[name=general]

# The non-commutative signature of a path

Carlo Bellingeri ⋆ ⋆ Technische Universität Berlin, Straße des 17. Juni 135,
10623 Berlin  and  Nicolas Gilliers † † Universität Greifswald Institut für
Mathematik und Informatik Walther-Rathenau-Str. 47 17489 Greifswald, Germany.

###### Abstract.

We introduce a class of operators associated to the signature of a smooth path
$X$ with values in a $C^{\star}$ algebra $\mathcal{A}$. These operators serve
as basis of Taylor expansions of solutions to controlled differential
equations of interest in non-commutative probability. They are defined by
fully contracting iterated integrals of $X$, seen as tensors, with the product
of $\mathcal{A}$. Were it considered that partial contractions should be
included, we explain how these operators yield a trajectory on a group of
representations of a combinatorial Hopf monoid. To clarify the role of partial
contractions, we build an alternative group-valued trajectory whose increments
embody full-contractions operators alone. We obtain therefore a notion of
signature, which seems more appropriate in non-commutative probability.

###### Key words and phrases:

Signature, Rough paths, Hopf monoid, Non-commutative probability, Operads,
Properads, Duoidal categories

###### 2020 Mathematics Subject Classification:

18M60,18M80,60L10,46L89

###### Contents

  1. 1 Introduction
    1. 1.1 The rough paths approach
    2. 1.2 Motivation and previous works
    3. 1.3 Outline
  2. 2 Algebraic structure on levelled forests
    1. 2.1 Levelled trees and forests
    2. 2.2 A first algebraic structure on levelled forests
    3. 2.3 Hopf monoid of levelled forests
  3. 3 Iterated integrals of a path as operators
    1. 3.1 Full and partial contractions operators
    2. 3.2 Chen relation
    3. 3.3 Integration by part properties
  4. 4 Group of non-commutative signatures
    1. 4.1 Algebra of faces-contractions
    2. 4.2 Group acting on faces contractions
    3. 4.3 The non-commutative signature of a path.
  5. 5 Appendix

##  1\. Introduction

This work intends to explore a direction suggested in [DS13] and aiming rough
paths principles to study the following class of differential equations

$\mathrm{d}Y_{t}=a(Y_{t})\cdot{\rm d}X_{t}\cdot b(Y_{t})\,,\quad
Y_{0}\in\mathcal{A}\,.$ (1)

In the above equation, the driving path $X\colon[0,1]\to\mathcal{A}$ takes
values in a $C^{\star}$-algebra $(\mathcal{A},\cdot,\star,\|\cdot\|)$ and
$a,b:\mathbb{\mathcal{A}}\to\mathbb{\mathcal{A}}$ are two polynomial functions
or Fourier transforms of regular measures with exponential moments, see [BS01,
DS13].

This paper is the first of two whose objectives are to introduce a new notion
of geometric rough paths, tailored to the class of equations (1). In this
work, we focus on the algebra underlying Taylor expansions of solutions to
equations (1), discarding other crucial aspects (such as measurability).

###  1.1. The rough paths approach

In the nineties [Lyo98], T.J. Lyons proposed the appropriate mathematical
framework to study differential equations

${\rm d}Y_{t}=\sigma(Y_{t})\mathrm{d}X_{t}\,,\quad
Y_{0}=y_{0}\in\mathbb{R}^{d}.$ (2)

In (2), the solution $Y$ is a continuous path in $\mathbb{R}^{d}$,
$\sigma\colon\mathbb{R}^{d}\to\textrm{{{End}}}(\mathbb{R}^{n},\mathbb{R}^{d})$
is a smooth vector field and the driving path $X$ is Hölder continuous. If $X$
is smooth, standard differential calculus provides a rigourous interpretation
to $\eqref{eqn:controlleddiff}$. For paths with lower regularity, Young’s
theory of integration [You36] gives sense to equation (2) driven by a Hölder
regular path $X$ with exponent greater than $\frac{1}{2}$. Interesting
stochastic driving paths are too irregular for Young integration. For
instance, Brownian trajectories are only
$\frac{1}{2}-\varepsilon,\leavevmode\nobreak\ \varepsilon>0$ Hölder
continuous. Classical Itô integration supplements limitations of Young’s
theory and defines integrals driven by continuous semi-martingales as limits
_in probability_ of Riemann sums.

Rough path theory extends the standard rules of differential and integral
calculus to Hölder paths $X$ and provides a _pathwise_ interpretation to (2).
Let us add more details. Given a smooth field $\sigma$ and
$Y_{0}\in\mathbb{R}^{n}$ the solution map $\Phi:X\mapsto Y$ to equation (2) is
continuous with respect to the Lipschitz norm on the space of smooth driving
paths $X$. A fundamental observation is the following one: by applying
Picard’s iterations to (2), one quickly reckons that the solution map $\Phi$
is a linear functional of the entire _signature_ of $X$, that is the infinite
collection of tensors.

$\mathbb{X}_{s,t}=\left(1,X_{t}-X_{s},\int_{\Delta^{2}_{s,t}}\mathrm{d}X_{t_{1}}\otimes\mathrm{d}X_{t_{2}}\,,\cdots\,,\int_{\Delta^{n}_{s,t}}\mathrm{d}X_{t_{1}}\otimes\cdots\otimes\mathrm{d}X_{t_{n}}\,,\cdots\right),$
(3)

where $\Delta^{m}_{s,t}:=\\{t>t_{1}>\cdots>t_{m}>s\\}$ is the $m$-dimensional
simplex. Now, signatures of smooth paths support a one parameter family of
topologies with respect to $\Phi$ is continuous. Complete spaces for these
topologies contain Hölder paths together with the additional data of an
_abstract signature_. These abstract signatures are called _rough paths_ and
can alternatively be characterized by a set of algebraic and analytical
properties. Indeed, a rough path is in particular the data (along with Hölder
estimates) for each pair of times $s,t$ of an element $\mathbb{X}_{s,t}$ of a
_group_ $(G,\star)$ included in the completed tensor space of $\mathbb{R}^{n}$
with the property that for each triple of times $s,u,t\in[0,1]^{3}$

$\mathbb{X}_{s,t}=\mathbb{X}_{su}\star\mathbb{X}_{ut}\,.$ (4)

The relations (4) are usually called _Chen’s relation_ after Kuo-Tsai Chen
[Che54] and its secular work on homology of loop spaces. We refer the reader
to the monograph [FH20] for a detailed exposition on rough path theory.

###  1.2. Motivation and previous works

We choose to have a intrinsic –coordinate-free– approach to (1) and to work
consistently with the specific class of fields we consider, that is with the
algebra product. Rough paths theory on infinite-dimensional spaces is more
intricate because of several notions of tensor products between two Banach
algebras, see [LLQ02]. Considering the class of equations (1) the _projective
tensor product_ is the only reasonable one, since the algebra product is
always continuous with respect to this topology. This is not true for the
spatial (or injective) topology. This limitation strikes with the results
obtained in [Vic04, CDM01]. In these works, the author defines a rough path
(in fact, a Lévy area) over the free Brownian motion in the _spacial tensor
product_ by using free Itô calculus. Whereas it is possible [LV07] to show
existence of a free Lévy area (up to a infinitesimal loss in regularity) in
the projective tensor product, an explicit procedure is missing.

To circumvent this issue, A. Deya and R. Schott introduced in [DS13] a weaker
notion of Lévy area tailored to the class of equations (1) when the Hölder
scale lies in $(\frac{1}{3},\frac{1}{2}]$: the product Lévy area. This object
embodies the data on the small scale behaviour of the driving path $X$ only in
the directions required to give sense to (1). The starting point to define it
is a fine analysis of (2) with $X$ smooth and the expansion of the solution
$Y$ obtained by applying Picard iterations. Pick $A,B\in\mathcal{A}$ and
consider the following example

$\mathrm{d}Y_{t}=(A\cdot Y_{t})\cdot\mathrm{d}X_{t}\cdot(Y_{t}\cdot B),\quad
Y_{0}=1_{\mathcal{A}}\,.$ (5)

Writing the first two steps of the Picard Iteration, we obtain

$\displaystyle
Y_{t}=1_{\mathcal{A}}+\int_{\Delta^{1}_{s,t}}A\cdot\textrm{d}X_{t_{1}}\cdot B$
$\displaystyle+\int_{\Delta^{2}_{s,t}}A^{2}\cdot\textrm{d}X_{t_{1}}\cdot
B\cdot\textrm{d}X_{t_{2}}\cdot
B+\int_{\Delta^{2}_{s,t}}A\cdot\textrm{d}X_{t_{2}}\cdot
A\cdot\textrm{d}X_{t_{1}}\cdot B^{2}+R_{s,t}\,,$ (6)

where $R_{s,t}$ is a remainder term satisfying $|R_{s,t}|\lesssim|t-s|^{3}$.
The above equation hints at a control, at any order, of the small variations
of $Y$ by the following expressions

$\mathbb{X}_{st}^{\sigma}=\int_{\Delta^{n}(s,t)}A_{0}\cdot\mathrm{d}X_{t_{\sigma^{-1}(1)}}\cdot\cdots\cdot\mathrm{d}X_{t_{\sigma^{-1}(n)}}\cdot
A_{n}\,,\quad A_{0},\cdots,A_{n}\in\mathcal{A}.$ (7)

where $\sigma$ is a permutation of $\\{1,2,\ldots,n\\}$. The expressions in
(7) are values of a multilinear operator $\mathbb{X}^{\sigma}_{s,t}$, that we
call full contraction operator, depending on a choice of a permutation
$\sigma$. The solution of the equation (1) expands over the contracted
iterated integrals (7) in the way alluded to above under the constraints that
the Fourier transforms of $a$ and $b$ are bounded measures on the real line. A
product Lévy area is an abstraction of the order two full contraction
operators, the ones indexed by permutations of $\\{1,2\\}$.

We elaborate on the observation of A. Deya and R. Schott and extract important
algebraic and analytical properties of the multilinear operators (7) with the
objective of developing a rough theory for the class of equations (2) with
driving noise $X$ of arbitrary low Hölder regularity. To put it shortly, the
main outcome of this work is “yes, it is possible" and we explain why by
associating to the operators (7) a smooth trajectory over a group of
triangular morphisms on a algebra of operators we introduce.

The main difficulties lie in writing a Chen relation for the operators (7)
understood as a certain “algebraic rule" for computing
$\eqref{eqn:iteratedintegrals}$ over an interval knowing the values of
$\eqref{eqn:iteratedintegrals}$ over a subdivision of this interval. Consider
for instance the full contraction operator

$\mathbb{X}^{3}_{s,t}(A_{0},A_{1},A_{2},A_{3}):=\int_{\Delta^{3}_{s,t}}A_{0}\cdot{\rm
d}X_{t_{1}}\cdot A_{1}\cdot{\rm d}X_{t_{3}}\cdot A_{2}\cdot{\rm
d}X_{t_{2}}\cdot A_{3}$ (8)

Then the Chasles identity implies the following deconcatenation formula:

$\begin{split}\mathbb{X}^{3}_{s,t}(A_{0},A_{1},A_{2},A_{3})&=\mathbb{X}^{3}_{su}(A_{0},A_{1},A_{2},A_{3})+\mathbb{X}^{3}_{ut}(A_{0},A_{1},A_{2},A_{3})\\\
&+\int_{t_{3}\in\Delta^{1}_{ut}}\int_{(t_{1},t_{2})\in\Delta^{2}_{su}}A_{0}\cdot{\rm
d}X_{t_{1}}\cdot A_{1}\cdot{\rm d}X_{t_{3}}\cdot A_{2}\cdot{\rm
d}X_{t_{2}}\cdot A_{3}\\\
&+\int_{(t_{2},t_{3})\in\Delta^{2}_{ut}}\int_{t_{1}\in\Delta^{1}_{su}}A_{0}\cdot{\rm
d}X_{t_{1}}\cdot A_{1}\cdot{\rm d}X_{t_{3}}\cdot A_{2}\cdot{\rm
d}X_{t_{2}}\cdot A_{3}\,.\end{split}$

The term on the second line above can not be expressed by composing order two
full contraction operators. Instead, we can obtain it by _composing_ the
operator,

$(A_{0},\cdots,A_{4})\mapsto\int_{\Delta^{2}_{su}}A_{0}\cdot{\rm
d}X_{t_{1}}\cdot A_{1}\otimes A_{2}\cdot{\rm d}X_{t_{2}}\cdot
A_{3}\in\mathcal{A}\otimes\mathcal{A}$ (9)

with the following full contraction one

$(A_{0},A_{1})\mapsto\int_{\Delta^{1}_{ut}}A_{0}\cdot{\rm d}X_{t_{1}}\cdot
A_{1}.$

Thus a naive approach leads in fact to relations involving not only full
contractions but also partial contractions.

The main results of the paper are contained in the last Section, Definition
4.17 and Theorem 4.18. In this definition, we introduce the notion of non-
commutative signature of a smooth path and in our main Theorem we prove that,
as for the classical theory, it yields a trajectory in an algebraic group.

###  1.3. Outline

Besides the introduction, this article is divided in two additional sections.
In Section 2, we introduce a Hopf monoid of levelled forests, reminiscent of
the Malvenuto-Reutenauer Hopf algebra of permutations.

In Section 3.1, we define the partial and full contractions operators we
alluded to, see Definitions 3.1 and 3.6.

In Section 3.2 we prove a Chen relation for these operators, see Proposition
3.8. Next, we explain how this yields a path on a group of triangular algebra
morphisms on an algebra spanned by couples of a tree and a word.

In Section 3.3 we associate to the full and partial contractions operators a
path of representations on the Hopf monoid of levelled forests we introduced
in Section 2, see Theorem 3.21.

In Section 4.1, we adopt a slightly different point of view and let the
iterated integrals of a path acting on a set of operators we call _faces-
contractions_ , see Definition 4.1. This yields a certain triangular algebra
morphism, see Definition 4.17 that we relate to the one introduced in Section
3.2. In Proposition 4.1, we relate partial to full contractions operators.

In a forthcoming article, we continue to develop the theory. In particular, we
introduce geometric non-commutative rough paths, geometric non-commutative
controlled rough path, the operations of integration and composition and prove
existence of solutions to (2).

Acknowledgements: The authors are especially grateful to Kurusch Ebrahimi-Fard
for many enlightening discussions. CB is supported by the DFG Research Unit
FOR2402 and NG is funded by DAAD kurzstipendium.

Preliminaries.
In the following we denote by $\mathcal{A}$ a generic complex $C^{\star}$
algebra with product $\mu$, unity $\mathbf{1}$, norm $\|\cdot\|$ and
involution $\star_{\mathcal{A}}$. In order to deal with a topology on the
algebraic tensor product $\otimes$ which behaves correctly with $\mu$, we will
use the projective tensor product (see e.g. [Rya02]). Given two Banach spaces
$(E,\|\cdot\|_{E})$ and $(F,\|\cdot\|_{F})$, the projective norm of an element
$x\in E\otimes F$ is defined by

$\|x\|_{\vee}=\inf\left\\{\sum_{i}\|a_{i}\|_{E}\|b_{i}\|_{F}\colon
x=\sum_{i}a_{i}\otimes b_{i}\right\\}.$

We denote by $E\check{\otimes}F$ the completion of $E\otimes F$ for the
projective norm. One can check the following properties

$\|a\otimes b\|_{\vee}=\|a\|_{E}\|b\|_{F},\leavevmode\nobreak\
\|a_{\sigma(1)}\otimes\cdots\otimes
a_{\sigma(n)}\|_{\vee}=\|a_{1}\otimes\cdots\otimes a_{n}\|_{\vee}\,,$ (10)

for any permutation $\sigma$ on the set $\\{1,2,\ldots,n\\}$ and
$a_{1},\cdots,a_{n}\in E$. The definition of projective norm yields
immediately that the multiplication $\mu$ extends to a continuous map
$\mathcal{A}\check{\otimes}\mathcal{A}\to\mathcal{A}$ and, more generally, for
any given pair of $C^{\star}$ algebras $\mathcal{A}$, $\mathcal{B}$,
$\mathcal{A}\check{\otimes}\mathcal{B}$ is again a $C^{\star}$ algebra.
Looking this operation from a very general point of view, the projective
tensor product makes the category of complex $C^{\star}$ algebras a monoidal
category (see the Appendix 5). In order to lighten the notation, we will adopt
the symbol $\otimes$ to denote both the the projective tensor product between
$C^{\star}$ algebras and the algebraic tensor product for pure tensors.
Similarly, we will replace the product $\mu$ with a dot $\cdot$.

For $n\geq 1$ an integer, we denote by $\mathfrak{S}_{n}$ the set of
permutations of $[n]=:\\{1,\cdots,n\\}$. We use one-line notation for
permutations, writing $\sigma=(\sigma_{1},\sigma_{2},\ldots\sigma_{n})$, where
$\sigma_{i}:=\sigma(i)$. The neutral element of $\mathfrak{S}_{n}$ is also
denoted by $id_{n}$. Sometimes we may omit the commas and just write
$\sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{n}$. By abuse of notation, the only
permutation of $[0]:=\emptyset$ is denoted by $\mathsf{\emptyset}$, from which
we defines $\mathfrak{S}_{0}:=\\{\emptyset\\}$. Given two integers $a,b$, we
denote by ${\rm Sh}(a,b)$ the set of all shuffles of the two intervals
$\llbracket 1,a\rrbracket$ and $\llbracket a+1,a+b\rrbracket$, that is
$\sigma\in{\rm Sh}(a,b)$ if and only if $\sigma$ is non-decreasing on
$\llbracket 1,a\rrbracket$ and on $\llbracket a+1,a+b\rrbracket$.

