Loop of formal diffeomorphisms and F{à}a di Bruno coloop bialgebra
Alessandra Frabetti Ivan Shestakov







 



















 

 




























 



































































 






















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,1]Alessandra FrabettiCorresponding author: 

,2]Ivan P. ShestakovEmail: 

[1]Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, 
Institut Camille Jordan, F-69622 Villeurbanne, France 
[2]Instituto de Matématica e Estatistica, 
Universitade de São Paulo, 
Caixa Postal 66281, São Paulo, SP, 05315-970, Brazil


 
Loop of formal diffeomorphisms and Faà di Bruno coloop bialgebra
    [
    December 30, 2023
================================================================






We consider a generalization of (pro)algebraic loops defined on general 
categories of algebras and the dual notion of a coloop bialgebra suitable 
to represent them as functors. 
We prove that the natural loop of formal diffeomorphisms with associative
coefficients is proalgebraic, and we give the closed formulas of the
codivisions on its coloop bialgebra. 
This result provides a generalization of the Lagrange inversion formula to 
series with non-commutative coefficients, and a loop-theoretic explanation 
to the existence of the non-commutative Faà di Bruno Hopf algebra. 


MSC: 20N05, 14L17, 18D35, 16T30


Keywords: Loops; Algebraic groups; Non-associative algebras; 
Cogroups; Coloops; Formal series; Formal diffeomorphisms.   






§ INTRODUCTION




 §.§ Presentation and overview of the results


An affine proalgebraic group G is a representable functor in groups defined 
on the category of commutative associative algebras over a field .  
The algebra representing G is the commutative Hopf algebra [G] 
of regular functions. 
In this paper we consider two generalizations of proalgebraic groups, on one
side to representable functors on categories of non-commutative algebras, 
on the other side to functors taking values in non-associative groups with 
divisions, that is, loops. 

Our main motivation comes from two proalgebraic groups of formal series 
appearing in renormalization in quantum field theory: the group of invertible 
series with constant term equal to 1, represented by the Hopf algebra of 
symmetric functions, and that of formal diffeomorphisms tangent to the
identity, represented by the Faà di Bruno Hopf algebra.
Details on the role played by these series in quantum field theory are given
in a separate section below. 

Both types of series make sense with non-commutative coefficients, and both
representative Hopf algebras admit a non-commutative version <cit.>. 
We are interested in the relationship between the non-commutative algebras 
and the sets of series. For this, we first consider generalizations of 
proalgebraic groups to categories of non-commutative algebras.

Functors in groups on general categories have been studied by algebraic 
topologists in the late 50's.
D. Kan considered them on the category of groups <cit.>, and 
B. Eckmann and P. Hilton <cit.> 
introduced them on general categories. 
Their representative Hopf-type object is called a cogroup.
In a category with coproduct  and initial object, a cogroup 
is an object H endowed with a comultiplication, a counit and an antipode 
satisfying the usual properties of Hopf algebras, where the comultiplication 
takes values in H H instead of H H (which is not 
necessarily defined). 
Cogroups are then generalizations of commutative Hopf algebras which,
unlike quantum groups in the case of associative algebras, 
preserve the functorial properties and the adjoint constructions. 
They have proved to be very fruitful in homotopy theory, where they appear as
special H-spaces <cit.>, as shown by I. Berstein <cit.>. 
A comprehensive study of cogroups in many varieties of algebras can be found 
in G. Bergman and A. Hausknecht's book <cit.>. 

Not all proalgebraic groups admit an extention to non-commutative algebras.
For instance, while the group of invertible formal series naturally 
extends as a proalgebraic group to the category of associative algebras, 
the group of formal diffeomorphism does not. 
We show, on this example, that the extention of the functor is sometimes 
possible if we regard the original group as a loop. 

Loops are multiplicative sets with unit and with a left and a right division 
instead of two-sided inverses. 
They first appeared, with some extra properties, in the work of R. Moufang 
<cit.> on alternative rings, that is, rings where the associator 
(a,b,c) = (ab)c-a(bc) is skew-symmetric. For an excellent historical review
on loops, see <cit.>. 
Associative loops are groups. 
Similarly to Lie groups, the tangent space of a smooth loop carries 
a particular algebraic structure called a Sabinin algebra 
<cit.>, which reduces to a Mal'cev algebra 
<cit.> for smooth Moufang loops. 
The notion of universal enveloping algebra has been extended to 
Sabinin algebras by I. Shestakov, U. U. Umirbaev <cit.> 
and J. Mostovoy, J. M. Pérez-Izquierdo <cit.>. 


In this paper we consider functors in loops on a general category 
with coproduct and initial object and call their representative objects 
coloops in . 
We specialise  to be a variety of algebras over a field  
to have a reasonable notion of generalized (pro)algebraic loop. 
The first simple example is the extention of the functors of invertible
elements in a unital algebra and that of unitary elements in a unital
involutive algebra. 
As expected, the largest category on which these functors are representable as 
loops turn out to be respectively that of alternative and of alternative 
involutive algebras (Prop. <ref> and Prop. <ref>). 
We also show that the loop of unitary elements in the Cayley-Dickson extention 
of an involutive algebra is not representable on non-commutative algebras 
(Prop. <ref>), even if examples of such loops exist. 
Then we turn to loops of formal series with coefficients in a non-commutative 
algebra. 
First we consider the set of invertible series (with constant term equal to
1). The algebra of series with coefficients in an alternative algebra is
alternative. 
Surprisingly, in contrast to the previous results, we find that the set of 
invertible series is a proalgebraic loop on all algebras, not necessarily 
alternative (Thm. <ref>). 
Finally, our main result concerns the natural loop of formal diffeomorphisms 
(tangent to the identity) with associative coefficients. 
We show that it is proalgebraic, and give the closed formulas of the
codivisions on its representative Faà di Bruno coloop bialgebra 
(Def. <ref> and Thm. <ref>). 
For this, we express the co-operations in terms of some recursive operators 
defined on any positively graded algebra (Thm. <ref>), 
which extend the natural pre-Lie product of the Witt Lie algebra 
(cf. <cit.>) but not as a multibrace product 
(cf. <cit.>), and which turn out to be very rich in combinatorial 
properties. 
The coefficients appearing in the divisions show up sequences of integer
numbers typical of the Lagrange inversion formula (as Catalan numbers) and some 
new ones, that we call (labeled) Lagrange coefficients 
(Def. <ref> and <ref>). 
This result is a generalization of the Lagrange inversion formula to 
series with non-commutative coefficients, and gives a loop-theoretic 
explanation to the existence of the non-commutative Faà di Bruno Hopf 
algebra <cit.>. 





 §.§ Motivation: formal series in quantum field theory


The main object of study in perturbative quantum field theory are the
correlation functions of the fields describing some elementary particles, 
from which one can compute the probability amplitude of any event involving
the particles.
These quantities are asymptotic series in the powers of a measurable parameter
λ, such as the electric charge, called the coupling constant. 
For instance, for a self-interacting field ϕ with coupling λ and 
mass m, the k-point correlation function is a series 

    G^(k)(x_1,...,x_k) = ⟨ϕ(x_1)⋯ϕ(x_k)⟩ 
    = ∑_n=0^∞ G^(k)_n(x_1,...,x_k;m,ħ) λ^n
 
where the nth coefficient is a finite sum of amplitudes of suitable Feynman 
graphs with k fixed external legs, which depend on the mass m 
and on the Plank constant's ħ, and n is related to the number of 
internal vertices of the graph. 

