The non-commutative signature of a path
TTHE NON-COMMUTATIVE SIGNATURE OF A PATH
CARLO BELLINGERI (cid:63)
AND NICOLAS GILLIERS † Abstract.
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 appear naturally in Taylor expansionsof solutions to controlled differential equations of interest in non-commutative probabilitytheory. They are built by fully – and partially – contracting iterated integrals of X , seenas tensors, with the product of A . We explain how these operators yield a trajectory ona convolution group of representations of a Hopf monoid of leveled forests we introduce.Finally, if considering only full contraction operators we build a trajectory on a group oftriangular operators, thereby obtaining a notion of signature for a path relevant for non-commutative probability. Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Hopf monoid of leveled binary forests . . . . . . . . . . . . . . . . . . . . .
3. Iterated integrals of a path as operators . . . . . . . . . . . . . . . . . . . .
4. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction
This work intends to broaden a path, opened by A. Deya and R. Schott in [6], towardsapplication of rough path principles for studying the following class of differential equations d Y t = a ( Y t ) · d X t · b ( Y t ) , Y ∈ A , (1.1)where the driving path X : [0 , → A has values in a C (cid:63) -algebra ( A , · , ∗ , (cid:107)·(cid:107) ) and a, b : A → A are either polynomials, either stemming from functions on the real line that are Fouriertransforms of regular measures with exponential moments, see [2, 6]. This restriction is keyto the present work since derivatives at any order of such functions are represented as elementsin tensor products of A . Date : February 24, 2021.2020
Mathematics Subject Classification.
Key words and phrases. signature, rough path, Hopf monoid, shuffle, non-commutative probability, planartrees, operads, properads, duoidal categories. a r X i v : . [ m a t h . OA ] F e b HE NON-COMMUTATIVE SIGNATURE OF A PATH 2
The rough paths approach.
Introduced in the middle nineties [11], T.J. Lyons pro-vided the appropriate mathematical framework to study differential equations under the form dY t = σ ( Y t ) [d X t ] , Y = y ∈ R n . (1.2)In equation (1.2), Y is a continuous paths in a finite dimensional state space R d , σ (the field)is a function from R d to the space End ( R n , R d ) of endomorphisms between the state spaceof the continuous driving noise X and the state space of the solution Y of (1.2). For asmooth driving path X , the equation (1.2) has a rigourous interpretation: the symbol d X stands for the (signed) measure whose repartition function is X itself. For paths with lowerHölder regularity, Young’s theory of integration [15] provides an interpretation to integrals ofthe form (cid:90) ts Y u d X u , ≤ s < t ≤ . (1.3)with X and Y two Hölder paths whose Hölder exponents sum to a number greater than one.Incidentally, (1.2) makes sense for path X with Hölder regularity greater than one half. Inprobability, there is a well-established theory to deal with equations (1.2) for certain stochasticprocesses, that is when X is a continuous semi-martingale (a Brownian motion, for example).These stochastic driving noises are too irregular for Young integration; Brownian paths are − ε Hölder continuous. Instead, the classical Itô integration gives a probabilistic understandingto 1.3, as the limit in probability of Riemann sums. Rough path theory extends the standardrules of differential and integral calculus to provide a meaningful pathwise interpretation to(1.2) for Hölder paths X with low regularity.In fact, given a smooth field σ and an initial point Y ∈ R n the usual solution map of theequation (1.2) Φ : X (cid:55)→ Y is not only continuous with respect to the Lipschitz norm on thespace of smooth driving paths X but one can define a family of topologies where Φ is stillcontinuous when we replace the input X with the data of its signature , the infinite collectionof tensors ( s, t ) → X st = (cid:32) , X t − X s , (cid:90) ∆ st d X t ⊗ d X t , · · · , (cid:90) ∆ nst d X t ⊗ · · · ⊗ d X t n , · · · (cid:33) , (1.4)where ∆ mst := { t > t > · · · > t m > s } is the m -dimensional simplex. The key observation isthe following one : by applying Picard iterations, one quickly reckon that the solution map Φ isa linear functional of the entire signature of X . Equivalently, the solution Y admits a Taylor-like expansion but with the signature of X as basis. One can thus consider the signature of X as the input of controlled differential equations. Now, signatures support a one parameterfamily of topologies, each making Φ a continuous map. Complete spaces for these topologiescontain Hölder paths together with the additional datas of abstract signatures. These abstractsignatures are called rough paths . Such object can alternatively be characterized by a setof algebraic and analytical properties. Indeed, a rough path is in particular the data (alongwith Hölder estimates) for each pair of times s, t of an element X st of a group ( G, (cid:63) ) includedin the completed tensor product such that for any s, u, t ∈ [0 , X st = X su (cid:63) X ut . (1.5)The relations (1.5) are usually called Chen’s relation after Kuo-Tsai Chen [5] and its secularwork on homology of loop spaces. The present work intends to give a similar description, thatis as a set of increments over a certain group, of contractions by an algebra product of iteratedintegrals of paths (in fact all their permutations) with values in a C (cid:63) algebra. These quantitiesact as basis to Taylor expansion of solutions to equations in the class . .1.2. Motivation and previous works.
This paper is the first of a series whose objective isto introduce a new notion of geometric rough paths, tailored to the class of equations (1.1)Such equations arise in non-commutative probability theory. Non-commutative stochasticintegration or quantum stochastic calculus has a long story, see e.g. [7] for stochastic calculus
HE NON-COMMUTATIVE SIGNATURE OF A PATH 3 on the Boson Fock space, or [2] for free stochastic calculus for the free Brownian motion. Inall cases, integration against the considered processes is defined as a careful extension of theusual Itô theory in a L setting. At the same time, even if it is possible to give a precisemeaning of (1.1) in a rough path setting when A is finite-dimensional, by essentially writingthe equations in coordinates, it is crucial to have a more intrinsic –coordinate-free– approachand to work consistently with the specific class of fields we consider, that with the algebraproduct.Rough paths theory on infinite-dimensional spaces is more intricate because tensor productsof Banach spaces are multiple, see [8]. Considering the class of equations (1.1) the projectivetensor product is the only reasonable, since the multiplication of the algebra is always con-tinuous with respect to this topology, which is not true for the spatial (or injective) topology.This limitation strikes with the results obtained in [14, 4]. Whereas an explicit probabilisticconstructions of rough paths over the free Brownian motion in the spacial tensor product is given in [4], a rather negative result entails non-existence of a Lévy area above the freeBrownian motion in the projective tensor product of the algebra.A. Deya and R. Schott introduced in [6] a weaker notion of rough paths, tailored to the classof equations (1.1) when the Hölder scale lies in (1 / , / . A product Lévy area [6] above apath X with Hölder regularity < γ ≤ is a weaker object embodying the data on the smallscale behaviour of the driving noise X in the directions required to give a meaning to (1.2),that is in the direction of the fields considered and all their derivatives. The starting point isthen a fine analysis of (1.2) if X is a smooth path and the expansion of the solution obtainedby applying Picard iterations. Consider this simple example, with A ∈ A and B ∈ A , d Y t = ( A · Y t ) · d X t · ( Y t · B ) , (1.6)The first two steps of the Picard Iteration give: Y t = Y s + (cid:90) ∆ st ( A · Y s ) · d X t · ( Y s · B )+ (cid:90) ∆ st ( A · Y s ) · d X t · ( Y s · B ) · d X t · ( Y s · B )+ (cid:90) ∆ st ( A · Y s ) · d X t · ( A · Y s ) · d X t · ( Y s · B ) + R st , (1.7)where R st is a remainder term involving triple products of the increments. The above equationhints at a control of the small variations of Y by the iterated integrals (cid:90) ∆ n ( s,t ) A · d X t σ − · · · · · d X t σ − n ) · A n , A , . . . , A n ∈ A . (1.8)This is in fact the case, for the class of equations we consider [6]. From these observations, weelaborate on the observation of A. Deya and R. Schott and extract important algebraic andanalytical properties of the multilinear operators (1.8) with the objective of developing a roughtheory 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 "yes, it is possible" and we explain why byassociating to the operators (1.8) a smooth trajectory over a group of triangular morphismson a algebra of operators we introduce.The main difficulties lie in writing a Chen relation for the operators (1.8) understood as acertain "algebraic rule" for computing (1.8) over an interval knowing the values of (1.8) overa subdivision of this interval. Consider for instance the full contraction operator X st ( A , A , A , A ) := (cid:90) ∆ st A · d X t · A · d X t · A · d X t · A (1.9) HE NON-COMMUTATIVE SIGNATURE OF A PATH 4
Then the Chasles identity implies the following deconcatenation formula: X st ( A , A , A , A ) = X st ( A , A , A , A ) + X ut ( A , A , A , A )+ (cid:90) t ∈ ∆ ut (cid:90) ( t ,t ) ∈ ∆ su A · d X t · A · d X t · A · d X t · A + (cid:90) ( t ,t ) ∈ ∆ ut (cid:90) t ∈ ∆ su A · d X t · A · d X t · A · d X t · A . The term on the second line above can not be expressed by composing order two full contrac-tion operators. Instead, we can obtaine it by composing the operator, ( A , . . . , A ) (cid:55)→ (cid:90) ∆ su A · d X t · A ⊗ A · d X t · A ∈ A ⊗ A (1.10)with the following full contraction one ( A , A ) (cid:55)→ (cid:90) ∆ ut A · d X t · A . Thus a naive approach leads in fact to relations involving not only full contractions but alsopartial contractions.The main result of the paper is loosely stated in the following Theorem.
Theorem 1.1.
Let X : [0 , → A be a smooth path. Then there exists a group ( G , ◦ ) , abi-graded set F and a function X : [0 , → G with the following properties: • For any s, u, t ∈ [0 , one has X st = X ut ◦ X su . (1.11) • For any s, t ∈ [0 , , X st has components a family of multilinear operators, { X fst } f ∈F , • The set F contains all permutations σ and X fst coincides with (1.8) when f = σ . • Given two elements X , Y ∈ G , X = Y ⇔ X σ = Y σ , for all permutations σ. We call the element X st the non-commutative signature of the path X and the relations (1.11) non-commutative Chen’s relations . The group G is identified in Proposition 3.26 and the function X is introduced in Definiton3.23. The countable set F is the set of planar leveled forets we introduce in Section 2.1.We now describe the outline of this paper. In Section 2, we introduce a Hopf monoid oflevelled forests, reminiscent of the Malvenuto Reuteneauer Poirier Hopf algebra. In Section3.1, we define the partial and full contractions operators we alluded to, see Definitions 3.1and 3.3. In Section 3.2 we prove a Chen relation for these operators, see Proposition 3.4.Next, we explain how this yields a path on a group of triangular algebra morphisms on analgebra spanned by couples of a tree and a word. In Section 3.4 we associate to the full andpartial contractions operators a path of representations on the Hopf monoid of levelled forestswe introduced in Section 2, see Theorem 3.14. In Section 3.5, we adopt a slightly differentpoint of view and let the iterated integrals of a path to act on a set of operators we call facescontractions , see Definition 3.15. This yields a certain triangular algebra morphism, seeDefinition 3.23 that we relate to the one introduced in Section 3.2. In Proposition 3.25, werelate partial to full contractions operators.In a forthcoming article, we continue to develop the theory. In particular, we introducegeometric non-commutative rough paths, geometric non-commutative controlled rough path,the operations of integration and composition and prove existence of solutions to (1.2). Acknowledgements : The authors are especially grateful to Kurusch Ebrahimi-Fard formany enlightening discussions. CB is supported by the DFG Research Unit FOR2402 andNG is funded by DAAD kurzstipend.
HE NON-COMMUTATIVE SIGNATURE OF A PATH 5
Preliminaries.
In the following we denote by A a generic complex C ∗ algebra with product µ , unity , norm (cid:107)·(cid:107) and involution ∗ . In order to deal with a topology on the algebraic tensor product ⊗ whichbehaves correctly with µ , we will use the projective tensor product (see e.g. [12]). Given twoBanach spaces ( E, (cid:107) · (cid:107) E ) and ( F, (cid:107) · (cid:107) F ) , the projective norm of an element x ∈ E ⊗ F isdefined by (cid:107) x (cid:107) ∨ = inf (cid:40)(cid:88) i (cid:107) a i (cid:107) E (cid:107) b i (cid:107) F : x = (cid:88) i a i ⊗ b i (cid:41) . We denote by E ˇ ⊗ F the completion of E ⊗ F for the projective norm. One can check thefollowing properties (cid:107) a ⊗ b (cid:107) ∨ = (cid:107) a (cid:107) E (cid:107) b (cid:107) F , (cid:107) a σ (1) ⊗ · · · ⊗ a σ ( n ) (cid:107) ∨ = (cid:107) a ⊗ · · · ⊗ a n (cid:107) ∨ , (1.12)for any permutation σ of order k and a , · · · , a n ∈ E . From the definition of projective tensorproduct, it follows easily that the multiplication map µ extends to a continuous map A ˇ ⊗A →A and, more generally, for any given couple of C ∗ algebras A , B , A ˇ ⊗B is again a C ∗ algebra.Looking this operation from a very general point of view, the projective tensor product makesthe category of complex C ∗ algebras a monoidal category (see the Appendix). In order tolighten the notation, we will adopt the symbol ⊗ to denote both the the projective tensorproduct between C ∗ algebras and the algebraic tensor product for pure tensors. Similarly, wewill replace the product µ with a dot · .For n ≥ an integer, we denote by S n the set of permutations of [ n ] = { , · · · , n } , which isdenoted by ( σ (1) · · · σ ( n )) . Given two integers a, b , we denote by Sh ( a, b ) the set of all shufflesof the intervals (cid:74) , a (cid:75) and [ a + 1 , a + b ] , σ ∈ S a + b , σ ∈ Sh ( a, b ) if and only if σ is non-decreasingon [1 , a ] and on [ a + 1 , a + b ] .2. Hopf monoid of leveled binary forests
Leveled trees and forests.
