On the Signature of a Path in an Operator Algebra

. We introduce a class of operators associated with the signature of a smooth path X with values in a C (cid:63) algebra A . These operators serve as the basis of Taylor expansions of solutions to controlled diﬀerential equations of interest in noncommutative probability. They are deﬁned by fully contracting iterated integrals of X , seen as tensors, with the product of 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 for noncommutative probability.


Introduction
This work intends to explore a direction suggested in [8] and aims to use paths principles for studying the following class of differential equations In the above equation, the driving path X : [0, 1] → A takes values in an unital C ⋆ -algebra (A, •, ⋆, ∥ • ∥) with unity 1 A and a, b : A → A are two polynomial functions or Fourier transforms of regular measures with exponential moments, see [3,8].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 (1.1).In this work, we focus on the algebra underlying Taylor expansions of solutions to equations (1.1), discarding other crucial aspects (such as measurability).

The rough paths approach
In the nineties [15], T.J. Lyons proposed the appropriate mathematical framework to study controlled differential equations In (1.2), the solution Y is a continuous path in R d , σ : R d → End R n , R d is a smooth vector field and the driving path X is Hölder continuous.If X is smooth, standard differential calculus provides a rigorous interpretation to (1.2).For paths with lower regularity, Young's theory of integration [23] gives sense to equation (1.2) driven by an Hölder regular path X with exponent greater than 1 2 .Interesting stochastic driving paths are too irregular for Young integration.For instance, Brownian trajectories are only 1  2 − ε, ε > 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 (1.2).Let us add more details.Given a smooth field σ and Y 0 ∈ R n the solution map Φ : X → Y to equation (1.2) 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 (1.2), one quickly reckons that the solution map Φ is a linear function of the entire signature of X, that is the infinite collection of tensors, where ∆ n s,t := {s < t 1 < • • • < t n < t} is the n-dimensional simplex.Signatures of smooth paths support a one-parameter family of topologies with respect to which Φ 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 a two parameters function (s, t) → X s,t with values in a group (G, •), included in the completed tensor space of R n , with the property that for each triple of times s, u, t ∈ [0, 1] 3 X s,t = X s,u ⋆ X u,t . ( The relations (1.4) are usually called Chen's relation after Kuo-Tsai Chen [7] and its secular work on the homology of loop spaces.We refer the reader to the monograph [10] for a detailed exposition of rough paths theory.

Motivation and previous works
We choose to have an intrinsic -coordinate-free -approach to (1.1) 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 [12].Considering the class of equations (1.1) 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 [6,22].In these works, the authors define 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 [16] to show the existence of a free Lévy area (up to an 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 [8] a weaker notion of Lévy area tailored to the class of equations (1.1) when the Hölder scale lies in 1  3 , 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 (1.1).The starting point to define it is a fine analysis of (1.2) with X smooth and the expansion of the solution Y obtained by applying Picard iterations.Pick A, B ∈ A and consider the following example (recall that • denotes the product of A) On the Signature of a Path in an Operator Algebra Writing the first two steps of the Picard Iteration, we obtain where R s,t is a remainder term satisfying |R s,t | ≲ |t − s| 3 .The above equation hints at a control, at any order, of the small variations of Y by the following expressions where σ is a permutation of {1, 2, . . ., n}.The expressions in (1.6) are values of a multilinear operator X σ s,t , that we call full contraction operator, depending on a choice of a permutation σ.The solution of the equation (1.1) expands over the contracted iterated integrals (1.6) 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 (1.6) with the objective of developing a rough theory for the class of equations (1.2) with driving noise X of arbitrary low Hölder regularity.To put it shortly, the main outcome of this work is a positive answer for that and we explain it by associating to the operators (1.6) a smooth trajectory over a group of triangular morphisms on an algebra of operators.
The main difficulties lie in writing a Chen relation for the operators (1.6) understood as a certain "algebraic rule" for computing (1.6) over an interval knowing the values of (1.6) over a subdivision of this interval.Consider for instance the full contraction operator Then the Chasles identity implies the following deconcatenation formula: 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 , . . ., A 4 ) → with the following full contraction one Thus a naive approach leads in fact to relations involving not only full contraction operators but also partial contractions.A remark on the terminology: we employ the term "contraction" to indicate that the operators reduce the degree of an input tensor, and "full" to indicate that it does so maximally.
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 noncommutative signature of a smooth path and in our main Theorem 4.18, we prove that, as for the classical theory, it yields a trajectory in a certain group.
Theorem 1.1.There exists a group (G, •) such that for each algebra-valued smooth trajectory X : [0, 1] → A there exists a map X : ∆ (2) → G from the two-dimensional simplex ∆ 2 with the following properties: • For any triple s < u < t ∈ [0, 1] 3 one has X s,t = X u,t • X s,u . (1.7) • For any pair 0 < s < t < 1, X s,t has a set of coordinates {X s,t (f ), f ∈ F} where the set F contains all permutations σ and X s,t (f ) is a certain bounded operator acting on folded projective tensor products of A which coincides with (1.6) when f = σ.
We call the element X s,t the noncommutative signature of the path X and the relations (1.7) noncommutative Chen's relations.Remark 1.2.We will define G as a set of representations of a certain algebra supported by trees with decorated leaves.The result that we want to prove in this work is purely algebraic and does not state any analytical property of X, which could be expected from the knowledgeable reader.We will in a separate work address integration theory against an irregular path drawn in A, and will gather at this time the relevant analytical context.

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 contraction 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 face-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 4.3, we relate partial-to full contraction operators.
In a forthcoming article, we continue to develop the theory.In particular, we introduce geometric noncommutative rough paths, geometric noncommutative controlled rough paths, and the operations of integration and composition.

Notations
In the following we denote by A a generic complex C ⋆ algebra with product µ, unity 1, norm ∥ • ∥ and involution ⋆.By definition, (A, ∥ • ∥) is a Banach algebra, the multiplication µ and the involution ⋆ are continuous with respect to ∥ • ∥, and In order to deal with a topology on the algebraic tensor product ⊗ which behaves correctly with µ, we will use the projective tensor product (see, e.g., [19]).Given two Banach spaces (E, ∥ • ∥ E ) and (F, ∥ • ∥ F ), the projective norm of an element x ∈ E ⊗ F is defined by We denote by E ⊗F the completion of E ⊗F for the projective norm.One can check the following properties for any permutation σ on the set {1, 2, . . ., n} and a 1 , . . ., a n ∈ E. The definition of projective norm yields immediately that the multiplication µ extends to a continuous map A ⊗A → A and, more generally, for any given pair of C ⋆ algebras A, B, A ⊗B is again a C ⋆ algebra.From a broader perspective, the projective tensor product makes the category of complex C ⋆ algebras a symmetric monoidal category (see Appendix A).In order to lighten the notation, we will adopt the symbol ⊗ to denote both the projective tensor product between C ⋆ algebras and the algebraic tensor product for pure tensors.Similarly, we will replace the product µ with a dot •.
For n ≥ 1 an integer, we denote by S n the set of permutations of [n] := {1, . . ., n}.We use one-line notation for permutations, writing σ = (σ 1 , σ 2 , . . ., σ n ), where σ i := σ(i).The neutral element of S n is also denoted by id n .Sometimes we may omit the commas and just write σ = σ 1 σ 2 • • • σ n .By abuse of notation, the only permutation of [0] := ∅ is denoted by ∅, from which we defines S 0 := {∅}.Given two integers a, b, we denote by Sh(a, b) the set of all shuffles of the two intervals 1, a and a + 1, a + b , that is σ ∈ Sh(a, b) if and only if σ is non-decreasing on 1, a and on a + 1, a + b .

Algebraic structure on levelled forests
The objective of the present section is to introduce the main combinatorial tool that will be used in this work: the levelled trees and forests.We will review their main properties and introduce new algebraic structures to them.