The computation of the correlation functions gives rise to some divergent 
integrals, or equivalently to the ill-defined product of singular
distributions. 
Giving a meaning to such terms requires a renormalization procedure, 
which globally amounts to suitably multiplying and composing the correlation 
functions with some others series, called renormalization factors, 
obtained by assembling the counterterms needed to cure each divergence 
<cit.>, <cit.>.
Given an ambient algebra A, typically  or the algebra 
((ε)) of Laurent series in a regularization parameter 
ε, in renomalization theory there appear two groups of formal 
series in the variable λ and coefficients in A: 


  * the set 
(A)={ a(λ)=∑_n≥ 0 a_n λ^n | 
a_0=1, a_n∈ A} 
of invertible series, endowed with the pointwise multiplication 
(ab)(λ)=a(λ) b(λ) and the unit 1(λ)=1, 
which represent the Green's functions (up to an invertible factor) 
and the renormalization factors;   


  * the set (A)={ a(λ)=∑_n≥ 0 a_n λ^n+1 | a_0=1, a_n∈ A} 
of formal diffeomorphisms, endowed with the composition law 
(a∘ b)(λ)=a( b(λ)) 
and the unit e(λ)=λ, which represent the bare coupling constants
(i.e. the coupling constants before the renormalization is performed). 

Dyson's renormalization formulas <cit.> are modeled by the 
semi-direct product (A)⋉(A), endowed with the law 

    (a_1,b_1) · (a_2,b_2) 
    = (a_1∘ a_2, (b_1∘ a_2) b_2 ),

where a_1,a_2∈(A) and b_1,b_2∈(A), which is well defined 
because formal diffeomorphisms act on invertible series from the right, 
by composition. 

These groups are proalgebraic on commutative algebras, so they are perfectly
described by their re*resentative Hopf algebra. Physically, this means 
that the overall renormalization procedure (except the scheme which says how 
to compute the counterterms) is independent of the chosen field theory,
whenever the latter leads to commutative amplitudes.  
The recent results on the Renormalization Hopf algebras, initiated by A. Connes 
and D. Kreimer <cit.>, show even a stronger 
result: co-operations dual to the multiplication and the composition of series 
exist even on Hopf algebras generated by Feynman graphs, which contain the
coordinate ring of the usual groups of power series. In other words, there 
exist proalgebraic groups of series expanded over Feynman graphs, or over
various types of trees, which project onto the groups  qnd 
and which turn out to be extremely efficient in handling the combinatorial
content of renormalization 
procedures <cit.>. 

The toy model ϕ^3 theory used by Connes-Kreimer is a scalar field theory 
and leads to the commutative algebra A= of amplitudes.  
However, interesting physical situations involve non-commutative algebras. 
In fact, Feynman amplitudes are complex numbers for single scalar fields, 
the coupling constants and the renormalization factors, but they are
4× 4 complex matrices for the fermionic or bosonic fields, and may be
represented by higher order matrices for theories involving several interacting
fields. 
In this case, forcing the final counterterms to be scalar, as imposed by the 
fact that the renormalization factors act on the (scalar) Lagrangian,
prevents us from describing the renormalization in a functorial way,
as shown by the results in <cit.>, where the Hopf
algebra does not represent a functorial group on A=M_4().  
In order to preserve this functoriality, there is a need to understand Dyson's 
formulas for sets of series (A) and (A) also when A is not a 
commutative algebra. This is the motivation for the present work. 


Acknowledgments. 
The authors warmly thank José María Pérez-Izquierdo and Jacob Mostovoy 
for the interesting discussions and suggestions related to the topic of this 
paper, and Jiang Zeng for pointing out an alternative combinatorial argument.
The authors warmly thank the anonymous referee for his clever comments on
some key steps of our construction. 

This work was partially supported by the COFECUB Project MA 157-16, by the
Brazilian Ministry of Science and Technology grant CNPq 303916/2014-1 and by 
the LABEX MILYON (ANR-10-LABX-0070) of Université́ de Lyon, within the
program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the
French National Research Agency (ANR).










§ LOOPS AND COLOOPS




 §.§ Loops and functors in loops



A loop is a non-empty set Q endowed with a multiplication 
Q× Q⟶ Q, (a,b)⟼ a· b, 
a (two-sided) unit 1∈ Q, a left division 
\:Q× Q⟶ Q and a right division 
:Q× Q⟶ Q satisfying the cancellation properties

    a· (a\ b) = b,         a\ (a· b) = b, 
    
    
    (a b)· b = a,         (a· b) b = a.

Given two loops Q and Q', a homomorphism of loops 
f:Q⟶ Q' is of course a map which preserves the 
multiplication, and therefore the unit and the divisions.   

The multiplication in a loop Q is not necessarily associative, if it is 
associative then the loop is a group. 
Any element a in a loop Q has a right inverse 1 a 
and a left inverse a\ 1, which do not necessarily coincide 
and do not necessarily determine the divisions, 
in the sense that they do not satisfy the identities 

    (a\ 1)· b = a\ b           
    b· (1 a) = b a

for any a,b∈ Q, which hold in any group. 

Examples of loops which are not groups are known since a long time, see 
for instance <cit.> or <cit.>. 
For finite loops, the multiplication table is a Latin square, and the number
of non isomorphic loops is known up to order 11 (cf. <cit.>).
For instance, the subset {± 1, ± i, ± j, ± k} of the hyperbolic
quaternions <cit.> (where i^2=j^2=k^2=1 and ij=k=-ji, jk=i=-kj,
ki=j=-ik) forms a finite loop.
Among infinite loops, two well known examples are the set of invertible
octonions and that of unitary octonions, which is homeomorphic to ^7. 


Denote by  the category of loops and let F:⟶_* 
be the forgetful functor to the category of pointed sets. 
As for functors in groups, a functor Q:⟶ on a
given category  is said to be representable if the composite
functor F Q is representable, cf. <cit.>. 
This means that Q is naturally isomorphic to a hom-set functor 
A ↦_(H,A) for a given object H in , and implies
that the loop operations on any loop Q(A) are determined by dual
co-operations on H, by convolution. 
Following an established terminology on cogroups, the representative object 
H can then be called a coloop in .
A reasonable notion of (pro)algebraic loop is obtained for  being
a variety of algebras over a field , its representative coloop then being
a sort of bialgebra. 

In this section we describe coloops in an axiomatic way. In the next sections 
we give some easy examples of algebraic and non-algebraic loops on 
associative and non-associative algebras, and then study extensively the loop 
of invertible series and that of formal diffeomorphisms.





 §.§ Coloops in general categories



Given a category , Yoneda Lemma says that the category of representable
functors from  to , with natural transformations,
is equivalent to . The equivalence is realized by the contravariant
Yoneda functor Y from  to the functor category ^, 
defined on any object H in  by the functor Y(H)=_(H, ), 
and on any map ϕ:H'→ H by the natural transformation 
Y(ϕ):Y(H)→ Y(H'):α↦α ϕ 
(cf. <cit.> for details). 
In this section we characterise the subcategory of  equivalent to 
representable functors from  to . 


The cartesian product of two functors Y(H_1) and Y(H_2) is known to be 
represented by the categorical coproduct H_1 H_2, i.e. 
Y(H_1)× Y(H_2)=Y(H_1 H_2), and the constant functor to the base 
point is known to be represented by an initial object I, i.e. it is 
of the form Y(I). We recall the categorical notations about the coproduct, 
the initial object and some related categorical maps 
we need to define coloops. 


The coproduct in a category  is a bifunctor  defined on
two objects A and B as the unique object A B together with two
maps i_1:A→ A B and i_2:B→ A B
satisfying the following universal property:
for any maps f:A→ C and g:B→ C, there exists a unique
map ⟨ f,g ⟩: A B → C such that
⟨ f,g ⟩  i_1 = f and ⟨ f,g ⟩  i_2 = g. 
On two maps f:A→ A' and g:B→ B', the bifunctor is
defined as the map
f g = ⟨ i_1'f,i_2'g ⟩:A B→ A' B'.
The coproduct can be extended to several objects and maps with similar
universal constructions, and turns out to be an associative bifunctor,
in the sense that
(A B) C = A (B C)= A B C
for any three objects and
(f g) h = f (g h)= f g h
for any three maps in . 