##  2\. Algebraic structure on levelled forests

The objective of the present section is to introduce the main combinatorial
tool that will be used through this work: the levelled trees and forests. We
will review their main properties and we will provide the construction of new
algebraic structures over it.

###  2.1. Levelled trees and forests

In the literature, one broadly finds several equivalent representations of a
permutation, such as a bijection of a finite sets or finite words without
repetitions on positive integers. We will mainly use the last one and a third
-tree like - graphical representation, presented in different variants in the
literature such as [SF99, pp. 23-24], [BW97, Definition 9.9] or [AS06, pp.
478]. We will follow the versions used by Loday and Ronco in [LR98, Section
2.4] and Forcey, Lauve and Sottile in [FLS10, Section 2.2.1].

First, recall that a _planar rooted tree_ is a planar graph with no cycles and
one distinguished vertex which we call the root. We oriented every tree from
bottom to top: the target of an edge is the vertex further to the root. In
this orientation, each vertex of a tree has at most one incoming edge (the
root is the only vertex with no incoming edge) and at most two outcoming
edges.

A _leaf_ of a tree is a vertex with no outcoming edges. The _degree_ of a tree
is the number of its leaves, we denote it by $\tau$ is $\tau$ is a planar
tree. An _internal vertex_ of a tree is a vertex that is not a leaf. The set
of internal vertices of a tree $\tau$ is denoted by $\mathbb{V}(\tau)$ and we
set $\|\tau\|:=|\mathbb{V}(t)|$. The set $\mathbb{V}(\tau)$ of internal
vertices of a planar tree $\tau$ is equipped with a partial order
$\leq_{\tau}$: if $u,v$ are two vertices of $\tau$, we write $u\leq_{\tau}v$
if there is an oriented path of edges of $t$, moving away from the root, from
$u$ to $v$. The poset $(\mathbb{V}(\tau),\leq_{\tau})$ has one minimum (the
root of $\tau$) and several maxima (the leaves of $\tau$).

A _planar binary tree_ is a planar rooted tree for which every internal node
has two children. A _levelled binary tree_ (or simply _levelled tree_) is a
binary tree $\tau$ together with a linear extension of the poset
$(\mathbb{V}(\tau),\leq_{\tau})$. Levelled trees are also called _ordered
binary tree_ (see [AS06]). By definition, a levelled tree with degree one is
the root tree (see Fig 1). Also, notice that the root tree has no internal
vertices and correspond to levelled tree $(\,\leavevmode\hbox to4.98pt{\vbox
to6.95pt{\pgfpicture\makeatletter\hbox{\hskip 2.49023pt\lower-2.49023pt\hbox
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} \pgfsys@invoke{\lxSVG@closescope
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,\emptyset)$ where
$\emptyset$ denotes the unique function from the empty set to the empty set.

We denote by ${\textrm{{{LT}}}}_{n}$ the set of levelled trees with $n$
_internal vertices_ , ${\textrm{{{LT}}}}(n)$ the set of levelled trees with
$n$ leaves, and ${\textrm{{{LT}}}}:=\cup_{n\geq 0}\,{\textrm{{{LT}}}}(n)$. The
complex span of LT is a graded vector space, its homogeneous component of
degree $n\geq 1$ is the linear span of trees with $n$ leaves
${\textrm{{{LT}}}}(n)$.

We justify now the terminology for labelled trees. Following [PS08] (pp. 7), a
_level function_ on a tree $t$ is a surjective increasing map

$\lambda:\left(\mathbb{V}(\tau),\leq_{\tau}\right)\to A,$

where $A$ is a totally ordered set. If $\tau$ is a planar binary tree and
$A=[\|\tau\|]$, then the pair $(\tau,\lambda)$ corresponds precisely to a
levelled tree. If $v$ is an internal vertex of $t$, we say that $v$ has level
$\lambda(v)$.

The following result seems to be folklore. For a proof, see [LR98]
(Proposition 2.3).

######  2.1 Proposition ([LR98]).

For every integer $n\geq 0$, the set of levelled trees with $n+1$ leaves is in
bijection with the set of permutations $\mathfrak{S}_{n}$.

The bijection associates to any levelled tree $(\tau,\lambda)$ with $n+1$
leaves a permutation $\sigma=\sigma(\tau,\lambda)\in\mathfrak{S}_{n}$ as
follows. Label the leaves of $\tau$ with $0,1,2,\ldots,n$ (in this order),
from left to right. For each $1\leq i\leq n$, let $v_{i}$ be the vertex which
lies in between the leaves $i-1$ and $i$. Then
$\sigma:=\sigma_{1}\sigma_{2}\cdots\sigma_{n}$, with
$\sigma_{i}:=\lambda(v_{i})$, see Figure below.

Figure 1.  Examples of levelled trees $(\tau,\lambda)$ in LT and their
associated permutations in $\mathfrak{S}_{1}$, $\mathfrak{S}_{2}$ and
$\mathfrak{S}_{3}$.

A _generation of a tree_ is a set of nodes at the same distance from the root.
We number a generation with the distance of the vertices of this generation to
the root.

When illustrating a levelled tree $(\tau,\lambda)$, it will be convenient to
emphasize the levelling (the map $\lambda$) of the tree. To do so, we _locate_
every vertex $v$ of $t$ in its corresponding level $\lambda(v)$ by adding
straight edges to $\tau$, if necessary (see figure 2). The resulting unlabeled
tree is such that every vertex has one or two children, and for every
generation (vertices at the same distance to the root) of the tree there is an
_unique_ vertex with two children. We call such a tree a _sparse quasi-binary
tree_. All operations introduced in this section have a convenient pictorial
description using sparse quasi-binary tree.

Figure 2.  Example of a levelled tree and its associated sparse quasi-binary
tree.

In summary, we have three equivalent ways to identify the same object:

$\begin{array}[]{c}\text{Permutations}\\\
\sigma:[n]\to[n]\end{array}\quad\longleftrightarrow\quad\begin{array}[]{c}\text{Levelled
trees}\\\ \text{with }n+1\text{
leaves}\end{array}\quad\longleftrightarrow\quad\begin{array}[]{c}\text{Sparse
quasi-binary trees}\\\ \text{with }n+1\text{ generations}\end{array}$

We use the symbol $t$ (with super and subscript) for a levelled tree presented
as either as a permutation $\sigma$ or a pair $(\tau,\lambda)$. The
representation of $t$ as a sparse quasi-binary tree will only be used in
drawings.

Levelled trees are not sufficient for our purposes. We will extend now the
notion of levelled trees to _forests_. A _planar forest_ is a word (a non-
commutative monomial) on planar trees.

In the following, we denote by ${\sf nt}(\varphi)$ the number of trees in the
forest $\varphi$, $|\varphi|$ the total number of leaves in the forest and we
set $\|\varphi\|$ equal to the number of internal vertices of the forests. If
all trees of $\varphi$ are binary trees, then $\|\varphi\|=|\varphi|-{\sf
nt}(\varphi)$. The poset $(\mathbb{V}(\varphi),\leq_{\varphi})$ of ordered
vertices of $f$ is the union of the posets of vertices of the trees in $f$.

In the following we will just consider planar forests of binary trees. The
notion of level function for a tree is naturally extended to any forest. This
allows to consider the following analog of levelled binary trees to binary
forests.

######  2.2 Definition (Levelled planar binary forests LF).

A _levelled planar binary forest_ $f$ (or simply a _levelled forest_) is a
pair $(\varphi,\lambda)$ formed by a binary forest $\varphi$ and an increasing
bijection

$\lambda:(\mathbb{V}(\varphi),\leq_{\varphi})\to[||\varphi||].$

We denote the set of planar binary forests by LF.

Figure 3.  A sparse quasi-binary forest with four trees.

The _degree_ of a levelled planar forest
$(\varphi,\lambda)\in{\textrm{{{LF}}}}$ is the number of leaves of $\varphi$
and is denote by $|\varphi|$. If $n\geq 1$ and $m\geq 1$, we denote by
${\textrm{{{LF}}}}(m,n)$ the set of levelled forests with $n$ leaves and $m$
trees. This allows to define a bigraduation on the set LF.

Recall that a generation of a tree is a set of internal vertices at the same
distance from the root. This notion extends to any forest. We denote by
${\textrm{{{LF}}}}_{n}$ the set of levelled trees with $n$ generations. A
levelled forest $(\varphi,\lambda)$ can be pictured as a forest of quasi-
binary trees, each with same number of generations, in the same way as a
explained before for levelled trees, where the internal vertices are ordered
vertically by adding straight edges according to $\lambda$, see Fig. 3. In
this representation, there is a unique internal vertex with two children among
all vertices of the forest on the same generation. We call such a forest a
_sparse quasi-binary forest_.

We now introduce now several (classical) operations on levelled trees and
levelled forests.

If $n\in\mathbb{N}$, _weak composition_ of $n$ is a sequence
$c=(c_{1},c_{2},\ldots,c_{k})$ of non-negative integers with
$n=c_{1}+\cdots+c_{k}$. If we restrict each $c_{i}$ to be non-zero, then $c$
is called a _composition_ of $n$. To a weak composition $c$, we associate the
multiset $I(c):=\\{c_{1},c_{1}+c_{2},\ldots,c_{1}+c_{2}+\cdots+c_{k-1}\\}$.
This gives a bijection between weak compositions of $n$ and multisubsets of
$\\{0\\}\cup[n]$. If $c$ is a composition, $I(c)$ is a set and we obtain a
bijection between compositions of $n$ and subsets of $[n-1]$. Weak
compositions of $n$ are partially ordered by _refinement_. The cover relations
are of the form

$(c_{1},\ldots,c_{i}+c_{i+1},\ldots,c_{k})\lessdot(c_{1},\ldots,c_{i},c_{i+1},\ldots,c_{k}).$

We write $c\vDash n$ if $c$ is a composition of $n$, and $c\vDash_{0}n$ if $c$
is a weak composition of $n$. The length $k$ of a (weak) composition
$(c_{1},\cdots,c_{k})$ is the number of parts of the composition.

The bijection between levelled trees and permutations used in the proposition
(2.1) extends to words $w=w_{1}w_{2}\cdots w_{r}\in\mathbb{N}^{*}$ without
repetition of letters; the associated labelled binary tree $t$ has now a level
function
$\lambda:\left(\mathbb{V}(t),\leq_{t}\right)\to\\{w_{1},w_{2},\ldots,w_{r}\\}$.

Every levelled forest $(f,\lambda)\in{\textrm{{{LF}}}}(n+k,k)$ gives rise to a
pair $(\sigma,c)$, where $\sigma\in\mathfrak{S}_{n}$ is obtained by
concatenating the non-empty words corresponding to each tree in $f$ (from left
to right) under the above described bijection, and $c\vDash_{0}n$ is the weak
composition of length $k$ obtained by tracking the number of vertices of each
tree in the forest $f$. Reciprocally any pair $(\sigma,c)$ gives raise to a
levelled planar binary forest, using the bijection between non-repeating words
and labelled planar trees. We call _split permutation_ a pair $(\sigma,c)$
with $\sigma\in\mathfrak{S}_{n}$ and $c\vDash_{0}n$.

$\begin{array}[]{c}\text{Split permutations}\\\ \sigma:[n]\to[n]\\\
c\vDash_{0}n\end{array}\quad\longleftrightarrow\quad\begin{array}[]{c}\text{Levelled
forests}\\\ \text{with }n+1\text{
leaves}\end{array}\quad\longleftrightarrow\quad\begin{array}[]{c}\text{Sparse
quasi-binary forests}\\\ \text{with }n+1\text{ generations}\end{array}$

As for levelled trees, we use the symbol $f$ to denote a levelled forest
presented either as a pair ($\varphi,\lambda)$ or as a split permutation
$(\sigma,c)$. The presentation of $f$ as a sparse quasi-binary forest will be
used in the drawings only.

Figure 4. A split permutation $((4132,(0,1,2,1))$ drawn as a levelled forest
and as a sparse quasi-binary forest.

The next definition introduces the notion of _vertical splitting_ for levelled
forests. Informally, a vertical splitting of consists in breaking vertical a
tree in the forest into two tree following a path from a leaf to the root of
that tree.

######  2.3 Definition (Vertical splitting of levelled forests).

Pick a levelled forest $f$ presented as a split permutation
$(\sigma,c)\in\mathfrak{S}_{n}$, $c\vDash_{0}n$ and $d\vDash_{0}n$ such that
$c\lessdot d$, we define the vertical splitting of $\curlyvee_{d}((\sigma,c))$
by $(\sigma,d)$.

We introduce inverse operations to splitting. The first one takes every trees
in the respresentation of a levelled forest as a sparse quasi-binary forest
and glue all together the trees of that forest along their external paths of
edges. In terms of split permutaton, this operation correspond to the
projection,

$\tau^{\flat}:=\sigma,\leavevmode\nobreak\ f=(\sigma,c)\in{\textrm{{{LF}}}}{}$
(11)

We will also need local gluing operation gluing two consecutive trees in the
represention of a levelled tree as a sparse quasi-binary tree, one again those
operations are most effectively written in terms of split permutation. We set
for any $f=(\sigma,c)\in{\textrm{{{LF}}}}$ and $1\leq i\leq{\sf nt}(f)-1$

${|\\!|_{\raisebox{-2.84526pt}{\scalebox{0.8}{$i$}}}}f={|\\!|_{\raisebox{-2.84526pt}{\scalebox{0.8}{$i$}}}}(\sigma,c):=(\sigma,(c_{1},\cdots,c_{i}+c_{i+1},\cdots,c_{{\sf
nt}(f)}))\,.$ (12)

Figure 5. The forests $\curlyvee_{(0,1,1,1,1)}((4132),(0,1,2,1)))$ and
${|\\!|_{\raisebox{-2.84526pt}{\scalebox{0.8}{$3$}}}}((4132),(0,1,2,1))$

We consider _horizontal analog_ to the above operations of vertical splitting
and gluing. For any word $w=w_{1}w_{2}\cdots w_{r}$ where each letter takes
value in $\mathbb{N}^{*}$ and $A\subseteq\text{alph}(w)$, the alphabet
generated by the different letters contained in $w$, we define $w\cap A$ as
the the word obtained from $w$ by erasing the letters which are not in $A$. We
write $w^{\prime}\subseteq w$ if there exists $A\subseteq\text{alph}(w)$ such
that $w^{\prime}=w\cap A$. In this case, we say that $w^{\prime}$ is a
_subword_ of $w$. We use now the definition of subword to define the notion of
subtree and subforest. Let $t\in{\textrm{{{LT}}}}(n)$ a levelled binary tree,
represented as a permutation $\sigma$. A _levelled subtree_ (or just
_subtree_) of $t$ is a levelled binary tree $t^{\prime}$ with associated
permutation $\sigma^{\prime}$ of the form $\sigma^{\prime}=\sigma\cap[p]$, for
$0\leq p\leq n$.

In this case, we write $\tau^{\prime}\subseteq\tau$. In terms of sparse quasi-
binary trees, $\tau^{\prime}$ is a subtree of $\tau$ if there exists $0\leq
p\leq\|\tau\|$ such that $\tau^{\prime}$, seen as a quasi-binary tree,
coincides with the quasi-binary tree associated with $\tau$ by erasing all
vertices on generations strictly bigger than $p$.

Figure 6.  On the first line: levelled forest $f=(213,(2,1))$, drawn as a
quasi-binary tree. On the second line: all levelled subforests included in
$f$.

This notion extends to levelled forests. Pick a levelled forest
$f=(\sigma,(c_{1},\cdots,c_{k}))\in{\textrm{{{LF}}}}{}$ and denote by
$(\sigma_{1},\cdots,\sigma_{k})$ the restrictions of $\sigma$ to the parts of
$c$:

$\sigma_{i}=\sigma_{]c_{1}+\cdots+c_{i-1},c_{1}+\cdots+c_{i}]}\,,\leavevmode\nobreak\
1\leq i\leq k$

with the convention that $\sigma_{\emptyset}=\emptyset$. A _levelled forest_
$f^{\prime}=(\sigma^{\prime},(c^{\prime}_{1},\cdots,c^{\prime}_{k}))$ with
same number of trees as $f$ (the composition $c^{\prime}$ has the same number
of parts as $c$) is a subforest of $f$ is there exists $p\leq\|f\|$ such that

$\sigma^{\prime}_{i}=\sigma_{i}\cap[p]\,.$

We write in this case $f^{\prime}\subset f$. In terms of levelled forest,
writing $(f,\lambda)$ and $(f^{\prime},\lambda^{\prime})$, then we have
$(f^{\prime},\lambda^{\prime})\subseteq(f,\lambda)$ if $f_{\alpha},f_{\beta}$
have the same number of trees and the labelling $\lambda_{\alpha}$ restricts
to $\lambda_{\beta}$ on the internal vertices of $f_{\beta}$. Equivalently,
considering $\alpha$ and $\beta$ as quasi-binary forests,
$\beta\subseteq\alpha$ if and only if $\beta$ comprise all nodes of $\alpha$
up to a certain generation of the quasi-binary forest $\alpha$.

