the definition of X, we know

$H_{m}{\mbox{\cyrs X}}H_{n}\subseteq\mathbb{Q}H_{m+n}$

for $m\geq 0,n\geq 0$, which means $({{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}},{\mbox{\cyrs X}})$ is a graded algebra.

Since $\pi:\mathbb{Q}{F^{ch}}\longrightarrow{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}$ keeps grading, that is $\pi(F_{k})\subseteq H_{k}$, hence

${\Delta_{\geq
1}}(H_{k})=(\pi\otimes\pi)\Delta(F_{k})\subseteq\bigoplus_{p+q=k}\pi(\mathbb{Q}F_{p})\otimes\pi(\mathbb{Q}F_{q})\subseteq\bigoplus_{p+q=k}\mathbb{Q}H_{p}\otimes\mathbb{Q}H_{q},$

so $({{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}},{\mbox{\cyrs X}},{\bf
1},{\Delta_{\geq 1}},\varepsilon_{\geq 1})$ is a connected graded bialgebra,
which means $({{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}},{\mbox{\cyrs X}},{\bf
1},{\Delta_{\geq 1}},\varepsilon_{\geq 1})$ is a Hopf algebra. ∎

###  4.2. Operators on the shuffle algebra

Modeled on partial derivatives on $\mathbb{Q}{F^{ch}}$, for any
$i\in\mathbb{Z}_{\geq 1}$, we define a linear map

$\delta_{i}:{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\longrightarrow{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$

as

$\displaystyle\delta_{i}({\bf 1})$ $\displaystyle=0,$ (27)
$\displaystyle\delta_{i}([s_{1},\cdots,s_{k}])$
$\displaystyle=\left\\{\begin{array}[]{ccc}\sum_{j=1}^{i}s_{j}[s_{1},\cdots,s_{j}+1,\cdots,s_{i},\cdots,s_{k}],&i\leq
k\\\ 0,&i>k\end{array}\right..$

######  Remark 4.1.

We take $\delta_{i}=0$ when $i\leq 0$ for convenience.

Let $p_{i,j}=\delta_{i}-\delta_{j}$, similar to Equation (13), we have

(28) $\displaystyle[s_{1},s_{2},\cdots,s_{k}]=\frac{p_{1,0}^{s_{1}-1}\circ
p_{2,1}^{s_{2}-1}\cdots\circ
p_{k,{k-1}}^{s_{k}-1}}{(s_{1}-1)!(s_{2}-1)!\cdots(s_{k}-1)!}[1,1,\cdots,1].$

By direct calculation, we have

######  Lemma 4.3.

For $i,j\geq 0$,

$\delta_{i}\circ\delta_{j}=\delta_{j}\circ\delta_{i}.$

######  Definition 4.1.

For a linear operator $A$ on ${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$, and
$j\in\mathbb{Z}_{\geq 0}$, we define the shifted tensor

$A\text{ⓢ}\delta_{j}:{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\otimes{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\longrightarrow{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\otimes{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$

by

$(A\text{ⓢ}\delta_{j})([s_{1},\cdots,s_{k}]\otimes[\vec{t}])=A([s_{1},\cdots,s_{k}])\otimes\delta_{j-k}([\vec{t}]).$

######  Lemma 4.4.

For $i,j\in\mathbb{Z}_{\geq 0}$,

$(A\text{ⓢ}\delta_{i})\circ(B\text{ⓢ}\delta_{j})=(B\text{ⓢ}\delta_{j})\circ(A\text{ⓢ}\delta_{i}),$

where $A=1$, $B=1$ or $A=\delta_{s},B=\delta_{t}$.

###### Proof.

We only need to prove the conclusion on the basis elements of
${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$.