An initial object in  is an object I together with a unique map
u_A:I→ A on any object, which commutes with any map
f:A→ B, that is, f u_A = u_B. 
Then, there are canonical isomorphisms A I ≅ A ≅ I A
given by the universal maps

    φ_1:=i_1:A ⟶ A  I
            ψ_1=⟨_A,u_A ⟩ :A  I ⟶ A, 
    
       φ_2:=i_2:A ⟶ I  A
            ψ_2=⟨ u_A,_A ⟩ :I  A ⟶ A.

In particular, we have I I≅ I, ⟨ u_A,u_A ⟩ = u_A 
and therefore also u_A B = u_A u_B.

For any objects A and B, there is a canonical symmetry operator 
τ_A,B= ⟨ i_2,i_1 ⟩:A B→ B A 
such that τ_A,B^-1=τ_B,A. Note that B A=A B 
as objects in , but the maps i_1 and i_2 are inverted. 
The twist τ is precisely the map which identifies A B and 
B A as universal objects.
To sum up, 
(,,I,τ) is a strict symmetric monoidal category. 

Furthermore, for any A, there exists a canonical folding map 
μ_A=⟨_A,_A ⟩:A A → A such that, 
for any maps f,g:A→ B, we have 

    ⟨ f,g⟩ = μ_B  (f g).

It follows that μ preserves the unit, i.e.
μ_A (u_A u_A) = ⟨ u_A,u_A ⟩ =u_A, 
that it is associative, i.e. 
μ_A(μ_A_A) = μ_A(_Aμ_A), 
and that it is commutative, i.e. μ_A τ_A,A=μ_A. 
It also follows that μ commutes with any map f:A→ B in , 
i.e. μ_B  (f f) = f μ_A. 
To sum up, we can say that any object (A,μ_A,u_A) is a commutative 
monoid in , with respect to the monoidal product , 
and that any map f:A→ B in  is a morphism of monoids. 
Finally, one can prove that the folding map on A B is 
given by μ_A B = 
(μ_A μ_B)  (_Aτ_B,A_B). 



Let us call coloop in  an object H endowed with the following 
maps in : 


  * a comultiplication :H⟶ H H; 


  * a counit ε:H⟶ I satisfying the counitary 
property

    (ε) Δ = φ_2
             
    (ε) Δ = φ_1 ,

where φ_1:H→ H I and 
φ_2:H→ I H are the canonical isomorphisms;  


  * a right codivision δ_r:H⟶ H H 
satisfying the right cocancellation properties

    (μ)  (δ_r)  = i_1 
             
    (μ)  (Δ) δ_r = i_1 ,

where i_1:H→ H H can be factorized as 
i_1=( u) φ_1; 


  * a left codivision δ_l:H⟶ H H 
satisfying the left cocancellation properties  

    (μ)  (δ_l)  = i_2 
             
    (μ)  (Δ) δ_l = i_2 ,

where i_2:H→ H H can be factorized as 
i_2=(u) φ_2. 

If H and H' are two coloops in , we say that a map 
f:H⟶ H' is a homomorphism of coloops if 
it commutes with the coproducts, the counits and the codivisions. 



Let H be a coloop in . 


  * The codivisions verify the identities 

    μ δ_r = u ε        μ δ_l = u ε ,

and the following partial counitality properties 

    (ε) δ_r = φ_1 
            
    (ε) δ_l = φ_2 .



  * We can define a right antipode S_r:H⟶ H
and a left antipode S_l:H⟶ H by setting 

    S_r := ψ_2  (ε) δ_r 
             
    S_l := ψ_1  (ε) δ_l  ,

where ψ_1=⟨,u_H⟩ : H I → H and 
ψ_2=⟨ u_H,⟩ : I H → H are isomorphisms. 
The antipodes satisfy the following left and right 5-terms 
identities 

    μ  (S_r )  = u ε        μ  ( S_l)  = u ε.




These properties are easily verified. A proof using tangle diagrams is 
given in the Appendix. 



Let  be a category with coproduct and initial object. 
Then the Yoneda functor is a contravariant equivalence of categories 
from the category of coloops in  to that of covariant representable 
functors Q:⟶. 



We follow the ideas of Eckmann-Hilton <cit.>, who 
characterized the subcategories of  equivalent to the category 
of representable functors from  respectively to the category  
of unital multiplicative sets, called unital magmas in <cit.>, 
and to the category  of groups[
  Eckmann-Hilton require  to have zero-maps, we replace them with an
  initial object.]. 

Let us first prove that the Yoneda functor, applied to coloops in , 
gives rise to a functor in loops. 
On a given coloop H, let us call Q=Y(H). We define the multiplication 
and the divisions on each set Q(A)=_(H,A) as usual convolution 
with the coproduct and the codivisions in H, namely

    α·β   = ⟨α,β⟩ Δ
    = μ_A  (αβ) Δ
    αβ   = ⟨α,β⟩ δ_r
    = μ_A  (αβ) δ_r
    
    α\β   = ⟨α,β⟩ δ_l
    = μ_A  (αβ) δ_l ,

for any α,β∈ Q(A). The unit in Q(A) is given, as usual,
by the map 1_A=u_A ε, and the left and right inverses of
α are then easily described as
α\ 1 = α  S_l and 1α = α  S_r. 
Then, using the cocancellation identities (<ref>) and 
(<ref>), and because μ_A is associative and commutes 
with -maps, it is easy to verify that the divisions given by 
(<ref>) satisfy the cancellation properties 
(<ref>) and (<ref>). 

Now fix a homomorphism of coloops ϕ: H'⟶ H, and call 
Q=Y(H), Q'=Y(H') and Φ=Y(ϕ). Yoneda Lemma tells us already that 
Φ is given on an object A by the map 

    Φ_A:     Q(A)⟶ Q'(A) 
       α⟼Φ_A(α) = α ϕ,

and that, for any f:A→ B, Φ acts on the map 
Q(f):Q(A)→ Q(B) given by Q(f)(α)= f α as a natural 
transformation, i.e. 

    Φ_B(Q(f)(α)) 
    = f α ϕ 
    = Q'(f)(Φ_A(α)).

It is then easy to verify that Φ_A is a homomorphism of loops, that is, 
for any α,β∈ Q(A), we have

    Φ_A(α·β)     = Φ_A(α)·Φ_A(β),

and similarly for the other co-operations. 

Viceversa, let us describe how a functor in loops Q gives rise to a coloop
structure on its representative object H. 
Suppose that the covariant functor Q is represented by an object H, 
i.e. Q=Y(H), that the set Q(A) is a loop for any A in , 
and that for any map f:A→ B the induced map 
Q(f):Q(A)→ Q(B) given by α↦ Q(f)(α)=f α 
is a loop homomorphism. 
We use repeatedly the fact that, given α,β∈ Q(A), for the
composite maps f (α·β), f α, f β∈ Q(B) we have

    f (α·β)    = Q(f)(α·β)
    = Q(f)(α)· Q(f)(β)
    = (f α)· (f β)

and similarly for the operations  and \.
Seeing i_1,i_2:H→ H H as elements of the loop
Q(H H), we define the comultiplication and the codivisions on H by 

    Δ = i_1· i_2,     δ_r = i_1  i_2,     δ_l = i_1 \ i_2

and the counit ε as the unit 1_I in Q(I).
It follows that the antipodes are the inverses of the identity map, 
S_r = 1_H _H and S_l = _H\ 1_H. 