Levelled trees and forests
In the literature, one broadly finds several equivalent representations of a permutation, such as a bijection of a finite set or a finite word 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 [20, pp. 23-24], [4, Definition 9.9] or [2, p. 478].We will follow the versions used by Loday and Ronco in [13,Section 2.4] and Forcey, Lauve and Sottile in [9, 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 |τ | if τ 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 τ is denoted by V(τ ) and we set ∥τ ∥ := |V(τ )|.The set V(τ ) of internal vertices of a planar tree τ is equipped with a partial order ≤ τ : if u, v are two vertices of τ , we write u ≤ τ v if there is an oriented path of edges of t, moving away from the root, from u to v. The poset (V(τ ), ≤ τ ) has one minimum (the root of τ ) and several maxima (the leaves of τ ).
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 τ together with a linear extension of the poset (V(τ ), ≤ τ ).Levelled trees are also called ordered binary trees (see [2]).By definition, a levelled tree with degree one is the root tree (see Figure 1).Also, notice that the root tree has no internal vertices and corresponds to levelled tree ( , ∅) where ∅ denotes the unique function from the empty set to the empty set.
We denote by LT(n) the set of levelled trees with n leaves, and LT := ∪ n≥1 LT(n).The complex span of LT is a graded vector space, and its homogeneous component of degree n ≥ 1 is the linear span of LT(n).
We justify now the terminology for levelled trees.Following [18, p. 7], a level function on a tree t is a surjective increasing map where A is a totally ordered set.If τ is a planar binary tree and A = [∥τ ∥], then the pair (τ, λ) corresponds precisely to a levelled tree.If v is an internal vertex of t, we say that v has level λ(v).
The following result seems to be folklore.For proof of this result, see [13,Proposition 2.3].

Proposition 2.1 ([13]
).For every integer n ≥ 0, the set of levelled trees with n + 1 leaves is in bijection with the set of permutations S n .
The bijection associates to any levelled tree (τ, λ) with n + 1 leaves a permutation σ = σ(τ, λ) ∈ S n as follows.Label the leaves of τ with 0, 1, 2, . . ., n (in this order), from left to right.For each 1 ≤ i ≤ n, let v i be the vertex which lies in between the leaves i − 1 and i.Then

N. Gilliers and C. Bellingeri
where A is a totally ordered set.If τ is a planar binary tree and A = [ τ ], then the pair (τ, λ) corresponds precisely to a levelled tree.If v is an internal vertex of t, we say that v has level λ(v).
The following result seems to be folklore.For proof of this result, see [LR98] (Proposition 2.3).

Proposition 1 ([LR98]
).For every integer n ≥ 0, the set of levelled trees with n + 1 leaves is in bijection with the set of permutations S n .
The bijection associates to any levelled tree (τ, λ) with n + 1 leaves a permutation σ = σ(τ, λ) ∈ S n as follows.Label the leaves of τ with 0, 1, 2, . . ., n (in this order), from left to right.For each 1 ≤ i ≤ n, let v i be the vertex which lies in between the leaves i − 1 and i.Then When illustrating a levelled tree (τ, λ), it will be convenient to emphasize the levelling (the map λ) of the tree without the use of labels on the vertices.To do so, we position each vertex v of the tree τ at the level λ(v); it is represented by a dot with y-coordinate λ(v).We add straight edges to τ (see figure 2) so that the level λ(v) is populated with |λ(v)| vertices (in particular, on the first level, we find the root of the tree).In the resulting tree, all vertices placed on the same level n have an equal distance to the root.
Notice that the resulting unlabeled tree is such that every vertex has either one or two children, and there When illustrating a levelled tree (τ, λ), it will be convenient to emphasize the levelling (the map λ) of a tree without the use of labels on the vertices.To do so, we position each vertex v of the tree τ at the level λ(v); it is represented by a dot with y-coordinate λ(v).We add straight edges to τ (see Figure 2) so that the level λ(v) is populated with |λ(v)| vertices (in particular, on the first level, we find the root of the tree).In the resulting tree, all vertices placed on the same level n have an equal distance to the root.
Notice that the resulting unlabeled tree is such that every vertex has either one or two children, and there is a 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 trees.When illustrating a levelled tree (τ, λ), it will be convenient to emphasize the levelling (the map λ) of the tree without the use of labels on the vertices.To do so, we position each vertex v of the tree τ at the level λ(v); it is represented by a dot with y-coordinate λ(v).We add straight edges to τ (see figure 2) so that the level λ(v) is populated with |λ(v)| vertices (in particular, on the first level, we find the root of the tree).In the resulting tree, all vertices placed on the same level n have an equal distance to the root.
Notice that the resulting unlabeled tree is such that every vertex has either one or two children, and there is a 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 trees.In summary, we have three equivalent ways to identify the same object: ←→ Levelled trees with n + 1 leaves ←→ Sparse quasi-binary trees with n + 1 generations We use the symbol t (with super and subscript) for a levelled tree presented either as a permutation σ or as a pair (τ, λ).The representation of t as a sparse quasi-binary tree will only be used in drawings.In summary, we have three equivalent ways to identify the same object: ←→ Levelled trees with n + 1 leaves ←→ Sparse quasi-binary trees with n + 1 generations.
We use the symbol t (with super and subscript) for a levelled tree presented either as a permutation σ or as a pair (τ, λ).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 noncommutative monomial) on planar trees.
In the following, we denote by nt(φ) the number of trees in the forest φ, |φ| the total number of leaves in the forest and we set ∥φ∥ equal to the number of internal vertices of the forests.If all trees of φ are binary trees, then ∥φ∥ = |φ| − nt(φ).The poset (V(φ), ≤ φ ) 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 considering the following analogue of levelled binary trees to binary forests.Definition 2.2 (levelled planar binary forests LF).A levelled planar binary forest f (or simply a levelled forest) is a pair (φ, λ) formed by a binary forest φ and an increasing bijection We denote the set of planar binary forests by LF.
The non-commutative signature 7 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 nt(ϕ) the number of trees in the forest ϕ, |ϕ| the total number of leaves in the forest and we set ϕ equal to the number of internal vertices of the forests.If all trees of ϕ are binary trees, then ϕ = |ϕ| − nt(ϕ).The poset (V(ϕ), ≤ ϕ ) 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 considering the following analogue of levelled binary trees to binary forests.Definition 1 (Levelled planar binary forests LF).A levelled planar binary forest f (or simply a levelled forest) is a pair (ϕ, λ) formed by a binary forest ϕ and an increasing bijection We denote the set of planar binary forests by LF.The degree of a levelled planar forest (ϕ, λ) ∈ LF is the number of leaves of ϕ and is denoted by |ϕ|.If n ≥ 1 and m ≥ 1, we denote by LF(m, n) the set of levelled forests with n leaves and m trees.This allows defining a bigraduation on the set LF.The degree of a levelled planar forest (φ, λ) ∈ LF is the number of leaves of φ and is denoted by |φ|.If n ≥ 1 and m ≥ 1, we denote by LF(m, n) the set of levelled forests with n leaves and m trees.This allows defining a bigraduation on the set LF.
A generation of a levelled tree is a set of internal vertices on the same level, that is at the same distance from the root in the sparse quasi-binary tree representation (we thus take into account the labelling of the vertices).This notion extends to any forest.A levelled forest (φ, λ) can be pictured as a forest of quasi-binary trees, each with equal number of generations, in the same way as explained before for levelled trees, where the internal vertices are ordered vertically by adding straight edges according to λ, see Figure 3.In this representation, there is a unique internal vertex with two children among all vertices of the forest of the same generation.We call such a forest a sparse quasi-binary forest.
We now introduce several (classical) operations on levelled trees and levelled forests.
To a weak composition c, we associate the multiset We write c ⊨ n if c is a composition of n, and c ⊨ 0 n if c is a weak composition of n.The length k of a (weak) composition (c 1 , . . ., 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 • • • w r ∈ N * without repetition of letters; the associated levelled tree t has now a level function λ : (V(τ ), ≤ t ) → {w 1 , w 2 , . . ., w r }.
Every levelled forest (f, λ) ∈ LF(n + k, k) gives rise to a pair (σ, c), where σ ∈ 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 ⊨ 0 n is the weak composition of length k obtained by tracking the number of internal vertices of each tree in the forest f .Reciprocally any pair (σ, c) yields a levelled planar binary forest, using the bijection between non-repeating words and levelled planar trees.We call split permutation a pair (σ, c) with σ ∈ S n and c ⊨ 0 n: As for levelled trees, we use the symbol f to denote a levelled forest presented either as a pair (φ, λ) or as a split permutation (σ, c).The presentation of f as a sparse quasi-binary forest will be used in the drawings only.