Consider a word $w=w_{1}w_{2}\cdots w_{r}$ where each letter takes value in
$\mathbb{N}^{*}$ and $A\subseteq\text{alph}(w)$. Let
$w(A)=\\{i^{w}_{A}(1)<\cdots<i^{w}_{A}(q)\\}$ be the increasing sequence of
indices of the letters of $w$ in $A$ and define the composition ${\sf
wc}(w,A)\vDash_{0}$

${\sf wc}(w,A):=(|]1,i^{w}_{A}(1)]|,\ldots,\leavevmode\nobreak\
|]i^{w}_{A}(i),i^{w}_{A}(i+1)]|,\leavevmode\nobreak\ |]i^{w}_{A}(q),r]|)\,.$

For example, if $r=9$ and $w=w_{1}\cdots w_{9}$, let $A=\\{2,3,4,7,9\\}$. Then
$[9]\setminus A=\\{1,5,6,8\\}$, $K(A)=\\{\\{1\\},\\{5,6\\},\\{8\\}\\}$.
Therefore, $\textsf{wc}(w,A)=(1,0,0,0,2,0,1,0)$. Thanks to this notion, we
introduce an horizontal splitting operation.

######  2.4 Definition (Horizontal splitting of trees).

Pick $\tau$ a levelled tree seen as a permutation $\sigma$ and an integer
$0\leq p\in\|\sigma\|$. The _horizontal splitting_ of $\sigma$ at $p$ is the
couple

$\succ_{p}\\!(t):=\left(\sigma\cap[p],(\sigma\cap\\{p+1,\ldots,n\\},\textsf{wc}(\sigma,[p]))\right)\,,$

formed by the subword of $\sigma$ containing the letters in $[p]$, and the
pair $(\sigma\cap\\{p+1,\ldots,n\\},\textsf{wc}(\sigma,[p]))$ representing the
sequence of subwords of $\sigma$ obtained from $\sigma$ after erasing the
letters in $[p]$. We call the first component of $\succ_{p}$ the _lower
component_ of the cut and the second component the _upper component_.

For instance, $\succ_{2}(25143)=(21;\,(543),(0,1,0,2))$. Horizontal splitting
acts on the sparse quasi-binary tree representation by detaching the first
lower $p$ generations (we include all edges connected to the vertices of the
$p^{th}$ generation. The resulting levelled tree form the lower component of
the cut. The generations above $p$ yields a the upper component of the cut.

Figure 7. Example of an horizontal splitting of a sparse quasi-binary tree

This operation extends to levelled forests in a straightforward manner.

######  2.5 Definition (Horizontal splitting of levelled forests).

Let $f=(\sigma,c)$ be a forest in ${\textrm{{{LF}}}}(k,n)$, with
$\sigma\in\mathfrak{S}_{n-1}$ and $(c_{1},\cdots,c_{k})\vDash_{0}n-1$ a weak
composition of length $k$. Once again, denote by $\sigma_{i}$, $1\leq
i\leq\|k\|$ the restriction of $\sigma$ to the parts of $c$. The _horizontal
splitting of the levelled forest_ $f$ at the level $p$ is the forest

$\displaystyle{\succ}_{p}(f)=({\succ}_{p}(f)_{-},{\succ}_{p}(f)_{+}):=({\succ}_{p}(\sigma_{1})_{-}\cdots{\succ}_{p}(\sigma_{k})_{-},{\succ}_{p}(\sigma_{1})_{+}\cdots{\succ}_{p}(\sigma_{k})_{+})$

For convenience, we use the shorter notations

$f_{-}^{p}:=\succ_{p}(f)^{-},\quad f_{+}^{p}:=\succ_{p}(f)^{+}$

Consider for example the forest
$\curlyvee_{\\!(1,1,3,0,1)}(542631)=(5,4,263,1)$. Since
$\succ_{3}\\!(5)=(\emptyset;(5))$, $\succ_{3}\\!(4)=(\emptyset;(4))$,
$\succ_{3}\\!(263)=(23;(\emptyset,6,\emptyset))$,
$\succ_{3}\\!(\emptyset)=(\emptyset;(\emptyset))$ and
$\succ_{3}\\!(1)=(1;(\emptyset))$, we obtain

${\succ}_{3}(5,4,263,1)=((\emptyset,\emptyset,23,\emptyset,1);(5,4,\emptyset,6,\emptyset,\emptyset,\emptyset))\,.$

An horizontal cut of a levelled forest $f$ outputs a pair of forests, the
lower component being a subforest of $f$. And reciprocally, one can check that
a subforest $f^{\prime}\subset f$ yields an unique horizontal cut of $f$, the
cutting level $k$ being the number of internal nodes of $f^{\prime}$
(equivalently the number of generations) We choose then to index horizontal
cuts of a levelled forest by its set of subforests augmented with the root
tree.

######  2.6 Definition.

For any given levelled forest $f$ and subforest $f^{\prime}\subset f$ we
denote by $f\backslash f^{\prime}$ the upper component of the horizontal cut
induced by $f^{\prime}$. We say that the levelled forest $f$ is _compatible_
with the levelled forest $f^{\prime}$ if the number of trees of $f$ matches
the number of leaves of $f^{\prime}$ in the representation of $f$ and
$f^{\prime}$ as sparse quasi-binary trees or as labelled trees.

By definition of horizontal splitting one has immediately the identity
$\succ_{||f^{\prime}||}f=(f^{\prime},f\backslash f^{\prime})\,.$ Moreover If
both levelled forests are given as split permutations,
$f=(\sigma,(c_{1},\cdots,c_{k}))$, $\sigma\in\mathfrak{S}(n)$ and
$f^{\prime}=(\sigma^{\prime},(c^{\prime}_{1},\cdots,c^{\prime}_{q}))$,
$\sigma^{\prime}\in\mathfrak{S}(n^{\prime})$ compatibility means that the
number $k$ of parts of $c_{1}$ is equal to $n^{\prime}+q={\sf
nt}(f^{\prime})+\|f^{\prime}\|$.

Given two permutations $\sigma=\sigma_{1}\cdots\sigma_{n}\in\mathfrak{S}_{n}$
and
$\sigma^{\prime}=\sigma^{\prime}_{1}\cdots\sigma^{\prime}_{n^{\prime}}\in\mathfrak{S}_{n^{\prime}}$
we denote by $\sigma\otimes\sigma^{\prime}$ the shifted concatenation of
$\sigma$ and $\sigma^{\prime}$,

$\sigma\otimes\sigma^{\prime}=\sigma_{1}\cdots\sigma_{n}(\sigma(1)+n)\cdots(\sigma(n)+n^{\prime})$

The following operation inverse to horizontal splitting is better understood
in terms labelled forests

######  2.7 Definition (Horizontal gluing).

Pick two compatible levelled forests $f=(\tau,\lambda)\in{\textrm{{{LF}}}}{}$
and $f^{\prime}=(\tau^{\prime},\lambda^{\prime})\in{\textrm{{{LF}}}}{}$, we
define the horizontal gluing the levelled forests
$f\\#f^{\prime}=(\phi\\#\phi^{\prime},\lambda^{\prime\prime})$ where

  1. (1)

$\phi\\#\phi^{\prime}$ is the planar forest obtained by successively
superimposing a root of a tree in $\phi^{\prime}$ to a leaf of $\phi$.

  2. (2)

The labelling $\lambda^{\prime\prime}$ restricts to $\lambda$ on the internal
vertices of $\phi$ in $\phi\\#\phi^{\prime}$ and to the labelling
$\lambda^{\prime}$ translated by $\|f\|$ on the internal vertices of
$\phi^{\prime}$ in $\phi\\#\phi^{\prime}$.

In terms of sparse quasi-binary trees and forests, horizontal gluing
corresponds to stacking the sparse quasi-binary trees representing
$f^{\prime}$ above the one representing $f$. Writing this operation in the
representation of levelled forests as split permutations is cumbersome and is
left to the reader, see also the figure below.

Figure 8. Horizontal gluing of a tree and a forest.

###  2.2. A first algebraic structure on levelled forests

We briefly recall important Hopf algebraic structure on permutations. The
vector space $\bigoplus_{n\,\geq\,0}\mathbb{C}\mathfrak{S}_{n}$ has a graded
Hopf algebra structure, called $\mathfrak{S}\textsf{Sym}$, introduced in
[MR95], and also considered in [GKL+95], where it is called the algebra of
free quasi-symmetric functions. This Hopf algebra is non-commutative, non-
cocommutative, graded and self-dual. In the following we introduce the product
$\shuffle$ dual to the aforementionned coproduct of the Hopf algebra of non-
commutative symmetric functions. First, for two permutations
$\alpha\in\mathfrak{S}_{n}$ and $\beta\in\mathfrak{S}_{p}$, we introduce

$\textsf{Sh}(\alpha,\beta):=\\{\sigma\,\in\,\mathfrak{S}_{n+p}\,:\,{\rm
std}(\sigma\cap\\{1,\ldots,n\\})=\alpha,\leavevmode\nobreak\ {\rm
std}(\sigma\cap\\{n+1,\ldots,n+p\\})=\beta\\}.$

the set of _shuffles_ of $\alpha$ and $\beta$. In the above equation ${\rm
std}$ denotes the standardization map: ${\rm std}(\sigma\cap\\{a_{1}<\cdots
a_{k}\\})$ is the image of $\sigma\cap\\{a_{1}<\cdots a_{k}\\}$ by the map
$a_{k}\mapsto k$.

In particular, if $\text{id}_{n}$ and $\text{id}_{p}$ are the identity
permutations of the symmetric groups $\mathfrak{S}_{n}$ and
$\mathfrak{S}_{p}$, respectively, we let

$\text{Sh}(n,p):=\sf{Sh}(\text{id}_{n},\text{id}_{p}).$

Given a permutation $\tau\in\mathfrak{S}(n)$ and integer $p\geq 1$ we denote
by $\tau+p$ the bijection of $\\{p+1,\cdots,p+n\\}$ defined by

$(\sigma+p)(p+k)=\sigma(k)+p$

With $\sigma\in\mathfrak{S}(p)$, $\sigma\otimes(\tau+p)\in\mathfrak{S}(n+p)$
is the permutation equal to $\sigma$ on $\\{1,\cdots,p\\}$ and $(\tau+p)$ on
$\\{p+1,\cdots,p+n\\}$.

The _shuffle product of two permutations_ $\sigma\in\mathfrak{S}_{p}$ and
$\tau\in\mathfrak{S}_{n}$ is defined by

$\sigma\shuffle\tau:=\sum_{\alpha\in{\rm
Sh}(p,n)}\alpha\circ[\sigma\otimes(\tau+p)].$

where we have used the explicit notation $\circ$ for the composition of
permutations. We extend the shuffle product from levelled trees (permutations)
to levelled forests. If the permutations $\sigma$ and $\tau$ are presented as
sparse quasi-binary trees, computing their shuffle product is done by adding
straight edges at the bottom of $\tau$, then gluing this tree to the right of
$\sigma$ (we identify the outer paths of edges) and finallly shuffling
vertically the generations.

######  2.8 Definition (Shuffle product of levelled planar forests).

Let $f=(\sigma,(c_{1},\cdots,c_{k})$ and
$g=(\tau,(c^{\prime}_{1},\cdots,c^{\prime}_{q}))$ be two levelled forests, we
define the shuffle product of $f$ and $g$ by

$f\shuffle
g=(\sigma\shuffle\tau,(c_{1},\cdots,c_{k}+c_{1}^{\prime},\cdots,c^{\prime}_{q})).$

######  2.9 Example.

We present in detail the product $12\shuffle(12,(1,1))$, presented as sparse
quasi-binary forests in Figure 9.

Figure 9. Shuffle product of levelled forests

We decorate the branching nodes with two different colors to track the two
original permutations in the product. The result of the product is given by
the following sum of sparse quasi-binary forests (see Fig. 10).

Figure 10.  The product in Fig. 9 expanded over sparse quasi-binary forests.

We notice that by construction of $\shuffle$ all terms contained in the sum
the blue and red dots preserve the same ordering of generations in the initial
factors.

A permutation $\sigma\in\mathcal{S}_{n}$ acts on the left of a levelled forest
$f=(\sigma,c)\in{\textrm{{{LF}}}}$ with $n$ generations as follows

$\sigma\cdot(\tau,c):=(\sigma\circ\tau,c).$

We denote by $c_{n}$ the permutation
$(n,1)(n-1,2)\cdots(n-\lfloor\frac{n}{2}\rfloor,\lfloor\frac{n}{2}\rfloor)$.
We use the right action of $c_{n}$ to define a involution on
${\textrm{{{LF}}}}_{n}$, which is the _horizontal mirror symmetric of a
forest_

$\begin{array}[]{cccc}\theta:&\mathbb{C}{\textrm{{{LF}}}}&\to&\mathbb{C}{\textrm{{{LF}}}}\\\
&f=(\sigma,c)&\mapsto&(\sigma c_{\|f\|}^{-1},c_{n},\cdots,c_{1})\end{array}$
(13)

######  2.10 Proposition.

$(\mathbb{C}{\textrm{{{LF}}}},\shuffle,\theta)$ is an involutive algebra.

###### Proof.

The result follows as a direct a consequence of the following two facts: the
left and right actions of $\mathfrak{S}_{n}$ on ${\textrm{{{LF}}}}_{n}$
commute and $c_{n+m}=\tau_{n,m}\circ(c_{n}\otimes(c_{m}+n))$ (we add $n$ to
all letters of $c_{m}$ and concatenate the resulting word to the one
representing $c_{n}$) where $\tau_{n,m}$ is the shuffle in ${\rm Sh}(n,m)$
determined by $\tau_{n,m}(1)=m+1,\tau_{n,m}(n)=m+n$. ∎

We will sometimes refer to $\theta$ as the _horizontal involution_ , for
obvious reasons, to distinguish it from a second involution permuting
vertically the generations of a levelled forest that we define below.

###  2.3. Hopf monoid of levelled forests

In this section we introduce a Hopf algebraic structure on the bicollection of
spanned by levelled forests and denoted $\mathcal{L}\mathcal{F}$,

$\mathcal{L}\mathcal{F}(n,m)=\mathbb{C}{\textrm{{{LF}}}}(m,n)\quad n,m\geq 1.$
(14)

In addition, we set $\mathcal{L}\mathcal{F}(0,0)=\mathbb{C}$,
$\mathcal{L}\mathcal{F}(0,n)=\mathcal{L}\mathcal{F}(m,0)=0$, $n,m\geq 1$ and
we denote by $\mathcal{L}\mathcal{T}$ the collection spanned by levelled
binary trees

$\mathcal{L}\mathcal{T}(n)=\mathbb{C}{\textrm{{{LT}}}}(n)\,,\quad n\geq 1.$
(15)

This Hopf algebra is an object in the category of bicollections endowed with
the vertical tensor product $\varobar$. In general, as it is briefly explained
in the Appendix 5, the two-folded vertical tensor product $A\varobar A$ of a
monoid $A$ in the monoidal category (Coll2, $\varobar$) is not a monoid in the
same category. Owing to the fact that the monoid generated by
$\mathcal{L}\mathcal{F}$ in $(\mathrm{Coll}_{2},\varobar)$ is _symmetric_ , in
particular $\mathcal{L}\mathcal{F}\varobar\mathcal{L}\mathcal{F}$ is a monoid
in a natural way, it makes sense to require compatibility between a product
and a co-product on $\mathcal{L}\mathcal{F}$. We write the unit
$\mathbb{C}_{\varobar}$ for the vertical tensor product as

$\mathbb{C}_{\varobar}=\bigoplus_{n\geq 0}\mathbb{C}1_{n}.$

Recall that we denote by $|f|$ the number of leaves of a levelled forest $f$
and ${\sf nt}(f)$ the number of trees in $f$.

We begin with the definition of the coproduct acting on the bicollection
${\sf\mathcal{L}\mathcal{F}}$ of levelled forests. Let $f$ be a levelled
forest. Let $f^{\prime}$ be a levelled subforest of $f$ (recall that
$f^{\prime}$ contains the roots of all trees in $f$). By definition of the
forest $f\backslash f^{\prime}$, the number of outputs of the forest
$f\backslash f^{\prime}$ is equal to the number of inputs of the forest
$f^{\prime}$ (the number of trees of $f\backslash f^{\prime}$ matches the
number of leaves of $f^{\prime}$), the following makes senses

$\Delta(f)=\sum_{f^{\prime}\subset f}f^{\prime}\varobar f\backslash
f^{\prime},\leavevmode\nobreak\ f\in\mathcal{L}\mathcal{F}\,.$ (16)

This operation is a genuine coproduct with respect to the vertical ternsor
product.

######  2.11 Proposition.

The morphism
$\Delta:\mathcal{L}\mathcal{F}\rightarrow\mathcal{L}\mathcal{F}\varobar\mathcal{L}\mathcal{F}$
is coassociative

$(\Delta\varobar\mathrm{id}_{\mathcal{L}\mathcal{F}})\circ\Delta=(\mathrm{id}_{\mathcal{L}\mathcal{F}}\varobar\Delta)\circ\Delta$
(17)

and the moprhism
$\varepsilon\colon\mathcal{L}\mathcal{F}\rightarrow\mathbb{C}_{\varobar}$
given by

$\varepsilon(f)=\begin{cases}1_{n}&\text{if}\,\,f=\underbrace{\bullet\cdots\bullet}_{n\;\text{times}}\\\
0&\text{otherwise.}\end{cases}$

is the counity for $\Delta$. i.e.