Let us show that these maps give a coloop structure to H, and that the
functor Q↦ H is inverse to the Yoneda one, H↦ Q=Y(H). 
For any α,β∈ Q(A), we apply (<ref>) to A=H H, 
B=A, f=⟨α,β⟩:H H→ A and
to the elements α=i_1, β=i_2 of Q(H H), and get

    ⟨α,β⟩ Δ    = ⟨α,β⟩  (i_1· i_2)
    
        = (⟨α,β⟩  i_1) ·
    (⟨α,β⟩  i_2)
    = α·β

and similarly for the operations  and \. 
Now apply Q to a unit map u_A:I→ A.
Since Q(u_A):Q(I)→ Q(A) is a homomorphism of loops, it preserves
the units, and therefore, for ε=1_I∈ Q(I), we have

    Q(u_A)(ε) = u_A ε = 1_A.

In particular we have u_H ε = 1_H, and therefore, using
(<ref>), we have 

    ⟨ u_Hε,_H⟩ Δ    = 1_H·_H = _H.

On the other side, we have 

    ⟨ u_Hε,_H⟩ Δ    = ⟨ u_H,_H⟩  (ε) Δ
        = ψ_2  (ε) Δ,

and we obtain the equality ψ_2  (ε) Δ =. 
Since ψ_2 is the inverse map to φ_2, we obtain 
(ε) Δ = φ_2, which proves (<ref>). 
Let us show equalities (<ref>).
Firstly, we have trivially that

    ( u_H)  i_1 = ⟨ i_1,i_2  u_H⟩  i_1 = i_1.

Secondly, note that δ_r · i_2 = (i_1 i_2)· i_2 = i_1,
therefore 

    (μ)  (δ_r) Δ    = ⟨ i_1,i_2μ⟩ ⟨ i_1 δ_r,i_2 ⟩ 
    (i_1· i_2) 
        = ⟨ i_1,i_2μ⟩  (δ_r · i_2) 
        = ⟨ i_1,i_2μ⟩  i_1 = i_1.

Finally, since
⟨α,β⟩ (i_1 i_2) = αβ 
and (i_1· i_2) i_2 = i_1, we also have 

    (μ)  (Δ) δ_r 
        = ⟨ i_1,i_2μ⟩ ⟨ i_1 Δ,i_2 ⟩ 
    (i_1 i_2) 
        = ⟨ i_1,i_2μ⟩ ((i_1· i_2)  i_2) 
        = ⟨ i_1,i_2μ⟩  i_1 = i_1.

The same arguments apply to the left codivision. 





The relationship between coloops and cogroups is straightforward. 
As usual, a coloop H is coassociative if 

    ()     = () ,

and H is cocommutative if τ =. 

We say that H has the left and right coinverse property 
if the codivisions are determined by the antipodes, that is, 

    δ_l = (S_l) . 
            δ_r = ( S_r)

These identities correspond to the analogues (<ref>) in the 
loop Q=Y(H). 

Furthermore, an antipode on H is a map S:H→ H satisfying 
the 5-terms identity 

    μ   (S )   
    = μ   ( S)   
    = u ε .

This can happen if and only if S_r=S_l=:S. Note that the unicity of the
antipode satisfying (<ref>) does not imply that the coinverse
properties (<ref>) are verified. A counterexample is given by the
coloop of formal diffeomorphisms, cf. Section <ref>. 

A cogroup in a category  is an object H endowed with a coassociative 
comultiplication Δ, a counit ε satisfying the counitary 
property (<ref>) and an antipode S satisfying the 5-terms identity 
(<ref>), cf. <cit.>.
It follows from the dual statement on loops and groups (cf. <cit.>), that



If H is a coassociative coloop, then it is a cogroup. 


In the Appendix we prove with tangles that coassociativity implies 
that the left and the right antipodes coincide, and therefore H has an 
antipode satisfying the coinverse property. 






 §.§ (Pro)algebraic loops


Let  be a variety of unital algebras over a field , that is, the 
subcategory of vector spaces over  which collects all algebras of a certain 
type, given by a set of operations of various arities, including the unit 
of arity 0, defined by a set of identities (cf. <cit.> ch. V). 
For instance,  can be the category of -algebras, where  
is an algebraic operad with (0)={  }
(cf. <cit.>). 

Then,  has a coproduct and an initial object 
(cf. <cit.> ch. IX), therefore we can apply to  the results 
of the previous section. 
More precisely, the initial object is given by the trivial unital algebra . 
Suppose that in  there are operations p of arity n>0, and let 
A̅ denote the subalgebra of an algebra A determined by such
operations. Then, given two algebras A and B in , the coproduct
A B is the quotient of the free algebra
(A̅⊕B̅) (which always exists, cf. <cit.> ch. V)
by the ideal generated by the identities

    p_(A⊕ B)(a_1,...,a_n) = p_A(a_1,...,a_n) ∈A̅, 
        
        p_(A⊕ B)(b_1,...,b_n) = p_B(b_1,...,b_n) ∈B̅, 
        
        1_A = 1_B = 1_(A⊕ B),

for all the operations p admitted in . 
The universal properties of  follow from the universal properties 
of the free algebra (A̅⊕B̅). 




  * In the category _ of unital commutative and associative algebras over 
, the free algebra _(V) on a vector space V is the 
symmetric algebra S(V), and the coproduct of two algebras A and B 
is the tensor product A B. 


  * In the category _ of unital associative algebras over , 
the free algebra _(V) is the tensor algebra T(V), and the 
coproduct[
In associative algebras, the coproduct is usually called free product 
and denoted by ⋆.
] 
A B of two algebras A and B is the tensor algebra 
T(A̅⊕B̅) modulo relations (<ref>), 
which mean that a a' = a  a whenever a and a' are both in 
A̅ or both in B̅. As a vector space, we then have

    A B = ⊕⊕_n≥ 1 [nA̅⊗B̅⊗A̅⊗⋯ ⊕ nB̅⊗A̅⊗B̅⊗⋯],

and the multiplication in A B is given by the 
concatenation modulo the above relations. 
For instance, if we denote the multiplication in A B by , 
we have

    (b a) (b' a' b”) 
    = b a b' a' b”
        (b a) (a' b' a”)
    = b (a a') b' a”.



  * Let _ be the category of unital algebras (not necessarily associative, 
also called magmatic) over . The free unital algebra on a vector space V 
is the tensor algebra with parenthesizing T{V}, and the coproduct 
A B of two algebras A and B is the quotient of 
T{A̅⊕B̅} modulo relations 
(<ref>), which again mean that a a' = a  a 
whenever a and a' are both in A̅ or both in B̅. 
The multiplication in A B is the concatenation with parenthesis. 


  * An alternative algebra is an algebra A such that the associator 
(a,b,c) = (ab)c-a(bc) is skew-symmetric, that is 

    (b,a,c) = -(a,b,c)          (a,c,b) = -(a,b,c)

for any a,b,c∈ A. This is equivalent to requiring that 

    (ab)b = a(bb)          (aa)b = a(ab)

for any a,b∈ A. 
Unital alternative algebras over a field  form a subcategory of _, 
denoted by _, which is a variety with initial object . 
The coproduct A B of two unital alternative algebras is the quotient 
of the coproduct in _ by the relations (<ref>). 
For details see <cit.> or <cit.>. 


  * In a category , an (anti) involution is a unary linear operation 
^*:A⟶ A such that 

    (a^*)^* = a
             
    (a_1 a_2)^* = a_2^* a_1^*,

for any a,a_1,a_2∈ A. Each of the four previous categories of algebras 
can be considered with involution, and denote by ^*. For such algebras, 
the initial object and the coproduct are the same as in , the involution 
on A B is automatically defined from the involutions on A and B 
by properties (<ref>). 
Note that, in _ and in _, the parenthesizing of a word 
a_1⋯ a_n is inverted from left to right by the involution, 
together with the single letters of the word. 