N. Gilliers and C. Bellingeri
Every levelled forest (f, λ) ∈ LF(n + k, k) gives rise to a pair (σ, c), where σ ∈ S n is obtained by concatenating the non-empty words corresponding to each tree in f (from left to right) under the abovedescribed bijection, and c 0 n is the weak composition of length k obtained by tracking the number of internal vertices of each tree in the forest f .Reciprocally any pair (σ, c) yields a levelled planar binary forest, using the bijection between non-repeating words and levelled planar trees.We call split permutation a pair (σ, c) with σ ∈ S n and c 0 n.

Split permutations
As for levelled trees, we use the symbol f to denote a levelled forest presented either as a pair (ϕ, λ) or as a split permutation (σ, c).The presentation of f as a sparse quasi-binary forest will be used in the drawings only.The next definition introduces the notion of vertical splitting for levelled forests.Informally, a vertical splitting of a levelled forest f consists in breaking f into two forests, each bordered by a chosen path of edges in f , starting at a leaf of f and ending at a root of a tee in f .The first forest (resp.the second) is on the right (resp.one the left) of this path.The next definition introduces the notion of vertical splitting for levelled forests.Informally, a vertical splitting of a levelled forest f consists in breaking f into two forests, each bordered by a chosen path of edges in f , starting at a leaf of f and ending at a root of a tee in f .The first forest (resp.the second) is on the right (resp.one the left) of this path.We introduce inverse operations to splitting.The first one takes every tree in the representation of a levelled forest as a sparse quasi-binary forest and glues all together the trees of that forest along their external paths of edges.In terms of split permutation, this operation corresponds to the projection, We will also need a local operation gluing two consecutive trees in the representation of a levelled tree as a sparse quasi-binary tree, once again those operations are most effectively written in terms of split permutation.We set for any f = (σ, c) ∈ LF and 1 The next definition introduces the notion of vertical splitting for levelled forests.Informally, a vertical splitting of a levelled forest f consists in breaking f into two forests, each bordered by a chosen path of edges in f , starting at a leaf of f and ending at a root of a tee in f .The first forest (resp.the second) is on the right (resp.one the left) of this path.
Definition 2 (Vertical splitting of levelled forests).Pick a levelled forest f presented as a split permutation (σ, c) ∈ S n , c 0 n and d 0 n such that c d, we define the vertical splitting d ((σ, c)) of (σ, c) following d by (σ, d).
We introduce inverse operations to splitting.The first one takes every tree in the representation of a levelled forest as a sparse quasi-binary forest and glues all together the trees of that forest along their external paths of edges.In terms of split permutation, this operation corresponds to the projection, We will also need a local operation glueing two consecutive trees in the representation of a levelled tree as a sparse quasi-binary tree, once again those operations are most effectively written in terms of split permutation.We set for any f = (σ, c) ∈ LF and (2.2) We consider horizontal analogues to the above operations of vertical splitting and glueing.For any word w = w 1 w 2 • • • w r where each letter takes value in N * and A ⊆ alph(w), the alphabet generated by the different We consider horizontal analogues to the above operations of vertical splitting and gluing.For any word w = w 1 w 2 • • • w r where each letter takes value in N * and A ⊆ alph(w), the alphabet generated by the different letters contained in w, we define w ∩ A as the word obtained from w by erasing the letters which are not in A. We write w ′ ⊆ w if there exists A ⊆ alph(w) such that w ′ = w ∩ A. In this case, we say that w ′ is a subword of w.We use now the definition of subword to define the notion of subtree and subforest.Let t ∈ LT(n) a levelled binary tree, represented as a permutation σ.A levelled subtree (or just subtree) of t is a levelled binary tree t ′ with associated permutation σ ′ of the form In this case, we write τ ′ ⊆ τ .In terms of sparse quasi-binary trees, τ ′ is a subtree of τ if there exists 0 ≤ p ≤ ∥τ ∥ such that τ ′ , seen as a quasi-binary tree, coincides with the quasi-binary tree associated with τ by erasing all vertices on generations strictly bigger than p.
This notion extends to levelled forests.Pick a levelled forest f = (σ, (c 1 , . . ., c k )) ∈ LF and denote by (σ 1 , . . ., σ k ) the restrictions of σ to the parts of c: with same number of trees as f (the composition c ′ has the same number of parts as c) is a subforest of f is there exists p ≤ ∥f ∥ such that levelled binary tree t with associated permutation σ of the form σ = σ ∩ [p], for 0 ≤ p ≤ n.
In this case, we write τ ⊆ τ .In terms of sparse quasi-binary trees, τ is a subtree of τ if there exists 0 ≤ p ≤ τ such that τ , seen as a quasi-binary tree, coincides with the quasi-binary tree associated with τ by erasing all vertices on generations strictly bigger than p.
with same number of trees as f (the composition c has the same number of parts as c) is a subforest of f is there exists p ≤ f such that We write in this case f ⊂ f .In terms of levelled forest, writing (f, λ) and (f , λ ), then we have (f , λ ) ⊆ (f, λ) if f α , f β have the same number of trees and the labelling λ α restricts to λ β on the internal vertices of f β .Equivalently, considering α and β as quasi-binary forests, β ⊆ α if and only if β comprise all nodes of α up to a certain generation of the quasi-binary forest α.
Consider a word w = w 1 w 2 • • • w r where each letter takes value in N * and A ⊆ alph(w).Let w(A) = {i w A (1) < • • • < i w A (q)} be the increasing sequence of indices of the letters of w in A and define the composition wc(w, A) 0 r wc(w, A) For example, if r = 10 and w = 1293548851, let A = {2, 3, 4, 7, 9}.Then [9] \ A = {1, 5, 6, 8}, w(A) = {2 < 3 < 4 < 6}.Therefore, wc(w, A) = (1, 1, 1, 2, 4).Thanks to this notion, we introduce the horizontal splitting of a levelled tree.Definition 3 (Horizontal splitting of trees).Pick τ a levelled tree seen as a permutation σ and an integer 0 ≤ p ∈ σ .The horizontal splitting of σ at p is the couple  We write in this case f ′ ⊂ f .In terms of levelled forest, writing (f, λ) and (f ′ , λ ′ ), then we have (f ′ , λ ′ ) ⊆ (f, λ) if f α , f β have the same number of trees and the labelling λ α restricts to λ β on the internal vertices of f β .Equivalently, considering α and β as quasi-binary forests, β ⊆ α if and only if β comprise all nodes of α up to a certain generation of the quasi-binary forest α.
Consider a word w = w 1 w 2 • • • w r where each letter takes value in N * and A ⊆ alph(w).Let be the increasing sequence of indices of the letters of w in A and define the composition wc(w, A) ⊨ 0 r wc(w, A) For example, if r = 10 and w = 1293548851, let A = {2, 3, 4, 7, 9}.Then [9] ). Thanks to this notion, we introduce the horizontal splitting of a levelled tree.This operation extends to levelled forests in a straightforward manner.Definition 2.5 (horizontal splitting of levelled forests).Let f = (σ, c) be a forest in LF(k, n), with σ ∈ S n−1 and (c 1 , . . ., c k ) ⊨ 0 n − 1 a weak composition of length k.Once again, denote by σ i , 1 ≤ i ≤ ∥k∥ the restriction of σ to the parts of c.The horizontal splitting of the levelled forest f at the level p is the forest For convenience, we use the shorter notations Consider for example the forest ⋎ A 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 ′ ⊂ f yields a unique horizontal cut of f , the cutting level k being the number of internal nodes of f ′ (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.Definition 2.6.For any given levelled forest f and subforest f ′ ⊂ f we denote by f \f ′ the upper component of the horizontal cut induced by f ′ .We say that the levelled forest f is compatible with the levelled forest f ′ if the number of trees of f matches the number of leaves of f ′ in the representation of f and f ′ as sparse quasi-binary trees or as levelled trees.