$(\varepsilon\varobar\mathrm{id}_{\mathcal{L}\mathcal{F}})\circ\Delta=(\mathrm{id}_{\mathcal{L}\mathcal{F}}\varobar\varepsilon)\circ\Delta=\mathrm{id}\,.$
(18)

###### Proof.

Let $f$ be a levelled forest, to show coassociativity we notice that

$((\Delta\varobar\mathrm{id}_{\mathcal{L}\mathcal{F}})\circ\Delta)(g)=\sum_{\begin{subarray}{c}f^{\prime\prime},f^{\prime},f\\\
f^{\prime\prime}\\#f^{\prime}\\#f=g\end{subarray}}f^{\prime\prime}\varobar
f^{\prime}\varobar
f=((\mathrm{id}_{\mathcal{L}\mathcal{F}}\varobar\Delta)\circ\Delta)(g)\,.$
(19)

Equation (18) is trivial. ∎

We proceed now with the definition of a vertical product on levelled forests.

######  2.12 Definition (Monoidal product on levelled forests).

Given two forests $f$ and $f^{\prime}$ with ${\sf nt}(f^{\prime})=|f|$, we
define $\nabla(f\varobar f^{\prime})$ as the sum of forests obtained by first
stacking $f^{\prime}$ up to $f$ and then shuffling the generations of
$f^{\prime}$ with the generations of $f$ (see Section 2.1 for the definition
of the action of a permutation on the generations of a forest),

$\nabla(f\varobar
f^{\prime})=\sum_{s\in\textrm{{{Sh}}}(\|f\|,\|f^{\prime}\|)}s\cdot(f\,\\#\,f^{\prime}).$
(20)

Associativity of the product $\nabla$ is easily checked. The unit
$\eta:\mathbb{C}_{\varobar}\rightarrow\mathcal{L}\mathcal{F}$ is defined by
$\eta(1_{m})=\,\leavevmode\hbox to4.98pt{\vbox
to6.95pt{\pgfpicture\makeatletter\hbox{\hskip 2.49023pt\lower-2.49023pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,^{m}$. Let $n\geq
1$, recall that we denote by $c_{n}$ the maximal element for the Bruhat order
in $\mathfrak{S}_{n}$:

$c_{n}=n(n-1)\cdots 321$

For example, $c_{1}=1$, $c_{2}=21$, $c_{3}=321$, $c_{4}=4321$. Given these
notion, we state the main theorem of the section

######  2.13 Theorem.

$(\mathcal{L}\mathcal{F},\nabla,\eta,\Delta,\varepsilon)$ is a conilpotent
Hopf algebra in the category $(\mathrm{Coll}_{2}$,
$\varobar,\mathbb{C}_{\varobar})$.

To achieve this result we introduce an explicit antipode map.

######  2.14 Definition.

Pick $n,m\geq 1$ two integers. Let $f\in\mathcal{L}\mathcal{F}(n,m)$ be a
levelled forest and define its _vertical mirror symmetric_
$f^{\star}\in\mathcal{L}\mathcal{F}(n,m)$ by

$f^{\star}=c_{\|f\|}\cdot f.$

We extend $\star$ as a conjugate-linear morphism on the bicollection
$\mathcal{L}\mathcal{F}$.

######  2.15 Proposition.

Let $f$ be a levelled forest. The map
$S:\mathcal{L}\mathcal{F}\rightarrow\mathcal{L}\mathcal{F}$ defined by

$S(f)=(-1)^{\|f\|}f^{\star}$ (21)

is an antipode:
$\nabla\circ(S\varobar\mathrm{id}_{\mathcal{L}\mathcal{F}})\circ\Delta=\nabla\circ(\mathrm{id}_{\mathcal{L}\mathcal{F}}\varobar
S)\circ\Delta=\varepsilon\circ\eta$.

###### Proof.

Let $a,b$ be two integers greater than one. Set $n=a+b$. The set of shuffles
$\textrm{{{Sh}}}(a,b)$ is divided into two mutually disjoint subsets, the set
of shuffles sending $a$ (the subset $\textrm{{{Sh}}}(a,b)_{+}$) to $n$ and the
set of shuffles that do not (resp. $\textrm{{{Sh}}}(a,b)_{-}$).

Recall that if $f$ is a forest then $f^{k}_{-}$ denotes the forest obtained by
extracting the $k$ first lowest generations of $f$ and $f^{k}_{+}$ denotes the
forest obtained by extracting the $k$ highest generations of $f$. By
definition, one has:

$\nabla(f^{\prime}\varobar f\backslash
f^{\prime})=\sum_{s\in\textrm{{{Sh}}}(\|f^{\prime}\|,\|f\backslash
f^{\prime}\|)}s\cdot(f^{\prime}\,\\#\,(f\backslash
f^{\prime})^{\star}),\leavevmode\nobreak\ f^{\star}=c_{\|f\|}\cdot f.$

The following relation is easily checked and turn to be the cornerstone of the
proof:

$\tilde{s}\circ(c_{n}\otimes\mathrm{id}_{m})=s\circ(c_{n+1}\otimes\mathrm{id}_{m-1}),\leavevmode\nobreak\
s\in\textrm{{{Sh}}}(m-1,n+1)_{-},$ (22)

with $\tilde{s}$ the unique shuffle in $\textrm{{{Sh}}}(m,n)_{+}$ such that
$\tilde{s}(m)=n+m,\leavevmode\nobreak\ \tilde{s}(i)=s(i)$. Set
$\bar{S}(f)=(-1)^{\|f\|}f^{\star}$. We prove by induction that $S=\bar{S}$.
Assume that $S(f)=\bar{S}(f)$ for any forest $f$ with at most $N\geq 1$
generations and pick a forest $f$ with $N+1$ generations. Then, from the
induction hypothesis we get:

$S(f)+f+(\mathrm{id}\varobar\bar{S})\circ\bar{\Delta}(f)=0.$
$\displaystyle\nabla\circ(\mathrm{id}\varobar\bar{S})\circ\bar{\Delta}(f)$
$\displaystyle=\sum_{f^{\prime}\subset f}(-1)^{\|f\backslash
f^{\prime}\|}\sum_{s\in\textrm{{{Sh}}}(\|f^{\prime}\|,\|f\backslash
f^{\prime}\|)}s\cdot\left[f^{\prime}\,\\#\,(f\backslash
f^{\prime})^{\star}\right]$
$\displaystyle=\sum_{k=1}^{\|f\|-1}(-1)^{k}\sum_{s\in\textrm{{{Sh}}}(\|f\|-k,k)}s\cdot\left[(f_{-}^{\|f\|-k}\,\\#\,(f_{+}^{k})^{\star}\right].$

We divide the sum over the set Sh$(\|f\|-k,k)$ into two sums. The first sums
ranges over the subset $\textrm{{{Sh}}}(\|f\|-k,k)_{+}$ and the second one
ranges overs $\textrm{{{Sh}}}(\|f\|-k,k)_{-}$. Then, we gather the sums over
$\textrm{{{Sh}}}(\|f\|-k,k)_{+}$ and $\textrm{{{Sh}}}(\|f\|-k+1,k-1)_{-}$:

$\displaystyle\nabla\circ(\mathrm{id}\varobar\bar{S})\circ\bar{\Delta}=\sum_{k=2}^{\|f\|-2}(-1)^{k}\sum_{s\in\textrm{{{Sh}}}(\|f\|-k,k)_{+}}s\cdot\left[f^{\|f\|-k}_{-}\,\\#\,(f_{+}^{k})^{\star}\right]$
$\displaystyle\hskip
99.58464pt-(-1)^{k}\sum_{s\in\textrm{{{Sh}}}(\|f\|-k+1,k-1)_{-}}s\cdot\left[f^{\|f\|-k+1}_{-}\,\\#\,(f_{+}^{k-1})^{\star})\right]$
$\displaystyle+(-1)\sum_{s\in\textrm{{{Sh}}}(1,\|f\|-1)_{-}}s\cdot\left[f^{\|f\|-1}_{-}\,\\#\,(f_{+}^{1})^{\star}\right]+(-1)^{\|f\|-1}\sum_{s\in\textrm{{{Sh}}}(\|f\|-1,1)_{+}}s\cdot\left[f^{1}_{-}\\#(f_{+}^{\|f\|-1})^{\star}\right]$

Using equation (22), the right hand side of the last equation is equal to

$\displaystyle\nabla\circ(\bar{S}\varobar\mathrm{id})\circ\bar{\Delta}=$
$\displaystyle=0-\sum_{s\in\textrm{{{Sh}}}(1,\|f\|-1)_{-}}s\cdot\left[f^{1}_{-}\,\\#\,(f_{+}^{\|f\|-1})^{\star}\right]+(-1)^{\|f\|-1}\sum_{s\in\textrm{{{Sh}}}(1,\|f\|-1)_{+}}s\cdot\left[f^{\|f\|-1}_{-}\,\\#\
(f_{+}^{1})^{\star}\right]$ $\displaystyle=-f+(-1)^{\|f\|-1}f^{\star}\,.$

∎

We defined the three structural morphisms $\nabla,\Delta,S$. To turn LF into a
Hopf monoid, we have to check compatibility between the coproduct $\Delta$ and
the product $\nabla$; the coproduct $\Delta$ should be an morphism of the
monoid $(\mathcal{L}\mathcal{F},\nabla$). This only makes sense provided that
we can define a product on the tensor product
$\mathcal{L}\mathcal{F}\varobar\mathcal{L}\mathcal{F}$.

Recall that if $f$ is a levelled forest and $0\leq k\leq\|f\|$, one denotes by
$f_{-}^{k}$ the levelled subforest of $f$ corresponding to the $k$ generations
at the bottom of $f^{\prime}$: $t_{f_{-}^{k}}$ is the planar subforest of
$t_{f}$ with set of internal vertices the set of internal vertices of $f$
labeled by an integer less than $k$ and for leaves the vertices (including the
leaves) of $t_{f}$ connected to one of the latter internal vertices. The
levelled forest $f_{+}^{k}$ is obtained similary by extracting the $k$ top
generations of $f^{\prime}$.

With $p,q\geq 1$ two integers, we denote by $\tau_{p,q}$ the shuffle in ${\rm
Sh}(p,q)$ satisfying $\tau_{p,q}(1)=q+1$ and $\tau_{p,q}(p)=p+q$.

######  2.16 Definition.

Define the braiding map

${\sf
K}:\mathcal{L}\mathcal{F}\varobar\mathcal{L}\mathcal{F}\rightarrow\mathcal{L}\mathcal{F}\varobar\mathcal{L}\mathcal{F}$

by, for $g$ and $f$ levelled forests such that $f\varobar
g\in\mathcal{L}\mathcal{F}\varobar\mathcal{L}\mathcal{F}$,

${\sf K}(f\varobar
g)=\left(\tau_{\|f\|,\|g\|}\cdot(f\\#g)\right)_{-}^{\|g\|}\varobar\left(\tau_{\|f\|,\|g\|}\cdot(f\\#g)\right)_{+}^{\|f\|}.$

We pictured in Fig. 11 examples of the action of the braiding map on pairs of
levelled forests.

Figure 11. Actions of the braiding map

We defined the braiding map ${\sf K}$ as acting on
$\mathcal{L}\mathcal{F}\varobar\mathcal{L}\mathcal{F}$. We extend ${\sf K}$ as
a $2$-functor on the product of the monoid generated by
$\mathcal{L}\mathcal{F}$ in (Coll2,$\varobar$). This means in particular that
for integers $p,q\geq 1$, we define a bicollection morphism

${\sf K}_{p,q}:\mathcal{L}\mathcal{F}^{\varobar
p}\varobar\mathcal{L}\mathcal{F}^{\varobar
q}\rightarrow\mathcal{L}\mathcal{F}^{\varobar
q}\varobar\mathcal{L}\mathcal{F}^{\varobar p}.$

Pick $f_{1}\varobar\cdots\varobar f_{p}\in\mathcal{L}\mathcal{F}^{\varobar p}$
and $g_{1}\varobar\cdots\varobar g_{q}\in\mathcal{L}\mathcal{F}^{\varobar q}$.

We define the levelled forest $\succ_{c}h$ with $c\vDash_{0}n$ to be the
element in $\mathcal{L}\mathcal{F}^{\varobar p}$ obtained by the following
iterative application of horizontal splittings:

$\succ^{-}_{c}\,=\,\succ_{c_{1}}^{-}(\cdots(\succ_{c_{1}+\cdots+c_{n-1}}^{-}(h))\cdots)$
$\displaystyle{\sf K}_{p,q}(f_{1}\varobar\cdots\varobar f_{p}\varobar
g_{1}\varobar\cdots\varobar g_{q})$ $\displaystyle\hskip
85.35826pt=\leavevmode\nobreak\
\succ^{-}_{\|g_{1}\|,\cdots,\|g_{p}\|,\|f_{1}\|,\cdots,\|f_{q}\|}\big{(}\tau_{\|f_{1}\|+\cdots+\|f_{p}\|,\|g_{1}\|+\cdots+\|g_{q}\|}\cdot
f_{1}\\#\cdots\\#g_{q}\big{)}$

The collection of morphisms ${\sf K}_{p,q}$ yields a $2$-functor on the
category with objects $\mathcal{L}\mathcal{F}^{\otimes p}$ but with restricted
classes of morphisms.

######  2.17 Definition.

Let $p,q\geq 0$ be integers and $\varphi:\mathcal{L}\mathcal{F}^{\otimes
p}\to\mathcal{L}\mathcal{F}^{\otimes q}$, we say that $\varphi$ is _gluing
equivariant_ if $\varphi$ commutes with the operations
$|\\!|_{\raisebox{-2.84526pt}{\small$i$}}$,namely, with
$f_{1}\varobar\cdots\varobar f_{m}\in\mathcal{L}\mathcal{F}^{\varobar m}$,

$\varphi(|\\!|_{\raisebox{-2.84526pt}{\small$i$}}[f_{1}\varobar\cdots\varobar
f_{p}])=|\\!|_{\raisebox{-2.84526pt}{\small$i$}}[\varphi(f_{1}\varobar\cdots\varobar
f_{p})],1\leq i\leq k$

where

$|\\!|_{\raisebox{-2.84526pt}{\small$i$}}[f_{1}\varobar\cdots\varobar
f_{p}]\,:=\,\succ^{-}_{\|f_{1}\\},\cdots,\|f_{p}\|}|\\!|_{\raisebox{-2.84526pt}{\small$i$}}[f_{1}\sharp\cdots\sharp
f_{p}]$

We denote by ${\rm Hom}_{eq.}(p,q)$ the class of all gluing equivariant
morphisms between $\mathcal{L}\mathcal{F}^{\varobar p}$ and
$\mathcal{L}\mathcal{F}^{\varobar q}$. Note that the identity morphisms are
gluing equivariant and that the composition of two gluing equivariant
morphisms is gluing equivariant. Also, for each $p,q\geq 0$, $K_{p,q}$ is
gluing equivariant.

######  2.18 Proposition.

The monoid generated by the bicollection $\mathcal{L}\mathcal{F}$ in
$(\mathrm{Coll}_{2},\varobar)$ with morphisms restricted to the gluing
equivariant morphisms is a symmetric monoidal category with symmetry
constraints $({\sf K}_{p,q})_{p,q\geq 0}$,

${\sf K}_{p,q}\circ{\sf K}_{q,p}=\mathrm{id}\mathrm{\leavevmode\nobreak\
and\leavevmode\nobreak\ }(\mathrm{id}_{\mathcal{L}\mathcal{F}^{\varobar
q}}\varobar{\sf K}_{p,r})\circ({\sf
K}_{p,q}\varobar\mathrm{id}_{\mathcal{L}\mathcal{F}^{\varobar r}})={\sf
K}_{p,q+r}$

###### Proof.

Both assertions are trivial and rely on the following relations between the
permutations $\tau_{p,q}$, $p,q\geq 0$:

$\tau_{p,q}\circ\tau_{q,p}=\mathrm{id},(\mathrm{id}_{q}\otimes\tau_{p,r})\circ(\tau_{p,q}\otimes\mathrm{id}_{r})=\tau_{p,q+r},\leavevmode\nobreak\
p,q,r\geq 0.$

∎

Using the above defined symmetry constraint $K$, we can endow the two-folds
tensor product $\mathcal{L}\mathcal{F}\varobar{\textrm{{{LT}}}}$ with an
algebra product:

$(\nabla\varobar\nabla)\circ(\mathrm{id}\varobar{\sf
K}\varobar\mathrm{id}):\mathcal{L}\mathcal{F}^{\varobar
4}\rightarrow\mathcal{L}\mathcal{F}^{\varobar 2}$

######  2.19 Proposition.

The two bicolletion morphisms
$\Delta\colon\mathcal{L}\mathcal{F}\to\mathcal{L}\mathcal{F}\varobar\mathcal{L}\mathcal{F}$
and
$\nabla\colon\mathcal{L}\mathcal{F}\varobar\mathcal{L}\mathcal{F}\rightarrow\mathcal{L}\mathcal{F}$
are vertical algebra morphisms. With
$\nabla^{(2)}=\nabla\circ(\nabla\varobar\mathrm{id})=\nabla\circ(\mathrm{id}\varobar\nabla)$,
this means that

$\nabla^{(2)}=\nabla^{(2)}\circ(\mathrm{id}\varobar{\sf
K}\varobar\mathrm{id}),\quad(\nabla\varobar\nabla)\circ(\mathrm{id}\varobar{\sf
K}\varobar\mathrm{id})\circ(\Delta\varobar\Delta)=\Delta\circ\nabla.$

######  2.20 Remark.

We can rephrase the fact that $\nabla$ is an algebra morphism by saying that
$(\mathcal{L}\mathcal{F},\nabla)$ is, in fact, a commutative algebra.