It is obvious that

$(A\text{ⓢ}\delta_{i})\circ(B\text{ⓢ}\delta_{j})({\bf 1}\otimes{\bf
1})=0=(B\text{ⓢ}\delta_{j})\circ(A\text{ⓢ}\delta_{i})({\bf 1}\otimes{\bf 1}).$

For $(s_{1},\cdots,s_{m})\in\mathbb{Z}_{\geq 1}^{m}$,

$\displaystyle(A\text{ⓢ}\delta_{i})\circ(B\text{ⓢ}\delta_{j})({\bf
1}\otimes[s_{1},\cdots,s_{m}])$ $\displaystyle=$
$\displaystyle(A\text{ⓢ}\delta_{i})(B({\bf
1})\otimes\delta_{j}([s_{1},\cdots,s_{m}]))=(A\circ B)({\bf
1})\otimes(\delta_{i}\circ\delta_{j})([s_{1},\cdots,s_{m}]))$ $\displaystyle=$
$\displaystyle(B\text{ⓢ}\delta_{j})(A({\bf
1})\otimes\delta_{i}([s_{1},\cdots,s_{m}]))=(B\text{ⓢ}\delta_{j})\circ(A\text{ⓢ}\delta_{i})({\bf
1}\otimes[s_{1},\cdots,s_{m}]),$

and

$\displaystyle(A\text{ⓢ}\delta_{i})\circ(B\text{ⓢ}\delta_{j})([s_{1},\cdots,s_{m}]\otimes{\bf
1})$ $\displaystyle=$
$\displaystyle(B\text{ⓢ}\delta_{j})\circ(A\text{ⓢ}\delta_{i})([s_{1},\cdots,s_{m}]\otimes{\bf
1})=0.$

For $(s_{1},\cdots,s_{m})\in\mathbb{Z}_{\geq 1}^{m}$,
$(t_{1},\cdots,t_{n})\in\mathbb{Z}_{\geq 1}^{n}$, by Lemma 4.3,

$\displaystyle(A\text{ⓢ}\delta_{i})\circ(B\text{ⓢ}\delta_{j})([s_{1},\cdots,s_{m}]\otimes[t_{1},\cdots,t_{n}])$
$\displaystyle=$
$\displaystyle(A\text{ⓢ}\delta_{i})(B([s_{1},\cdots,s_{m}])\otimes\delta_{j-m}([t_{1},\cdots,t_{n}]))$
$\displaystyle=$ $\displaystyle(A\circ
B)([s_{1},\cdots,s_{m}])\otimes(\delta_{i-m}\circ\delta_{j-m})([t_{1},\cdots,t_{n}])$
$\displaystyle=$ $\displaystyle(B\circ
A)([s_{1},\cdots,s_{m}])\otimes(\delta_{j-m}\circ\delta_{i-m})([t_{1},\cdots,t_{n}])$
$\displaystyle=$
$\displaystyle(B\text{ⓢ}\delta_{j})(A([s_{1},\cdots,s_{m}])\otimes\delta_{i-m}([t_{1},\cdots,t_{n}]))$
$\displaystyle=$
$\displaystyle(B\text{ⓢ}\delta_{j})\circ(A\text{ⓢ}\delta_{i})([s_{1},\cdots,s_{m}]\otimes[t_{1},\cdots,t_{n}]),$

for $s,t\leq m$, otherwise both sides will be $0$. ∎

######  Lemma 4.5.

If a linear operator $\triangle:{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\longrightarrow{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\otimes{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$ satisfies the following
conditions

  1. (1)

$\triangle({\bf 1})={\bf 1}\otimes{\bf 1}$;

  2. (2)

$\triangle([\underbrace{1,\cdots,1}_{k}]):=\sum_{j=0}^{k}[\underbrace{1,\cdots,1}_{j}]\otimes[\underbrace{1,\cdots,1}_{k-j}]$;

  3. (3)

for any $i\in\mathbb{Z}_{\geq 1}$, $\delta_{i}$ commutes with $\triangle$ in
the following sense

$(1\text{ⓢ}\delta_{i}+\delta_{i}\otimes
1)\circ\triangle=\triangle\circ\delta_{i};$

then it is unique.

###### Proof.

For two linear maps $\triangle_{1}$, $\triangle_{2}$ satisfying the
conditions, obviously, $\triangle_{1}({\bf 1})=\triangle_{2}({\bf 1})$, so we
prove $\triangle_{1}=\triangle_{2}$ by induction on the weight $n$ of basis
elements of ${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$.