By definition of horizontal splitting one has immediately the identity
Moreover, if both levelled forests are given as split permutations, f = (σ, (c 1 , . . ., c k )), σ ∈ S(n) and f ′ = (σ ′ , (c ′ 1 , . . ., c ′ q )), σ ′ ∈ S(n ′ ) compatibility means that the number k of parts of c 1 is equal to The following operation, inverse to horizontal splitting, is better understood in terms of levelled forests.
2. The labelling λ ′′ restricts to λ on the internal vertices of ϕ in ϕ#ϕ ′ and to the labelling λ ′ translated by ∥f ∥ on the internal vertices of ϕ ′ in ϕ#ϕ ′ .
In terms of sparse quasi-binary trees and forests, horizontal gluing corresponds to stacking the sparse quasi-binary trees representing f ′ 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.by f on the internal vertices of φ in φ#φ .
In terms of sparse quasi-binary trees and forests, horizontal glueing corresponds to stacking the sparse quasi-binary trees representing f 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.

A first algebraic structure on levelled forests
We briefly recall important Hopf algebraic structures on permutations.The vector space n ≥ 0 CS n has a graded Hopf algebra structure, called F QSym, 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, noncocommutative, graded and self-dual.In the following, we introduce the product ¡ dual to the aforementioned coproduct of the Hopf algebra of non-commutative symmetric functions.First, for two permutations α ∈ S n and β ∈ S p , we introduce the set of shuffles of α and β.In the above equation std denotes the standardization map: In particular, if id n and id p are the identity permutations of the symmetric groups S n and S p , respectively, we let Sh(n, p) := Sh(id n , id p ).
Given a permutation τ ∈ S(n) and integer p ≥ 1 we denote by τ + p the bijection of {p + 1, • • • , p + n} defined by The shuffle product of two permutations σ ∈ S p and τ ∈ S n is defined by where we have used the explicit notation • for the composition of permutations.We extend the shuffle product from levelled trees (permutations) to levelled forests.If the permutations σ and τ are presented as sparse quasi-binary trees, computing their shuffle product is done by adding straight edges at the bottom of τ , then glueing this tree to the right of σ (we identify the outer paths of edges) and finally shuffling vertically the generations.

A first algebraic structure on levelled forests
We briefly recall important Hopf algebraic structures on permutations.The vector space n≥0 CS n has a graded Hopf algebra structure, called F QSym, introduced in [17], and also considered in [11], where it is called the algebra of free quasi-symmetric functions.This Hopf algebra is noncommutative, non-cocommutative, graded and self-dual.In the following, we introduce the product ¡ dual to the aforementioned coproduct of the Hopf algebra of noncommutative symmetric functions.First, for two permutations α ∈ S n and β ∈ S p , we introduce Sh(α, β) := {σ ∈ S n+p : std(σ ∩ {1, . . ., n}) = α, std(σ ∩ {n + 1, . . ., n + p}) = β}, the set of shuffles of α and β.In the above equation std denotes the standardization map: In particular, if id n and id p are the identity permutations of the symmetric groups S n and S p , respectively, we let Sh(n, p) := Sh(id n , id p ).
Given a permutation τ ∈ S(n) and integer p ≥ 1 we denote by τ + p the bijection of {p + 1, . . ., p + n} defined by With σ ∈ S(p), σ ⊗ (τ + p) ∈ S(n + p) is the permutation equal to σ on {1, . . ., p} and (τ + p) on {p + 1, . . ., p + n}, The shuffle product of two permutations σ ∈ S p and τ ∈ S n is defined by where we have used the explicit notation • for the composition of permutations.We extend the shuffle product from levelled trees (permutations) to levelled forests.If the permutations σ and τ are presented as sparse quasi-binary trees, computing their shuffle product is done by adding straight edges at the bottom of τ , then gluing this tree to the right of σ (we identify the outer paths of edges) and finally shuffling vertically the generations.Definition 2.8 (shuffle product of levelled planar forests).Let f = (σ, (c 1 , . . ., c k )) and g = (τ, (c ′ 1 , . . ., c ′ q )) be two levelled forests, we define the shuffle product of f and g by We present in detail the product 12 ¡ (12, (1, 1)), presented as sparse quasi-binary forests in Figure 9.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).We notice that by the construction of ¡ all terms contained in the sum the blue and red dots preserve the same ordering of generations in the initial factors.
A permutation σ ∈ S n acts on the left of a levelled forest f = (σ, c) ∈ LF with n generations as follows We denote by c n the permutation (n, 1) ).We use the right action of c n to define a involution on LF(n − 1), which is the horizontal mirror symmetric of a forest We will sometimes refer to θ 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.Example 2.9.We present in detail the product 12 ¡ (12, (1, 1)), presented as sparse quasibinary forests in Figure 9.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 Figure 10).
be two levelled forests, we define the shuffle product of f and g by We present in detail the product 12 ¡ (12, (1, 1)), presented as sparse quasi-binary forests in Figure 9.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).We notice that by the construction of ¡ all terms contained in the sum the blue and red dots preserve the same ordering of generations in the initial factors.
A permutation σ ∈ S n acts on the left of a levelled forest f = (σ, c) ∈ LF with n generations as follows We denote by c n the permutation (n, 1) ).We use the right action of c n to define a involution on LF(n − 1), which is the horizontal mirror symmetric of a forest We will sometimes refer to θ 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.We will sometimes refer to θ 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.

Hopf monoid of levelled forests
In this section, we introduce a Hopf algebraic structure on the bicollection of spanned by levelled forests and denoted LF, In addition, we set LF(0, 0) = C, LF(0, n) = LF(m, 0) = 0, n, m ≥ 1 and we denote by LT the collection spanned by levelled binary trees This Hopf algebra is an object in the category of bicollections endowed with the vertical tensor product .In general, as is briefly explained in the Appendix A, the two-folded vertical tensor product A A of a monoid A in the monoidal category (Coll 2 , ) is not a monoid in the same category.Owing to the fact that the monoid generated by LF in (Coll 2 , ) is symmetric, in particular, LF LF is a monoid in a natural way, it makes sense to require compatibility between a product and a coproduct on LF.We write the unit C for the vertical tensor product as Recall that we denote by |f | the number of leaves of a levelled forest f and nt(f ) the number of trees in f .We begin with the definition of the coproduct acting on the bicollection LF of levelled forests.Let f be a levelled forest.Let f ′ be a levelled subforest of f (recall that f ′ contains the roots of all trees in f ).By definition of the forest f \f ′ , the number of outputs of the forest f \f ′ is equal to the number of inputs of the forest f ′ (the number of trees of f \f ′ matches the number of leaves of f ′ ), the following makes senses This operation is a genuine coproduct with respect to the vertical tensor product.
Proposition 2.11.The morphism ∆ : LF → LF LF is coassociative and the morphism ε : LF → C given by is the counity for ∆, i.e., Proof .Let f be a levelled forest, to show coassociativity we notice that