A coloop H in a variety of unital algebras  is called a 
coloop -bialgebra. 
Its associated functor in loops Q=Y(H) is then called an 
algebraic loop on  if H is a finitely generated algebra, 
and a proalgebraic loop on  if H is not finitely generated. 
In this case, it is an inductive limit of finitely generated coloop 
-bialgebras. 


There are not many known examples of algebraic loops on non-commutative
algebras, but some of them are quite special.    
In section <ref>, we give two easy examples of algebraic groups 
on commutative algebras which can be extended as groups to associative algebras 
(the groups of invertible elements and that of unitary ones), 
and another one which can not be extended to a functor on associative algebras
even as a loop (the Cayley-Dickson loop). 
Viceversa, in section <ref> we give the example of 
a proalgebraic group which can be extended to associative algebras only as a 
proalgebraic loop (the loop of formal diffeomorphisms).

Finally, the two groups of invertible and unitary elements can be extended to
alternative algebras if we regard them as algebraic loops, and in section
<ref> we also give an example of an algebraic group
which can be extended to all associative algebras as a group,
and to non-associative algebras as a loop (the loop of invertible series).

While the functoriality of the first examples is straightforward, for the
two loops of formal series it is not. The group of formal diffeomorphisms
is a local approximation of the most simple group of smooth diffeomorphisms
on a manifold, and the existence of a proalgebraic version on associative
algebras is a new step in the study of non-commutative geometry.
In particular, its existence as a proalgebraic loop allows us to consider
a physical “renormalization loop” to replace the standard group
<cit.>, which could be applied in any perturbative theory
when the function rings have to be replaced by tensor algebras, as in
<cit.>. 


All known examples of algebraic groups and loops on non-commutative algebras
have free underlying algebra structure. The fact that this should hold in any
category (under certain completeness hypothesis) has not been proved,
but it was proved for cogroups in several categories:
by D. Kan <cit.> in the category of groups, 
by I. Berstein <cit.> (and later reproved by J. Zhang in <cit.>) 
in the category of graded connected associative algebras, 
and by B. Fresse <cit.> in the category of complete algebras over any 
operad. 
For coloops, this result is proved by G. Bergman and A.O. Hausknecht 
<cit.> in the category of graded connected associative 
rings.


Before giving the examples, we mention two maps which allow us to compare 
coloop and cogroup bialgebras to usual Hopf algebras. 
A coloop -bialgebra has the operations 
p:H^⊗  n⟶ H from , 
and the categorical folding map μ:H H→ H needed 
to describe the coloop axioms, which can be iterated on n copies of H. 
In general, there is no relationship between these two types of operations,
since H^⊗  n need not be an algebra in . 


Assume that  is a category of algebras such that, for any 
-algebras A and B, the tensor product A B is again 
an -algebra with componentwise operations 

    p^(n)_A⊗ B(a_1⊗ b_1,⋯,a_n⊗ b_n) = 
    p^(n)_A(a_1,⋯,a_n)⊗ p^(n)_B(b_1,⋯,b_n)

and unit 1_A⊗ B=1_A 1_B.



For any n≥ 2 and for any n algebras A_k, with k=1,...,n, 
we call canonical projection of A_1⋯ A_n
onto A_1⋯ A_n the algebra homomorphism

    π:= ⟨ j_1,...,j_n ⟩:
    A_1⋯ A_n ⟶ A_1⋯ A_n

induced by the injective algebra maps
j_k:A_k→ A_1⋯ A_n given by 

    j_k(a_k)=1_A_1⋯ a_k⋯ 1_A_n.

The map π reorders the elements of A_1⋯ A_n and 
then multiplies them within each A_k to get elements in 
A_1⋯ A_n. For instance, if we denote by a^(k) an 
element a∈ A_k seen in the coproduct A_1⋯ A_n, we have 

    π(a^(1) b^(2) c^(1) d^(2)) = (ac) (bd).

This map is surjective, because a preimage of any 
a_1 a_2⋯ a_n∈ A_1⋯ A_n 
by π is given by 
a_1^(1) a_2^(2)⋯ a_n^(n)∈ A_1⋯ A_n. 

Note that, when all A_k coincide and we are given an operation p 
of arity n, the map p π: A^ n⟶ A 
is not, in general, an algebra homomorphism (because p is not), 
and therefore it surely differs from the folding map
μ= ⟨_A,...,_A ⟩. 
In fact, μ multiplies the elements of A in the order they appear 
in A^ n (it is a concatenation), 
while p π first reorders the factors in A^ n with π,
as explained above, then multiplies them (it is a componentwise 
operation).  




On the other side, for any n≥ 2 and for any n algebras A_k, with
k=1,...,n, there are categorical maps 
i_k:A_k→ A_1⋯ A_n. 
For any operation p of arity n in , we call 
canonical inclusion of A_1⋯ A_n in 
A_1⋯ A_n the linear map 
ι_p:A_1⋯ A_n ⟶
A_1⋯ A_n
defined by 

    ι_p(a_1⋯ a_n) := p_(i_1(a_1),...,i_n(a_n)),

where p_:(A_1⋯ A_n)^ n→ 
A_1⋯ A_n 
denotes the operation p on the coproduct algebra
A_1⋯ A_n. 
It follows from the definition of  that this map is injective. 

Note that ι_p is not, in general, an algebra homomorphism, because 
the operation p in  is not. 
However, when all A_k coincide (say, with A), the map ι_p allows us
to recover the operation p_A:A^ n→ A from the folding map
μ, in the sense that μ ι_p=p_A, because 

    μ p_(i_1(a_1),...,i_n(a_n)) = p_A(a_1,...,a_n)
 
for any a_1,...,a_n∈ A. 



When the map ι_p is well defined, we have
π ι_p=_A_1⊗⋯⊗ A_n.



Denote by p_⊗ the operation p on the tensor algebra
A_1⋯ A_n. Since π is an algebra homomorphism,
for any a_k∈ A_k, with k=1,...,n, we have 

    π ι_p(a_1⋯ a_n)
       = π  p_(i_1(a_1),...,i_n(a_n)) 
    = p_⊗(π(i_1(a_1)),...,π(i_n(a_n)))  
    
        = p_(⟨ j_1,...,j_n ⟩(i_1(a)),...,
    ⟨ j_1,...,j_n ⟩(i_n(a_n)))
    
        = p_⊗(j_1(a_1),...,j_n(a_n))
    
        = a_1⋯ a_n.




These maps allow us in particular to compare the coloop bialgebra representing 
some loop to other types of bialgebras related to it which appear in the 
literature. In particular, the universal enveloping algebra of the Sabinin 
algebra associated to the loop has been studied in 
<cit.>. 
Because of the axioms, it is clear that the graded dual of this universal 
enveloping algebra does not coincide with the bialgebra H^⊗ induced 
by a coloop bialgebra H, nor in _ nor in _.


Finally, let us use these maps to compare associative coloop bialgebras and 
Hopf algebras. Let H be a coloop bialgebra in _. 
Denote by H^⊗ the algebra H endowed with the usual co-operations 

    ^⊗=π , δ_r^⊗=π δ_r, δ_l^⊗=π δ_r: 
    H^⊗⟶ H^⊗⊗ H^⊗,

the counit ε and the antipodes S_r, S_l, which are all still 
algebra homomorphisms on H^⊗. 


 
If  is coassociative, then ^⊗ is coassociative.
Moreover, we have 

    S_r= (ε⊗) δ_r^⊗         
    S_l= (⊗ε) δ_l^⊗.