For $n=1$,

$\triangle_{1}([1])={\bf 1}\otimes[1]+[1]\otimes{\bf 1}=\triangle_{2}([1]).$

Assume that $\triangle_{1}([\vec{s}])=\triangle_{2}([\vec{s}])$ for elements
with weight $n\geq 1$, let us prove the equality for
$\vec{s}=(s_{1},\cdots,s_{k})$ and $|\vec{s}|=n+1$. If $k=n+1$, then
$s_{i}=1,i=1,\cdots,k$, so $\triangle_{1}([\vec{s}])=\triangle_{2}([\vec{s}])$
by the second condition. If $n+1>k$, assume $j\ (1\leq j\leq k)$ is the least
number such that $s_{j}\geq 2$, then

$\displaystyle\triangle_{1}([s_{1},\cdots,s_{j},\cdots,s_{k}])=\triangle_{1}\frac{\delta_{j}-\delta_{j-1}}{s_{j}-1}([s_{1},\cdots,s_{j}-1,\cdots,s_{k}])$
$\displaystyle=$
$\displaystyle\frac{1\text{ⓢ}(\delta_{j}-\delta_{j-1})+(\delta_{j}-\delta_{j-1})\otimes
1}{s_{j}-1}\triangle_{1}([s_{1},\cdots,s_{j}-1,\cdots,s_{k}])$
$\displaystyle=$
$\displaystyle\frac{1\text{ⓢ}(\delta_{j}-\delta_{j-1})+(\delta_{j}-\delta_{j-1})\otimes
1}{s_{j}-1}\triangle_{2}([s_{1},\cdots,s_{j}-1,\cdots,s_{k}])$
$\displaystyle=$
$\displaystyle\triangle_{2}([s_{1},\cdots,s_{j},\cdots,s_{k}]).$

Hence we have the uniqueness. ∎

###  4.3. Characterization of the coproduct on the shuffle algebra

In this part, we characterize the induced coproduct ${\Delta_{\geq 1}}$ in
terms of the commutativity of the linear map with the linear operators
$\delta_{i}$, $i\in\mathbb{Z}_{\geq 1}$.

######  Lemma 4.6.

  1. (1)

For the constant function $1\in{F^{ch}}$,

$\pi\circ{\partial}_{i}(1)=\delta_{i}\circ\pi(1).$

  2. (2)

For ${\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\in{F^{ch}}$, and any $i\geq 0$,

$\pi\circ{\partial}_{\ell_{i}}({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}})=\delta_{i}\circ\pi({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}).$

###### Proof.

Item(1) For the constant function $1\in{F^{ch}}$,
$\pi\circ{\partial}_{i}(1)=0$, $\delta_{i}\circ\pi(1)=\delta_{i}({\bf 1})=0$.

Item(2) For any ${\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\in{F^{ch}}$,

  * •

if $i\leq k$,

$\displaystyle\pi\circ{\partial}_{\ell_{i}}({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}})$
$\displaystyle=\pi(\sum_{j=1}^{i}s_{j}{\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{j}+1,\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{i}},\cdots,x_{\ell_{k}}\end{array}]}})$
$\displaystyle=\sum_{j=1}^{i}s_{j}[s_{1},\cdots,s_{j}+1,\cdots,s_{k}]$
$\displaystyle=\delta_{i}([s_{1},\cdots,s_{k}])$
$\displaystyle=\delta_{i}\circ\pi({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}).$

  * •

if $i>k$,

$\displaystyle\pi\circ{\partial}_{\ell_{i}}({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}})=0=\delta_{i}\circ\pi({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}).$

∎

######  Lemma 4.7.

Let $B=1,A=1$ or $B={\partial}_{\ell_{j}},A=\delta_{j}$, then for Chen
fractions ${\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}$,
${\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}}$ with distinct
$\ell_{1},\cdots,\ell_{m+k}$, and any $i,j\geq 0$, we have

$\displaystyle(\pi\otimes\pi)(B\otimes{\partial}_{\ell_{i}})(1\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle(A\text{ⓢ}\delta_{i})(\pi\otimes\pi)(1\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}}),$
$\displaystyle(\pi\otimes\pi)(B\otimes{\partial}_{\ell_{i}})({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes 1)$ $\displaystyle=$
$\displaystyle(A\text{ⓢ}\delta_{i})(\pi\otimes\pi)({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes 1),$
$\displaystyle(\pi\otimes\pi)(B\otimes{\partial}_{\ell_{i}})({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle(A\text{ⓢ}\delta_{i})(\pi\otimes\pi)({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}}),$