Equation (2.4) is trivial. ■
We proceed now with the definition of a vertical product on levelled forests.
Definition 2.12 (monoidal product on levelled forests).Given two forests f and f ′ with nt(f ′ ) = |f |, we define ∇(f f ′ ) as the sum of forests obtained by first stacking f ′ up to f and then shuffling the generations of f ′ with the generations of f (see Section 2.1 for the definition of the action of a permutation on the generations of a forest), The associativity of the product ∇ is easily checked.The unit η : C → LF is defined by η(1 m ) = m .Let n ≥ 1, recall that we denote by c n the maximal element for the Bruhat order in S n : For example, c 1 = 1, c 2 = 21, c 3 = 321, c 4 = 4321.Given these notions, we state the main theorem of the section Theorem 2.13.(LF, ∇, η, ∆, ε) is a conilpotent Hopf algebra in the category (Coll 2 , , C ).
To achieve this result we introduce an explicit antipode map.Definition 2.14.Pick n, m ≥ 1 two integers.Let f ∈ LF(n, m) be a levelled forest and define its vertical mirror symmetric f ⋆ ∈ LF(n, m) by We extend ⋆ as a conjugate-linear morphism on the bicollection LF.
Proposition 2.15.Let f be a levelled forest.The map S : LF → LF defined by 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 The following relation is easily checked and turns to be the cornerstone of the proof: with s the unique shuffle in Sh(m, n) + such that s(m) = n+m, s(i) = s(i).Set S(f ) = (−1) ∥f ∥ f ⋆ .We prove by induction that S = S. Assume that S(f ) = S(f ) for any forest f with at most N ≥ 1 generations and pick a forest f with N + 1 generations.Then, from the induction hypothesis we get We divide the sum over the set Sh(∥f ∥ − k, k) into two sums.The first sums ranges over the subset Sh(∥f ∥ − k, k) + and the second one ranges overs Sh(∥f ∥ − k, k) − .Then, we gather the sums over Sh(∥f ∥ − k, k) + and Sh(∥f Using equation (2.6), the right-hand side of the last equation is equal to We defined the three structural morphisms ∇, ∆, S. To turn LF into a Hopf monoid, we have to check compatibility between the coproduct ∆ and the product ∇; the coproduct ∆ should be a morphism of the monoid (LF, ∇).This only makes sense provided that we can define a product on the tensor product LF LF.
Recall that if f is a levelled forest and 0 ≤ k ≤ ∥f ∥, one denotes by f k − the levelled subforest of f corresponding to the k generations at the bottom of f ′ : t f k − is the planar subforest of t f with a set of internal vertices the set of internal vertices of f labelled 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 similarly by extracting the k top generations of f ′ .With p, q ≥ 1 two integers, we denote by τ p,q the shuffle in Sh(p, q) satisfying τ p,q (1) = q + 1 and τ p,q (p) = p + q.Definition 2.16.Define the braiding map K : LF LF → LF LF by, for g and f levelled forests such that f g ∈ LF LF, We pictured in Figure 11 examples of the action of the braiding map on pairs of levelled forests.
Using equation (2.12), the right hand side of the last equation is equal to We defined the three structural morphisms ∇, ∆, S. To turn LF into a Hopf monoid, we have to check compatibility between the coproduct ∆ and the product ∇; the coproduct ∆ should be a morphism of the monoid (LF, ∇).This only makes sense provided that we can define a product on the tensor product LF LF.
Recall that if f is a levelled forest and 0 ≤ k ≤ f , one denotes by f k − the levelled subforest of f corresponding to the k generations at the bottom of f : t f k − is the planar subforest of t f with a set of internal vertices the set of internal vertices of f labelled 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 similarly by extracting the k top generations of f .With p, q ≥ 1 two integers, we denote by τ p,q the shuffle in Sh(p, q) satisfying τ p,q (1) = q + 1 and τ p,q (p) = p + q.Definition 10.Define the braiding map K : LF LF → LF LF by, for g and f levelled forests such that f g ∈ LF LF, We pictured in Fig. 11 examples of the action of the braiding map on pairs of levelled forests.We defined the braiding map K as acting on LF LF.We extend K as a 2-functor on the product of the monoid generated by LF in (Coll 2 , ).This means in particular that for integers p, q ≥ 1, we define a bicollection morphism K p,q : LF p LF q → LF q LF p .
Pick f 1 • • • f p ∈ LF p and g 1 • • • g q ∈ LF q .We define the levelled forest ≻ c h with c ⊨ 0 n to be the element in LF p obtained by the following iterative application of horizontal splittings: The collection of morphisms K p,q yields a 2-functor on the category with objects LF ⊗p but with restricted classes of morphisms.First, it is not difficult to see that K = K 1,1 is an involution and therefore that K p,q is an involution too, for any p, q ≥ 1.It follows from the fact that given f and g two levelled forests, Definition 2.17.Let p, q ≥ 0 be integers and φ : LF ⊗p → LF ⊗q , we say that φ is gluing equivariant if φ commutes with the operations where We denote by Hom eq (p, q) the class of all gluing equivariant morphisms between LF p and LF 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 ≥ 0, K p,q is gluing equivariant.
Proposition 2.18.The monoid generated by the bicollection LF in (Coll 2 , ) with morphisms restricted to the gluing equivariant morphisms is a symmetric monoidal category with symmetry constraints (K p,q ) p,q≥0 , Proof .Both assertions are trivial and rely on the following relations between the permutations τ p,q , p, q ≥ 0: τ p,q • τ q,p = id, (id q ⊗ τ p,r ) • (τ p,q ⊗ id r ) = τ p,q+r , p, q, r ≥ 0. ■ Using the above-defined symmetry constraint K, we can endow the two-fold tensor product LF LT with an algebra product: Proposition 2.19.The two bicollection morphisms ∆ : LF → LF LF and ∇ : LF LF → LF are vertical algebra morphisms.With ∇ (2) = ∇ • (∇ id) = ∇ • (id ∇), this means that Remark 2.20.We can rephrase the fact that ∇ is an algebra morphism by saying that (LF, ∇) 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 ≤ i ≤ 3), For the second assertion, we write first For each integer 1 ≤ k ≤ ∥f ∥, we split the set of shuffles Sh(∥f ∥, ∥g∥) according to the cardinal q of the set • τk,q with τk,q the unique shuffle that sends the interval ∥f ∥ + 1, ∥f ∥ + q to the interval k − q + 1, k and fixes the interval Notice that τk,q = τ k−q,q and τk,q The case ∥f ∥ + 1 ≤ k ≤ ∥f ∥ + ∥g∥ is similar, we split the set of shuffles Sh(∥f ∥, ∥g∥) according to the cardinal of the set s −1 ( k + 1, ∥f ∥ + ∥g∥ ) ∩ 1, ∥f ∥ ) and omitted for brevity.Finally, we obtain for ∆ • ∇(f g) the expression: By collecting altogether the statements of Proposition 2.19 (proving compatibility between the product ∇ and the coproduct ∆) and Proposition 2.15 proves Theorem 2.13.Notice that compatibility between the coproduct and coproduct makes sense because the monoid generated by LF is symmetric (for the symmetry constraint K) as stated in Proposition 2.18.

Iterated integrals of a path as operators
Let us fix a smooth path, X : [0, 1] → 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.