If  is coassociative, the two terms 

    (^⊗⊗) ^⊗ 
    = (π⊗) π_(H H) H   ()

and 

    (⊗^⊗) ^⊗ 
    = (⊗π) π_H (H H)   ()

coincide, because  is coassociative and because the two maps 
(π⊗)⊗π_(H H) H 
and (⊗π)⊗π_H (H H) coincide with the 
standard projection 
π:H H H ⟶ H⊗ H⊗ H. 
Therefore ^⊗ is coassociative. 

For any a∈ H, the term δ_r(a)∈ H^(1) H^(2) 
is a finite sum of products of elements of H^(1) and of H^(2) 
in alternative order. 
The right antipode S(a)_r=(ε) δ_r(a) 
turns all the factors belonging to H^(1) into scalars, which can then be 
positioned on the lefthand side of all the remaining elements belonging to 
H^(2). Therefore the result is the same that we obtain if we first 
reorder the factors in H^(1) all at the leftmost position by 
applying δ_r(a)^⊗. 
Same with S_l by putting all the scalars on the rightmost position. 


Note however that S_r and S_l do not necessarily satisfy the left 
and right 5-terms identities for ^⊗ on H^⊗, because 

    m (S_r) ^⊗ = m π (S_r) i ^⊗ 
    = μ ι π (S_r) ι π

and ι π is not the identity map on H H. 
Therefore, even if H is a cogroup bialgebra, H^⊗ is not necessarily 
a Hopf algebra. 





§ COLOOPS OF INVERTIBLE AND UNITARY ELEMENTS
 




 §.§ Loop of invertible elements
 

In this section we give an example of an abelian algebraic group which can be 
extended to associative algebras as a group, to alternative algebras as a loop, 
but not to non-associative algebras, even as a loop. 


Let  be a field. 
For any unital commutative algebra A over , the set 

    I(A) = { a∈ A |  }

is the abelian group of invertible elements in A. 
The functor I is represented on _ by the commutative 
(and cocommutative) Hopf algebra of Laurent polynomials = [x,x^-1], 
with co-operations 

    Δ (x) = x ⊗ x,     ε (x) = 1,      S(x) = x^-1.

In fact, elements a∈ I(A) are in bijection with algebra homomorphisms
α:→ A such that α(x)=a and α(x^-1)=a^-1.
Then, if α and β give respectively
the elements a and b, their convolution product coincides with the product
in A, because we have

    (α·β)(x)     = μ_A (α⊗β) Δ(x)
      = α(x) β(x) = ab, 
    α^-1(x)     = α S(x) = α (x^-1)=a^-1.

We show that the functor I admits an extention to associative algebras as 
a group, and that it admits an extention to non-associative algebras, as a loop, 
only on alternative algebras.  



We call invertible coloop bialgebra on  the 
associative algebra = [x,x^-1] endowed with the following co-operations 
with values in the coproduct  of the category _: 

    (x) = x^(1)  x^(2)      (x^-1) = (x^-1)^(2)  (x^-1)^(1),
    
       ϵ(x) = 1       ϵ(x^-1) = 1 ,
    
       δ_r(x) = x^(1)  (x^-1)^(2)      δ_r(x^-1) = x^(2)  (x^-1)^(1) ,
    
       δ_l(x) = (x^-1)^(1)  x^(2)      δ_l(x^-1) = (x^-1)^(2)  x^(1) ,

where x^(k)=i_k(x) is the generator x seen in the kth copy of 
of the coproduct algebra , for k=1,2, and similarly
for (x^-1)^(k)=i_k((x^-1)^(k)). 

It follows that there is a two-sided antipode given by S(x)=x^-1 and 
S(x^-1)=x. 



The algebra  is a cogroup bialgebra in _ and represents, 
for any associative algebra A, the group 

    I(A) = __(,A)

of invertible elements of A. 
Moreover, the group I(A) is abelian if A is commutative. 



The axioms of a coloop bialgebra for the codivisions are easily verified.
For instance, the computations

    (μ)(δ_r) Δ (x)
          = (μ)(δ_r)(x^(1) x^(2)) 
        = (μ)(x^(1) (x^-1)^(2) x^(3)) 
        = x^(1) (x^-1)^(2) x^(2)
        = x^(1) = i_1(x) 
    
    (μ)(Δ) δ_r (x)
          = (μ)(Δ)(x^(1) (x^-1)^(2)) 
        = (μ)(x^(1) x^(2) (x^-1)^(3)) 
        = x^(1) x^(2) (x^-1)^(2)
        = x^(1) = i_1(x)

and the analogue computations for x^-1 prove the cocancellation
(<ref>) for δ_r. 
The first claim is then ensured by the fact that  is coassociative. 
In fact,

    (Δ) Δ (x)     = x^(1)  x^(2)  x^(3)
      = (Δ) Δ (x),

and similarly for x^-1. 
Thus, I(A) is a group by Theorem <ref> and Proposition
<ref>. 

The fact that the group I(A) is abelian if A is commutative 
is less evident because  is not cocommutative. In fact, we have 

    τ (x) = τ(x^(1)  x^(2)) = x^(2)  x^(1)≠ x^(1)  x^(2) = (x).

It is however true because the generators x and x^-1 are group-like, and 
therefore the commutativity of the convolution product only depends on that 
of the multiplication in A. 



The group I(A) is the simplest algebraic group at all: it describes
invertible elements in an associative algebra A whatever is the nature of
A, that is, without making use of any internal structure of A. 
The simplest non-trivial example is the group I(M_n()) = GL_n(),
which is recovered as the set of M_n()-valued algebra homomorphisms
on  without using the non-homogeneous relation (A)≠ 0
which defines invertible matrices (or, more precisely, the relation
(A)=t where t determines a new scalar invertible generator
of the coordinate ring). 




The algebraic group I can be extended as a loop to a variety of algebras 
⊂_ if  is a subcategory of alternative algebras
_ admitting coproduct and initial object. 
In particular, it is an algebraic loop on _. 



If I could be extended as an algebraic loop to , its representative 
coloop bialgebra should be the algebra = [x,x^-1] with 
co-operations defined on generators as in Def. <ref> but 
taking values in the coproduct  of the category . 
This algebra is not a coloop bialgebra in , because the 
codivisions do not satisfy the cocancelation properties 
(<ref>) and (<ref>). 
In fact, the element 

    (μ) (δ_r) Δ(x) 
        = (x^(1)  (x^-1)^(2))  x^(2)

can not coincide with i_1(x) = x^(1) in . 
However, the conditions under which the cocancellation properties hold, all 
similar to the one above, are guaranteed in the category of alternative 
algebras, where (ab)b^-1=a=b^-1(ba) for any a and any invertible b
(cf. <cit.>). 



The octonions Ø form an alternative algebra, therefore one can apply I
to Ø. The set I(Ø) of invertible octonions is a well
known Moufang loop (cf. <cit.>), that is, it is a loop satisfying
the Moufang identities 

    a(b(ca)) = ((ab)a)c          (ab)(ca)     = (a(bc))a 
    
    a(b(cb)) = ((ab)c)b          (ab)(ca)     = a((bc)a
 
for any elements a,b,c. 






 §.§ Loop of unitary elements
 


Consider now involutive algebras A, and the subgroup of I(A) made of 
unitary elements in A, namely

    U(A) = { a∈ A | a a^*=1 },

when A is commutative. Exactly as for I, the functor U is represented 
on _^* by the commutative Hopf algebra = [x,x^* | x x^*=1 ], 
with co-operations 

    Δ (x) = x ⊗ x,     ε (x) = 1,      
    S(x) = x^* .