###### Proof.

For $B=1$, $A=1$, by Lemma 4.6,

$\displaystyle(\pi\otimes\pi)(1\otimes{\partial}_{\ell_{i}})(1\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})={\bf
1}\otimes\pi{\partial}_{\ell_{i}}({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle{\bf
1}\otimes\delta_{i}\pi({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})=(1\text{ⓢ}\delta_{i})(\pi\otimes\pi)(1\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}}),$

and for $B={\partial}_{\ell_{j}},A=\delta_{j}$,

$\displaystyle(\pi\otimes\pi)({\partial}_{\ell_{j}}\otimes{\partial}_{\ell_{i}})(1\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})=0$ $\displaystyle=$
$\displaystyle(\delta_{j}\text{ⓢ}\delta_{i})(\pi\otimes\pi)(1\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}}).$

By definition of ${\partial}_{\ell_{i}}$ and $\delta_{i}$, it is clear that

$\displaystyle(\pi\otimes\pi)(B\otimes{\partial}_{\ell_{i}})({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes 1)=0$ $\displaystyle=$
$\displaystyle(A\text{ⓢ}\delta_{i})(\pi\otimes\pi)({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes 1).$

By lemma 4.6,

$\pi\circ B({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}})=A\circ\pi({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}),$

so

$\displaystyle(\pi\otimes\pi)(B\otimes{\partial}_{\ell_{i}})({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle(\pi\otimes\pi)\big{(}B({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}})\otimes{\partial}_{\ell_{i}}({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})\big{)}$ $\displaystyle=$
$\displaystyle
A([s_{1},\cdots,s_{k}])\otimes\pi{\partial}_{\ell_{i}}({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}}).$

let $p_{1}=\ell_{k+1}$, $\cdots$, $p_{m}=\ell_{m+k}$, that is
$\ell_{i}=p_{i-k}$, then

$\displaystyle(\pi\otimes\pi)(B\otimes{\partial}_{\ell_{i}})({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle
A([s_{1},\cdots,s_{k}])\otimes\pi{\partial}_{p_{i-k}}({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{p_{1}},\cdots,x_{p_{m}}\end{array}]}}).$

By Lemma 4.1,

$\displaystyle(A\text{ⓢ}\delta_{i})(\pi\otimes\pi)({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle
A([s_{1},\cdots,s_{k}])\otimes\delta_{i-k}\pi({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle
A([s_{1},\cdots,s_{k}])\otimes\delta_{i-k}\pi({\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{p_{1}},\cdots,x_{p_{m}}\end{array}]}}).$

By lemma 4.6, we have

$\displaystyle(\pi\otimes\pi)(B\otimes{\partial}_{\ell_{i}})({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle(A\text{ⓢ}\delta_{i})(\pi\otimes\pi)({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}}\otimes{\scriptsize{[\begin{array}[]{c}t_{1},\cdots,t_{m}\\\
x_{\ell_{k+1}},\cdots,x_{\ell_{m+k}}\end{array}]}}).$

∎

######  Lemma 4.8.

The coproduct ${\Delta_{\geq 1}}:{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\longrightarrow{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\otimes{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$ commutes with $\delta_{i}$:

$(1\text{ⓢ}\delta_{i}+\delta_{i}\otimes 1){\Delta_{\geq 1}}={\Delta_{\geq
1}}\delta_{i}.$

###### Proof.

It is obvious that $(1\text{ⓢ}\delta_{i}+\delta_{i}\otimes 1){\Delta_{\geq
1}}({\bf 1})={\Delta_{\geq 1}}\delta_{i}({\bf 1})=0$.