Full and partial contraction operators
In what follows, we will intensively use the identifications in the previous sections between levelled trees, and levelled forests and their corresponding permutation and split permutations.In what follows, for any couple of Bananch spaces A, B we use the notation Hom(A, B) to denote the set of linear continuous maps between A and B. Definition 3.1.For any integer n ≥ 1 and levelled tree σ in LT(n), we define the full contraction of X along σ as the map X σ : [0, 1] 2 → Hom A ⊗n , A given for any A 1 , . . ., A n ∈ A by where σ is identified with a permutation in S n−1 when n ≥ 2 and X • s,t = id A .
Remark 3.2.The above definition may be misleading since the identity (3.1) defines a linear map on the algebraic tensor product, whereas we used ⊗ to denote the projective tensor product.However, the algebraic tensor product is a dense subspace of A ⊗n and we interpret X σ s,t as the unique continuous operator extending the values in (3.1).Similar considerations apply throughout the paper.
If linearly extended to the vector space spanned by all levelled trees (or equally permutations), the map σ → X σ s,t yields naturally a morphism between the collection LT in (2.2) and the endomorphism collection End A given by see Appendix A. The partial contraction operators, that we now introduce, extend σ → X σ s,t to a morphism between LF to many-to-many operators, i.e., elements of Hom Vect C A ⊗n , A ⊗m , m < n.
To properly define them, we denote by End 2 A the bicollection of noncommutative polynomials on multilinear maps on A with values in A. That is using the notation in the Appendix A when n ≥ 1 and m ≥ 1 and the condition k A is endowed with a monoidal product ∇ End 2 A associated to the vertical tensor product .This operation extends the usual canonical operadic structure • on End A as a monoidal morphism, see Appendix A. For example, given two non-trivial elements v ∈ End We introduce also the double tensor algebra of A. We start from the unital tensor algebra

Elements of T (A) are linear combinations of words
and ∅ is the unity for concatenation of words.The double tensor algebra is given by Elements of T 2 (A) are represented as words of words.To distinguish the internal concatenation of T (A) and the second order concatenation, we use the symbol | for the concatenation product on T T (A) and the symbol 1 stands now for the unit of |.In the following, for any integers m ≥ 1 and n ≥ 1 we use the notation with the convention that A 0 = C∅.For both words on words in A and words on endomorphisms in A, we freely identify the sequence of vector spaces End 2 A (m, n) (resp.T 2 (A)(m, n)) with their direct sum.We will however make clear this distinction for other collections and bicollections.We relate T 2 (A) and End 2 A via an explicit representation.
Definition 3.3 (representation of the algebra T 2 (A)).We define a representation Op : T 2 (A) → End 2 A of the algebra T 2 (A), | extending the following values, for The representation Op has one crucial property.By definition, Op is compatible with the concatenation product | on T 2 (A).As explained, End 2 A is endowed with a vertical monoidal structure ∇ End 2

A
. The same kind of structure exists on T 2 (A).Indeed, T (A) can be endowed with an operadic structure •, that we call words insertions.Given a word a 1 ⊗ • • • ⊗ a n ∈ T (A) and w 0 , . . ., w n ∈ T (A), one defines One can check that • satisfies the associativity and unitality constraints of an operadic composition.We then extend this operadic composition as a horizontal monoidal morphism and define this way an associative product Then Op is compatible with respect to the products ∇ End 2 A and ∇ T 2 (A) .That is 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 and apply Op to the result, We can apply first Op to w 1 and w 2 and compose together the resulting operators, Substituting to Y 0 ⊗ Y 1 ⊗ Y 2 ⊗ Y 3 the right-hand side of the last equation, we recover (3.3), The vector space T (A) is the natural space wherein the signature of a smooth path X takes values, see (1.3).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 (A) with length n ≥ 1.Let c = (c 1 , . . ., c k ) be a composition of n.The composition (c 1 , . . ., c k ) yields a splitting of w: we define the element Definition 3.5.For any integer n ≥ 1 and levelled tree σ in LT(n), we denote by X σ the map where σ is identified with a permutation in S n−1 when n ≥ 2 and X • s,t = ∅.For any levelled forest f = (σ, c) ∈ LF(m, n) we denote by X f the application X f s,t = X σ s,t c .Definition 3.6.For any integers n ≥ 1, n ≥ m ≥ 1 and any levelled forest f = (σ, c) ∈ LF(m, n), we define the partial-contraction of X along the forest f as a map Example 3.7.Let us calculate the partial contraction associated with the levelled forest f in Figure 3.In this case, n = 5 and the word on words representing f is We associate it with the formal expression The first term on the left of the above expression corresponds to the first tree in f , with two vertices labelled 1 and 3.It is followed on its right by a straight tree, yielding the first ∅.We then interleave a 10-tuple (A 0 , . . ., A 9 ) of elements in A between each dX t i , replacing the empty letter ∅ by one of the A's, Finally, we integrate over ∆ n s,t and obtain the following formula for X f s,t (A 0 , . . ., A 9 ),

Chen relation
In this section, we use from time to time the symbol to improve readability.We describe how the concatenation of paths lifts to the full and partial contractions operators, that is we write an extension of Chen identity over iterated integral, see [7] for these operators.
Proposition 3.8 (Chen relation).For any forest f ∈ LF and any three times (s, u, t) ∈ [0, 1] 3 one has Written in term of the notations introduced in (2.3) and the map X s,t : LF → End 2 A defined by X s,t (f ) = X f s,t , the equation (3.4) becomes Example 3.9.Before writing the proof, we check equation (3.4) 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 X f s,t is given by Using the standard properties of Lebesgue integration, we can easily write At the same time, we list all subforests in f ′ ⊂ f in Figure 6 together with f \ f ′ .Proposition 3.8 implies the equality , which is exactly (3.5).
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 ∈ LF since Op is a representation of the word-insertions operad (see equatio (3.2)).We prove the identity by induction on the generation of f and with • the operation ∇T2 (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 ∆ n+1 s,t according to s < u < t one has where where i is the order of the i th tree in the forest f whose root is decorated by 1.By construction of f \f 1 , this forest has only N generations and the recursive hypothesis to the forest f \f 1 implies .
We insert this last relation into equation (3.6) to get the identity . ■ Remark 3.10.We apply the above formula to the levelled tree f which is a right comb tree obtained by grafting corollas with two leaves with each other, always on their rightmost node.By cutting such a tree at a certain level, we obtain on one hand a smaller comb tree and on the other hand, we obtain a levelled forest with only straight trees, except for the last one, the rightmost, which is also a comb tree.By denoting comb n the comb tree with n internal nodes, we thus get for a tuple A 0 , . . ., A n ∈ A, We denote by FS the graded vector space equal to the direct sum of all vector spaces in the collection FS.Notice that elements of A are 0-ary operators in the collection FS and, for example, the above formula for L gives L(U Proposition 3.14.FS = (FS, L, 1 ⊗ 1) is an operad.
Proof .The following proposition holds and rests on the associativity of the product on A. ■ In the collection FS, 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 many inputs as it has of 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 Figure 13.
The word 1 ⊗ 1 acts as the unit for L.
We denote by FS the graded vector space equal to the direct sum of all vector spaces in the collection FS.Notice that elements of A are 0-ary operators in the collection FS and, for example, the above formula for L gives Proposition 10.FS = (FS, L, 1 ⊗ 1) is an operad.
Proof .The following proposition holds and rests on the associativity of the product on A.
In the collection FS, 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 many inputs as it has of 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.We denote by LT # the set of levelled trees graded by the numbers of faces, LT # (n) the set of levelled trees with n faces, and LT (A) the space LT(A) seen as a graded vector space with Notice that the endomorphism Xs,t we defined in the previous section satisfies: We also set for any levelled tree τ ∈ LT, The space LT # (A) is equipped with an involution LT # (A) , defined by We denote by LT # the set of levelled trees graded by the numbers of faces, LT # (n) the set of levelled trees with n faces, and LT ♯ (A) the space LT(A) seen as a graded vector space with LT # (A)(n) = CLT # (n) ⊗ FS(n).Notice that the endomorphism Xs,t we defined in the previous section satisfies: We also set for any levelled tree τ ∈ LT, The space LT # (A) is equipped with an involution ⋆ LT # (A) , defined by (3.9) The graded vector space LT # (A) yields a collection LT # (A) by setting the space n-ary operators LT # (A) equal to LT # (A)(n).We set abusively For any U, A ∈ LT # (A), we introduce the notation Observe that U • A is only linear on U , not on A.
We introduce for any face-contractions operators m = τ ∈LT m τ the face-contraction norm where ∥ • ∥ is the usual operator norm induced by A. From the notation ♯(( we also denote by ♯ the morphism of graded vector spaces Notice that the operator ♯(( Its output can be computed by drawing a sparse quasi-binary tree τ and placing A 1 , . . ., A τ up to the leaves of τ and the dX t i on the unique vertex with two children on the i th generation of τ .Whereas in the previous section the arguments of the multilinear operators we considered were located on the leaves, in this section they are located on the faces.Some operations we defined on trees can be push-forward via ♯ to define a proper unital Banach algebra with involution.We denote these operations with similar notation as the operations defined over LT # (A).
We call ¡ and ⋆ LT # (FC) (m) the shuffle product on face-contractions and the involution on facecontractions.
Moreover, ♯ is a morphism of unital Banach algebra, i.e., one has the identities where V ∈ FC(p), W i ∈ FC(n i ), 1 ≤ i ≤ p and the symbol • in the right-hand side of the above equation stands for functional composition in End A , We set FC= FC, L, id A .Notice that with this definition, the map ♯ is a morphism between the operads FS and FC, namely, for A 1 , . . ., A p+1 ∈ A and W 1 , . . ., W p ∈ FS