The objective of the present section is to introduced thecombinatorial tool that will be used through this work; the algebra of leveled trees, isomor-phic to the Malvenuto-Reutenauer-Poirier algebra. In the literature, one broadly finds twoequivalent representations of a permutation, either as a word on integers or as a bijectionfrom a certain interval of integers. We use a third – tree like – graphical representation of apermutation introduced in [9] by Ronco and Loday.First, recall that a planar rooted tree is a planar graph with no cycles and one distin-guished vertex we call the root. A tree is oriented from top to bottom : the target of edgeis the vertex closest to the root (for the graph distance). In this orientation, each vertex ofa tree as at most one outgoing edge (the root is the only vertex with no outgoing edge) andseveral inputs. A leaf of a tree is a vertex with no incoming edges. An internal vertex ofa tree is a vertex that is not a leaf. A planar binary tree is a tree for which every internalnode has two inputs.Pick t a planar rooted tree. The set of internal vertices of τ is equipped with a partialorder ≺ τ . Pick v, w two vertices of t , we write v ≺ t w if there is an oriented path of edges of t from w to v .The set of internal vertices of a tree t is denote V ( t ) and we set | V ( t ) | = (cid:107) t (cid:107) . Notice thatfor binary trees we have (cid:107) t (cid:107) = | t | − . Definition 2.1 ( leveled planar binary tree LT ) . A leveled planar binary tree (orsimply a leveled tree) is a pair ( t, g ) with t a planar binary tree and g an increasing function g : ( V ( τ ) , ≺ τ ) → [ (cid:107) τ (cid:107) ] We denote the set of planar binary tree by LT .Notice that the root tree has no internal vertices and correspond to leveled tree ( , ∅ ) wherehere ∅ denotes the unique function from the empty set to the empty set. The degree of a HE NON-COMMUTATIVE SIGNATURE OF A PATH 6 leveled planar tree τ ∈ LT is the number of its leaves. The complex span of LT is a gradedvector space, its homogeneous component of degree n ≥ is the linear span of trees with n leaves. By definition, a leveled tree with degree one is the root tree (see Fig 1). We denoteby LT n the set of leveled trees with n generations and LT ( n ) the set of leveled trees with n leaves. Proposition 2.2 ([9]) . Let n be an integer greater than one. The set of leveled planary rootedbinary trees LT n +1 is in bijection with the set of permutations S n . We use throughout this work a convenient graphical representation of a leveled binary tree τ = ( t, f ) . It consists in associating to τ a planar tree τ (cid:48) that is obtained by vertically orderingthe internal vertices of t , according to t by adding straight edges. A generation of such a treeis the set of all internal vertices at the same distance from the root. Each generation of τ (cid:48) hasexactly one vertex with two inputs, see Fig. 1. This graphical presentation turns effective todescribe certain operations that we introduced in the next sections on leveled planar forests. Figure 1.
Examples of leveled trees τ = ( t, f ) (we have drawn τ (cid:48) , see above)in LT and their associated word. We obtain the corresponding binary trees t by contracting all the straight edges.Leveled trees are not sufficient for our purposes, we need leveled planar forests that weintroduce now.A planar forest is a word (a non-commutative monomial) on planar trees. In the following,we denote by nt ( f ) the number of trees in the forest f , | f | the total number of leaves in theforest and we set (cid:107) f (cid:107) equal to the number of internal vertices of the forests. If all trees of f are binary trees, then (cid:107) f (cid:107) = | f | − nt ( f ) . The poset ( V ( f ) , ≺ f ) of ordered internal vertices of f is the union of the posets of internal vertices of the trees in f . Definition 2.3 ( leveled planar binary forests LF ) . A leveled planar binary forest f (orsimply a leveled forest) is a pair ( t f , (cid:96) f ) of a planar forest t f and a increasing bijection g : V ( f ) → [ || f || ] , v ≺ f w ⇔ g ( v ) < g ( w ) . We denote the set of planar binary tree by LF . Figure 2.
A leveled forest with five trees.The degree of a leveled planar forest f ∈ LT is the number of leaves of t f . The set LF is bi-graded, the homogeneous set of degrees n ≥ and m ≥ is LT ( n, m ) of leveled forests HE NON-COMMUTATIVE SIGNATURE OF A PATH 7 with n leaves and m trees. Notice that the leveled forests with the same numbers of leavesand trees are the leveled forests f with t f a forest of root trees. Finally, we denote by LF n the set of leveled trees with n generations.A leveled forest f = ( t f , t g ) can be pictured as planar forest in the same way as a explainedbefore for leveled trees, the internal vertices are ordered vertically by adding straight edgesaccording to g , see Fig. 2. We denote this forest by f (cid:48) .Above, we settled a bijection between leveled trees in LT and permutations. We can thustransfer the actions of a permutation σ in S n by right and left multplication on S n to left andright actions on leveled trees with n generations. We now explain how this action extends toleveled planar forests.First, we define bijection between leveled forests in LF and pairs of a permutation and aninterval partition (that may contain empty blocks).Pick a leveled forest f = ( t f , (cid:96) f ) . To each planar decorated forest with n internal vertices τ = ( t, (cid:96) | V ( τ )) with t a tree in the planar forest f , we associate a (possibly empty) word [ f ] τ with entries [ n ] ; it is obtained by reading from left to right the labels of the tree.If τ is the root tree then we associate to this tree the empty word ∅ . In that way we builda word on words [ f ] = [ f ] τ · · · [ f ] τ nt ( f ) . For example, the word on words associated with theleveled planar forest in Fig. 2 is | ∅ | | | ∅ . It is easily seen that this correspondance between leveled forests with n internal vertices awords on words, each with entries in [ n ] such that each integer appears only once is bijective.A permutation σ ∈ S (cid:107) f (cid:107) left acts on the word [ f ] by evaluating this permutation on eachletter of each word in [ f ] . We denote by σ · f the forest associated with the word σ · [ f ] .For example, the permutation (1 , , acts on the word on words representing the leveledforest in Fig 2 by (1 , , · (13 | ∅ | | | ∅ ) = 23 | ∅ | | | ∅ . We define a right action of σ on f commuting with the left one in the next section.We thus have three presentation of a leveled forest f , either a couple ( t f , (cid:96) f ) , either as abinary planar forest with straight edges, f (cid:48) , either as a word [ f ] . With t f = t · · · t nt ( f ) , wewrite f = f · · · f p with f i = ( t i , (cid:96) f | V ( t i ) ) .2.2. Operations on leveled forests. If α = ( t α , f α ) and β = ( t β , f β ) are two leveled trees,we write β ⊂ α and say that β is a leveled subtree of α if1. t β contains the root and is a subtree of t α , in particular V ( t β ) ⊂ V ( t α ) ,2. t α ( v ) = t β ( v ) , v ∈ V ( t β ) .Similarly, if β and α are two leveled forests, we write β ⊂ α if the two forests have the sumnumber of trees, so α = α · · · α p , β = β · · · β p for a certain integer p ≥ and β i ⊂ α i for ≤ i ≤ p . Figure 3.
On the first line a leveled forest and on the second line the leveledsubforests included.
Definition 2.4 ( Vertical cutting and gluing of forests ) . For α and β two leveled forestsas above, with β ⊂ α , we define the cut of α by β as the leveled forest α \ β = ( t α \ β , (cid:96) α \ β ) with HE NON-COMMUTATIVE SIGNATURE OF A PATH 8 t α \ β is the planar forest obtained by erasing all edges and internal vertices that belongsto t β from t α ,2. noticing that V ( t α \ β ) = V ( t α ) \ V ( t β ) , one defines (cid:96) α \ β ( v ) = (cid:96) t α ( v ) −(cid:107) β (cid:107) for v ∈ V ( t α \ β ) .The operation opposite to cutting is gluing and is denoted . Pictorially, we glue aleveled forest α up to β , by stacking the forest α up to the forest β if the number of treesin t α matches the numbers of leaves of t β . More formally, we define β α = ( t β α , (cid:96) β α if nt ( α ) = | β | by1. t β α is the grafting of the forest t α up to the forest t β ,2. noticing that V ( t β α ) = V ( t α ) (cid:116) V ( t β ) , (cid:96) β α ( v ) = (cid:96) α ( v ) for v ∈ V ( α ) , (cid:96) β α ( v ) = (cid:96) β ( v ) for v ∈ V ( t β ) Above, we introduced binary operations on leveled forests, that are vertical cutting anggluing. We introduce their horizontal counterpart, that are unary operations.
Definition 2.5 ( Horizontal gluing and cutting of forests ) . Let f ∈ LF a leveled forest,we denote by f (cid:91) ∈ LT the leveled tree corresponding to the planar forest (with straight edges)obtained by concatenating all together the trees in the forest f (cid:48) along their external edges.Otherwise stated, the word representation [ f (cid:91) ] of f (cid:91) is obtained by concatenating all words in [ f ] . Let τ be a leveled tree and ( n , . . . , n k ) a composition of n , n i ≥ and (cid:80) ki =1 = n . Wedefine τ n ,...,n k the leveled forest which is represented by the word [ τ ] · · · [ τ ] n | · · · | [ τ ] n + n k − +1 · · · [ τ ] n + ··· + n k . with the convention that the chunk [ τ ] n + ··· + n i − · · · [ τ n + ··· + n i ] = ∅ if n i = 0 . Figure 4.
On the first line, the right tree is obtained by gluing all trees inthe forest f along their external edges. On the second line, the two rightmostforest are horizontal cuts of the leftmost forests with parameters (1 , and (1 , , , respectively.Let f = f · · · f p a leveled forest. Then a simple drawing show that for all subforest f (cid:48) ⊂ f ,one has f \ f (cid:48) = (cid:104) f (cid:91) \ ( f (cid:48) (cid:91) ) (cid:105) || f || ,..., (cid:107) f p (cid:107) , f (cid:48) ⊂ f. We can design more general cutting and gluing operations, such as for example gluing of asubset of trees of a leveled forests. These operations will be needed only in the last section,where they are defined.2.3.
Shuffle product on leveled forests.
In this section, we denote by (cid:104)(cid:104) n (cid:105)(cid:105) the set ofwords on words with entries in [ n ] , for example | , | , |∅ ∈ (cid:104)(cid:104) n (cid:105)(cid:105) . A permutation σ ∈ S n acts on the left of a word w with entries in [ n ] : one simply applies σ on each letter in w . Theaction of σ extends as a morphism of (cid:104)(cid:104) n (cid:105)(cid:105) for the concatenation product · .Owing to the correspondance between leveled forests and words on words, the above de-fined action of the permutation σ induces an action on the set LF n of leveled forests with n HE NON-COMMUTATIVE SIGNATURE OF A PATH 9 generations. For example, the permutation (1 , , acts on the word on words representingthe leveled forest in Fig 2 by (1 , , · (13 | ∅ | | | ∅ ) = 23 | ∅ | | | ∅ . With f = f · · · f p and g = g · · · g q two leveled forests, we denote by f × g the leveled forestdefined by1. t f × g is the planar forest that start (when read from the left) with the trees t f , . . . , t f p − followed by the tree obtained by grafting t g to the rightmost leaf of f p and end withthe trees g · · · g p ,2. (cid:96) f × g ( v ) = f ( v ) , v ∈ V ( f ) , (cid:96) f × g ( v ) = g ( v ) + (cid:107) α (cid:107) , v ∈ V ( g ) .The operations × is better understood with the help of the word representation of a leveledforest, [ f × g ] is the word [ f ] · · · | ([ f ] nt ( f ) · [ g ] ) | · · · [ g ] q . The binary operations × extends toas an unital bilinear associative product on C [ LT ] , its unit is the empty word ∅ . Definition 2.6 ( Shuffle product of leveled planar forests).
Let f and g be two leveledforests, define the shuffle product of f and g by f (cid:1) g = (cid:88) s ∈ Sh ( (cid:107) f (cid:107) , (cid:107) f (cid:48) (cid:107) ) s · ( f × g ) . and extends it bilinearly to C [ LF ] . Remark . The product (cid:1) restricted to LT is the shuffle product of Malvenuto-Reutenauer.The product (cid:1) crosses degrees : the number of leaves of f (cid:1) g is | f | + | g | − and the numberof trees is nt ( f ) + nt ( g ) . From associativity of × and well known facts about shuffle, (cid:1) isassociative and unital; its unit is the root tree.In addition to the left action of a permutation σ ∈ S n on a leveled forest with n generations,one can also define a right action of σ that consists in permuting horizontally the generations,instead of vertically. In terms of the word on word representing a forest f , this means [ f ] · σ = [[ f (cid:91) ] · σ ] (cid:107) f (cid:107) ,..., (cid:107) f nt ( f ) (cid:107) := [ f (cid:91)σ (1) · · · f (cid:91)σ ( (cid:107) f (cid:107) ) ] (cid:107) f (cid:107) ,..., (cid:107) f nt ( f ) (cid:107) (2.1)We denote by c n the permutation ( n, n − , · · · ( n − (cid:98) n (cid:99) , (cid:98) n (cid:99) ) . We use the right actionof c n to define a involution on LF n , which is the horizontal mirror symmetric of a forest : C [ LF ] → C [ LF ] f (cid:55)→ ( f (cid:91) · c (cid:107) f (cid:107) ) (cid:107) f ( nt ( f ) (cid:107) ,..., (cid:107) f (cid:107) (2.2) Proposition 2.8. ( C [ LF ] , (cid:1) , ) is an involutive algebra.Proof. It is a direct a consequence of the following two facts : the left and right actions of S n on LF n commute and c n + m = τ n,m ◦ c n ⊗ c m where τ n,m is the shuffle in Sh ( n, m ) determinesby τ n,m (1) = m + 1 , τ n,m ( n ) = m + n . (cid:3) We will sometimes refer to as the horizontal involution , for obvious reasons, to dis-tinguish it from a second involution permuting vertically the generations of a forest that wedefine below.2.4.