###### Proof.

We begin with the first assertion. Pick $f_{1},f_{2},f_{3},f_{4}$ compatible
levelled forests (the number of inputs of $f_{i}$ matches the number of
outputs of $f_{i+1}$, $1\leq i\leq 3$),

$\displaystyle(\nabla^{(2)}\circ{\sf K})(f_{1}\varobar f_{2}\varobar
f_{3}\varobar f_{4})$ $\displaystyle=\sum_{s\in{\rm
Sh}(\|f_{1}\|,\|f_{3}\|,\|f_{2}\|,\|f_{4}\|)}s\cdot\left[f_{1}\\#\left(\tau_{\|f_{2}\|,\|f_{3}\|}\cdot(f_{2}\\#f_{3})\right)_{-}^{\|f_{3}\|}\\#\left(\tau_{\|f_{2}\|,\|f_{3}\|}\cdot(f_{2}\\#f_{3})\right)_{+}^{\|f_{2}\|}\\#f_{4}\right]$
$\displaystyle=\sum_{s\in{\rm
Sh}(\|f_{1}\|,\|f_{3}\|,\|f_{2}\|,\|f_{4}\|)}\left(s(\mathrm{id}\otimes\tau_{\|f_{2}\|,\|f_{3}\|})\right)\cdot\Big{(}f_{1}\\#f_{2}\\#f_{3}\\#f_{4}\Big{)}$
$\displaystyle=\sum_{s\in{\rm
Sh}(\|f_{1}\|,\|f_{2}\|,\|f_{3}\|,\|f_{4}\|)}s\cdot\Big{(}f_{1}\\#f_{2}\\#f_{3}\\#f_{4}\Big{)}=\nabla^{(2)}(f_{1}\varobar
f_{2}\varobar f_{3}\varobar f_{4}).$

For the second assertion, we write first:

$\displaystyle(\Delta\circ\nabla)(f\varobar g)=\sum_{1\leq
k\leq\|f\|+\|g\|}\sum_{s\in{\rm
Sh}(k,\|f\|+\|g\|-k)}(s\cdot(f\\#g))_{-}^{k}\varobar(s\cdot(f\\#g))_{+}^{\|f\|+\|g\|-k}$

For each integer $1\leq k\leq\|f\|$, we split the set of shuffles ${\rm
Sh}(\|f\|,\|g\|)$ according to the cardinal $q$ of the set $s^{-1}(\llbracket
1,k\rrbracket)\cap\llbracket\|f\|+1,\|f\|+\|g\|\rrbracket$. Then a shuffle
$s\in{\rm Sh}(\|f\|,\|g\|)$ $s=(s_{1}\otimes s_{2})\circ\tilde{\tau}_{k,q}$
with $\tilde{\tau}_{k,q}$ the unique shuffle that sends the interval
$\llbracket\|f\|+1,\|f\|+q\rrbracket$ to the interval $\llbracket
k-q+1,k\rrbracket$ and fixes the interval
$\llbracket\|f\|+q+1,\|f\|+\|g\|\rrbracket$.

$\sum_{\begin{subarray}{c}1\leq k\leq\|f\|,1\leq q\leq\|g\|,\\\ 1\leq q\leq
k\end{subarray}}\sum_{\begin{subarray}{c}s_{1}\in{\rm Sh}(k-q,q),\\\
s_{2}\in{\rm Sh}(\|f\|-(k-q)\|g\|-q)\end{subarray}}((s_{1}\otimes
s_{2})\circ\tilde{\tau}_{k,q})\cdot(f\\#g))_{-}^{k}\varobar((s_{1}\otimes
s_{2})\circ\tilde{\tau}_{k,q}\cdot(f\\#g))_{+}^{\|f\|+\|g\|-k}$

Notice that $\tilde{\tau}_{k,q}=\tau_{k-q,q}$ and

$\tilde{\tau}_{k,q}\cdot(f\,\\#\,g)=f_{-}^{k-q}\\#(\tau_{\|f\|-(k-q),q}\cdot(f_{+}^{\|f\|-(k-q)}\,\\#\,g_{-}^{q}))\\#g_{+}^{\|g\|-q}.$

It follows that

$\displaystyle(s_{1}\otimes
s_{2})\circ\tilde{\tau}_{k,q}\cdot(f\\#g))_{-}^{k}$
$\displaystyle=\left((s_{1}\otimes\mathrm{id})\cdot
f_{-}^{k-q}\\#(\tau_{\|f\|-(k-q),q}\cdot(f_{+}^{\|f\|-(k-q)}\,\\#\,g_{-}^{q}))\\#g_{+}^{\|g\|-q}\right)^{k}_{-}$
$\displaystyle=s_{1}\cdot f_{-}^{k-q}\\#(\tau_{\|f\|-(k-q),q}\cdot
f_{+}^{\|f\|-(k-q)}\\#g_{-}^{q})_{-}^{q}.$

Similar computations show that

$((s_{1}\otimes
s_{2})\circ\tilde{\tau}_{k,q}\cdot(f\\#g))_{+}^{\|f\|+\|g\|-k}=s_{2}\cdot(\tau_{\|f\|-(k-q),q}\cdot(f_{+}^{\|f\|-(k-q)}\,\\#\,g_{-}^{q}))_{+}^{\|f\|-(k-q)}\\#g_{+}^{\|g\|-q})\,.$

The case $\|f\|+1\leq k\leq\|f\|+\|g\|$ is similar, we split the set of
shuffles Sh$(\|f\|,\|g\|)$ according to the cardinal of the set
$s^{-1}(\llbracket k+1,\|f\|+\|g\|\rrbracket)\cap\llbracket
1,\|f\|\rrbracket$) and omitted for brevity. Finally, we obtain for
$\Delta\circ\nabla(f\varobar g)$ the expression:

$\displaystyle\sum_{\begin{subarray}{c}1\leq k\leq\|f\|,\\\ 1\leq
q\leq\|g\|\end{subarray}}\sum_{\begin{subarray}{c}s_{1}\in{\rm Sh}(k,q)\\\
s_{2}\in{\rm Sh}(\|f\|-k,\|g\|-q)\end{subarray}}$ $\displaystyle
s_{1}\cdot(f_{-}^{k}\\#(\tau_{\|f\|-k,q}\cdot(f_{+}^{\|f\|-k}\,\\#\,g_{-}^{q}))_{-}^{q})\varobar$
$\displaystyle\varobar
s_{2}\cdot(\tau_{k,q}\cdot(f_{+}^{\|f\|-k}\,\\#\,g_{-}^{q}))_{+}^{\|f\|-k}\\#g_{+}^{\|g\|-q})\,,$

which is easily seen to be equal to
$(\nabla\varobar\nabla)\circ(\mathrm{id}\varobar{\sf
K}\varobar\mathrm{id})\circ(\Delta\varobar\Delta)(f\varobar g)$. ∎

##  3\. Iterated integrals of a path as operators

Let us fix a smooth path, $X:[0,1]\rightarrow\mathcal{A}$. In this section, we
use the algebraic tools developed previously and introduce _partial-_ and
_full-contraction_ operators. These operators are indexed by levelled forests
and provide a different perspective on the iterated integrals of $X$, as a
representation of the monoid of levelled forest introduced in the previous
section (see Theorem 2.13), rather than as a sequence of tensors.

###  3.1. Full and partial contractions operators

In what follows, we will intensively use the identifications in the previous
sections between levelled trees, levelled forests and their corresponding
permutation and split permutations.

######  3.1 Definition.

For any integer $n\geq 1$ and levelled tree $\sigma$ in
${\textrm{{{LT}}}}(n)$, we define the _full contraction of $X$ along $\sigma$_
as the map $\mathbb{X}^{\sigma}\colon[0,1]^{2}\to{\rm Hom}_{{\rm
Vect}_{\mathbb{C}}}(\mathcal{A}^{\otimes n},\mathcal{A})$ given for any
$A_{1}\,,\cdots\,,A_{n}\in\mathcal{A}$ by

$\mathbb{X}_{s,t}^{\sigma}(A_{1}\otimes\cdots\otimes
A_{n})=\int_{\Delta_{s,t}^{n-1}}A_{1}\cdot\mathrm{d}X_{t_{\sigma(1)}}\cdot
A_{2}\cdots\mathrm{d}X_{t_{\sigma(n-1)}}\cdot A_{n}\,,$ (23)

where $\sigma$ is identified with a permutation in $\mathfrak{S}_{n-1}$ when
$n\geq 2$ and $\mathbb{X}_{s,t}^{\bullet}=\mathrm{id}_{\mathcal{A}}$.

######  3.2 Remark.

The above definition may be misleading since the identity (23) defines a
linear map on the algebraic tensor product, whereas we used $\otimes$ to
denote the projective tensor product. However, the algebraic tensor product is
a dense subspace of $\mathcal{A}^{\otimes n}$ and we interpret
$\mathbb{X}^{\sigma}_{s,t}$ as the unique continuous operator extending the
values in (23). Similar considerations apply throughout the paper.

If linearly extended to the vector space spanned by all levelled trees (or
equally permutations), the map $\sigma\mapsto\mathbb{X}^{\sigma}_{s,t}$ yields
naturally a morphism between the collection $\mathcal{L}\mathcal{T}$ in (15)
and and the endomorphism collection ${\rm End}_{\mathcal{A}}$ given by

${\rm End}_{\mathcal{A}}(n)={\rm Hom}_{{\rm
Vect}_{\mathbb{C}}}(\mathcal{A}^{\otimes n},\mathcal{A})\quad n\geq 1\,,$

see Appendix 5. The partial contraction operators, that we now introduce,
extend $\sigma\mapsto\mathbb{X}^{\sigma}_{s,t}$to a morphism between
$\mathcal{L}\mathcal{F}$ to many-to-many operators, i.e. elements of ${\rm
Hom}_{\mathrm{Vect}_{\mathbb{C}}}(\mathcal{A}^{\otimes n},\mathcal{A}^{\otimes
m})$ $m<n$.

To properly define them, we denote by ${\rm End}^{2}_{\mathcal{A}}$ the
bicollection of _non-commutative polynomials on multilinear maps on
$\mathcal{A}$ with values in_ $\mathcal{A}$. That is using the the notation in
the Appendix 5

${\rm End}_{\mathcal{A}}^{2}(m,n):=T({\rm
End}_{\mathcal{A}})(m,n)=\bigoplus_{\begin{subarray}{c}k_{1}+\cdots+k_{m}=n\end{subarray}}{\rm
End}_{\mathcal{A}}(k_{1})\otimes\cdots\otimes{\rm End}_{\mathcal{A}}(k_{m})$

when $n\geq 1$ and $m\geq 1$ and the condition $k_{1}+\cdots+k_{m}=n$ is
satisfied for some integer $k_{1},\cdots,k_{m}\geq 1$. Moreover we set ${\rm
End}^{2}_{\mathcal{A}}(0,0)=\mathbb{C}$ and ${\rm
End}^{2}_{\mathcal{A}}(m,n)=0$ otherwise. The bicollection ${{\rm
End}_{\mathcal{A}}^{2}}$ is endowed with a monoidal product $\nabla_{{\rm
End}^{2}_{\mathcal{A}}}$ associated to the vertical tensor product $\varobar$.
This operation extends the usual canonical operadic structure $\circ$ on ${\rm
End}_{\mathcal{A}}$ as a monoidal morphism, see Appendix 5. For example, given
two non-trivial elements $v\in{\rm End}_{\mathcal{A}}^{2}(k,n)$,
$v=v_{1}\otimes\cdots\otimes v_{k}$ with and $u\in{\rm
End}_{\mathcal{A}}^{2}(m,k)$, $u=u_{1}\otimes\cdots\otimes u_{m}$ one has

$\nabla_{{\rm End}^{2}_{\mathcal{A}}}(u\varobar v)=(u_{1}\circ
v_{1}\otimes\cdots\otimes
v_{k_{1}}))\otimes\cdots\otimes(u_{m}\circ(v_{k-k_{m}+1}\otimes\cdots\otimes
v_{k}))\,.$

We introduce also the the double tensor algebra of $\mathcal{A}$. We start
from the unital tensor algebra

$\bar{T}(\mathcal{A})=\mathbb{C}\emptyset\oplus\bigoplus_{n\geq
1}\mathcal{A}^{\otimes n}\,.$

Elements of $\bar{T}(\mathcal{A})$ are linear combination of words
$w=a_{1}\cdots a_{k}:=a_{1}\otimes\cdots\otimes a_{k}$ and $\emptyset$ is the
unity for concatenation of words. The double tensor algebra is given by

$\bar{T}^{2}(\mathcal{A}):=\bar{T}(\bar{T}(\mathcal{A}))=\mathbb{C}1\oplus\bigoplus_{n\geq
1}\bar{T}(\mathcal{A})^{\otimes n}\,.$

Elements of $\bar{T}^{2}(\mathcal{A})$ are represented as words of words. To
distinguish the internal concatenation of $\bar{T}(\mathcal{A})$ and the
second order concatenation, we use the symbol $|$ for the concatenation
product on $\bar{T}(\bar{T}(\mathcal{A}))$ and the symbol $1$ stands now for
the unit of $|$. In the following, for any integers $m\geq 1$ and $n\geq 1$ we
use the notation

$\bar{T}^{2}(\mathcal{A})(m,n):=\bigoplus_{\begin{subarray}{c}n_{1}+\cdots+n_{m}=n\\\
n_{i}\geq
1\end{subarray}}\mathcal{A}^{\otimes(n_{1}-1)}\otimes\cdots\otimes\mathcal{A}^{\otimes(n_{m}-1)}$

with the convention that $\mathcal{A}^{0}=\mathbb{C}\emptyset$. For both words
on words in $\mathcal{A}$ and words on endomorphisms in $\mathcal{A}$, we
freely identify the sequence of vector spaces ${{\rm
End}_{\mathcal{A}}^{2}}(m,n)$ (resp. $\bar{T}^{2}(\mathcal{A})(m,n)$) with
their direct sum. We will however make clear this distinctions for other
collections and bicollections. We relate $\bar{T}^{2}(\mathcal{A})$ and ${\rm
End}_{\mathcal{A}}^{2}$ via an explicit representation.

######  3.3 Definition (Representation of the algebra
$\bar{T}^{2}(\mathcal{A})$).

We define a representation
$\mathrm{Op}:\bar{T}^{2}(\mathcal{A})\rightarrow\mathrm{End}^{2}_{\mathcal{A}}$
of the algebra $(\,\bar{T}^{2}(\mathcal{A}),\,|\,)$ extending the following
values, for $A_{1}\otimes\cdots\otimes A_{n}\in
T(\mathcal{A}),\leavevmode\nobreak\ X_{i}\in\mathcal{A}$,

$\displaystyle{\mathrm{Op}}(A_{1}\otimes\cdots\otimes
A_{n})(X_{1}\otimes\cdots\otimes X_{n})=X_{0}\cdot A_{1}\cdot\cdots\cdot
A_{n}\cdot X_{n},$ $\displaystyle{\rm Op}(\emptyset)(X_{1})={\rm
id}_{\mathcal{A}}.$

The representation Op has one crucial property. By definition, ${\rm Op}$ is
compatible with the concatenation product $|$ on $\bar{T}^{2}(\mathcal{A})$.
As explained, ${\rm End^{2}_{\mathcal{A}}}$ is endowed with a vertical
monoidal structure $\nabla_{{\rm End}^{2}_{\mathcal{A}}}$. The same kind of
structure exists on $\bar{T}^{2}(\mathcal{A})$. Indeed, $\bar{T}(\mathcal{A})$
can be endowed with an operadic structure $\circ$, that we call _words-
insertions_. Given a word $a_{1}\otimes\cdots\otimes
a_{n}\in\bar{T}(\mathcal{A})$ and $w_{0},\cdots,w_{n}\in\bar{T}(\mathcal{A})$,
one defines

$a_{1}\cdots a_{n}\circ(w_{0}\otimes\cdots\otimes w_{n}):=w_{0}\otimes
a_{1}\otimes w_{1}\cdots w_{n-1}\otimes a_{n}\otimes w_{n}.$ (24)

One can check that $\circ$ satisfy asssociatiy and unitaly constraints of an
operadic composition. We then extend this operadic composition as an
horizontal monoidal morphism and define in this way an associative product

$\nabla_{\bar{T}^{2}(\mathcal{A})}:\bar{T}^{2}(\mathcal{A})\varobar\bar{T}^{2}(\mathcal{A})\to\bar{T}^{2}(\mathcal{A})\,.$

Then ${\rm Op}$ is compatible with respect to the products $\nabla_{{\rm
End}_{\mathcal{A}}^{2}}$ and $\nabla_{\bar{T}^{2}(\mathcal{A})}$. That is

$(\nabla_{\rm End^{2}_{\mathcal{A}}}\varobar\nabla_{\rm
End^{2}_{\mathcal{A}}})\circ({\rm Op}\varobar{\rm Op})={\rm
Op}\circ\nabla_{\bar{T}^{2}(\mathcal{A})}\,.$ (25)

######  3.4 Example.

Pick $a_{1},a_{2}\in\mathcal{A}$ and $b_{1},b_{2},b_{3}\in\mathcal{A}$ and
consider

$w_{1}=a_{1}|a_{2},\quad w_{2}=b_{1}b_{2}|\emptyset|b_{3}|\emptyset$

The two words $w_{1}$ and $w_{2}$ are compatible, since $w_{1}$ has four
inputs and $w_{2}$ has four outputs, we compose them together

$\nabla_{\bar{T}^{2}(\mathcal{A})}(w_{1}\varobar
w_{2})=b_{1}b_{2}a_{1}|b_{3}a_{2}.$ (26)

and apply Op to the result,

${\rm Op}(b_{1}b_{2}a_{1}|b_{3}a_{2})(X_{0},\cdots,X_{6})=X_{0}\cdot
b_{1}\cdot X_{1}\cdot b_{2}\cdot X_{2}\cdot a_{1}\cdot X_{3}\otimes X_{4}\cdot
b_{3}\cdot X_{5}a_{2}\cdot X_{6}.$ (27)

We can apply first ${\rm Op}$ to $w_{1}$ and $w_{2}$ and compose together the
resulting operators,

$\displaystyle{\rm Op}(w_{1})(Y_{0},Y_{1},Y_{2})=Y_{0}\cdot a_{1}\cdot
Y_{1}\otimes Y_{2}\cdot a_{2}\cdot Y_{3},$ $\displaystyle{\rm
Op}(w_{2})(X_{0},\cdots,X_{6})=X_{0}\cdot b_{1}\cdot X_{1}\cdot b_{2}\cdot
X_{2}\otimes X_{3}\otimes X_{4}\cdot b_{3}\cdot X_{5}\otimes X_{6}$

Substituting to $Y_{0}\otimes Y_{1}\otimes Y_{2}$ the right-hand side of the
last equation, we recover (27).