Group acting on faces-contractions
From the algebra structure defined on LT # (FC) and the properties the morphism ♯ : LT # (A) → LT # (FC), we will also introduce a group which plays the same role of G(A) in (3.14) (the group where the maps Xs,t takes value) at the level of face-contractions.
To achieve this, we first rewrite the vector space LT # (FC) in an equivalent way so that we can speak of components.Using the identification between permutations and levelled trees from Proposition 2.1 and the intrinsic product of S n , for any levelled tree τ with ∥τ ∥ = n and σ ∈ S ∥τ ∥ the map which sends Each map ϕ σ is continuous and has inverse given by ϕ σ −1 .Combining the action of the maps {ϕ σ : σ ∈ S ∥τ ∥ }, we introduce the map ϕ : and by extension of ϕ to LT # (FC) we obtain a continuous isomorphism where for each n ≥ 0, c n is the levelled tree represented by the permutation id n .For brevity, we use the notation LT # (FC)(n) := LT # (FC)(c n ).Also, a generic element of the tensor product FC(n) ⊗ CLT n will be denoted mτ (we omit the symbol ⊗) where m ∈ FC(n) and τ ∈ LT n .
Definition 4.5.Let X : LT # (FC) → LT # (FC) be an endomorphism of LT # (FC) and τ, τ ′ ∈ LT # (FC) a couple of levelled trees.We define the components of X as the set of continuous linear maps defined by the relation We remark that for any given family {X (τ ′ , τ ), τ, τ ′ ∈ LT # (FC)} like above formula (4.1) defines actually a graded endomorphism of LT # (FC).Moreover, for any given graded endomorphism X : LT # (FC) → LT # (FC) the component X (τ ′ , τ ) can be computed on any m ∈ FC(∥τ ∥) as where (•) τ ′ is the natural projection on the component associated with τ ′ as the right factor.
With the notations introduced so far, and omitting conjugation by ϕ, fora an operator X ∈ End LT # (FC) , one writes Remark 4.6.The involution ⋆ LT # (FC) defined in the previous section induced through ϕ an involution on n≥0 FC(c n ) ⊗ CLT n , denoted by the same symbol, one has In the next definition, we introduce a specific class of operators.
Proof .Pick X , Y ∈ U C (FC), we have to show that where the last equality follows from the simple observation that θ(f \f ′ ) = θ(f )\θ(f ′ ) for any pair of forests f ′ ⊂ f .We deduce that For any pair of forests f ′ ⊂ f , ■ We now further restrict the group that will support signatures of smooth valued paths.It is defined by a set of equations on components of an operator in U C indexed by levelled forests.To specify these equations, we resort to operations on levelled forests defined in Section 2.1.
Pick a integer n ≥ 1.Let f = (σ, c), σ ∈ S n , c ⊨ 0 n be a levelled forest with n generations and nt(f ) trees.For any subset I ⊂ [nt(f ) − 1] of integers we denote by f I the levelled forest obtained by gluing the trees f i in f at positions i ∈ I along their external edges, We set also The integers ℓ f j , 1 ≤ j ≤ nt(f ) − 1 index spaces between consecutive sparse quasi-binary trees in the levelled forest f , which can also be considered as faces of f (in addition to the ones delimited by two consecutive leaves of a tree in f ).
Recall also that elements of A are considered as face-contractions operators with 0 input.Pick mFC(p) a face-contractions operator with arity p, m 1 , . . ., m q , 1 ≤ g ≤ p other face-contractions operators and a sequence of integers 1 ≤ i 1 < • • • < i q ≤ p.In the definition below, we denote by the operator obtained by composing m j with the i th j input of m, the remaining inputs are filled with the identity.Definition 4.13.We define G(FC) ⊂ U C (FC) as the subset comprising all operators X ∈ U C (FC) satisfying for any quadruple where X 1 , . . ., X q ∈ A. In addition, we denote by G ⋆ (FC) the subset of self-adjoint operators in G(FC) for the involution Θ defined in Proposition 4.12, • Y i j +s−j • A ℓ f i j +s +1 , 1 < s < i j+1 − i j , (3) Z ℓ f j +t = A ℓ f j +t−j , 1 < t < ℓ f j − ℓ f j+1 .
Proof .Pick two endomorphisms X and Y in G(FC).Pick f a levelled forest and I = {i 1 < • • • < i q } ⊂ 1, nt(f ) − 1 .We prove that X • Y ∈ G(FC).We already know that X • Y ∈ U C (FC), hence it will be sufficient to prove (4.3) for X • Y. Pick m ∈ LT # (FC) as in Definition 4.13, one has Owing to associativity of •, we have ℓ f The statement follows by noticing that

The noncommutative signature of a path
We are now ready to state the main Definition of the notion of the signature of a smooth path that is adapted to the class of equations (1.1).
Theorem 4. For any smooth path X the operators X s,t satisfy the following properties: 1. X s,t ∈ G(FC), 2. X s,t = X u,t • X s,u , 3. • Xs,t = X s,t • .
Besides, if X is a self-adjoint trajectory then X s,t is a morphism of algebras.Recall that Xs,t is defined in equation (3.7).
Theorem 4.18.For any smooth path X the operators X s,t satisfy the following properties: (1) X s,t ∈ G(FC), (2) X s,t = X u,t • X s,u , Definition A.3.We call a (reduced) collection P a sequence of complex vector spaces2 {P (n)} n≥1 .A morphism between two collections P , Q is a sequence of linear maps {ϕ(n)} n≥1 with ϕ(n) : P (n) → Q(n), n ≥ 0. For any couple of morphisms between collections we define the composition of morphisms by composing each component.We denote the category of collections by Coll.
The category Coll has a natural monoidal structure ⊙ over it: for any couple of collections P and Q and morphisms f , g we define Denoting by C ⊙ the collection We keep the notation ⊙ for the monoidal operation.It is common in the literature to denote the morphism ρ by •, i.e., for every k ≥ 1, p ∈ P (k) and q i ∈ Q(n i ) for i = 1, . . ., k, Moreover, for any 1 ≤ i ≤ k and q i ∈ Q(n i ) we use also the notation • i to denote partial composition p • i q := p • η P (1)(1) ⊗i−1 ⊗ q ⊗ η P (1)(1) ⊗k−i , where η P (1) : C → P (1).Since the maps ρ(n) n≥1 carry multiple inputs and give back one output, it is common in the literature to call them many-to-one operators.A classical example to understand this definition is given by the endomorphism operad End V of a complex vector space V with elements End V (n) = Hom Vect C V ⊗n , V , n ≥ 1, η End V (1)(1) = id V , and operadic composition where n = n 1 + • • • + n k .In fact, it is possible to generalise the notion of an operad to model composition between many-to-many operators, that is operators with multiple in-and outputs.This leads us to define the category of bicollections.
Definition A.5.We call a bicollection a two parameters family of complex vector spaces P = {P (n, m)} n,m≥0 .
A morphism between two bicollections P , Q is a sequence of linear maps {ϕ(n, m)} n,m≥0 with ϕ(n, m) : P (n, m) → Q(n, m).For any couple of morphisms between bicollections we define the composition of morphisms by composing each component.We denote the category of bicollections by Coll 2 .
The category of bicollections is endowed with two compatible monoidal structures.
Definition A.6.For any couple of bicollections P and Q and morphisms f , g we define the horizontal tensor product ⊖ as follows: (P ⊖ Q)(n, m) := We recall that the same structure takes also the name of double monoid in the literature, see, e.g., [1].