Hopf monoid of leveled forests.
In this section we introduce a Hopf algebraic struc-ture on the bicollection of spanned by leveled forests and denoted LF , LF ( n, m ) = C [ LF ( n, m )] , n, m ≥ . In addition, we set LF (0 ,
0) = C , LF (0 , n ) = LF ( m,
0) = 0 , n, m ≥ and LT for thecollection spanned by leveled binary trees. This Hopf algebra is an object in the category ofbicollections endowed with the vertical tensor product (cid:9) . In general, as it is briefly explainedin the Appendix, the two-folded vertical tensor product A (cid:9) A of a monoid A in the monoidalcategory (Coll , (cid:9) ) is not a monoid in the same category. Owing to the fact that the monoidgenerated by LF in (Coll , (cid:9) ) is symmetric , in particular LF (cid:9) LF is a monoid in a natural HE NON-COMMUTATIVE SIGNATURE OF A PATH 10 way, it makes sense to require compatibility between a product and a co-product on LF . Wewrite the unit C (cid:9) for the vertical tensor product as C (cid:9) = (cid:77) n ≥ C n . Recall that we denote by | f | the number of leaves of a leveled forest f and nt ( f ) the numberof trees in f .We begin with the definition of the coproduct acting on the bicollection LF of leveledforests. Let f be a leveled forest. Let f (cid:48) be a leveled subforest of f (recall that f (cid:48) contains theroots of all trees in f ). By definition of the forest f \ f (cid:48) , the number of outputs of the forest f \ f (cid:48) is equal to the number of inputs of the forest f (cid:48) (the number of trees of f \ f (cid:48) matchesthe number of leaves of f (cid:48) ), the following makes senses ∆( f ) = (cid:88) f (cid:48) ⊂ f f (cid:48) (cid:9) f \ f (cid:48) , f ∈ LF (2.3) Proposition 2.9 (Coproduct) . The bicollections morphism ∆ :
LF → LF (cid:9) LF is coasso-ciative: (∆ (cid:9) id LF ) ◦ ∆ = (id LF (cid:9) ∆) ◦ ∆ (co-ass.) ε : LF → C (cid:9) , ε ( n ) = 1 n , ε ( f ) = 0 , f (cid:54) = n , (counit) ( ε (cid:9) id LF ) ◦ ∆ = (id LF (cid:9) ε ) ◦ ∆ = id . Proof.
Let f be a leveled forest, to show coassociativity we notice that: ((∆ (cid:9) id LF ) ◦ ∆)( g ) = (cid:88) f (cid:48)(cid:48) ,f (cid:48) ,ff (cid:48)(cid:48) f (cid:48) f = g f (cid:48)(cid:48) (cid:9) f (cid:48) (cid:9) f = ((id LF (cid:9) ∆) ◦ ∆)( g ) . (2.4)Hence the above properties follow. (cid:3) We proceed now with the definition of a vertical product ∇ : LF (cid:9) LF → LF . Given twoforests f and f (cid:48) with nt ( f (cid:48) ) = | f | , we define ∇ ( f (cid:9) f (cid:48) ) as the sum of forests obtained by firststacking f (cid:48) up to f and then shuffling the generations of f (cid:48) with the generations of f (seeSection 2.1 for the definition of the action of a permutation on the generations of a forest), ∇ ( f (cid:9) f (cid:48) ) = (cid:88) s ∈ Sh ( (cid:107) f (cid:107) , (cid:107) f (cid:48) (cid:107) ) s · ( f f (cid:48) ) . (2.5)Associativity of the product ∇ is easily checked. The unit η : C (cid:9) → LF is defined by η (1 m ) = m . Let n ≥ , recall that we denote by c n the maximal element for the Bruhatorder in S n : c n = (cid:89) p ∈ (cid:74) , (cid:98) n (cid:99) (cid:75) ( p, n − p ) . For example, c = (1) , c = (12) , c = (1 , , c = (14)(23) . Given these notion, we state themain theorem of the section Theorem 2.10. ( LF , ∇ , η, ∆ , ε ) is a conilpotent Hopf algebra in the category (Coll , (cid:9) , C (cid:9) ) . To achieve this result we introduce an explicit antipode map.
Definition 2.11.
Pick n, m ≥ two integers. Let f ∈ LF ( n, m ) be a leveled forest anddefine its vertical mirror symmetric f (cid:63) ∈ LF ( n, m ) by f (cid:63) = c (cid:107) f (cid:107) · f. We extend (cid:63) as a conjugate-linear morphism on the bicollection LF . HE NON-COMMUTATIVE SIGNATURE OF A PATH 11
Proposition 2.12.
Let f be a leveled forest. The map S : LF → LF defined by S ( f ) = ( − (cid:107) f (cid:107) f (cid:63) (2.6) is an antipode: ∇ ◦ ( S (cid:9) id LF ) ◦ ∆ = ∇ ◦ (id LF (cid:9) S ) ◦ ∆ = ε ◦ η .Proof. Let a, b be two integers greater than one. Set n = a + b . The set of shuffles Sh ( a, b ) isdivided 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 ) − ).Recall that if f is a forest then f k − denotes the forest obtained by extracting the k first lowestgenerations of f and f k + denotes the forest obtained by extracting the k highest generationsof f . By definition, one has: ∇ ( f (cid:48) (cid:9) f \ f (cid:48) ) = (cid:88) s ∈ Sh ( (cid:107) f (cid:48) (cid:107) , (cid:107) f \ f (cid:48) (cid:107) ) s · ( f (cid:48) f \ f (cid:48) ) (cid:63) ) , f (cid:63) = c (cid:107) f (cid:107) · f. The following relation is easily checked and turn to be the cornerstone of the proof: ˜ s ◦ ( c n ⊗ id m ) = s ◦ ( c n +1 ⊗ id m − ) , s ∈ Sh ( m − , n + 1) − , (2.7)with ˜ s the unique shuffle in Sh ( m, n ) + such that ˜ s ( m ) = n + m, ˜ s ( i ) = s ( i ) . Set ¯ S ( f ) =( − (cid:107) f (cid:107) f (cid:63) . We prove by induction that S = ¯ S . Assume that S ( f ) = ¯ S ( f ) for any forest f with at most N ≥ generations and pick a forest f with N + 1 generations. Then, from theinduction hypothesis we get: S ( f ) + f + (id (cid:9) ¯ S ) ◦ ¯∆( f ) = 0 . ∇ ◦ (id (cid:9) ¯ S ) ◦ ¯∆( f ) = (cid:88) f (cid:48) ⊂ f ( − (cid:107) f \ f (cid:48) (cid:107) (cid:88) s ∈ Sh ( (cid:107) f (cid:48) (cid:107) , (cid:107) f \ f (cid:48) (cid:107) ) s · (cid:2) f (cid:48) f \ f (cid:48) ) (cid:63) (cid:3) = (cid:107) f (cid:107)− (cid:88) k =1 ( − k (cid:88) s ∈ Sh ( (cid:107) f (cid:107)− k,k ) s · (cid:104) ( f (cid:107) f (cid:107)− k − f k + ) (cid:63) (cid:105) . We divide the sum over the set Sh ( (cid:107) f (cid:107) − k, k ) into two sums. The first sums ranges over thesubset Sh ( (cid:107) f (cid:107) − k, k ) + and the second one ranges overs Sh ( (cid:107) f (cid:107) − k, k ) − . Then, we gather thesums over Sh ( (cid:107) f (cid:107) − k, k ) + and Sh ( (cid:107) f (cid:107) − k + 1 , k − − : ∇ ◦ (id (cid:9) ¯ S ) ◦ ¯∆ = (cid:107) f (cid:107)− (cid:88) k =2 ( − k (cid:88) s ∈ Sh ( (cid:107) f (cid:107)− k,k ) + s · (cid:104) f (cid:107) f (cid:107)− k − f k + ) (cid:63) (cid:105) − ( − k (cid:88) s ∈ Sh ( (cid:107) f (cid:107)− k +1 ,k − − s · (cid:104) f (cid:107) f (cid:107)− k +1 − f k − ) (cid:63) ) (cid:105) + ( − (cid:88) s ∈ Sh (1 , (cid:107) f (cid:107)− − s · (cid:104) f (cid:107) f (cid:107)− − f ) (cid:63) (cid:105) + ( − (cid:107) f (cid:107)− (cid:88) s ∈ Sh ( (cid:107) f (cid:107)− , + s · (cid:104) f − f (cid:107) f (cid:107)− ) (cid:63) (cid:105) Using equation (2.7), the right hand side of the last equation is equal to: ∇ ◦ ( ¯ S (cid:9) id) ◦ ¯∆ = 0 − (cid:88) s ∈ Sh (1 , (cid:107) f (cid:107)− − s · (cid:104) f − f (cid:107) f (cid:107)− ) (cid:63) (cid:105) + ( − (cid:107) f (cid:107)− (cid:88) s ∈ Sh (1 , (cid:107) f (cid:107)− + s · (cid:104) f (cid:107) f (cid:107)− − f ) (cid:63) (cid:105) = − f + ( − (cid:107) f (cid:107)− f (cid:63) This ends the proof. (cid:3)
HE NON-COMMUTATIVE SIGNATURE OF A PATH 12
We defined the three structural morphisms ∇ , ∆ , S . To turn LF into a Hopf monoid, wehave to check compatibility between the coproduct ∆ and the product ∇ ; the coproduct ∆ should be an morphism of the monoid ( LF , ∇ ). This only makes sense provided that we candefine a product on the tensor product LF (cid:9) LF .Recall that if f is a leveled forest and ≤ k ≤ (cid:107) f (cid:107) , one denotes by f k − the leveled subforestof f corresponding to the k generations at the bottom of f (cid:48) : 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 internalvertices. The leveled forest f k + is obtained similary by extracting the k top generations of f (cid:48) .With p, q ≥ 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 2.13.
Define the braiding map K : LF (cid:9) LF → LF (cid:9) LF by, for g and f leveled forests such that f (cid:9) g ∈ LF (cid:9) LF , K ( f (cid:9) g ) = (cid:0) τ (cid:107) f (cid:107) , (cid:107) g (cid:107) · ( f g ) (cid:1) (cid:107) g (cid:107)− (cid:9) (cid:0) τ (cid:107) f (cid:107) , (cid:107) g (cid:107) · ( f g ) (cid:1) (cid:107) f (cid:107) + . We pictured in Fig. 5 examples of the action of the braiding map on pairs of leveled forests.
Figure 5.
Actions of the braiding mapWe defined the braiding map K as acting on LF (cid:9) LF . We extend K as a -functor onthe product of the monoid generated by LF in (Coll , (cid:9) ). This means in particular that forintegers p, q ≥ , we define a bicollection morphism K p,q : LF (cid:9) p (cid:9) LF (cid:9) q → LF (cid:9) q (cid:9) LF (cid:9) p . Pick f (cid:9) · · · (cid:9) f p ∈ LF (cid:9) p and g (cid:9) · · · (cid:9) g q ∈ LF (cid:9) q . We first stack vertically theforests f , . . . , g q yielding a leveled forest f · · · g q . The permutation τ (cid:107) f (cid:107) + ··· + (cid:107) f p (cid:107) , (cid:107) g (cid:107) + ··· + (cid:107) g q (cid:107) then act on the generation of f · · · g q . Finally, we define h n ,...,n p with n + · · · + n p = (cid:107) h (cid:107) to be the element in LF (cid:9) p obtained by recursively extracting generations of f , starting withthe first n generations at the bottom, continuing with the next n generations and so on, weset K p,q ( f (cid:9) · · · (cid:9) f p (cid:9) g (cid:9) · · · (cid:9) g q ) = ( τ (cid:107) f (cid:107) + ··· + (cid:107) f p (cid:107) , (cid:107) g (cid:107) + ··· + (cid:107) g q (cid:107) · f · · · g q ) (cid:107) g (cid:107) ,..., (cid:107) g p (cid:107) , (cid:107) f (cid:107) ,..., (cid:107) f q (cid:107) (2.8) Proposition 2.14.
The monoid generated by the bicollection LF in (Coll , (cid:9) ) is a symmetricmonoidal category with symmetry constraints ( K p,q ) p,q ≥ , K p,q ◦ K q,p = id and (id LF (cid:9) q (cid:9) K p,r ) ◦ ( K p,q (cid:9) id LF (cid:9) r ) = K p,q + r Proof.
Both assertions are trivial and rely on the following relations between the permutations τ p,q , p, q ≥ : τ p,q ◦ τ q,p = id , (id q ⊗ τ p,r ) ◦ ( τ p,q ⊗ id r ) = τ p,q + r , p, q, r ≥ . (cid:3) HE NON-COMMUTATIVE SIGNATURE OF A PATH 13
Using the above defined symmetry constraint K , we can endow the two-folds tensor product LF (cid:9) LT with an algebra product: ( ∇ (cid:9) ∇ ) ◦ (id (cid:9) K (cid:9) id) : LF (cid:9) → LF (cid:9) Proposition 2.15.