Let us call unitary coloop bialgebra on  the associative algebra 
= [x,x^* | x x^*=1 ] endowed with the co-operations defined 
on generators exactly as those in Def. <ref>, 
where the generator x^-1 is replaced by x^*. 


As for the invertible coloop bialgebra, one can prove that 


The algebra  is a cogroup bialgebra in _^* and represents, 
for any involutive associative algebra A, the group 

    U(A) = __^*(,A)

of unitary elements of A. 
Moreover, the group U(A) is abelian if A is commutative. 



For =, this functor allows us to describe several groups of 
unitary matrices. 


  * Applied to the algebra M_n(), if we take the transposition of matrices as 
involution, it gives U() = {1,-1}, and the orthogonal group 
U(M_n())=O(n) for n>1.  


  * On M_n(), we take as involution the complex conjugate of the transposition. 
Then U()=U(1) ≅^1 and U(M_n())=U(n) is the unitary 
group. 


  * Let $̋ be the algebra of quaternions, spanned overby1and by
three imaginary unitsi,j,kwhich anticommute with each other.
The conjugate of a quaternionq=a+b i+c j+d kis the quaternionq^*=a-b i-c j-d k. The conjugation is an involution, and the
real number q = √(q q^*) = √(q^* q) = √(a^2+b^2+c^2+d^2)defines a multiplicative norm on$̋. Then, the functor U applied to $̋
gives the subgroupU()̋ ≅Sp(1) ≅SU(2) ≅^3ofI()̋consisting of unit norm quaternions. 

On the set of matricesM_n()̋, we take as involution the quaternionic
conjugate of the transposition. ThenU(M_n()̋) ≅U(n,)̋ ≅Sp(n)is the compact symplectic group,
also called the hyperunitary group. 


Again exactly as for the invertible coloop bialgebra, one can prove the 
next result. 


The algebraic group U can be extended as a loop to a variety of algebras 
⊂_^* if  is a subcategory of involutive 
alternative algebras _^* admitting coproduct and initial object. 
In particular, U is an algebraic loop on _^*. 

An alternative algebra of octonions Ø=Ø(α,β,γ) is spanned
over a field  of characteristic not 2 by e_0=1 and by seven imaginary
units e_i, for i=1,...,7, with an involved table of multiplication
(cf. <cit.>, <cit.>, <cit.>).
Over the field  there are two non-isomorphic octonion algebras:
the classical division Cayley octonions Ø and the split matrix
Cayley-Dickson algebra ()<cit.>, also known as the Zorn
vector-matrix algebra <cit.>. The last one may be defined over an
arbitrary commutative ring.
The conjugate of an octonion q=∑_i=0^7a_ie_i is the octonion
q^*=a_0e_0-∑_i=1^7a_ie_i. Again, the conjugation is an involution,
and the scalar n(q)=qq^*=q^*q defines a multiplicative norm on Ø
(and an isotropic quadratic form on ()).

Then, for the classical Cayley octonions Ø, the set U(Ø) is the Moufang
subloop of the loop I(Ø) consisting of unit norm octonions, which is  
homeomorphic to the sphere S^7, while for the matrix Cayley-Dickson algebra
() the loop U(()) is not compact. 
The loops U(Ø) and U(()) can be compared to the groups
U()̋≅^3 and U()≅ SL_2() obtained respectively for
division quaternions $̋ and for split quaternions.  


 §.§ Unitary Cayley-Dickson loops


In this section we give an example of a loop which is not algebraic on 
associative algebras. 
 

Letbe a field andjdenote an imaginary unit.  
For any involutive commutative algebraAover, the setU_CD(A) = { a+b j ∈ A+A j | a a^*+b b^*=1 }gives the group of unitary elements in the Cayley-Dickson algebraA+A jwith multiplication(a+b j) (c+dj) = (ac-d^*b)+(da+bc^*) j ,unit1, and involution(a+b j)^* = a^*-b j. 

The functorA⟼ U_CD(A)is representable on_^*, 
by the commutative Hopf algebra= [x,x^*,y,y^* | x x^*+y y^*=1 ]with co-operations 

    Δ (x) = x x - y y^*       Δ (y) = x y + y x^*,  
    
       ε (x) = 1       ε (x) = 0, 
    
        S(x) = x^*        S(y) = -y .

The algebraic group U_CD can not be extended as an algebraic loop to the
category of involutive associative algebras. 

If U_CD could be extended to an algebraic loop to _^*, its 
representative coloop bialgebra  should be an associative algebra 
generated by x, x^*, y and y^* submitted to conditions which give 
x x^*+y y^*=1 if the variables commute. 
The co-operations should then be defined on generators exactly as in the 
commutative case, but taking values in the coproduct 
of the category _^*. 

The conditions x x^*=x^*x and x x^*+y^*y=1 are enough to guarantee that 
the algebra  has a well defined comultiplication, a counit and an 
antipode satisfying the 5-terms relations. 
However, the codivisions, defined according to the coinverse properties 
(<ref>) as 

    δ_r(x) = x^(1)  (x^*)^(2) + (y^*)^(2)  y^(1)      δ_r(y) = - y^(2)  x^(1) + y^(1)  x^(2)
       δ_l(x) = (x^*)^(1)  x^(2) + (y^*)^(2)  y^(1)      δ_l(y) = y^(2)  (x^*)^(1) - y^(1)  (x^*)^(2) ,

satisfy the cocancellation identities (<ref>) and  
(<ref>) if and only if 

x^(1) (x^*x)^(2) = (x^*x)^(2) x^(1)         
y^(1) (y^*y)^(2) = (y^*y)^(2) y^(1)
in . This could happen for two reasons. 
The first is that the map n:⟶ given by 
n(a) = a a^* = a^*a has scalar values, i.e. its image is in 
u()⊂. 
This is the case if  is a  composition algebra,
cf. <cit.>. 
But composition algebras do not have a categorical coproduct. 
The second possibility to verify these conditions is that the identity
a^(1) b^(2) = b^(2) a^(1) holds in  for any 
elements a,b∈. This means that = and 
therefore it is only possible in the category _^*. 
 
In agreement with this result, namely that the construction U_CD is not
functorial on associative algebras, there are few examples of loops arising 
as sets of unitary elements in the Cayley-Dickson algebra constructed 
on an associative algebra. For instance, we can consider the associative
algebras of matrices A=M_n(K) with entries in involutive algebras K over
the field =, with involution given by the transposition of the matrices
plus the involution of their matrix elements.
The unitary elements in A+Aj are preserved by divisions if A is a
composition algebra, and matrix algebras, in general, are not. So, in general,  
U_CD(A) is not a loop. There are few exceptions: 

  * 
The set U_CD(M_n()) is a loop for n=1,2. 
For n=1 (when A= is commutative) it is an abelian group
U_CD() = U() ≅^1, and for n=2 the loop U_CD(M_2())
coincides with the loop U(()) from Example
<ref>, since M_2()+M_2()j≅() is a
matrix Cayley-Dickson algebra.


  *  
The set U_CD(M_n()) is a loop for n=1,2.
For n=1 (when A= is commutative) it is a group
U_CD() = U()̋≅^3, and for n=2 the loop U_CD(M_2())
coincides with the loop U(()) of unital elements in the split
matrix Cayley-Dickson algebra over the complex numbers .


  * 
The set U_CD(M_n()̋) is a Moufang loop only for n=1, and we have 
U_CD()̋≅ U(Ø) ≅^7. For n>1, the set U_CD(M_n()̋)
is not a loop because M_n()̋ is not a composition algebra. 