Figure 1 :
Figure 1: Examples of levelled trees (τ, λ) in LT and their associated permutations in S 1 , S 2 and S 3 .

Figure 1 .
Figure 1.Examples of levelled trees (τ, λ) in LT and their associated permutations in S 1 , S 2 and S 3 .

Figure 1 :
Figure 1: Examples of levelled trees (τ, λ) in LT and their associated permutations in S 1 , S 2 and S 3 .

Figure 2 :
Figure 2: Example of a levelled tree and its associated sparse quasi-binary tree, before adding straight edges in the centre and after, on the right.

Figure 2 .
Figure2.Example of a levelled tree and its associated sparse quasi-binary tree, before adding straight edges in the centre and after, on the right.

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

Figure 3 .
Figure 3.A sparse quasi-binary forest with four trees.
This gives a bijection between weak compositions of n and multisubsets of {0} ∪ [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 , . . ., c i + c i+1 , . . ., c k ) ⋖ (c 1 , . . ., c i , c i+1 , . . ., c k ).

Definition 2 . 3 (
vertical splitting of levelled forests).Pick a levelled forest f presented as a split permutation (σ, c) ∈ S n , c ⊨ 0 n and d ⊨ 0 n such that c ⋖ d, we define the vertical splitting ⋎ d ((σ, c)) of (σ, c) following d by (σ, d).

Figure 6 :
Figure 6: On the first line: levelled forest f = (213, (2, 1)), drawn as a quasi-binary forest.On the second line: all levelled subforests included in f .This notion extends to levelled forests.Pick a levelled forest f = (σ, (c 1 , • • • , c k )) ∈ LF and denote by (σ 1 , • • • , σ k ) the restrictions of σ to the parts of c: formed by the subword of σ containing the letters in [p], and the pair (σ∩{p+1, . . ., n}, wc(σ, [p])) representing the sequence of subwords of σ obtained from σ after erasing the letters in [p].We call the first component of p the lower component of the cut and the second component the upper component.

Definition 2 . 4 (
horizontal splitting of trees).Pick τ a levelled tree seen as a permutation σ and an integer 0 ≤ p ∈ ∥σ∥.The horizontal splitting of σ at p is the couple ≻ p (t) := σ ∩ [p], (σ ∩ {p + 1, . . ., n}, wc(σ, [p])) , formed by the subword of σ containing the letters in [p], and the pair (σ∩{p+1, . . ., n}, wc(σ, [p])) representing the sequence of subwords of σ obtained from σ after erasing the letters in [p].We call the first component of ≻ p the lower component of the cut and the second component the upper component.For instance, ≻ 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 forms the lower component of the cut and the generations above it yield the upper component of the cut.10 N. Gilliers and C. Bellingeri

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

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

Figure 8 :
Figure 8: Horizontal glueing of a tree and a forest.

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

Figure 9 :
Figure 9: Shuffle product of levelled forests

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

) Proposition 2 .
(CLF, ¡,θ) is an involutive algebra.Proof .The result follows as a direct a consequence of the following two facts: the left and right actions of S n on LF(n − 1) commute and c n+m = τ n,m • (c n ⊗ (c m + n)) (we add n to all letters of c m and concatenate the resulting word to the one representing c n ) where τ n,m is the shuffle in Sh(n, m) determined by τ n,m (1) = m + 1, τ n,m (n) = m + n.

Figure 9 :
Figure 9: Shuffle product of levelled forests

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

) Proposition 2 .
(CLF, ¡,θ) is an involutive algebra.Proof .The result follows as a direct a consequence of the following two facts: the left and right actions of S n on LF(n − 1) commute and c n+m = τ n,m • (c n ⊗ (c m + n)) (we add n to all letters of c m and concatenate the resulting word to the one representing c n ) where τ n,m is the shuffle in Sh(n, m) determined by τ n,m (1) = m + 1, τ n,m (n) = m + n.

Figure 10 .
Figure 10.The product in Figure 9 expanded over sparse quasi-binary forests.We notice that by the construction of ¡ all terms contained in the sum the blue and red dots preserve the same ordering of generations in the initial factors.A permutation σ ∈ S n acts on the left of a levelled forest f = (σ, c) ∈ LF with n generations as followsσ • (τ, c) := (σ • τ, c).We denote by c n the permutation (n, 1)(n − 1, 2) • • • n − ⌊ n 2 ⌋, ⌊ n 2 ⌋ .Weuse the right action of c n to define a involution on LF(n − 1), which is the horizontal mirror symmetric of a forest θ : CLF → CLF, f = (σ, c) → σc −1 ∥f ∥ , c n , . . ., c 1 .Proposition 2.10.(CLF, ¡,θ) is an involutive algebra.Proof .The result follows as a direct a consequence of the following two facts: the left and right actions of S n on LF(n − 1) commute and c n+m = τ n,m • (c n ⊗ (c m + n)) (we add n to all letters of c m and concatenate the resulting word to the one representing c n ) where τ n,m is the shuffle in Sh(n, m) determined by τ n,m (1) = m + 1, τ n,m (n) = m + n. ■

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

Figure 11 :
Figure 11: Actions of the braiding map.Figure 11.Actions of the braiding map.

Figure 11 .
Figure 11: Actions of the braiding map.Figure 11.Actions of the braiding map.

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

Figure 13 .
Figure 13.The faces of a levelled tree are indicated with arrows.
only depend on the trees τ α , τ β and the shuffle s + .Summing over all shuffles s + , we get α\τ α ¡ β\τ β .The statement of the lemma follows by computing the sum over s − .■ N. Gilliers and C. Bellingeri

Definition 4 . 7 .
For any integer k ≥ 1 and Y 1 , . . ., Y k ∈ A we introduce the operator

Figure 14 :
Figure 14: face-contractions of a levelled forest
to check that the triple (Coll, ⊙, C ⊙ ) is a monoidal category.If the vectors spaces of the collections P and Q above are Banach algebras, then we might use in place of the algebraic tensor product ⊗ the projective one ⊗.An operad is a monoid in the monoidal category (Coll, ⊙, C ⊙ ): Definition A.4.A non-symmetric operad (or simply an operad) is a monoid in the monoidal category (Coll, ⊙, C ⊙ ), i.e., a triple (P, ρ, η P ) of the following objects P ∈ Ob(Coll), ρ : P ⊙ P → P, η P : C ⊙ → P, satisfying the properties (ρ ⊙ id P ) • ρ = (id P ⊙ ρ) • ρ and (η P ⊙ id P ) • ρ = (id P ⊙ η P ) • ρ = id P .