The vector space $\bar{T}(\mathcal{A})$ is the natural space where take value
the signature of a smooth path $X$, see (3). To define partial contractions we
need to implement the freedom of permutations and the double tensor algebra
inside the usual signature. Let $w$ be a word in $T(\mathcal{A})$ with length
$n\geq 1$. Let $c=(c_{1},\cdots,c_{k})$ be a composition of $n$. The
composition $(c_{1},\cdots,c_{k})$ yields a splitting of $w$: we define the
element $\left[w\right]_{(n_{1},\cdots,n_{k})}\in\bar{T}^{2}(\mathcal{A})$ by

$\left[w\right]_{(n_{1},\cdots,n_{k})}=w_{1}\cdots w_{n_{1}}|w_{n_{1}+1}\cdots
w_{n_{1}+n_{2}}|\cdots|w_{n_{1}+\cdots+n_{k-1}+1}\cdots
w_{n_{1}+\cdots+n_{k}},$

with the convention $w_{c_{1}+\cdots+c_{i-1}+1}\cdots
w_{c_{1}+\cdots+c_{i-1}+c_{i}}=\emptyset$ if $c_{i}=0$.

######  3.5 Definition.

For any integer $n\geq 1$ and levelled tree $\sigma$ in
${\textrm{{{LT}}}}(n)$, we denote by $X^{\sigma}$ the map
$X^{\sigma}\colon[0,1]^{2}\to\bar{T}(A)$ given by

$X_{s,t}^{\sigma}=\int_{\Delta_{s,t}^{n-1}}\mathrm{d}X_{t_{\sigma(1)}}\otimes\cdots\otimes\mathrm{d}X_{t_{\sigma(n-1)}}\,,$
(28)

where $\sigma$ is identified with a permutation in $\mathfrak{S}_{n-1}$ when
$n\geq 2$ and $\mathbb{X}_{s,t}^{\bullet}=\emptyset$. For any levelled forest
$f=(\sigma,c)\in{\textrm{{{LF}}}}(m,n)$ we denote by $X^{f}$ the application
$X^{f}_{s,t}=\big{[}{{X}}_{s,t}^{\sigma}\big{]}_{c}$.

######  3.6 Definition.

For any integers $n\geq 1$, $n\geq m\geq 1$ and any levelled forest
$f=(\sigma,c)\in{\textrm{{{LF}}}}(m,n)$, we define the _partial-contraction of
$X$ along the forest $f$_ as a map
$\mathbb{X}^{f}\colon[0,1]^{2}\to\mathrm{Hom}(\mathcal{A}^{\otimes
n},\mathcal{A}^{\otimes m})$ given by

$\mathbb{X}_{s,t}^{f}=\mathrm{Op}\big{(}X^{f}_{s,t}\big{)}\,.$ (29)

######  3.7 Example.

Let us calculate the partial contraction associated to the levelled forest $f$
in Figure 3. In this case $n=5$ and the word on words representing $f$ is

$f=(13245,(2,0,2,1,0))$

We associate to it the formal expression

$dX_{t_{1}}dX_{t_{3}}\,\otimes\,\emptyset\otimes\,dX_{t_{2}}dX_{t_{4}}\,\otimes\,dX_{t_{5}}\,\otimes\,\emptyset.$

We then interleave a $10-$tuple $(A_{0}\,,\cdots,A_{9})$ of elements in
$\mathcal{A}$ between each expression $dX_{t_{i}}$, eventually replacing the
empty letter $\emptyset$,

$A_{0}\cdot dX_{t_{1}}\cdot A_{1}\cdot dX_{t_{3}}\cdot
A_{2}\,\otimes\,A_{3}\otimes\,A_{4}\cdot dX_{t_{2}}\cdot A_{5}\cdot
dX_{t_{4}}\cdot A_{6}\,\otimes\,A_{7}\cdot dX_{t_{5}}\cdot
A_{8}\,\otimes\,A_{9}$

Integrating over $\Delta^{n}_{s,t}$ we obtain the explicit form of
$\mathbb{X}_{s,t}^{f}(A_{0}\,,\cdots,A_{9})$,

$\int_{\Delta_{s,t}^{5}}\left(A_{0}\cdot dX_{t_{1}}\cdot A_{1}\cdot
dX_{t_{3}}\cdot A_{2}\,\right)\otimes\,A_{3}\otimes\left(\,A_{4}\cdot
dX_{t_{2}}\cdot A_{5}\cdot dX_{t_{4}}\cdot
A_{6}\,\right)\otimes\left(\,A_{7}\cdot dX_{t_{5}}\cdot
A_{8}\right)\,\otimes\,A_{9}\,.$

###  3.2. Chen relation

In this section, we use from time to time the symbol $\circ$ in place of
$\nabla_{\bar{T}^{2}(\mathcal{A})}$ or $\nabla_{\rm End^{2}_{\mathcal{A}}}$ to
improve readability. We describe how concatenation of paths lift to the full
and partial contractions operators, that is we write an extension of Chen
identity over iterated integral, see [Che54] for these operators.

######  3.8 Proposition (Chen relation).

For any forest $f\in{\textrm{{{LF}}}}$ and any three times
$(s,u,t)\in[0,1]^{3}$ one has

$\mathbb{X}^{f}_{s,t}=\sum_{f^{\prime}\subset f}\nabla_{\rm
End_{\mathcal{A}}^{2}}\left[\mathbb{X}_{u,t}^{f^{\prime}}\varobar\mathbb{X}^{f\backslash
f^{\prime}}_{s,u}\right].$ (30)

Written in term of the notations introduced in (16) and the map
$\mathbb{X}_{s,t}\colon\mathcal{L}\mathcal{F}\rightarrow\mathrm{End}^{2}_{\mathcal{A}}$
defined by $\mathbb{X}_{s,t}(f)=\mathbb{X}_{s,t}^{f}$, the equation (30)
becomes

${\mathbb{X}}_{s,t}=\nabla_{\mathrm{End}^{2}(\mathcal{A})}\circ(\mathbb{X}_{u,t}\varobar\mathbb{X}_{s,u})\circ\Delta\,.$
(31)

######  3.9 Example.

Before writing the proof, we check equation (30) on an explicit example given
by the levelled forest $f=(213,(2,1))$ in Figure 6, to see how the operations
combine themselves. In that case, the operator $\mathbb{X}^{f}_{s,t}$ is given
by

$\mathbb{X}^{f}_{s,t}(A_{0},\cdots,A_{5})=\int_{\Delta_{s,t}^{3}}\left(A_{0}\cdot
dX_{t_{2}}\cdot A_{1}\cdot dX_{t_{1}}\cdot
A_{2}\,\right)\otimes\,\left(A_{3}\cdot dX_{t_{3}}\cdot A_{4}\right)\,.$

Using the standard properties of Lebesgue integration, we can easily write

$\begin{split}\mathbb{X}^{f}_{s,t}=\mathbb{X}^{f}_{s,u}+\mathbb{X}^{f}_{u,t}&+\int_{t_{1}\in\Delta_{ut}^{1}}\int_{(t_{2},t_{3})\in\Delta^{2}_{su}}\left(A_{0}\cdot
dX_{t_{2}}\cdot A_{1}\cdot dX_{t_{1}}\cdot
A_{2}\,\right)\otimes\,\left(A_{3}\cdot dX_{t_{3}}\cdot A_{4}\right)\\\
&+\int_{(t_{1},t_{2})\in\Delta_{ut}^{2}}\int_{t_{3}\in\Delta^{1}_{su}}\left(A_{0}\cdot
dX_{t_{2}}\cdot A_{1}\cdot dX_{t_{1}}\cdot
A_{2}\,\right)\otimes\,\left(A_{3}\cdot dX_{t_{3}}\cdot
A_{4}\right)\,.\end{split}$ (32)

At the same time, we list all subforests in $f^{\prime}\subset f$ in Figure 6
together with $f\setminus f^{\prime}$.

$\begin{array}[]{ |c|c| }\hline\cr f&f\setminus f^{\prime}\\\
\hline\cr(\emptyset,(0,0))&(213,(2,1))\\\ (1,(1,0))&(23,(1,1))\\\
(21,(2,0))&(3,(0,1))\\\ (213,(2,1))&(\emptyset,(0,0))\\\ \hline\cr\end{array}$
Figure 12.  The subforests in Figure 6 presented as words on words in the
first column. In the second column, the cut of $f$ by each of these forests.

Proposition 3.8 states the equality

$\mathbb{X}^{f}_{s,t}=\mathbb{X}^{f}_{s,u}+\mathbb{X}^{f}_{u,t}+\mathbb{X}^{(1,(1,0))}_{u,t}\circ\mathbb{X}^{(23,(1,1))}_{s,u}+\mathbb{X}^{(21,(2,0))}_{u,t}\circ\mathbb{X}^{(3,(0,1))}_{s,u}\,,$
(33)

which is exactly (32).

###### Proof.

It is sufficient to show the identity when $s<u<t$. The statement of the
proposition is implied by the same statement but for the iterated integrals
$X^{f}_{s,t}$, $f\in{\textrm{{{LF}}}}$ since Op is a representation of the
word-insertions operad (see equation (25)). We prove then the identity

$X^{f}_{s,t}=\sum_{f^{\prime}\subset f}X_{u,t}^{f^{\prime}}\circ
X^{f\backslash f^{\prime}}_{s,u}$

by induction on the generation of $f$ and with $\circ$ the operation
$\nabla_{\rm\bar{T}^{2}(\mathcal{A})}$. The initialization is done for forests
with $0$ generations. Assume that the results as been proved for forests
having at most $N$ generations and let $f$ be a forest with $N+1$ generations.
Splitting the simplex $\Delta_{s,t}^{n+1}$ according to $s<u<t$ one has

$\displaystyle X_{s,t}^{f}=\int_{s}^{t}d{X}_{t_{1}}\circ
X_{s,t_{1}}^{f\backslash f_{1}}+\int_{u}^{t}d{X}_{t_{1}}\circ
X_{s,t_{1}}^{f\backslash f_{1}}=X_{s,u}^{f}+\int_{u}^{t}d{X}_{t_{1}}\circ
X_{s,t_{1}}^{f\backslash f_{1}}\,,$ (34)

where $f_{1}={\succ}_{1}(f)^{-}$ and $\circ$ is the word insertion of the
element

$\mathrm{d}X_{t_{1}}=\emptyset^{\otimes
i-1}\otimes\mathrm{d}X_{t_{1}}\otimes\emptyset^{|f|-i}\,,$

where $i$ is the order of the $i^{th}$ tree in the forest $f$ whose root is
decorated by $1$. By construction of $f\backslash f_{1}$, this forest has only
$N$ generations and the recursive hypothesis to the forest $f\backslash f_{1}$
implies

$X_{s,t_{1}}^{f\backslash f_{1}}=\sum_{f^{\prime\prime}\subset f\backslash
f_{1}}X_{u,t_{1}}^{f^{\prime\prime}}\circ X^{(f\backslash f_{1})\backslash
f^{\prime\prime}}_{s,u}\,.$

We insert this last relation into equation (34) to get the identity

$\int_{u}^{t}\mathrm{d}X_{t_{1}}\circ
X_{u,t_{1}}^{f^{\prime}}=\sum_{f^{\prime\prime}\subset f\backslash
f_{1}}\int_{u}^{t}\mathrm{d}X_{t_{1}}\circ
X_{u,t_{1}}^{f^{\prime\prime}}\circ\left[X^{(f\backslash f_{1})\backslash
f^{\prime\prime}}_{s,u}\right]=\sum_{f^{\prime}\subset
f\,f^{\prime}\neq\emptyset}X_{u,t}^{f^{\prime}}\circ X^{(f\backslash
f^{\prime})}_{s,u}.$

∎

######  3.10 Remark.

If we choose for the levelled tree $f$ the right comb tree, that is the tree
obtained by grafting corollas with two leaves with each others, always on the
rightmost nodes, we find back the classical Chen identity [Che54]. In fact, by
cutting such a tree we obtain a smaller comb tree and a levelled forest with
only straight trees, except for the last one which is a comb tree.

To the family of operators
$\\{\mathbb{X}^{f}_{s,t}\,,\;f\in{\textrm{{{LT}}}}\\}$, we now associate a
family of endomorphisms on

${\textrm{{{LT}}}}(\mathcal{A}):=\bigoplus_{n\geq 0}\mathcal{A}^{\otimes
n}\otimes\bigoplus_{\begin{subarray}{c}\tau\in{\textrm{{{LT}}}}\\\
|\tau|=n\end{subarray}}\mathbb{C}[\tau]$

For the remaining part of the article, we use the lighter notations

$a\otimes\tau=a\tau\in\mathcal{A}^{\otimes|\tau|}\otimes\mathbb{C}\tau\,,\quad{\textrm{{{LT}}}}(\mathcal{A})(\tau):=\mathcal{A}^{\otimes|\tau|}\tau\subset{\textrm{{{LT}}}}(\mathcal{A}).$

Although it is not yet clear if it is possible to associate to the full and
partial contractions operators a path on a certain convolution group of
representations, our statement of the Chen relation makes clear that any
prospective deconcatenation product $\Delta$ should act on a tree by cutting
it in all possible ways, generations after generations. In section 2.3 we
prove this cutting operation yields a comonoid in
$(\mathcal{L}\mathcal{F},\varobar)$.

From Proposition 3.8 we immediately deduce the following properties.

######  3.11 Proposition.

The family of maps $\bar{\mathbb{X}}_{s,t}$ defined by

$\begin{array}[]{cccc}\bar{\mathbb{X}}_{s,t}:&{\textrm{{{LT}}}}(\mathcal{A})&\rightarrow&{\textrm{{{LT}}}}(\mathcal{A}),\\\
&a\,\tau&\mapsto&\displaystyle\sum_{\tau^{\prime}\subset\tau}\bar{\mathbb{X}}_{s,t}^{\tau\backslash\tau^{\prime}}(a)\,\tau^{\prime}\end{array}$
(35)

have the following properties:

  1. 1.

for every levelled tree $\tau\in{\textrm{{{LT}}}}$

$(\bar{\mathbb{X}}_{s,t}-\mathrm{id})({\textrm{{{LT}}}}(\mathcal{A})(\tau))\subset\bigoplus_{\tau^{\prime}\subsetneq\tau}{\textrm{{{LT}}}}(\mathcal{A})(\tau^{\prime})\,.$

  2. 2.

for any $(s,u,t)\in[0,1]^{3}$ one has the so-called _non-commutative Chen’s
relations_ holds

$\bar{\mathbb{X}}_{s,t}=\bar{\mathbb{X}}_{u,t}\circ\bar{\mathbb{X}}_{s,u}.$

###### Proof.

It is sufficient to show only point 2 for any $s<u<t$, since point 1 is
trivial. For any given and $A\,\tau\in{\textrm{{{LT}}}}(\mathcal{A})$, to the
Chen’s relation in Proposition 3.8 implies

$\displaystyle\bar{\mathbb{X}}_{s,t}(A\,\tau)=\sum_{\tau^{\prime}\subset\tau}\bar{\mathbb{X}}_{s,t}^{\tau\backslash\tau^{\prime}}(A)\cdot\tau^{\prime}$
$\displaystyle=\sum_{\tau^{\prime}\subset\tau}\sum_{\tau^{\prime\prime}\subset\tau\backslash\tau^{\prime}}\bar{\mathbb{X}}_{u,t}^{\tau^{\prime\prime}}(\bar{\mathbb{X}}_{s,u}^{(\tau\backslash\tau^{\prime})\backslash\tau^{\prime\prime}}(A_{1}\otimes\cdots\otimes
A_{|\tau|}))\tau^{\prime}$
$\displaystyle=\sum_{\tau^{\prime}\subset\tau}\sum_{\tau^{\prime\prime}\subset\tau\backslash\tau^{\prime}}\bar{\mathbb{X}}_{u,t}^{\tau^{\prime\prime}}(\bar{\mathbb{X}}_{s,u}^{(\tau\backslash(\tau^{\prime\prime}\sharp\tau^{\prime})}(A_{1}\otimes\cdots\otimes
A_{|\tau|}))\tau^{\prime}\,.$

By performing the change of variable $g=f^{\prime\prime}\sharp f^{\prime}$,
$g^{\prime}=f^{\prime}$, we obtain

$\begin{split}\bar{\mathbb{X}}_{s,t}(A_{1}\otimes\cdots\otimes
A_{|f|}\,f)&=\sum_{g\subset f}\sum_{g^{\prime}\subset
g}\bar{\mathbb{X}}_{u,t}^{g\backslash
g^{\prime}}(\bar{\mathbb{X}}_{s,u}^{f\backslash g}(A_{1}\otimes\cdots\otimes
A_{|f|}))\,g^{\prime}\\\
&=(\bar{\mathbb{X}}_{u,t}\circ\bar{\mathbb{X}}_{s,t})(A_{1}\otimes\cdots\otimes
A_{|f|}\,f)\,.\end{split}$

∎

###  3.3. Integration by part properties

Another important property of iterated integral are integration by part
formulae, e.g. the identity

$\int_{\Delta^{2}_{s,t}}\mathrm{d}X_{t_{1}}\otimes\mathrm{d}X_{t_{2}}+\int_{\Delta^{2}_{s,t}}\mathrm{d}X_{t_{2}}\otimes\mathrm{d}X_{t_{1}}=(X_{t}-X_{s})\otimes(X_{t}-X_{s})\,.$

We investigate the consequences of such identities at the level of full-
partial contractions and their related associated endomorphisms
$\bar{\mathbb{X}}_{s,t}$. These relations imply a specific "compatibility
condition" with respect to a certain product on
${\textrm{{{LT}}}}(\mathcal{A})$ defined from the shuffle operations on trees
in Definition 2.8. We start with the consequences on partial contractions.