The two bicolletion morphisms ∆ :
LF → LF (cid:9) LF and ∇ : LF (cid:9) LF →LF are vertical algebra morphisms. With ∇ (2) = ∇ ◦ ( ∇ (cid:9) id) = ∇ ◦ (id (cid:9) ∇ ) , this means that ∇ (2) = ∇ (2) ◦ (id (cid:9) K (cid:9) id) , ( ∇ (cid:9) ∇ ) ◦ (id (cid:9) K (cid:9) id) ◦ (∆ (cid:9) ∆) = ∆ ◦ ∇ . Remark . 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 , f , f , f compatible leveled forests (thenumber of inputs of f i matches the number of outputs of f i +1 , ≤ i ≤ ), ( ∇ (2) ◦ K )( f (cid:9) f (cid:9) f (cid:9) f )= (cid:88) s ∈ Sh ( (cid:107) f (cid:107) , (cid:107) f (cid:107) , (cid:107) f (cid:107) , (cid:107) f (cid:107) ) s · (cid:104) f (cid:0) τ (cid:107) f (cid:107) , (cid:107) f (cid:107) · ( f f ) (cid:1) (cid:107) f (cid:107)− (cid:0) τ (cid:107) f (cid:107) , (cid:107) f (cid:107) · ( f f ) (cid:1) (cid:107) f (cid:107) + f (cid:105) = (cid:88) s ∈ Sh ( (cid:107) f (cid:107) , (cid:107) f (cid:107) , (cid:107) f (cid:107) , (cid:107) f (cid:107) ) (cid:0) s (id ⊗ τ (cid:107) f (cid:107) , (cid:107) f (cid:107) ) (cid:1) · (cid:16) f f f f (cid:17) = (cid:88) s ∈ Sh ( (cid:107) f (cid:107) , (cid:107) f (cid:107) , (cid:107) f (cid:107) , (cid:107) f (cid:107) ) s · (cid:16) f f f f (cid:17) = ∇ (2) ( f (cid:9) f (cid:9) f (cid:9) f ) . For the second assertion, we write first: (∆ ◦ ∇ )( f (cid:9) g ) = (cid:88) ≤ k ≤(cid:107) f (cid:107) + (cid:107) g (cid:107) (cid:88) s ∈ Sh ( k, (cid:107) f (cid:107) + (cid:107) g (cid:107)− k ) ( s · ( f g )) k − (cid:9) ( s · ( f g )) (cid:107) f (cid:107) + (cid:107) g (cid:107)− k + For each integer ≤ k ≤ (cid:107) f (cid:107) , we split the set of shuffles Sh ( (cid:107) f (cid:107) , (cid:107) g (cid:107) ) according to thecardinal q of the set s − ( (cid:74) , k (cid:75) ) ∩ (cid:74) (cid:107) f (cid:107) + 1 , (cid:107) f (cid:107) + (cid:107) g (cid:107) (cid:75) . Then a shuffle s ∈ Sh ( (cid:107) f (cid:107) , (cid:107) g (cid:107) ) s = ( s ⊗ s ) ◦ ˜ τ k,q with ˜ τ k,q the unique shuffle that sends the interval (cid:74) (cid:107) f (cid:107) + 1 , (cid:107) f (cid:107) + q (cid:75) tothe interval (cid:74) k − q + 1 , k (cid:75) and fixes the interval (cid:74) (cid:107) f (cid:107) + q + 1 , (cid:107) f (cid:107) + (cid:107) g (cid:107) (cid:75) . (cid:88) ≤ k ≤(cid:107) f (cid:107) , ≤ q ≤(cid:107) g (cid:107) , ≤ q ≤ k (cid:88) s ∈ Sh ( k − q,q ) ,s ∈ Sh ( (cid:107) f (cid:107)− ( k − q ) (cid:107) g (cid:107)− q ) (( s ⊗ s ) ◦ ˜ τ k,q ) · ( f g )) k − (cid:9) (( s ⊗ s ) ◦ ˜ τ k,q · ( f g )) (cid:107) f (cid:107) + (cid:107) g (cid:107)− k + Notice that ˜ τ k,q = τ k − q,q and ˜ τ k,q · ( f g ) = f k − q − τ (cid:107) f (cid:107)− ( k − q ) ,q · ( f (cid:107) f (cid:107)− ( k − q )+ g q − )) g (cid:107) g (cid:107)− q + . It follows that ( s ⊗ s ) ◦ ˜ τ k,q · ( f g )) k − = (cid:16) ( s ⊗ id) · f k − q − τ (cid:107) f (cid:107)− ( k − q ) ,q · ( f (cid:107) f (cid:107)− ( k − q )+ g q − )) g (cid:107) g (cid:107)− q + (cid:17) k − = s · f k − q − τ (cid:107) f (cid:107)− ( k − q ) ,q · f (cid:107) f (cid:107)− ( k − q )+ g q − ) q − . Similar computations show that (( s ⊗ s ) ◦ ˜ τ k,q · ( f g )) (cid:107) f (cid:107) + (cid:107) g (cid:107)− k + = s · ( τ (cid:107) f (cid:107)− ( k − q ) ,q · ( f (cid:107) f (cid:107)− ( k − q )+ g q − )) (cid:107) f (cid:107)− ( k − q )+ g (cid:107) g (cid:107)− q + ) . The case (cid:107) f (cid:107) + 1 ≤ k ≤ (cid:107) f (cid:107) + (cid:107) g (cid:107) is similar, we split the set of shuffles Sh ( (cid:107) f (cid:107) , (cid:107) g (cid:107) ) accordingto the cardinal of the set s − ( (cid:74) k + 1 , (cid:107) f (cid:107) + (cid:107) g (cid:107) (cid:75) ) ∩ (cid:74) , (cid:107) f (cid:107) (cid:75) ) and omitted for brevity. Finally,we obtain for ∆ ◦ ∇ ( f (cid:9) g ) the expression: (cid:88) ≤ k ≤(cid:107) f (cid:107) , ≤ q ≤(cid:107) g (cid:107) (cid:88) s ∈ Sh ( k,q ) s ∈ Sh ( (cid:107) f (cid:107)− k, (cid:107) g (cid:107)− q ) s · ( f k − τ (cid:107) f (cid:107)− k,q · ( f (cid:107) f (cid:107)− k + g q − )) q − ) (cid:9) s · ( τ k,q · ( f (cid:107) f (cid:107)− k + g q − )) (cid:107) f (cid:107)− k + g (cid:107) g (cid:107)− q + ) which is easily seen to be equal to ( ∇ (cid:9) ∇ ) ◦ (id (cid:9) K (cid:9) id) ◦ (∆ (cid:9) ∆)( f (cid:9) g ) . (cid:3) HE NON-COMMUTATIVE SIGNATURE OF A PATH 14 Iterated integrals of a path as operators
For the entire section, X : [0 , → A denotes a smooth path. We introduce a set ofmultilinear functions on A called partial- and full-contraction operators, indexed by leveledforests and pairs of times s < t . These operators are obtained by contracting a tuple ofelements of the algebra using the multiplication with an iterated integral of the path X , seeDefinition 3.1.3.1. Full and partial contractions operators.Definition 3.1.
Let n ≥ an integer and σ a permutation in S n . We introduce the map X σ : [0 , → Hom ( A ⊗ ( n +1) , A ) X σst ( A , . . . , A n ) = (cid:90) ∆ nst A · d X t σ (1) · · · d X t σ ( n ) · A n . (3.1)If linearly extended to the vector space spanned by all leveled trees (or equally permu-tations), σ (cid:55)→ X σst is a collection morphism, see Appendix 4. The above definition may bemisleading, recall that ⊗ denotes the projective tensor product, not the algebraic one. Itcontains the algebraic tensor product as a dense subspace and X σst is the unique continuousoperator extending the values prescribed by the above definition. The partial contraction op-erators correspond to leveled forests. We denote by End A the non-commutative polynomialson multilinear maps on A with values in A . More precisely, with End A the collection End A (1) = C · id A , End A ( n ) = Hom( A ⊗ n , A ) , n ≥ one has with the notation in use in the Appendix , End A = ¯ T (End A ) . As such
End is endowed with a product ∇ End this is induced by the functional compositionand extends the canonical operadic structure on End A as horizontal monoid morphism (seeAppendix 4). We have for two words u · · · u n and v · · · v p , where v i ∈ End( A ⊗ n i , A ) , n = n · · · n p , ∇ End (2) A ( v (cid:9) u ) = ( v ◦ ( u ⊗ · · · ⊗ u n )) · · · ( v p ◦ ( u n + ··· + n i − ⊗ · · · ⊗ u n + ··· + n i )) were ◦ stand for the functional composition.Recall that ¯ T ( A ) = C ∅ ⊕ (cid:76) n ≥ A ⊗ n . We use the notation | to denote the concatenationproduct on ¯ T (( A )) := ¯ T ¯ T ( A ) = C · ⊕ (cid:77) n ≥ ¯ T ( A ) ⊗ n . Definition 3.2 ( Representation of the algebra ¯ T (( A )) ). We define a representation
Op : ¯ T (( A )) → End A as extending the following values, for A ⊗ . . . ⊗ A n ∈ T ( A ) , X i ∈ A , Op( A · · · A n )( X ⊗ · · · ⊗ X n ) = X · A · · · · · A n · X n . Let w be a word in T ( A ) of length n . Let ( n , . . . , n k ) be a composition of n ; n i ≥ and n = (cid:80) i n i . The composition ( n , . . . , n k ) yields a splitting of w : we define the element [ w ] ( n ,...,n k ) ∈ T (( A )) by [ w ] ( n ,...,n k ) = w · · · w n | w n +1 · · · w n + n | · · · | w n + ··· + n k − +1 · · · w n + ··· + n k , with the convention w n + ··· + n i − +1 · · · w n + ··· + n i − + n i = ∅ if n i = 0 . Definition 3.3.
Let f be a leveled planar forest and < s, t < T two times. Define X fs,t ∈ ¯ T (( A )) and the partial contractions operators X fst by X fst = (cid:104) X f (cid:91) st (cid:105) ( (cid:107) f (cid:107) ,..., (cid:107) f nt ( f ) (cid:107) ) , X fst = Op( X fst ) . HE NON-COMMUTATIVE SIGNATURE OF A PATH 15
Let us finish with a remark on the representation Op . By definition, Op is compatible withthe concatenation product on ¯ T (( A )) . As explained, End A is endowed with a (cid:9) monoidalstructure ∇ End (2) A . The same kind of structure exists on ¯ T (( A )) . In fact, ¯ T ( A ) can be endowedwith an operadic structure ◦ , that we call words-insertions. Given a word a · · · a n ∈ ¯ T ( A ) and w , . . . , w n ∈ ¯ T ( A ) , a · · · a n ◦ ( w ⊗ · · · ⊗ w n ) = w a w · · · w n − a n w n (3.2)One can check that ◦ satisfy asssociatiy and unitaly constraints of an operadic composition.We then extend this operadic composition as an (cid:9) -monoidal morphism and define in this wayan associative product on ∇ ¯ T (( A )) : ¯ T (( A )) (cid:9) ¯ T (( A )) → ¯ T (( A )) . Then Op is compatible withrespect to the products ∇ End A and ∇ ¯ T (( A )) : ( ∇ End A (cid:9) ∇ End A ) ◦ (Op (cid:9) Op) = Op ◦ ∇ ¯ T (( A )) . (3.3)3.2. Chen relation.
In this section, we study first how concatenation of paths lift to thefull and partial contractions operators, that is we write a Chen identity for the latters. Inaddition, we introduce a two parameters family of endomorphisms, constitutive of a modelin the meaning of Hairer’s theory of regularity structure, acting on a direct sum over leveledtrees (or equivalently permutations). In this section, the symbol ◦ denotes alternatively the (cid:9) -composition ∇ ¯ T (( A )) or ∇ End A . Proposition 3.4 ( Chen relation).