For $(s_{1},\cdots,s_{k})\in\mathbb{Z}_{\geq 1}^{k}$, if $k\geq i$, for
distinct $l_{1},\cdots,l_{k}\in\mathbb{Z}_{\geq 1}$, by Lemma 4.7,

$\displaystyle{\Delta_{\geq 1}}\delta_{i}([s_{1},\cdots,s_{k}])$
$\displaystyle=\sum_{j=1}^{i}s_{j}{\Delta_{\geq
1}}([s_{1},\cdots,s_{j}+1,\cdots,s_{k}])$
$\displaystyle=\sum_{j=1}^{i}s_{j}(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{j}+1,\cdots,s_{k}\\\
x_{l_{1}},\cdots,x_{l_{j}},\cdots,x_{l_{k}}\end{array}]}})$
$\displaystyle=(\pi\otimes\pi)\Delta{\partial}_{l_{i}}({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{i},\cdots,s_{k}\\\
x_{l_{1}},\cdots,x_{l_{i}},\cdots,x_{l_{k}}\end{array}]}})$
$\displaystyle=(\pi\otimes\pi)(1\otimes{\partial}_{l_{i}}+{\partial}_{l_{i}}\otimes
1)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{i},\cdots,s_{k}\\\
x_{l_{1}},\cdots,x_{l_{i}},\cdots,x_{l_{k}}\end{array}]}})$
$\displaystyle=(1\text{ⓢ}\delta_{i}+\delta_{i}\otimes
1)(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{i},\cdots,s_{k}\\\
x_{l_{1}},\cdots,x_{l_{i}},\cdots,x_{l_{k}}\end{array}]}})$
$\displaystyle=(1\text{ⓢ}\delta_{i}+\delta_{i}\otimes 1){\Delta_{\geq
1}}([s_{1},\cdots,s_{k}]).$

If $k<i$, then ${\Delta_{\geq 1}}\delta_{i}([s_{1},\cdots,s_{k}])=0$, and

$\displaystyle(1\text{ⓢ}\delta_{i}+\delta_{i}\otimes 1){\Delta_{\geq
1}}([s_{1},\cdots,s_{k}])$ $\displaystyle=$
$\displaystyle(1\text{ⓢ}\delta_{i}+\delta_{i}\otimes
1)(\pi\otimes\pi)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle(\pi\otimes\pi)(1\otimes{\partial}_{\ell_{i}}+{\partial}_{\ell_{i}}\otimes
1)\Delta({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle(\pi\otimes\pi)\Delta{\partial}_{\ell_{i}}({\scriptsize{[\begin{array}[]{c}s_{1},\cdots,s_{k}\\\
x_{\ell_{1}},\cdots,x_{\ell_{k}}\end{array}]}})$ $\displaystyle=$
$\displaystyle 0.$

So we have the conclusion. ∎

######  Proposition 4.3.

A linear map $\Delta:{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\longrightarrow{{\mathcal{H}}_{\mathbb{Z}_{\geq
1}}}\otimes{{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$ satisfies the following
conditions

  1. (1)

$\Delta({\bf 1})={\bf 1}\otimes{\bf 1}$;

  2. (2)

$\Delta([\underbrace{1,\cdots,1}_{k}]):=\sum_{j=0}^{k}[\underbrace{1,\cdots,1}_{j}]\otimes[\underbrace{1,\cdots,1}_{k-j}]$;

  3. (3)

for any $i\in\mathbb{Z}_{\geq 1}$, $\delta_{i}$ commutes with $\Delta$ in the
following sense

$(1\text{ⓢ}\delta_{i}+\delta_{i}\otimes 1)\circ\Delta=\Delta\circ\delta_{i},$

is a homomorphism with respect to the shuffle product.

###### Proof.

By Lemma 4.8, ${\Delta_{\geq 1}}$ satisfies the conditions. So by Lemma 4.5,

$\Delta={\Delta_{\geq 1}},$

by Proposition 4.2, $\Delta$ is an algebra homomorphism. ∎

######  Remark 4.2.

This gives an indirect proof that a linear map satisfying the conditions in
the Proposition 4.3 is indeed a homomorphism with respect to the shuffle
product. It also characterizes the coproduct in
${{\mathcal{H}}_{\mathbb{Z}_{\geq 1}}}$.

Acknowledgments. This research is partially supported by the National Natural
Science Foundation of China (11890663, 12126354 and 11821001).

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