######  3.12 Proposition.

For any given couple of forests $f$ and $g$ with ${\sf nt}(g)=|f|$ and any
$(s,t)\in[0,1]^{2}$ one has

$\nabla_{\rm
End_{\mathcal{A}}^{2}}(\mathbb{X}^{f}_{s,t}\varobar\mathbb{X}^{g}_{s,t})=\sum_{s\in\textrm{{{Sh}}}(\|f\|,\|g\|)}\mathbb{X}_{s,t}^{s\cdot(f\,\\#\,g)}.$
(36)

Written in term of the notations introduced in (20) and the map
$\mathbb{X}_{s,t}\colon\mathcal{L}\mathcal{F}\rightarrow\mathrm{End}^{2}_{\mathcal{A}}$
defined by $\mathbb{X}_{s,t}(f)=\mathbb{X}_{s,t}^{f}$, the equation (36)
becomes

$\nabla_{\mathrm{End}^{2}(\mathcal{A})}(\mathbb{X}_{s,t}\varobar\mathbb{X}_{s,t})=\mathbb{X}_{s,t}\nabla\,.$
(37)

###### Proof.

Let us fix $s<t$. Writing the forests $f$ and $g$ as the split permutations
$f=(\sigma_{1},c_{1})$ and $g=(\sigma_{2},c_{2})$, it follows from the shuffle
identity for iterated integrals of $X$ that one has the identity

$\displaystyle X^{\sigma_{1}}_{s,t}\otimes X^{\sigma_{2}}_{s,t}=\sum_{s\in{\rm
Sh}(|f|-1,|g|-1)}\int_{\Delta^{|f|+|g|-2}_{s,t}}\mathrm{d}X_{t_{(s\circ\sigma)(1)}}\otimes\cdots\otimes\mathrm{d}X_{t_{(s\circ\sigma)(|f|+|g|-2))}}\,,$
(38)

where $\sigma=\sigma_{1}\otimes\sigma_{2}$. By applying the split permutation
of $f\\#g$ to both sides we deduce the identity

$\displaystyle X^{f\\#g}_{s,t}=\sum_{s\in{\sf
Sh}(|f|-1,|g|-1)}\int_{\Delta^{|f|+|g|-2}_{s,t}}X^{s\cdot(f\\#g)}_{s,t}\,.$
(39)

Composing with $\mathrm{Op}$ we conclude. ∎

We restate this identity at the level of $\bar{\mathbb{X}}_{s,t}$. This task
can be done by introducing an operadic composition $L$ on a collection of
words with entries in $\mathcal{A}$ different from before. Together with the
shuffle product on levelled trees, this operadic composition yields a
_structural_ map ${\sf L}$ on ${\textrm{{{LT}}}}^{\sharp}(\mathcal{A})$.
Further properties of ${\sf L}$ will turn central in better understanding the
composition of Taylor series for the fields $a,b$ in equation (2).

######  3.13 Definition (Faces substitution).

We define the collection of vector spaces $\mathcal{F}\mathcal{S}$ by

$\mathcal{F}\mathcal{S}(n)=\mathcal{A}^{\otimes n},\leavevmode\nobreak\ n\geq
1,.$

Next, define
$L:\mathcal{F}\mathcal{S}\circ\mathcal{F}\mathcal{S}\rightarrow\mathcal{F}\mathcal{S}$
as follows. Pick a word $U\in\mathcal{A}^{\otimes p+1}$ and words
$A^{i}\in\mathcal{A}^{\otimes m_{i}}$, $1\leq i\leq p$,
$A^{i}=(A^{i}_{(1)}\otimes\cdots A^{i}_{(m_{i})})$ and set

$\displaystyle L(U\otimes A^{1}\otimes\cdots\otimes A^{p})$
$\displaystyle:=\Big{(}U_{(1)}\cdot A^{1}_{(1)}\Big{)}\otimes
A^{1}_{(2)}\otimes\cdots\otimes\Big{(}A_{(m_{1})}^{1}\cdot U_{(2)}\cdot
A^{2}_{(1)}\Big{)}\otimes\cdots\otimes\Big{(}A^{p}_{(m_{p})}\cdot
U_{p}\Big{)}.$

The word $\mathbf{1}\otimes\mathbf{1}$ acts as the unit for $L$.

We denote by ${\sf FS}$ the graded vector space equal to the direct sum of all
vector spaces in the collection $\mathcal{F}\mathcal{S}$. Notice that elements
of $\mathcal{A}$ are $0$-ary operators in the collection
$\mathcal{F}\mathcal{S}$ and, for example, the above formula for $L$ gives
$L(U_{1}\otimes U_{(2)}\circ A)=U_{1}\cdot A\cdot U_{2}\in\mathcal{A}$, with
$U_{(1)}\otimes U_{(2)}\in\mathcal{A}^{\otimes 2}$.

######  3.14 Proposition.

${\rm FS}=(\mathcal{F}\mathcal{S},L,\mathbf{1}\otimes\mathbf{1})$ is an
operad.

###### Proof.

The following proposition holds and rests on associativity of the product on
$\mathcal{A}$. ∎

In the collection $\mathcal{F}\mathcal{S}$, a word with length $n$ is an
operator with $n-1$ entries, the inner gaps between the letters. So far, a
levelled tree was considered as an operator with as much inputs as it has
leaves. However, there is an alternative way to see such a tree as an
operator: by considering the _faces_ of the tree as inputs. A face is a region
enclosed between two consecutive leaves and delimited by two paths of edges
meeting at the least common ancestor, see Fig. 13.

Figure 13.  The faces of a levelled tree are indicated with arrows.

We denote by ${\textrm{{{LT}}}}^{\\#}$ the set of levelled trees graded by the
numbers of faces, ${\textrm{{{LT}}}}^{\\#}(n)$ the set of levelled trees with
$n$ faces, and ${\sf LT}^{\sharp}(\mathcal{A})$ the space ${\sf
LT}(\mathcal{A})$ seen as a graded vector space with ${\sf
LT}^{\\#}(\mathcal{A})(n)=\mathbb{C}{\sf
LT}^{\\#}(n)\otimes\mathcal{F}\mathcal{S}(n)$. Notice that the endomorphism
$\bar{\mathbb{X}}_{s,t}$ we defined in the previous section satisfies:

$(\bar{\mathbb{X}}_{s,t}-\mathrm{id})({\textrm{{{LT}}}}^{\\#}(\mathcal{A})(n))\subset\displaystyle\bigoplus_{k<n}{\textrm{{{LT}}}}^{\\#}(\mathcal{A})(k)\,.$

We also set for any levelled tree $\tau\in{\textrm{{{LT}}}}$,

${{\sf
LT}^{\\#}(\mathcal{A})}(\tau)=\mathbb{C}\tau\otimes\mathcal{FS}(\|\tau\|)\,.$

The space ${{\sf LT}^{\\#}(\mathcal{A})}$ is equipped with an involution
$\star_{{{\sf LT}^{\\#}(\mathcal{A})}}$, defined by

$\star_{{{\sf LT}^{\\#}(\mathcal{A})}}(A_{1}\otimes\cdots\otimes
A_{p}\,\tau)=A_{p}^{\star}\otimes\cdots\otimes A_{1}^{\star}\,\theta(\tau)$
(40)

The graded vector space ${\textrm{{{LT}}}}^{\\#}(\mathcal{A})$ yields a
collection $\mathcal{L}\mathcal{T}^{\\#}(\mathcal{A})$ by setting the space
$n$-ary operators $\mathcal{L}\mathcal{T}^{\\#}(\mathcal{A})$ equal to
${\textrm{{{LT}}}}^{\\#}(\mathcal{A})(n)$. We set abusively

${\textrm{{{LT}}}}^{\\#}(\mathcal{A})\circ{\textrm{{{LT}}}}^{\\#}(\mathcal{A}):=\bigoplus_{n\geq
0}\big{(}\mathcal{L}\mathcal{T}^{\sharp}(\mathcal{A})\circ\mathcal{L}\mathcal{T}^{\sharp}(\mathcal{A})\big{)}(n)$

For any $U,A\in{\textrm{{{LT}}}}^{\\#}(\mathcal{A})$, we introduce the
notation

$U\circ
A:=\sum_{\begin{subarray}{c}\tau,\tau_{1},\cdots,\tau_{\|\tau\|}\end{subarray}}U^{\tau}\,\tau\otimes
A^{\tau_{1}}\,\tau_{1}\otimes\cdots\otimes
A^{\tau_{\|f\|}}\,\tau_{\|\tau\|}\in{\textrm{{{LT}}}}^{\\#}(\mathcal{A})\circ{\textrm{{{LT}}}}^{\\#}(\mathcal{A}).$

Observe that $U\circ A$ is only linear on $U$, not on $A$.

######  3.15 Definition.

Define the morphism of graded vector spaces

${\sf
L}:{\textrm{{{LT}}}}^{\\#}(\mathcal{A})\circ{\textrm{{{LT}}}}^{\\#}(\mathcal{A})\rightarrow{\textrm{{{LT}}}}^{\\#}(\mathcal{A})$
$\displaystyle{\sf L}(U\otimes(A_{1}\otimes\cdots\otimes
A_{\|\alpha\|}))=\sum_{\begin{subarray}{c}\alpha\\\
\tau_{1},\cdots,\tau_{\|\alpha\|}\end{subarray}}L(U^{\alpha}\otimes
A^{\tau_{1}}_{1}\otimes\cdots\otimes
A^{\tau_{\|\alpha\|}}_{{\|\alpha\|}})\cdot\tau_{1}\shuffle\cdots\shuffle\tau_{\|\alpha\|},$
(41)

where

$U=\sum_{\alpha\in{\textrm{{{LT}}}}^{\\#}}U^{\alpha}\,\alpha,\leavevmode\nobreak\
A_{i}=\sum_{\tau_{i}\in{\textrm{{{LT}}}}^{\\#}}A_{i}^{\tau_{i}}\,\tau_{i}\,\in{\textrm{{{LT}}}}^{\\#}(\mathcal{A})\,.$

######  3.16 Lemma.

Let $\alpha$ and $\beta$ be two levelled trees in LT. For any
$A\in\mathcal{A}^{\otimes|\alpha|+|\beta|-1}$ one has

$\bar{\mathbb{X}}_{s,t}(A\,\alpha\shuffle\beta)=\sum_{\tau_{\alpha}\subset\alpha,\tau_{\beta}\subset\beta}\bar{\mathbb{X}}_{s,t}^{(\alpha\backslash\tau_{\alpha})\shuffle(\beta\backslash\tau_{\beta})}(A)\,\tau_{\alpha}\shuffle\tau_{\beta}\,.$
(42)

###### Proof.

The proof consists essentially in a re-summation. It stems from the definition
of the map $\bar{\mathbb{X}}_{s,t}$ that:

$\bar{\mathbb{X}}_{s,t}(A\cdot\alpha\shuffle\beta)=\sum_{\begin{subarray}{c}\tau\in\alpha\shuffle\beta\\\
\tau^{\prime}\subset\tau\end{subarray}}\bar{\mathbb{X}}_{s,t}^{\tau\backslash\tau^{\prime}}(A)\,\tau^{\prime}.$
(43)

Let $\tau\in\alpha\shuffle\beta$ a tree obtained by shuffling vertically the
generations of $\alpha$ and $\beta$ and pick $\tau^{\prime}\subset\tau$ a
subtree. Let $s$ be the shuffle in ${\rm Sh}(\sharp\alpha,\sharp\beta)$ such
that $\tau^{-1}=(\alpha\otimes\beta)\circ s^{-1}$. We associate to the pair
$(\tau,\tau^{\prime})$ a triple which consists in the tree $\tau$, and two
others trees $\tau^{\prime}_{\alpha}\subset\alpha$ and
$\tau^{\prime}_{\beta}\subset\beta$ satisfying

$\tau^{\prime}=(\tau^{\prime}_{\alpha}\otimes\tau^{\prime}_{\beta})\circ\tilde{s}^{-1},$

where $\tilde{s}$ is a shuffle in
$\textrm{{{Sh}}}\left(\|\tau^{\prime}_{\alpha}\|,\|\tau^{\prime}_{\beta}\|\right)$.
Such a permutation $\tilde{s}$ is unique, in fact it is obtained from $s^{1}$
by extracting the first $\|\tau^{\prime}\|$ letters of the word representing
$s^{-1}$, followed by standardization and finally inversion. Recall that
_standardization_ means that we translate the first $\|\tau^{\prime}\|$
letters representing $s^{-1}$, while maintaining their relative order to
obtain a word on integers in the interval $\llbracket
1,\sharp\tau^{\prime}\rrbracket$. It is clear that the map
$\phi:(\tau,\tau^{\prime})\mapsto(\tau,\tau^{\prime}_{\alpha},\tau^{\prime}_{\beta})$
is injective. Now, given
$\tau_{\alpha}\subset\alpha,\tau_{\beta}\subset\beta$, and two shuffles
$s_{-}\in\textrm{{{Sh}}}(\|\tau^{\prime}_{\alpha}\|,\|\tau^{\prime}_{\beta}\|)$,
$s_{+}\in\textrm{{{Sh}}}(\|\alpha\backslash\tau^{\prime}_{\alpha}\|,\|\beta\backslash\tau^{\prime}_{\beta}\|)$,
we define a third shuffle $s_{-+}$ in $\textrm{{{Sh}}}(\|\alpha\|,\|\beta\|)$
by requiring

$s_{-+}(i)=s_{-}(i),\leavevmode\nobreak\ 1\leq
i\leq\|\tau^{\prime}_{\alpha}\|,\leavevmode\nobreak\ \leavevmode\nobreak\
s_{-+}(\|\tau^{\prime}_{\alpha}\|+i)=s_{+}(i)+s_{-}(\|\tau^{\prime}_{\alpha}\|),\leavevmode\nobreak\
1\leq i\leq\|\tau_{\alpha}\backslash\tau^{\prime}_{\alpha}\|$

The map
$\delta:(\tau^{\prime}_{\alpha},\tau^{\prime}_{\beta},s_{+},s_{-})\mapsto(\tau,\tau^{\prime}_{\alpha},\tau^{\prime}_{\beta})$
with $\tau^{-1}=\alpha\otimes\beta\circ s_{-+}^{-1}$ is a bijection between
the image of $\phi$ and

$\mathcal{S}=\\{(\tau_{\alpha},\tau_{\beta},s_{+},s_{-}),\leavevmode\nobreak\
\tau_{\alpha}\subset\alpha,\tau_{\beta}\subset\beta,s_{-}\in\textrm{{{Sh}}}(\|\tau_{\alpha}\|,\|\tau_{\beta}\|),s_{+}\in\textrm{{{Sh}}}(\|\alpha\backslash\tau_{\alpha}\|,\|\beta\backslash\tau_{\beta}\|)\\}.$

We can thus rewrite the sum in the right hand side of (43) as follows:

$\displaystyle\sum_{\begin{subarray}{c}\tau\in\alpha\shuffle\beta\\\
\tau^{\prime}\subset\tau\end{subarray}}\bar{\mathbb{X}}_{s,t}^{\tau\backslash\tau^{\prime}}(A)\tau^{\prime}$
$\displaystyle=\sum_{\tau_{\alpha},\tau_{\beta},s_{+},s_{-}\in\mathcal{S}}\bar{\mathbb{X}}_{s,t}^{(\alpha\otimes\beta)\circ
s_{-+}\backslash\,(\tau^{\prime}_{\alpha}\otimes\tau^{\prime}_{\beta})\circ
s_{-}^{-1}}(\tau^{\prime}_{\alpha}\otimes\tau^{\prime}_{\beta})\circ
s_{-}^{-1}$

Now, we observe that the forest $(\alpha\otimes\beta)\circ
s_{-+}\backslash\,(\tau^{\prime}_{\alpha}\otimes\tau^{\prime}_{\beta})\circ
s_{-}^{-1}$ does only depend on the trees $\tau_{\alpha},\tau_{\beta}$ and the
shuffle $s_{+}$. Summing over all shuffles $s_{+}$, we get
$\alpha\backslash\tau_{\alpha}\shuffle\beta\backslash\tau_{\beta}$. The
statement of the Lemma follows by computing the sum over $s_{-}$. ∎

######  3.17 Definition (Product on the faces-substitution collection).

For any $A\in\mathcal{FS}(n)$ and $B\in\mathcal{FS}(m)$, we define their
product $A\cdot B$ by

$A\cdot B=A_{(1)}\otimes\cdots\otimes(A_{(n+1)}\cdot
B_{(1)})\otimes\cdots\otimes B_{(m+1)}.$

The product $\cdot$ is a product on the collection $\mathcal{FS}$ with unit
$1\in\mathcal{FS}(0)$, $A\cdot B\in\mathcal{FS}(n+m)$. Note that we use the
same symbol $\cdot$ for the above defined product and the product on the
algebra $\mathcal{A}$ for the reason that the former restricts to the latter
on $\mathcal{F}\mathcal{S}(0)$.