Let X : [0 , → A be a smooth n path. Let < s < u The statement of the proposition is implied by the same statement but for the iteratedintegrals X fst , f ∈ LF since ρ is a representation of the word-insertions operad. The initializa-tion is done for forests with generations. Assume that the results as been proved for forestshaving at most N generations and let f be a forest with N + 1 generations. X fst = (cid:90) us dX t ◦ X f \ f st + (cid:90) tu dX t ◦ X f \ f st = X fsu + (cid:90) tu dX t ◦ X f \ f st . (3.4)In the above formula, we use operadic composition in the coloured operad associated with theword insertion operad. We use the short notations: d X t = ∅ ⊗ i − ⊗ d X t ⊗ ∅ | f |− i ∈ W ⊗ · · · ⊗ W ⊗ · · · ⊗ W = ˆ W ,... ,... (3.5)with i the index of the tree in the forest f which has two nodes at its first generation. Also, X f \ f st is seen as an element of ˆ W ( n ) ⊗ · · · ⊗ ˆ W n i ,n i ⊗ · · · ⊗ ˆ W n k , where n i and n i are the twotrees left out by cutting out the root of ith tree in the forest f . For any subforest f (cid:48) of f \ f ,we use f (cid:48) n i ,n i to denote the subforest of f obtained by adding a root connecting together thetrees at position n and n . We apply the recursive hypothesis to the forest f \ f to get: X f \ f st = (cid:88) f (cid:48) ⊂ f \ f X f (cid:48) ut ◦ (cid:104) X ( f \ f ) \ f (cid:48) su (cid:105) = (cid:88) f (cid:48) ⊂ f \ f X f (cid:48) ut ◦ X f \ ( f (cid:48) ) n i ,n i su . We insert this last relation into equation (3.4) to get the result since, with (3.5), (cid:90) tu d X t ◦ X f (cid:48) u,t = X ( f (cid:48) ) n i ,n i u,t . Thereby obtaining the thesis. (cid:3) HE NON-COMMUTATIVE SIGNATURE OF A PATH 16 If we choose for the leveled tree f a comb tree, that is a tree obtained by grafting corollaswith two leaves with each others, always on the rightmost node, we find back the classicalChen identity. In fact, by cutting such a tree we obtain a smaller comb tree and a leveledforest with only straight trees, except for the last one which is a comb tree. To the family ofoperators { X fst , s < t, f ∈ LT } , we now associate a triangular endomorphism on LT ( A ) = (cid:77) τ ∈ LT A ⊗| τ | ⊗ C [ τ ] . For the remaining part of the article, we use the lighter notation a · τ = a ⊗ τ ∈ A ⊗| τ | ⊗ C [ τ ] In classical rough path theory, the signature of path x yields a path X : [0 , → G on thegroup of characters G on the shuffle Hopf algebra ( H, ∆ , (cid:1) , S ) as explained in the introduction.To such a path, we associate a path of invertible triangular endomorphisms of H , ¯ X · = id ⊗ X · ◦ ∆ . Whereas it is not clear yet if it is possible to associate to the full and partial contractionsoperators a path on a certain convolution group of representations, our statement of the Chenrelation makes clear that any prospective deconcatenation product ∆ should act on a treeby cutting it in all possible ways, generations after generations. In section 2.4 we prove thiscutting operation yields a comonoid in ( LF , (cid:9) ) . For the time being, we simply write downthe following formula for the model, ¯ X st : (cid:77) τ ∈ LT A ⊗| τ | −→ (cid:77) τ ∈ LT A ⊗| τ | ,a · τ (cid:55)−→ (cid:88) τ (cid:48) ⊂ τ X τ \ τ (cid:48) st ( a ) · τ (cid:48) (3.6)Of course, the map ¯ X st crosses degrees, and we write ¯ X st = id + (cid:80) ∞ k =1 ¯ X ( k ) st , with ¯ X ( k ) st : LT ( A ) → LT ( A ) a · τ (cid:55)→ (cid:88) τ (cid:48) ⊂ τ (cid:107) τ \ τ (cid:48) (cid:107) = k X τ \ τ (cid:48) st ( a ) · τ (cid:48) (3.7)Proposition 3.4 immediately implies the following one. Proposition 3.5. Let s < u < t < T be three times, then1. for every leveled tree τ ∈ LT ( ¯ X st ( A τ ) − id) (cid:40) (cid:77) τ (cid:48) ⊂ τ A τ (cid:48) ; 2. for any s, u, t the so-called non-commutative Chen’s relations holds ¯ X st = ¯ X ut ◦ ¯ X su (3.8) 3. for any k ≥ there exist a constant C > such that (cid:107) ¯ X kst (cid:107) ≤ (cid:107) X (cid:107) kLip k ! | t − s | k Proof. We prove point 2, the Chen relation, the others are trivial. Let s < u < t be threetimes and A f · f ∈ LF ( A ) . Pursuant to the Chen’s relation (Proposition 3.4), X st ( A f · f ) = (cid:88) f (cid:48) ⊂ f X f \ f (cid:48) st ( A f ) · f (cid:48) = (cid:88) f (cid:48) ⊂ f (cid:88) f (cid:48)(cid:48) ⊂ f \ f (cid:48) X f (cid:48)(cid:48) ut ( X ( f \ f (cid:48) ) \ f (cid:48)(cid:48) su ( A ⊗ · · · ⊗ A | f | )) · f (cid:48) = (cid:88) f (cid:48) ⊂ f (cid:88) f (cid:48)(cid:48) ⊂ f \ f (cid:48) X f (cid:48)(cid:48) ut ( X ( f \ ( f (cid:48)(cid:48) (cid:93)f (cid:48) ) su ( A ⊗ · · · ⊗ A | f | )) · f (cid:48) HE NON-COMMUTATIVE SIGNATURE OF A PATH 17 We perform the change of variable g = f (cid:48)(cid:48) (cid:93)f (cid:48) , g (cid:48) = f (cid:48) and we obtain X st ( A ⊗· · ·⊗ A | f | · f ) = (cid:88) g ⊂ f (cid:88) g (cid:48) ⊂ g X g \ g (cid:48) ut ( X f \ gsu ( A ⊗· · ·⊗ A | f | )) · g (cid:48) = ( X ut ◦ X st )( A ⊗· · ·⊗ A | f | · f ) . This concludes the proof. (cid:3) Geometric properties. In this section, we investigate consequences of the integrationby part formula, in terms of relation between full- and partial-contraction operators associatedwith the smooth path X over the same time interval and on the endomorphisms (a model) ¯ X st , s < t .To set the ground for the second part of our work in which we define composition of n-c.controlled rough path with smooth functions on A , we introduce a new operadic composition L on a collection of words with entries in A that is different from the composition of thewords-insertions operad. This operad encodes operations brought up by the Chain’ rule fora certain class of functions (the field a and b are part of). We should elaborate on this in aforthcoming article. Definition 3.6. We define the collection of vector spaces F S = ( F S (0) , F S (1) , F S (2) , . . . ) by F S ( n ) = A ⊗ n +1 , n ≥ . Next, define L : F S ◦ F S → F S as follows. Pick a word U ∈ A ⊗ n and words A i ∈ A ⊗ m i , ≤ i ≤ p and set L ( U ⊗ A ⊗ · · · ⊗ A p ) = (cid:16) U (1) · A (cid:17) ⊗ A ⊗ · · · ⊗ (cid:16) A m ) · U (2) · A (cid:17) ⊗ · · · ⊗ (cid:16) A p ( m p ) · U n (cid:17) . The word ⊗ acts as the unit for L .We denote by FS the graded vector space equal to the direct sum of all vector spaces inthe collection F S . Notice that elements of A are -ary operators in the collection F S and forexample, the above formula gives L ( U ⊗ U (2) ◦ A ) = U · A · U ∈ A . The following propositionholds and relies on associativity of the product on A . Proposition 3.7. FS = ( F S , L, ⊗ is an operad. In the collection F S , a word with length n is an operator with n − entries, the inner gapsbetween the letters. So far, a leveled tree was considered as an operator with as much inputsas it has leaves. However, there is an alternative way to see such a tree as an operator : byconsidering the faces of the tree as inputs. A face is a region enclosed between two consecutiveleaves and delimited by the two paths of edges meeting at the least common ancestor, seeFig. 6. We denote by LT the set of leveled trees graded by the numbers of faces, LT ( n ) theset of leveled trees with n faces, and LT (cid:93) ( A ) the space LT ( A ) seen as a graded vector spacewith LT ( A )( n ) = C (cid:2) LT ( n ) (cid:3) ⊗ F S ( n ) . Notice that the endomorphism ¯ X st we defined in Figure 6. Faces of a leveled tree.the previous section satisfies: ( ¯ X st − id)( LT ( A )( n )) ⊂ (cid:77) k Notation. The graded vector space LT ( A ) yields a collection LT ( A ) , simply by settingthe space n operators of LT ( A ) equal to LT ( A )( n ) . We set, abusively, LT ( A ) ◦ LT ( A ) = (cid:77) n ≥ (cid:0) LT (cid:93) ( A ) ◦ LT (cid:93) ( A ) (cid:1) ( n ) (3.9)For any U, A ∈ LT ( A ) , we set U ◦ A = (cid:88) τ,τ ,...,τ (cid:107) τ (cid:107) U τ · τ ⊗ A τ · τ ⊗ · · · ⊗ A τ (cid:107) f (cid:107) · τ (cid:107) τ (cid:107) ∈ LT ( A ) ◦ LT ( A ) . Definition 3.8. Define the morphism of collections L : LT ( A ) ◦ LT ( A ) → LT ( A ) L ( U ⊗ ( A ⊗ · · · ⊗ A (cid:107) α (cid:107) )) = (cid:88) ατ ,...,τ (cid:107) α (cid:107) L ( U α ⊗ A τ ⊗ · · · ⊗ A τ (cid:107) α (cid:107) (cid:107) α (cid:107) ) · τ (cid:1) · · · (cid:1) τ (cid:107) α (cid:107) , where we have the definition U = (cid:88) α ∈ LT U α · α, A i = (cid:88) τ i ∈ LT A τ i i · τ i . Lemma 3.9. Let α and β be two leveled trees in LT , and pick A ∈ A ⊗| α | + | β |− . ¯ X st ( A · α (cid:1) β ) = (cid:88) τ α ⊂ α,τ β ⊂ β X ( α \ τ α ) (cid:1) ( β \ τ β ) st ( A ) · τ α (cid:1) τ β , (3.10) Proof. The proof consists essentially in a re-summation. It stems from the definition of themap ¯ X st that: ¯ X st ( A · α (cid:1) β ) = (cid:88) τ ∈ α (cid:1) βτ (cid:48) ⊂ τ X τ \ τ (cid:48) st ( A ) · τ (cid:48) . (3.11)Let τ ∈ α (cid:1) β a tree obtained by shuffling vertically the generations of α and β and pick τ (cid:48) ⊂ τ a subtree. Let s be the shuffle in Sh ( (cid:93)α, (cid:93)β ) such that τ − = ( α ⊗ β ) ◦ s − . We associateto the pair ( τ, τ (cid:48) ) a triple which consists in the tree τ , and two others trees τ (cid:48) α ⊂ α and τ (cid:48) β ⊂ β satisfying τ (cid:48) = ( τ (cid:48) α ⊗ τ (cid:48) β ) ◦ ˜ s − , where ˜ s is a shuffle in Sh (cid:16) (cid:107) τ (cid:48) α (cid:107) , (cid:107) τ (cid:48) β (cid:107) (cid:17) . Such a permutation ˜ s is unique, in fact it is obtainedfrom s by extracting the first (cid:107) τ (cid:48) (cid:107) letters of the word representing s − , followed by stan-dardization and finally inversion. Recall that standardization means that we translate thefirst (cid:107) τ (cid:48) (cid:107) letters representing s − , while maintaining their relative order to obtain a word onintegers in the interval (cid:74) , (cid:93)τ (cid:48) (cid:75) .It is clear that the map φ : ( τ, τ (cid:48) ) (cid:55)→ ( τ, τ (cid:48) α , τ (cid:48) β ) is injective. Now, given τ α ⊂ α, τ β ⊂ β , andtwo shuffles s − ∈ Sh ( (cid:107) τ (cid:48) α (cid:107) , (cid:107) τ (cid:48) β (cid:107) ) , s + ∈ Sh ( (cid:107) α \ τ (cid:48) α (cid:107) , (cid:107) β \ τ (cid:48) β (cid:107) ) , we define a third shuffle s − + in Sh ( (cid:107) α (cid:107) , (cid:107) β (cid:107) ) by requiring s − + ( i ) = s − ( i ) , ≤ i ≤ (cid:107) τ (cid:48) α (cid:107) , s − + ( (cid:107) τ (cid:48) α (cid:107) + i ) = s + ( i ) + s − ( (cid:107) τ (cid:48) α (cid:107) ) , ≤ i ≤ (cid:107) τ α \ τ (cid:48) α (cid:107) The map δ : ( τ (cid:48) α , τ (cid:48) β , s + , s − ) (cid:55)→ ( τ, τ (cid:48) α , τ (cid:48) β ) with τ − = α ⊗ β ◦ s − − + is a bijection between theimage of φ and S = { ( τ α , τ β , s + , s − ) , τ α ⊂ α, τ β ⊂ β, s − ∈ Sh ( (cid:107) τ α (cid:107) , (cid:107) τ β (cid:107) ) , s + ∈ Sh ( (cid:107) α \ τ α (cid:107) , (cid:107) β \ τ β (cid:107) ) } . We can thus rewrite the sum in the right hand side of (3.11) as follows: (cid:88) τ ∈ α (cid:1) βτ (cid:48) ⊂ τ X τ \ τ (cid:48) st ( A ) τ (cid:48) = (cid:88) τ α ,τ β ,s + ,s − ∈S X ( α ⊗ β ) ◦ s − + \ ( τ (cid:48) α ⊗ τ (cid:48) β ) ◦ s − − s,t ( τ (cid:48) α ⊗ τ (cid:48) β ) ◦ s − − HE NON-COMMUTATIVE SIGNATURE OF A PATH 19 Now, we observe that the forest ( α ⊗ β ) ◦ s − + \ ( τ (cid:48) α ⊗ τ (cid:48) β ) ◦ s − − does only depend on the trees τ α , τ β and the shuffle s + . Summing over all shuffles s + , we get α \ τ α (cid:1) β \ τ β . The statementof the Lemma follows by computing the sum over s − . (cid:3) Definition 3.10. Next, with A ∈ FS ( n ) and B ∈ FS ( m ) , we define their product A · BA · B = A (1) ⊗ · · · ⊗ ( A ( n +1) · B (1) ) ⊗ · · · ⊗ B ( m +1) . The product · is a graded product on the collection FS with unit ∈ FS (0) , A · B ∈ FS ( n + m ) , A ∈ FS ( n ) , B ∈ FS ( m ) Remark . The product · has a very special form, namely: A · B = (1 ⊗ ⊗ ◦ ( A ⊗ B ) = L ((1 ⊗ ⊗ ⊗ ( A ⊗ B )) . and the relation L ( m ⊗ ( id FS ⊗ m )) = L ( m ⊗ ( m ⊗ id FS ) with m = 1 ⊗ ⊗ entails associativityof the product · . We say that m ∈ F S (2) is a multiplication in the operad ( F S , L ) . Inaddition, associativity of the operadic composition L results in the following distributivitylaw ( A · B ) ◦ C = ( A ◦ B ) · ( B ◦ C ) , A, B, C ∈ FS . Conjointly with the shuffle product on leveled trees, the product · brings in a graded algebraproduct (cid:1) : LT ( A ) ⊗ LT ( A ) → LT ( A ) , namely ( A · α ) (cid:1) ( B · β ) = ( A · B ) · α (cid:1) β, (sh)with unit · . Let f, g two leveled forests and A ∈ A ⊗| f | , B ∈ A ⊗| g | , the integration by partformula (cid:90) ∆ st d X t ⊗ d X t + (cid:90) ∆ st d X t ⊗ d X t = ( X t − X s ) ⊗ ( X t − X s ) implies for the iterated integrals of X : (cid:90) ∆ nst d X σ · t ⊗ (cid:90) ∆ mst d X σ · t = (cid:90) Delta n + mst d X σ (cid:1) σ · t which implies for the contractions operators the following relation X fst ( A ⊗ . . . ⊗ A | f | ) · X gst ( B ⊗ . . . ⊗ B | g | ) = X f (cid:1) gst (( A ⊗ . . . ⊗ A | f | ) · ( B ⊗ . . . ⊗ B | g | )) (3.12) Proposition 3.12. Let α and β be two leveled forests and pick A ∈ A | α | , B ∈ A ⊗| β | , ¯ X st (( A · α ) (cid:1) ( B · β )) = ¯ X st ( A · α ) (cid:1) ¯ X st ( B · β ) Proof. It is a simple consequence of the previous Proposition 3.9 and the shuffle relation forthe partial contraction operators (3.12) In fact, one has the trivial identities ¯ X st (( A · B ) · α (cid:1) β ) = (cid:88) τ α ⊂ α,τ β ⊂ β X α \ τ α (cid:1) β \ τ β st ( A · B ) · τ α (cid:1) τ β = (cid:88) τ α ⊂ α,τ β ⊂ β X α \ τ α ( A ) · X β \ τ β st ( B ) · τ α (cid:1) τ β = ¯ X st ( A ) · ¯ X st ( B ) Thereby obtaining the desired identity. (cid:3) Corollary 3.13 (Geometricity) . For all times < s < t < T , it holds that: L ◦ (id ◦ ¯ X st ) = ¯ X st ◦ L . (3.13) HE NON-COMMUTATIVE SIGNATURE OF A PATH 20 Proof. For the proof, we rely solely on Proposition 3.12. L ( U α ⊗ ¯ X st ( A β · β ) ⊗ · · · ⊗ ¯ X st ( A β (cid:93)α · β (cid:93)α ))= ¯ X st ( U α (1) ) (cid:1) ¯ X st ( A β · β ) (cid:1) ¯ X st ( U α (2) ) · · · ¯ X st ( A β (cid:93)α · β (cid:93)α ) (cid:1) ¯ X st ( U α ( | α | ) )= ¯ X st (cid:0) ( U α (1) ) (cid:1) ( A β · β ) (cid:1) ( U α (2) ) · · · ( A β (cid:93)α · β (cid:93)α ) (cid:1) ( U α ( | α | ) ) (cid:1) = ¯ X st ( L ( U α · α ⊗ A β · β ⊗ · · · ⊗ A β (cid:93)α · β (cid:93)α )) (cid:3) We denote by G ( A ) the group of triangular invertible algebra morphisms on LT ( A ) , G ( A ) = { X ∈ Hom Alg ( LT ( A ) , LT ( A )) : ( X − id)( LT ( A )( τ )) ⊂ (cid:77) τ (cid:48) ⊂ τ LT ( A )( τ (cid:48) ) } (3.14)and denote ¯ X st := X | st . Then, for all pairs of times s < t , X st ∈ G ( A ) , and ¯ X st = ¯ X ut ◦ X su .3.4. Representations of the monoid of leveled forests.Theorem 3.14. Let X : [0 , → A be a smooth path. With the notation introduced so far,define X st : LF → End A by X st ( f ) = X fst , f ∈ LF , then X st = ∇ End ( A ) ◦ ( X ut (cid:9) X su ) ◦ ∆ , ∇ End ( A ) ◦ ( X st (cid:9) X st ) = X st ◦ ∇ Proof. The first assertion follows directly from Proposition 3.4 and the definition of the co-product ∆ . The second one follows from the shuffle identity for iterated integrals of X (seenas tensors) that we recall here, with σ = σ ⊗ σ , σ ∈ S k , σ ∈ S l , (cid:90) ∆ kst (cid:90) ∆ lst d X u s − σ (1)) ⊗ · · · ⊗ d X u s − σ ( k + l )) = (cid:88) v ∈ Sh ( k,l ) (cid:90) ∆ k + lst d X u ( v ◦ s − ◦ σ )(1) ⊗ · · · ⊗ d X u ( v ◦ s − ◦ σ )( k + l )) (cid:3) Operators of faces contractions. The Taylor expansion of a solution Y of an equationin the class we consider only involves the full contraction operators, the operators built oniterated integrals associated to trees. However, we explain in the previous section that inorder to write the Chen relation for these operators, we have to consider partial contractionoperators indexed by forests. These operators appear as coefficients of an endomorphism ¯ X acting on LT ( A ) . These "coefficients" associated with forests can not be related to the"coefficients" associated with trees if A is truly infinite dimensional. More formally, X st (cid:55)→ X st | (3.15)is not injective. In this section, we explain how to remedy to this problem, and make operatorsassociated to forests "technical proxys" for the Chen relation, that can be constructed fromthe operators associated to leveled trees.To achieve this, we define a new collections F C of operators, that we call faces contrac-tions and remember that for such a collection we denote by FC the graded vector space FC = (cid:88) n ≥ F C ( n ) . Finally, to the iterated integrals of X we associate a trajectory on a subgroup F ( FC ) ofinvertible triangular algebra morphisms (with respect to a product we define) on LT ( FC ) .Recall that to keep notations contained, we identify leveled trees in LT and permutations. HE NON-COMMUTATIVE SIGNATURE OF A PATH 21 Definition 3.15. Let τ ∈ LT be a leveled tree with at least one generation and pick A ⊗· · · ⊗ A ( | τ | ) ∈ A | τ | and define the continuous linear map (cid:93) ( A ⊗ · · · ⊗ A | τ | · τ ) : A ⊗| τ | → A by, for X , . . . , X (cid:107) τ (cid:107) ∈ A , (cid:93) (( A ⊗ · · · ⊗ A | τ | ) · τ )( X , . . . , X (cid:107) τ (cid:107) ) = A · X τ (1) · · · · X τ ( (cid:107) τ (cid:107) ) · A | τ | . In addition, we set FC ( ) = A . We denote by FC ( τ ) the closure, with respect to the op-erator norm, in the Banach space of all multilinear maps on A of the space of all τ -facescontractions , specifically FC ( τ ) = Cl (cid:16)(cid:110) (cid:93) (( A ⊗ · · · ⊗ A | τ | ) · τ ) , A ⊗ · · · ⊗ A | τ | ∈ A | τ | (cid:111)(cid:17) . and set LT ( FC ) = (cid:77) τ ∈ LT FC ( τ ) . Remark . Notice that the operator (cid:93) ( A ⊗ · · · ⊗ A | τ | ) has (cid:107) τ (cid:107) = | τ | − inputs. It canbe pictorially represented by drawing the leveled tree τ and placing the A (cid:48) s up to the leavesof τ and the X (cid:48) s in the faces of τ ; X i is located on the i th generation of τ . Whereas in theprevious section, arguments of the multilinear operators we considered where located on theleaves, in this section they are located on the faces of a tree.Let n ≥ an integer and pick σ a permutation in S n . With τ σ the leveled tree associated to σ − , σ · τ σ is a comb tree associated to the identity permutation. The permutation σ acts onfaces contractions operators by sending (cid:93) ( A ⊗ · · · ⊗ A n · τ ) in FC ( τ ) to (cid:93) ( A ⊗ · · · ⊗ A n · σ · τ ) in FC ( σ · τ ) with (cid:107) τ (cid:107) = n . Thus σ : (cid:77) τ ∈ LT (cid:107) τ (cid:107) = n FC ( τ ) → (cid:77) τ ∈ LT (cid:107) τ (cid:107) = n FC ( σ · τ ) is a continuous invertible operator. Hence, the linear map φ : LT ( FC ) → (cid:77) τ ∈ LT ( FC ) FC (id (cid:107) τ (cid:107) ) ⊗ τ (cid:88) τ ∈ LT m τ (cid:55)→ (cid:88) τ ∈ LT τ − ( m τ ) ⊗ τ is a continuous ismorphism too. Hence, if we define the collection F C by setting for n ≥ aninteger F C ( n ) = FC (id n ) , we sees that LT ( FC ) is in fact the graded vector space associated to the Hadamard productof LT with F C . In the following, we denote by (cid:93) the endormorphsim of graded vector spaces (cid:93) : LT ( FC ) → LT ( A ) A ⊗ · · · ⊗ A ( | τ | ) (cid:55)→ (cid:93) ( A ⊗ · · · ⊗ A ( | τ | ) ) (3.16) Notation. Pick X an endomorphism of LT ( FC ) and define the components of X , X ( τ (cid:48) , τ ) : FC ( (cid:107) τ (cid:107) ) → FC ( (cid:107) τ (cid:48) (cid:107) ) , τ (cid:48) ⊂ τ by requiring that X ( m τ ) = (cid:88) τ (cid:48) ,τ φ (cid:16) X st ( τ (cid:48) , τ ) (cid:16) φ − ( m τ ) (cid:17) ⊗ τ (cid:48) (cid:17) , m τ ∈ FC ( τ ) . (3.17)It will be convenient in the following to discuss either on the components of X , either on therestrictions - corestrictions of X X | β | α : FC ( α ) → FC ( β ) , α, β ∈ LT HE NON-COMMUTATIVE SIGNATURE OF A PATH 22 We continue by defining a product on LT ( FC ) . Proposition 3.17. Let α, β ∈ LT two leveled trees and A ∈ A ⊗| α | , B ∈ A ⊗| β | , then for anytuple X , . . . , X (cid:107) α (cid:107) + (cid:107) β (cid:107) one has (cid:93) (( A · α ) (cid:1) ( B · β ))( X , . . . , X || α || + || β || )= (cid:88) s ∈ Sh ( (cid:107) α (cid:107) , (cid:107) β (cid:107) ) (cid:93) ( A · α )( X s (1) , . . . , X s ( (cid:107) α (cid:107) ) ) · (cid:93) ( B · β )( X s ( (cid:107) α (cid:107) +1) , . . . , X s ( (cid:107) α (cid:107) + (cid:107) β (cid:107) ) ) Besides, the following estimates holds (cid:107) (cid:93) ( A · α (cid:1) B · β ) (cid:107) ≤ ( (cid:93)α + (cid:93)β )! (cid:93)α ! (cid:93)β ! (cid:107) (cid:93)A · α (cid:107)(cid:107) (cid:93)A · β (cid:107) . Thanks to Proposition 3.17, there exists a product (cid:1) on LT ( FC ) for which (cid:93) is an algebramorphism. Definition 3.18 ( Shuffle product on faces contractions operators). Pick m α ∈ FC ( α ) and m β ∈ FC ( β ) two faces contractions operators and define m α (cid:1) m β := (cid:88) s ∈ Sh ( (cid:107) α (cid:107) , (cid:107) β (cid:107) ) m α ( X s (1) , . . . , X s ( (cid:107) α (cid:107) ) ) · m β ( X s ( (cid:107) α (cid:107) +1) , . . . , X s ( (cid:107) α (cid:107) + (cid:107) β (cid:107) ) ) we call the operation shuffle product on faces contractions The involution (cid:63) on A induces an involution on LT ( FC ) turning (cid:93) into a morphism ofinvolutive algebras. Pick α a leveled tree. Recall that ( α ) denotes the tree obtained byhorizontal mirror symmetry of α . Define for any contraction operator m α ∈ FC ( α ) the facescontractions operator (cid:63) ( m α ) in FC ( ( α )) by (cid:63) ( m α )( X ⊗ . . . ⊗ X (cid:107) α (cid:107) ) = (cid:63) A ( m α ( (cid:63) A ⊗(cid:107) α (cid:107) ( X ⊗ · · · ⊗ X (cid:107) α (cid:107) )))= (cid:63) A ( m α ( (cid:63) A ⊗(cid:107) α (cid:107) ( X (cid:63) (cid:107) α (cid:107) ⊗ · · · ⊗ X (cid:63) ))) , X ⊗ · · · ⊗ X (cid:107) α (cid:107) ∈ A ⊗(cid:107) α (cid:107) Proposition 3.19. The quadruple ( LT ( FC ) , (cid:1) , (cid:63), (cid:107) (cid:107) ) is a C (cid:63) - algebra.Proof. We only check compatibility between the involution (cid:63) and the product (cid:1) . Let α, β two leveled trees and pick m α ∈ FC ( α ) and m β ∈ FC ( β ) . Then (cid:63) ( m α (cid:1) m β )( X ⊗ · · · ⊗ X (cid:107) α (cid:107) + (cid:107) β (cid:107) )= (cid:88) s ∈ Sh ( (cid:107) α (cid:107) , (cid:107) β (cid:107) ) m β ( X (cid:63)s ( (cid:107) α (cid:107) + (cid:107) β (cid:107) ) , . . . , X (cid:63)s ( (cid:107) α (cid:107) +1) ) (cid:63) · m α ( X (cid:63)s ( (cid:107) α (cid:107) ) , . . . , X (cid:63) ) (cid:63) = (cid:88) s ∈ Sh ( (cid:107) α (cid:107) , (cid:107) β (cid:107) ) (cid:63) ( m β )( X s ( (cid:107) α (cid:107) +1) , . . . , X (cid:63)s ( (cid:107) α (cid:107) + (cid:107) β (cid:107) ) ) · (cid:63) ( m α )( X s (1) , . . . , X (cid:63) (cid:107) α (cid:107) )= (cid:63) ( m β ) (cid:1) (cid:63) ( m α ) . Therefore we prove the result. (cid:3) Remark . Definition 3.8 introduces an operadic composition on the collection F S . Wedefine an operadic composition, that we denote by the symbol ˜ L , on the collection F C of facescontractions operator induced by the canonical operadic structure on End A , ˜ L ( V ◦ ( W ⊗ · · · ⊗ W p )) = V ◦ ( W ⊗ · · · ⊗ W p ) , (3.18)where V ∈ FC ( p ) , W i ∈ FC ( n i ) 1 ≤ i ≤ p and the symbol ◦ in the right hand side of theabove equation stands for the composition in End A . We set FC = ( F C , ˜ L, id A ) . Notice thatwith this definition, the (cid:93) operator is a morphism between the operads FS and FC, namely,for A , . . . , A p +1 ∈ A and W , . . . , W p ∈ F S ˜ L ( (cid:93) ( A ⊗ · · · ⊗ A p +1 ) ◦ (cid:93)W ⊗ · · · ⊗ (cid:93)W p ) = (cid:93)L ( A ⊗ · · · ⊗ A p ◦ ( W ⊗ · · · ⊗ W p )) (3.19)We use the same formula (3.8) to define the endomorphism ˜L : LT ( FC ) → LT ( FC ) . HE NON-COMMUTATIVE SIGNATURE OF A PATH 23 Denote by T ( FC ) the group of invertible triangular algebra morphisms on ( LT ( FC ) , (cid:1) ) . T ( FC ) = { α ∈ End Alg ( LPBT ( FC )) : ( α − id)( FC ( τ )) ⊂ (cid:77) τ (cid:48) (cid:40) τ FC ( τ (cid:48) ) } , Let k ≥ an integer and pick k elements of A , A , . . . , A k ∈ A . Define the following operatorsacting on LT ( FC ) : L A ,...,A k : LT ( FC ) → LT ( FC ) (3.20)by, for m τ a faces-contraction operator in FC ( τ ) , τ ∈ LT and τ (cid:48) ∈ LT , L A ,...,A k | τ (cid:48) τ ( m )( X , . . . , X (cid:107) τ (cid:48) (cid:107) ) = m ( X , . . . , X (cid:107) τ (cid:48) (cid:107) , A , . . . , A k ) if (cid:107) τ (cid:107) = (cid:107) τ (cid:48) (cid:107) + k,L A ,...,A k | τ (cid:48) τ ( m )( X , . . . , X (cid:107) τ (cid:48) (cid:107) ) = 0 otherwise . Notice that the norm of such an operator satisfies (cid:107) L A ,...,A k (cid:107) ≤ (cid:107) A ⊗ · · · ⊗ A k (cid:107) . Hence, L k : A ⊗ k (cid:51) ( A ⊗ · · · ⊗ A k ) (cid:55)→ L A ,...,A k is well defined and continuous. We call P the closure for the operator norm of the direct sumof the ranges of the operators L k : P = C(cid:96) ( (cid:77) k ≥ Im( L k )) (3.21)The space P is a Banach algebra, since L A ,...,A k ◦ L B , ··· ,B q = L A ,...,B q . In addition, fromthe very definition of L A ,...,A k , L A ,...,A q ( τ (cid:48) , τ ) depends only on the forest τ \ τ (cid:48) . Lemma 3.21. Pick ˜ X and ˜ Y two operators in P . Then, by setting for all pairs of leveledforest f, f (cid:48) ∈ LF such that f (cid:9) f (cid:48) ∈ LF (cid:9) LF is well defined, (( ˜ X ⊗ ˜ Y ) , f (cid:9) f (cid:48) ) = ˜ X ( f ) ◦ ˜ X ( f (cid:48) ) . one has (( ˜ X ⊗ ˜ Y ) , f (cid:9) f (cid:48) ) = (( ˜ Y ⊗ ˜ X ) , K ( f (cid:9) f (cid:48) )) , f (cid:9) f (cid:48) ∈ LF (cid:9) LF . (3.22) Proof. Let A , . . . , A | f (cid:48) | ∈ A and call σ (resp. σ (cid:48) ) the permutation associated with f (cid:91) (resp. f (cid:48) (cid:91) ). We use the notation cb n for the right-comb tree associated with the identity permutation id n . Next, define s the permutation in S (cid:107) f (cid:107) + (cid:107) f (cid:48) (cid:107) by • s f (cid:9) f (cid:48) ( k ) = i , if the k th face of cb nt ( f ) ) f f (cid:48) (reading the faces from left to right) isthe i th face of f , • s f (cid:9) f (cid:48) ( k ) = (cid:107) f (cid:107) + i if the k th face of f f (cid:48) is the i th face of f (cid:48) , • s f (cid:9) f (cid:48) ( k ) = (cid:107) f (cid:107) + (cid:107) f (cid:48) (cid:107) + i if the k th face is the i th face of cb nt ( f ) .With K ( f (cid:9) f (cid:48) ) = f (cid:48) (1) (cid:9) f (1) , with (cid:107) f (cid:48) (cid:107) = (cid:107) f (cid:48) } and (cid:107) f (cid:107) = (cid:107) f (1) (cid:107) notice that f (cid:48) (1) (cid:91) = f (cid:48) (cid:91) , f (1) (cid:91) = f (cid:91) , s K ( f (cid:9) f (cid:48) ) = s f (cid:9) f (cid:48) . Notice that (( ˜ X ⊗ ˜ Y ) , f (cid:9) f (cid:48) ) is non-zero only of pair of forests f (cid:2) f (cid:48) with (cid:107) f (cid:107) = p and (cid:107) f (cid:48) (cid:107) = q . Pick two such forests f, f (cid:48) . Pick U , . . . , U nt ( f ) − ∈ A . Pick X = L X ,...,X p and Y = L Y ,...,Y q two operators in P and put Z = ( U , . . . , U nt ( f ) , X , . . . , X p , Y , . . . , Y q ) . Onehas (( ˜ X ⊗ ˜ Y ) , f (cid:9) f (cid:48) )( (cid:93) ( A ⊗ · · · ⊗ A | f (cid:48) | ))( U , . . . , U nt ( f ) − )= A · Z s − f (cid:9) f (cid:48) (1) ⊗ · · · ⊗ Z s − f (cid:9) f (cid:48) ( (cid:107) f (cid:107) + (cid:107) f (cid:48) (cid:107) ) · A | f (cid:48) | = (( ˜ Y st ⊗ ˜ X st ) , K ( f (cid:9) f (cid:48) ))( (cid:93) ( A ⊗ · · · ⊗ A | f (cid:48) | ))( U , . . . , U nt ( f ) − ) (cid:3) HE NON-COMMUTATIVE SIGNATURE OF A PATH 24 In the following, we use the notation T P ( FC ) = T ( FC ) ∩ P . (3.23)for the set of triangular algebra morphisms of ( LT ( FC ) , (cid:1) ) in the Banach algebra P . Proposition 3.22. Endow T P ( FC ) with the involution: (cid:63) ( E ) = (cid:63) ◦ E ◦ (cid:63) (3.24) Then, first, (cid:63) : T P ( FC ) → T P ( FC ) is well defined and (cid:63) is an algebra morphism.Proof. Direct consequence of the definition of the poduct (cid:1) . (cid:3) Definition 3.23. Pick X : [0 , → A a smooth path. Let < s < t < be two times anddefine a triangular endomorphism ˜ X st : LT ( FC ) → LT ( FC ) in P determined by, for m α ∈ FC ( α ) and α ∈ LT , ˜ X st ( m α ) = (cid:88) β ⊂ α ˜ X st | βα ( m α ) , ˜ X st | βα : FC ( α ) → FC ( β ) , ˜ X st | βα ( m α )( X , . . . , X (cid:107) β (cid:107) ) = (cid:90) s Faces contractions of a leveled forestIt is immediate to infer from equation (3.25) that the components of ˜ X st ( α, β ) and ˜ X st ( β (cid:48) , α (cid:48) ) are equal provided that β \ α = β (cid:48) \ α (cid:48) and we use the notation ˜ X st ( α \ β ) for the common value.To be complete, if f is a leveled forest, then ˜ X st ( f ) : FC ( (cid:107) f (cid:107) + nt ( f ) − → FC ( nt ( f ) − (3.26)Set for any time < t < , ˜ X st = (cid:88) τ ∈ LT ˜ X st || τ . Theorem 3.24. Let X be a smooth path. Then ˜ X st is an algebra morphism for any pair oftimes < s < t < and ˜ X st ∈ T P ( FC ) , X st = ˜ X ut ◦ ˜ X su . (3.27) Besides, (cid:93) ◦ ¯ X st = ˜ X st ◦ (cid:93), s, t ∈ [0 , and if X is a trajectory of self-adjoint operators then ˜ X st is a morphism of C (cid:63) -algebras.Proof. The two first assertion are direct consequence of the equivariance property (cid:93) ◦ ¯ X st =˜ X st ◦ (cid:93), s, t ∈ [0 , which is downward to check. (cid:3) HE NON-COMMUTATIVE SIGNATURE OF A PATH 25 Notice that Corollary 3.24 implies, together with 3.12 that ˜ X st ◦ ˜L = ˜L ◦ id ◦ ˜ X st , s, t ∈ [0 , . (3.28)Also, Lemma 3.21 implies that for two smooth paths X, Y : [0 , → A (( ˜ X st ⊗ ˜ Y st ) , f (cid:9) f (cid:48) ) = (( ˜ Y st ⊗ ˜ X st ) , K ( f (cid:9) f (cid:48) )) , f (cid:9) f (cid:48) ∈ LF (cid:9) LF , (3.29)Let f be a leveled forest with n generations and nt ( f ) forests. Denote by n i the numberof generations of the i th tree in f and set (cid:96) fj = (cid:80) ji =1 n i + j for ≤ j ≤ nt ( f ) − . In thefollowing proposition, we denote by ◦ the operadic composition ˜ L . Recall that elements of A are considered as faces contractions operators with inputs. Pick m is a faces contractionoperator of arity p , m , . . . , m q faces contraction operators and a sequece of integers ≤ i <. . . < i q ≤ p , we denote by m ◦ i ,...,i q m ⊗ · · · ⊗ m q (3.30)the operator obtained by connecting m j at the i thj input of m . Recall that we denote by f (cid:91) the forest obtained by gluing together along their external edges the trees of f . We define apartial gluing operations which consists in gluing a subset of trees of f together, and index byinterval of integers in (cid:74) , nt ( f ) − (cid:75) . We denote by f I , I ⊂ (cid:74) , nt ( f ) − (cid:75) the forest obtainedby gluing the trees of f with index contained in I along their external edges. Proposition 3.25. Let m ∈ FC ( n ) be a faces contraction operator, f a forest satisfying n = (cid:107) f (cid:107) + nt ( f ) − , I = { i < · · · < i p } ⊂ (cid:74) , nt ( f ) − (cid:75) an interval of integers and A , . . . , A nt ( f ) − elements of A . Then for any pair of times < s < t < , ˜ X st ( f )( m ) ◦ i ,...,i p ( A , . . . , A i p ) = ˜ X st ( f I )( m ◦ (cid:96) i ,...,(cid:96) ip A I ) (3.31) In particular, ˜ X st ( f )( m )( A , . . . , A nt ( f ) − ) = ˜ X st ( f (cid:91) )( m ◦ (cid:96) ,...,(cid:96) nt ( f ) − A ⊗ · · · ⊗ A nt ( f ) − ) (3.32)The above proposition implies that the endomorphism ˜ X st is characterized by its values onthe leveled trees. Partial contractions are thus technical proxys required to write the Chenrelation for the operator ˜ X st but bear no additional data on the small scale behaviour of X .This is compliant with the simple observation that expansion of a solution of an equation inthe class (1.2) does only involve full contractions.Besides the fact that for all pair of times s < t , ˜ X st is a triangular algebra morphisms,we observed to other properties : the first one, equation (3.22), stipulates exchange relationsbetween operators ˜ X . The second one is equation (3.31). Whereas it is immediate to definean abstract set of operators (without referring to the path X ) satisfying (3.31), it is moredifficult when it comes to (3.22).Denote by G ( FC ) the set of all triangular operators ˜ X : LT ( FC ) → LT ( FC ) in T P ( FC ) satisfying equation (3.31) and by G (cid:63) ( FC ) the subgroup of self-adjoint operators in G ( FC ) . Proposition 3.26. The sets G ( FC ) and G (cid:63) ( FC ) are sub-groups of the group of triangularinvertible endomorphisms of LT ( FC ) .Proof. Pick two endomorphisms ˜ X and ˜ Y in G ( FC ) . Pick f a leveled forest and I = { i < · · · i p } ⊂ (cid:74) , nt ( f ) − (cid:75) . We show that ˜ X ◦ ˜ Y ∈ G ( FC ) . One has (( ˜ X ◦ ˜ Y )( f ))( m ) ◦ i ,...,i p ( A I ) = (cid:88) f (cid:48) ⊂ f ˜ X ( f (cid:48) ) (cid:16) ˜ Y ( f \ f (cid:48) )( m ) (cid:17) ◦ i ,...,i p ( A I )= (cid:88) f (cid:48) ⊂ f ˜ X ( f (cid:48) I ) (cid:16) ˜ Y ( f \ f (cid:48) )( m ) ◦ (cid:96) f (cid:48) i , ··· ,(cid:96) f (cid:48) ip A I (cid:17) = (cid:88) f (cid:48) ⊂ f ˜ X ( f (cid:48) I )( ˜ Y ((( f \ f (cid:48) ) (cid:96) f (cid:48) i , ··· ,(cid:96) f (cid:48) ip )) (cid:16) m ◦ (cid:96) f(cid:96)f (cid:48) i ,...,(cid:96) f(cid:96)f (cid:48) ip ( A I ) (cid:17) HE NON-COMMUTATIVE SIGNATURE OF A PATH 26 Owing to associativity of ◦ , we have (cid:96) f(cid:96) f (cid:48) i , . . . , (cid:96) f(cid:96) f (cid:48) ip = (cid:96) f (cid:48) (cid:9) fi , . . . , (cid:96) f (cid:48) (cid:9) fi p . The statement followsby noticing that { (cid:16) f (cid:48) I , ( f \ f (cid:48) ) (cid:96) f (cid:48) i ,...,(cid:96) f (cid:48) ip (cid:17) , f (cid:48) ⊂ f } = { ( f (cid:48) , f \ f (cid:48) ) , f (cid:48) ⊂ f I } . (cid:3) Appendix We recall some definitions from the theory of operads and more generally, we underline herethe categorical notions we use in this work. The reader will find below, among other things,definitons of collections, operads, bi-collections and PROSs. All of concepts are standardin the algebraic literature, see e.g. the monographies [10, 1], but not very known betweennon-algebraists. Hence the need of this small appendix. For further details we refer to [13, 3].At the base of these definitions above lies the concept of monoidal category . In loosewords, it is a category C = ( Ob ( C ) , Mor ( C ))) equipped with an operation and a unityelement I ∈ Ob ( C ) . The operation associates to any couple of objects A, B ∈ Ob ( C ) aobject A B ∈ Ob ( C ) and to any couple of morphisms f : A → A (cid:48) , g : B → B (cid:48) a morphism f g : A B → A (cid:48) B (cid:48) in a functorial way. In order that ( C , , I ) is a monoidal category, theoperation must