######  3.18 Remark.

The product $\cdot$ has a very special form, namely:

$A\cdot B=(\mathbf{1}\otimes\mathbf{1}\otimes\mathbf{1})\circ(A\otimes
B)=L((\mathbf{1}\otimes\mathbf{1}\otimes\mathbf{1})\otimes(A\otimes B)).$

and the relation $L(m\otimes(id_{\sf FS}\otimes m))=L(m\otimes(m\otimes
id_{\sf FS})$ with $m=\mathbf{1}\otimes\mathbf{1}\otimes\mathbf{1}$ entails
associativity of the product $\cdot$. We say that
$m\in\mathcal{F}\mathcal{S}(2)$ is a _multiplication_ in the operad
$(\mathcal{F}\mathcal{S},L)$. In addition, associativity of the operadic
composition $L$ results in the following distributivity law

$(A\cdot B)\circ C=(A\circ B)\cdot(B\circ C),\leavevmode\nobreak\ A,B,C\in{\sf
FS}.$

Conjointly with the shuffle product on levelled trees, the product $\cdot$
brings in a graded algebra product
$\shuffle:{\textrm{{{LT}}}}^{\\#}(\mathcal{A})\otimes{\textrm{{{LT}}}}^{\\#}(\mathcal{A})\rightarrow{\textrm{{{LT}}}}^{\\#}(\mathcal{A})$,
namely

$(A\,\alpha)\shuffle(B\,\beta)=(A\cdot B)\,\alpha\shuffle\beta,$ (44)

with unit $\mathbf{1}\cdot\,\leavevmode\hbox to4.98pt{\vbox
to6.95pt{\pgfpicture\makeatletter\hbox{\hskip 2.49023pt\lower-2.49023pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{1.4143pt}{2.84544pt}\pgfsys@curveto{1.4143pt}{3.62654pt}{0.7811pt}{4.25975pt}{0.0pt}{4.25975pt}\pgfsys@curveto{-0.7811pt}{4.25975pt}{-1.4143pt}{3.62654pt}{-1.4143pt}{2.84544pt}\pgfsys@curveto{-1.4143pt}{2.06435pt}{-0.7811pt}{1.43114pt}{0.0pt}{1.43114pt}\pgfsys@curveto{0.7811pt}{1.43114pt}{1.4143pt}{2.06435pt}{1.4143pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ $}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,$. The above
relation on full-contractions operators yield compatibility of the
endomorphism $\bar{\mathbb{X}}_{s,t}$ with the product $\shuffle$ defined in
(44).

######  3.19 Proposition.

Let $\alpha$ and $\beta$ be two levelled trees and pick
$A\in\mathcal{A}^{|\alpha|},B\in\mathcal{A}^{\otimes|\beta|}$,

$\bar{\mathbb{X}}_{s,t}((A\,\alpha)\shuffle(B\,\beta))=\bar{\mathbb{X}}_{s,t}(A\,\alpha)\shuffle\bar{\mathbb{X}}_{s,t}(B\,\beta)$

###### Proof.

The result is a simple consequence of the previous Proposition 3.16 and the
shuffle relation for the partial contraction operators (42) In fact, one has
the trivial identities

$\displaystyle\bar{\mathbb{X}}_{s,t}((A\cdot B)\cdot\alpha\shuffle\beta)$
$\displaystyle=\sum_{\begin{subarray}{c}\tau_{\alpha}\subset\alpha,\tau_{\beta}\subset\beta\end{subarray}}{\bar{\mathbb{X}}}^{\alpha\backslash\tau_{\alpha}\shuffle\beta\backslash\tau_{\beta}}_{s,t}(A\cdot
B)\,\tau_{\alpha}\shuffle\tau_{\beta}$
$\displaystyle=\sum_{\begin{subarray}{c}\tau_{\alpha}\subset\alpha,\tau_{\beta}\subset\beta\end{subarray}}{\bar{\mathbb{X}}}^{\alpha\backslash\tau_{\alpha}}(A)\cdot{\bar{\mathbb{X}}}_{s,t}^{\beta\backslash\tau_{\beta}}(B)\,\tau_{\alpha}\shuffle\tau_{\beta}=\bar{\mathbb{X}}_{s,t}(A)\cdot\bar{\mathbb{X}}_{s,t}(B)\,.$

Thereby obtaining the desired identity. ∎

######  3.20 Corollary.

For all times $0<s<t<1$, it holds that:

${\sf
L}\,\circ\,(\mathrm{id}\circ\bar{\mathbb{X}}_{s,t})=\bar{\mathbb{X}}_{s,t}\,\circ\,{\sf
L}\,.$ (45)

###### Proof.

According to Proposition 3.19, one has the identity

$\displaystyle{\sf
L}(U^{\alpha}\otimes\bar{\mathbb{X}}_{s,t}(A^{\beta_{1}}\,\beta_{1})\otimes\cdots\otimes\bar{\mathbb{X}}_{s,t}(A^{\beta_{\sharp\alpha}}\,\beta_{\sharp\alpha}))$
$\displaystyle=\bar{\mathbb{X}}_{s,t}(U^{\alpha}_{(1)}\,\leavevmode\hbox
to4.98pt{\vbox to6.95pt{\pgfpicture\makeatletter\hbox{\hskip
2.49023pt\lower-2.49023pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{1.4143pt}{2.84544pt}\pgfsys@curveto{1.4143pt}{3.62654pt}{0.7811pt}{4.25975pt}{0.0pt}{4.25975pt}\pgfsys@curveto{-0.7811pt}{4.25975pt}{-1.4143pt}{3.62654pt}{-1.4143pt}{2.84544pt}\pgfsys@curveto{-1.4143pt}{2.06435pt}{-0.7811pt}{1.43114pt}{0.0pt}{1.43114pt}\pgfsys@curveto{0.7811pt}{1.43114pt}{1.4143pt}{2.06435pt}{1.4143pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ $}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)\shuffle\bar{\mathbb{X}}_{s,t}(A^{\beta_{1}}\,\beta_{1})\shuffle\bar{\mathbb{X}}_{s,t}(U^{\alpha}_{(2)}\,\leavevmode\hbox
to4.98pt{\vbox to6.95pt{\pgfpicture\makeatletter\hbox{\hskip
2.49023pt\lower-2.49023pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{1.4143pt}{2.84544pt}\pgfsys@curveto{1.4143pt}{3.62654pt}{0.7811pt}{4.25975pt}{0.0pt}{4.25975pt}\pgfsys@curveto{-0.7811pt}{4.25975pt}{-1.4143pt}{3.62654pt}{-1.4143pt}{2.84544pt}\pgfsys@curveto{-1.4143pt}{2.06435pt}{-0.7811pt}{1.43114pt}{0.0pt}{1.43114pt}\pgfsys@curveto{0.7811pt}{1.43114pt}{1.4143pt}{2.06435pt}{1.4143pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ $}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)\cdots\bar{\mathbb{X}}_{s,t}(A^{\beta_{\sharp\alpha}}\cdot\beta_{\sharp\alpha})\shuffle\bar{\mathbb{X}}_{s,t}(U^{\alpha}_{(|\alpha|)}\,\leavevmode\hbox
to4.98pt{\vbox to6.95pt{\pgfpicture\makeatletter\hbox{\hskip
2.49023pt\lower-2.49023pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{1.4143pt}{2.84544pt}\pgfsys@curveto{1.4143pt}{3.62654pt}{0.7811pt}{4.25975pt}{0.0pt}{4.25975pt}\pgfsys@curveto{-0.7811pt}{4.25975pt}{-1.4143pt}{3.62654pt}{-1.4143pt}{2.84544pt}\pgfsys@curveto{-1.4143pt}{2.06435pt}{-0.7811pt}{1.43114pt}{0.0pt}{1.43114pt}\pgfsys@curveto{0.7811pt}{1.43114pt}{1.4143pt}{2.06435pt}{1.4143pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ $}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)$
$\displaystyle=\bar{\mathbb{X}}_{s,t}\big{(}(U^{\alpha}_{(1)}\,\leavevmode\hbox
to4.98pt{\vbox to6.95pt{\pgfpicture\makeatletter\hbox{\hskip
2.49023pt\lower-2.49023pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{1.4143pt}{2.84544pt}\pgfsys@curveto{1.4143pt}{3.62654pt}{0.7811pt}{4.25975pt}{0.0pt}{4.25975pt}\pgfsys@curveto{-0.7811pt}{4.25975pt}{-1.4143pt}{3.62654pt}{-1.4143pt}{2.84544pt}\pgfsys@curveto{-1.4143pt}{2.06435pt}{-0.7811pt}{1.43114pt}{0.0pt}{1.43114pt}\pgfsys@curveto{0.7811pt}{1.43114pt}{1.4143pt}{2.06435pt}{1.4143pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ $}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)\shuffle(A^{\beta_{1}}\,\beta_{1})\shuffle(U^{\alpha}_{(2)}\,\leavevmode\hbox
to4.98pt{\vbox to6.95pt{\pgfpicture\makeatletter\hbox{\hskip
2.49023pt\lower-2.49023pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{1.4143pt}{2.84544pt}\pgfsys@curveto{1.4143pt}{3.62654pt}{0.7811pt}{4.25975pt}{0.0pt}{4.25975pt}\pgfsys@curveto{-0.7811pt}{4.25975pt}{-1.4143pt}{3.62654pt}{-1.4143pt}{2.84544pt}\pgfsys@curveto{-1.4143pt}{2.06435pt}{-0.7811pt}{1.43114pt}{0.0pt}{1.43114pt}\pgfsys@curveto{0.7811pt}{1.43114pt}{1.4143pt}{2.06435pt}{1.4143pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ $}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)\cdots(A^{\beta_{\sharp\alpha}}\cdot\beta_{\sharp\alpha})\shuffle(U^{\alpha}_{(|\alpha|)}\,\leavevmode\hbox
to4.98pt{\vbox to6.95pt{\pgfpicture\makeatletter\hbox{\hskip
2.49023pt\lower-2.49023pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{1.4143pt}{2.84544pt}\pgfsys@curveto{1.4143pt}{3.62654pt}{0.7811pt}{4.25975pt}{0.0pt}{4.25975pt}\pgfsys@curveto{-0.7811pt}{4.25975pt}{-1.4143pt}{3.62654pt}{-1.4143pt}{2.84544pt}\pgfsys@curveto{-1.4143pt}{2.06435pt}{-0.7811pt}{1.43114pt}{0.0pt}{1.43114pt}\pgfsys@curveto{0.7811pt}{1.43114pt}{1.4143pt}{2.06435pt}{1.4143pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ $}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,)\big{)}$
$\displaystyle=\bar{\mathbb{X}}_{s,t}({\sf L}(U^{\alpha}\,\alpha\otimes
A^{\beta_{1}}\,\beta_{1}\otimes\cdots\otimes
A^{\beta_{\sharp\alpha}}\,\beta_{\sharp\alpha}))\,.$

From which we deduce the equality. ∎

Summing up the result in this section and the previous one, we can actually
state the properties of $\bar{\mathbb{X}}_{s,t}$ by introducing a suitable
group of endormorphisms.

######  3.21 Theorem.

Denoting by $G(\mathcal{A})$ the group

$G(\mathcal{A})=\\{\bar{\mathbb{X}}\in\mathrm{Hom}_{\mathrm{Alg}}({{\sf
LT}^{\\#}(\mathcal{A})},{{\sf
LT}^{\\#}(\mathcal{A})}):(\bar{\mathbb{X}}-\mathrm{id})({{\sf
LT}^{\\#}(\mathcal{A})}(\tau))\subset\bigoplus_{\tau^{\prime}\subset\tau}{{\sf
LT}^{\\#}(\mathcal{A})}(\tau^{\prime})\\}\,,$ (46)

the endormorphisms $\bar{\mathbb{X}}_{s,t}$ satisfy the following properties

  1. (1)

$\bar{\mathbb{X}}_{s,t}\in G(\mathcal{A})$,

  2. (2)

for every time $(s,u,t)\in[0,1]^{3}$,
$\bar{\mathbb{X}}_{s,t}=\bar{\mathbb{X}}_{u,t}\circ\bar{\mathbb{X}}_{s,u}$.

##  4\. Group of non-commutative signatures

Looking again at the formal Peano-Picard expansion sketched in (6), we see a
sum of full contraction operators
$\\{\mathbb{X}^{\tau}_{s,t}\\}_{\tau\in{\textrm{{{LT}}}}}$ applied to generic
elements of the algebra $\mathcal{A}$, those operators appear thus as the
fundamental objects to generalise in a rough path setting. As explained in the
previous section, we were forced to consider partial contraction operators
indexed by levelled binary forests to write the Chen relation for these
operators. These operators appear as coefficients of an endomorphism
$\mathbb{X}$ acting on ${\textrm{{{LT}}}}({\sf\mathcal{A}})$. These
"coefficients" associated with forests can not be related to the
"coefficients" associated with trees if $\mathcal{A}$ is truly infinite
dimensional. A bit more formally, the application co-restricting an
endomorphism $\mathbb{X}_{s,t}$, for any pair of times $s<t$ to
$\mathbb{C}\,\left[\,\leavevmode\hbox to4.98pt{\vbox
to6.95pt{\pgfpicture\makeatletter\hbox{\hskip 2.49023pt\lower-2.49023pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{1.4143pt}{2.84544pt}\pgfsys@curveto{1.4143pt}{3.62654pt}{0.7811pt}{4.25975pt}{0.0pt}{4.25975pt}\pgfsys@curveto{-0.7811pt}{4.25975pt}{-1.4143pt}{3.62654pt}{-1.4143pt}{2.84544pt}\pgfsys@curveto{-1.4143pt}{2.06435pt}{-0.7811pt}{1.43114pt}{0.0pt}{1.43114pt}\pgfsys@curveto{0.7811pt}{1.43114pt}{1.4143pt}{2.06435pt}{1.4143pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ $}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right]$

$\bar{\mathbb{X}}(\cdot)\mapsto\sum_{\tau\in{\textrm{{{LT}}}}}\bar{\mathbb{X}}^{\tau}_{s,t}(\cdot)\,\leavevmode\hbox
to4.98pt{\vbox to6.95pt{\pgfpicture\makeatletter\hbox{\hskip
2.49023pt\lower-2.49023pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{1.4143pt}{2.84544pt}\pgfsys@curveto{1.4143pt}{3.62654pt}{0.7811pt}{4.25975pt}{0.0pt}{4.25975pt}\pgfsys@curveto{-0.7811pt}{4.25975pt}{-1.4143pt}{3.62654pt}{-1.4143pt}{2.84544pt}\pgfsys@curveto{-1.4143pt}{2.06435pt}{-0.7811pt}{1.43114pt}{0.0pt}{1.43114pt}\pgfsys@curveto{0.7811pt}{1.43114pt}{1.4143pt}{2.06435pt}{1.4143pt}{2.84544pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{2.84544pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{2.84544pt}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\hbox{{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{ }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\ $}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,$ (47)

is not injective, _we are lacking relations between partial and full
contractions operators._ Worth is the fact, that data of $\mathbb{X}$ is in
fact equivalent to the data of all iterated integrals of $X$, so that the
previous section is in fact a mere, though much more intricated reformulation
of the classical theory. Yet, we explain in this section how to get rid of
this partial contraction operators while maintaining a Chen relation for an
object comprising only full-contraction operators. These partial contraction
operators are turned into "technical proxys", that can be constructed from the
operators associated to levelled trees and bearing no additional information
on small scale behaviour of the paths, but allows for an efficient formulation
of the Chen relation.

###  4.1. Algebra of faces-contractions

To define a proper group where full contractions take value, we introduce a
new collection of operators, that we call _faces-contractions_. These
operators will replace words with entries in $\mathcal{A}$ of the previous
section.

######  4.1 Definition (Faces-contractions).

For any $\tau\in{\textrm{{{LT}}}}$, $\tau\neq\,\leavevmode\hbox to4.98pt{\vbox