satisfy two main properties, which emulate the tensor product operation onfinite dimensional vector spaces:1. (Associativity constraints) for any triple of objects A, B, C ∈ Ob ( C ) one has that theobject ( A B ) C is isomorphic to A ( B C ) in in a functorial way , that is there existsa natural isomorphism between the two functors ◦ ( id × ) and ◦ ( × id ) ;2. (Unitaly constraints) for any object A ∈ Ob ( C ) the objects A I and I A are (naturaly)isomorphic to A .The prototypical example is the category of finite dimensional vector spaces with monoidalproduct given by the tensor product of vector spaces. Another example is the category Set,the category of all sets with functions between sets as morphisms, with monoidal productgiven by the cartesian product of sets. Of interest in the present work is the -monoidalcategory of collections and bicollections that we now define.A monoid in a monoidal category is a categorical abstraction of a binary product on a set. Definition 4.1 (Monoid) . A monoid in a monoidal category ( C , ⊗ , I ) is a triple ( C, ρ, η ) with C ∈ Ob ( C ) , ρ : C ⊗ C → C , η : I → C meeting the constraints1. ρ ◦ ( ρ ⊗ id) = ρ ◦ id ⊗ ρ ,2. ρ ◦ ( η ⊗ id) = id Definition 4.2 (Comonoid) . A comonoid in a monoidal category ( C , ⊗ , I ) is a triple ( C, ∆ , ε ) with C ∈ Obj ( C ) , ∆ : C → C ⊗ C , ε : C → I meeting the constraints:1. ∆ ⊗ id ◦ ∆ = id ⊗ ∆ ◦ ∆ ,2. ε ⊗ id ◦ ∆ = id ⊗ ε ◦ ∆ Definition 4.3. We call a (reduced) collection P a sequence of complex vector spaces { P ( n ) } n ≥ . A morphism between two collections P, Q is a sequence of linear maps { φ ( n ) } n ≥ with φ ( n ) : P ( n ) → Q ( n ) , n ≥ . For any couple of morphisms between collections we definethe composition of morphisms by composing each component. We denote the category ofcollections by Coll . This monoidal category is particular in the sense that the monoidal product coincides with the cateogrialproduct. Such categories are called cartesian monoidal. The original definition involves vector spaces over a generic field but we consider only complex vectorspaces, in accordance with the structures presented so far. HE NON-COMMUTATIVE SIGNATURE OF A PATH 27 The category Coll has a natural monoidal structure (cid:12) over it: for any couple of collections P and Q and morphisms f, g we define ( P (cid:12) Q )( n ) := (cid:77) k ≥ n + ··· + n k = n P ( k ) ⊗ Q ( n ) ⊗ · · · ⊗ Q ( n k ) , ( f (cid:12) g )( n ) := (cid:77) k ≥ n + ··· + n k = n f ( k ) ⊗ g ( n ) ⊗ · · · ⊗ g ( n k ) . Denoting by C (cid:12) the collection C (cid:12) = (cid:40) C if n = 10 otherwise,it is straightforward to check that the triple (Coll , (cid:12) , C (cid:12) ) is a monoidal category. If the vectorsspaces of the collections P and Q above are Banach algebras, then we might use in place ofthe algebraic tensor product ⊗ the projective one ˆ ⊗ .An operad is a monoid in the monoidal category (Coll , (cid:12) , C (cid:12) ) : Definition 4.4. A non-symmetric operad (or simply an operad) is a monoid in the monoidalcategory (Coll , (cid:12) , C (cid:12) ) , i.e. a triple ( P, ρ, η P ) of the following objects P ∈ Ob ( Coll ) , ρ : P (cid:12) P → P , η P : C (cid:12) → P , satisfying the properties ( ρ (cid:12) id P ) ◦ ρ = ( id P (cid:12) ρ ) ◦ ρ and ( η P (cid:12) id P ) ◦ ρ = ( id P (cid:12) η P ) ◦ ρ = id P .We keep the notation (cid:12) for the monoidal operation. It is common in the literature todenote the morphism ρ by ◦ , i.e. for every k ≥ , p ∈ P ( k ) and q i ∈ Q ( n i ) for i = 1 , · · · , kp ◦ ( q ⊗ · · · ⊗ q n ) := ρ ( n + · · · + n k )( p ⊗ q · · · ⊗ q k ) . Moreover, for any ≤ i ≤ k and q i ∈ Q ( n i ) we use also the notation ◦ i to denote partialcomposition p ◦ i q := p ◦ ( η P (1)(1) ⊗ i − ⊗ q ⊗ η P (1)(1) ⊗ k − i ) , where η P (1) : C → P (1) . Since the maps ρ ( n ) n ≥ carry multiple inputs and give back oneoutput, it is common in the literature to call them many-to-one operators.In fact, it is possible to generalise the notion of an operad to model composition betweenmany-to-many operators, that is operators with multiple in- and outputs. This leads us todefine the category of bicollections. Definition 4.5. We call a bicollection a two parameters family of complex vector spaces P = { P ( n, m ) } n,m ≥ . A morphism between two bicollections P, Q is a sequence of linear maps { φ ( n, m ) } n,m ≥ with φ ( n, m ) : P ( n, m ) → Q ( n, m ) . For any couple of morphisms between bicollections we definethe composition of morphisms by composing each component. We denote the category ofbicollections by Coll .The category of bicollections is endowed with two compatible monoidal structures. Definition 4.6. For any couple of bicollections P and Q and morphisms f, g we define the horizontal tensor product (cid:9) as follows ( P (cid:9) Q )( n, m ) := (cid:77) n + n = nm + m = m P ( n , m ) ⊗ Q ( n , m ) , ( f (cid:9) g )( n, m ) := (cid:77) n + n = nm + m = m f ( n , m ) ⊗ g ( n , m ) . (4.1) HE NON-COMMUTATIVE SIGNATURE OF A PATH 28 together with the horizontal unity CCC (cid:9) = CCC (cid:9) ( m, n ) = (cid:40) C if n = m = 00 otherwise.We define also the vertical tensor product (cid:9) ( P (cid:9) Q )( n, m ) := + ∞ (cid:77) k =0 P ( n, k ) ⊗ Q ( k, m ) , ( f (cid:9) g )( n, m ) := + ∞ (cid:77) k =0 f ( n, k ) ⊗ g ( k, m ) . (4.2)together with the vertical unity CCC (cid:9) = CCC (cid:9) ( m, n ) = (cid:40) C if n = m otherwise.We refer to the triple ( Coll , (cid:9) , CCC (cid:9) ) and ( Coll , (cid:9) , CCC (cid:9) ) respectively as the category of hori-zontal bicollections and the vertical bicollections . Lemma 4.7. ( Coll , (cid:9) , CCC (cid:9) ) and ( Coll , (cid:9) , CCC (cid:9) ) are monoidal categories.Proof. This is simple computations, based on the fact that (Vect C , ⊗ , C ) is monoidal. (cid:3) Remark . We point at some core differences and similarities between the tensor product ofvector spaces, and the two tensor products we defined on bicollections. If V and W are twovector spaces, there exists an isomorphism of vector spaces S V,W : V ⊗ W → W ⊗ V . Theset { S ⊗ V,W , V, W ∈ Vect C } defines a natural transformation, called a symmetry contraint. Thevertical tensor product (cid:9) does not have such symmetry constraints, though we constructedsuch one but for the monoid generated by the bicollection LF . The horizontal tensor productis symmetric, if V and W are bicollections, S V,W : V (cid:9) W → W (cid:9) V, S (cid:9) ( V n ⊗ W m ) = S ⊗ ( V n ⊗ W m ) Call a category C a closed if for all objects A, B ∈ C the set of morphisms Hom C ( A, B ) is an object of C . The internal hom functor denoted [ A, − ] : C → C is defined by [ A, B ] = Hom C ( A, B ) , [ A, f ]( g ) = f ◦ g, f : B → C, g : A → B. A category C is a closed monoidal category if it is closed, monoidal and if the followingcompatibility holds: for all objects A, B, C ∈ C Hom C ( A, hom C ( B, C )) ∼ = Hom C ( A ⊗ B, C ) , with the isomorphism being natural in all three arguments. The category of finite dimensionalvector spaces with the usual tensor product is closed monoidal, owing to the fact that the setof linear maps between vector spaces is again a vector space and then using usual identifi-cation of bilinear maps with linear maps on the tensor product. Now, neither ( Coll , (cid:9) , CCC (cid:9) ) nor ( Coll , (cid:9) , CCC (cid:9) ) are closed monoidal. Indeed, they are not even closed, since there is nocanonical bigrading on the set of morphisms.There exists a functor from the category of collections to the category of bicollections, that isthe free horizontal monoid functor T : Coll → Coll , adjoint to the forgetful functor associatingto a a monoid ( P, γ, η ) for the horizontal tensor product (cid:9) the collection ( P (1 , n )) n ≥ . HE NON-COMMUTATIVE SIGNATURE OF A PATH 29 Definition 4.9. Let P = ( P n ) n ≥ be a collection, we define the word bicollection T ( P ) by T ( P )( m, n ) = (cid:77) k + ··· + k n = m P k ⊗ · · · ⊗ P k n , (4.3)when n ≥ and m ≥ . Moreover we set T ( P )(0 , 0) = C and T ( P )( m, 0) = T ( P )(0 , n ) = 0 . Proposition 4.10. Let C i , ≤ i ≤ be four bicollections, then there exists an explicitmorphism R C ,C ,C ,C : (cid:0) C (cid:9) C (cid:1) (cid:9) (cid:0) C (cid:9) C (cid:1) → ( C (cid:9) C ) (cid:9) ( C (cid:9) C ) . We call R C ,C ,C ,C the exchange law . Besides, if the bicollections C and C are equal andin the image of the free functor T , one has (cid:0) C (cid:9) W( C ) (cid:1) (cid:9) (cid:0) C (cid:9) W( C ) (cid:1) (cid:39) ( C (cid:9) C ) (cid:9) W( C ) . (4.4)The family of morphisms { R C ,C ,C ,C , C i ∈ Coll } defines a natural transformation (which is, in general, not an isomorphism) between the functors (cid:9) ◦ (cid:9) × (cid:9) and (cid:9) ◦ (cid:9) ×(cid:9) . In particular, for any quadruplet of morphisms f i : C i → D i , ≤ i ≤ , one hasthe following commutative diagram We denote by Alg (cid:9) (resp. CoAlg (cid:9) ) the category of all ( C (cid:9) C ) (cid:9) ( C (cid:9) C ) ( D (cid:9) D ) (cid:9) ( D (cid:9) D )( C (cid:9) C ) (cid:9) ( C (cid:9) C ) ( D (cid:9) D ) (cid:9) ( D (cid:9) D ) ( f (cid:9) f ) (cid:9) ( f (cid:9) f ) R C ,C ,C ,C R D ,D ,D ,D ( f (cid:9) f ) (cid:9) ( f (cid:9) f ) Figure 8. R is a natural transformationmonoids (resp. comonoids) in (Coll , (cid:9) , CCC (cid:9) ) , Alg (cid:9) (resp. CoAlg (cid:9) the category of monoids(resp. comonoids) in (Coll , (cid:9) , CCC (cid:9) ) . Proposition 4.11. [1, Prop. 6.3.5] The category (Alg (cid:9) , (cid:9) , CCC (cid:9) ) is a monoidal category. Indeed for any couple of horizontalalgebra ( A, m A (cid:9) , η A ) and ( B, m B (cid:9) , η B ) , the product m A (cid:9) B : A (cid:9) B → A (cid:9) B is defined m A (cid:9) B (cid:9) := ( m A (cid:9) (cid:9) m B (cid:9) ) ◦ R A,B,A,B , η A (cid:9) B = η A (cid:9) η B . (4.5) Moreover the bicollection CCC (cid:9) is a an horizontal monoid m (cid:9) (cid:9) : CCC (cid:9) (cid:9) CCC (cid:9) → CCC (cid:9) , η (cid:9) (cid:9) : CCC (cid:9) → CCC (cid:9) , (4.6) which are respectively a horizontal algebra and a horizontal unity.The category (CoAlg (cid:9) , (cid:9) , CCC (cid:9) ) is a monoidal category. Indeed for any couple of verticalcomonoid ( A, m A (cid:9) , η A ) and ( B, m B (cid:9) , η B ) , the product ∆ A (cid:9) B : A → A (cid:9) B is defined ∆ A (cid:9) B (cid:9) := R A,B,A,B ◦ ∆ A (cid:9) (cid:9) ∆ B (cid:9) , η A (cid:9) B = η A (cid:9) η B . (4.7) Moreover the bicollection CCC (cid:9) is a an horizontal monoid m (cid:9) (cid:9) : CCC (cid:9) (cid:9) CCC (cid:9) → CCC (cid:9) , η (cid:9) (cid:9) : CCC (cid:9) → CCC (cid:9) , (4.8) which are respectively a horizontal algebra and a horizontal unity. HE NON-COMMUTATIVE SIGNATURE OF A PATH 30 Definition 4.12. We call PROS a monoid in the monoidal category (Alg (cid:9) , (cid:9) , CCC (cid:9) ) . Thatis an horizontal monoid ( C, m C (cid:9) , η C (cid:9) ) , endowed with a couple of bicollections morphisms m C (cid:9) : C (cid:9) C → C , η C (cid:9) : CCC (cid:9) → C . defining a vertical monoidal structure on C . In addition, These morphisms are horizontalmorphisms.We recall that the same structure takes also the name of double monoid in the literature,see e.g. [1]. References [1] Marcelo Aguiar and Swapneel Arvind Mahajan. Monoidal functors, species and Hopf algebras , volume 29.American Mathematical Society Providence, RI, 2010.[2] Philippe Biane and Roland Speicher. 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