§ COLOOP OF INVERTIBLE SERIES


The group of invertible series (with constant term equal to1), is the set 
of formal series(A) = 
{ a(λ)=∑_n≥ 0 a_n λ^n | a_0=1, a_n∈ A }with coefficientsa_ntaken in a commutative algebraA, 
endowed with the pointwise multiplication(ab)(λ)=a(λ) b(λ), unit1(λ)=1, 
and where the inverse of a seriesa(λ)is found by recursion. 
It is an abelian proalgebraic group on, represented by the 
cocommutative Hopf algebra 

    = [x_n, n≥ 1]      (x_0=1) 
    (x_n)     = ∑_m=0^n x_m⊗ x_n-m

known as Hopf algebra of symmetric functions <cit.>. 

The functoradmits an evident extention to associative algebras 
as a functor in groups (but not abelian), represented by the cogroup bialgebra 
<cit.>
    = ⟨ x_n, n≥ 1 ⟩     (x_0=1) 
    (x_n)     = ∑_m=0^n x_m^(1) x_n-m^(2)

with antipode defined recursively. The projection of this bialgebra by the 
canonical mapπgiven in Def. <ref> coincides with 
the Hopf algebra of non-commutative symmetric functions (cf. <cit.>). 

In this section we show that the functorcan be extended to 
non-associative algebras, as a proalgebraic loop. 



 §.§ Loop of invertible series

Let A be a unital algebra and let λ be a formal variable. 
We call invertible series in λ with coefficients in A 
the formal series in the set 
(A)={ a=∑_n≥ 0 a_n λ^n | a_0=1, a_n∈ A }, 
 
endowed with the multiplication

a· b = 
∑_n≥ 0 ∑_m=0^n a_m b_n-m λ^n
 
and the unit e given by e_0=1 and e_n=0 for all n>1. 
For instance, 

    (a· b)_1     = a_1+b_1, 
     
    (a· b)_2     = a_2+a_1b_1+b_2, 
    
    (a· b)_3     = a_3+a_2b_1+a_1b_2+b_3.

For any unital algebra A, the set of invertible series (A) is a loop. 

It is clear that the series e is a unit for the given multiplication, so 
we only have to show that there exist a left and a right divisions satisfying 
the cancellation properties (<ref>) and 
(<ref>). 
Since the multiplication is completely symmetric in the the two variables, 
the proof for the two divisions is exactly the same. We do it for the right 
division. 

Given two series a=∑ a_n λ^n and b=∑ b_n λ^n, 
we define the right division a b=∑ (a b)_n λ^n 
so that (a b)ḃ = a, that is
∑_m=0^n (a b)_m b_n-m = a_n      .  
 
These equations are solved recusively from (a b)_0=1, and give 
the nth term

(a b)_n=a_n-∑_m=0^n-1 (a b)_m b_n-m. 

Let us then prove by induction that (a· b) b = a, that is, 
((a· b) b)_n = a_n for any n≥ 0. 
We have ((a· b) b)_0 = a_0=1 and, for any n≥ 1, 

    ((a· b) b)_n 
        = (a· b)_n-∑_m=0^n-1 (a· b)_m b_n-m
        = a_n+ ∑_m=0^n-1(a_m-(a· b)_m) b_n-m ,

so if we suppose that ((a· b) b)_m = a_m for any m≥ n-1, 
we have ((a· b) b)_n = a_n. 



For instance, for the right division we find 

    (a b)_1     = a_1-b_1, 
    
     
    (a b)_2     = a_2-a_1b_1-b_2+b_1b_1, 
    
     
    (a b)_3     = a_3-(a_1b_2+a_2b_1)+(a_1b_1)b_1-b_3+(b_1b_2+b_2b_1)-(b_1 b_1)b_1,

and for the left division we find

    (a\ b)_1     = b_1-a_1, 
    
     
    (a\ b)_2     = b_2-a_1b_1-a_2+a_1a_1, 
    
     
    (a\ b)_3     = b_3-(a_1b_2+a_2b_1)+a_1(a_1b_1)-b_3+(a_1a_2+a_2a_1)-a_1(a_1a_1).


 §.§ Coloop bialgebra of invertible series


For anyn≥ 1, letx_nbe a graded variable of degreen. 
ForX=_{x_n, n≥ 1}, the tensor algebraH=T(X)can be seen as 
the set of non-commutative polynomials in the variablesx_1,x_2,..., that 
we denote by⟨ x_n, n≥ 1⟩. It is then useful to denote 
the unit1ofHbyx_0. 

The unital associative coproduct algebraH His then the tensor 
algebraT(X^(1)⊕ X^(2))on two identical sets of variables, and
similarlyH H H = T(X^(1)⊕ X^(2)⊕ X^(3)). 
To simplify the notations, in this section we denote byx_n=x_n^(1),y_n=x_n^(2)andz_n=x_n^(3)the generators taken in the different copies
ofXin a coproduct algebra.
 

For any integern≥ 1and any1≤ℓ≤ n, let_n^ℓdenote the set of compositions ofnof lengthℓ, that is, 
the set of ordered sequences = (n_1,...,n_ℓ)such that 

    n_1+⋯ +n_ℓ=n,          n_1,...,n_ℓ≥ 1.

For instance, forℓ=1,2,3, we have 

    _1^1 = { (1) },     _2^1 = { (2) },   _2^2 = { (1,1) }, 
    
       _3^1 = { (3) },   _3^2 = { (2,1),(1,2) },   _3^3 = { (1,1,1) }.

Let us call  coloop bialgebra of invertible series the free unital algebra 
 = T{x_n | n≥ 1}
with the following graded co-operations: 

  * 
comultiplication : ⟶ given by

    (x_n)    = ∑_m=0^n x_m y_n-m ;

  * 
counit ε:⟶ given by 
ε(x_n) = δ_n,0; 


  * 
right codivision δ_r:⟶ 
given by

    δ_r(x_n)    = x_n-y_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1((((x_n_1-y_n_1) y_n_2)  y_n_3)⋯) y_n_ℓ+1,

where _n^ℓ+1 is the set of compositions of n of length ℓ+1, 
cf. (<ref>); 


  * 
left codivision δ_l:⟶ 
given by

    δ_l(x_n)    = y_n-x_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1
    x_n_1(⋯(x_n_2 (x_n_ℓ (y_n_ℓ+1-x_n_ℓ+1))) ).

The algebra  is a coloop bialgebra and represents the loop of 
invertible series as a functor :⟶. 

As a consequence, given an algebra A, a series 
a=∑_n≥ 0 a_n λ^n ∈(A) can be seen as an algebra 
homomorphism a:⟶ A defined on the generators of  
by a(x_n)=a_n, and the right and left division a b and 
a\ b are given at any order n by the following closed formulas: 

    (a b)_n     = μ_A (a b) δ_r(x_n) 
    
        = a_n-b_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1((((a_n_1-b_n_1) b_n_2)b_n_3)⋯)b_n_ℓ+1 , 
    
     
    (a\ b)_n     = μ_A (a b) δ_l(x_n) 
    
        = b_n-a_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1
    a_n_1(⋯(a_n_2 (a_n_ℓ (b_n_ℓ+1-a_n_ℓ+1))) ) .

The algebra  clearly represents the functor  with values in sets, 
and the comultiplication  represents the pointwise multiplication 
of series. 
The only thing which should be proved is that  is a coloop bialgebra 
with the given codivisions. 
The formulas for the left and for the right codivisions are perfectly 
symmetric, in the sense that δ_l=τ δ_r, 
so it suffices to give the details for one codivision. 
Let us then show that the right codivision satisfies the two equations 
(<ref>). 

Concerning the first one, we have 

    (δ_r) (x_n) 
        = δ_r(x_n) + z_n + ∑_m=1^n-1δ_r(x_m) z_n-m,

which is an element of , and since
μ:⟶ multiplies the variables y and z
(in the order they appear) and puts the result in the right-hand side copy of
 in , we have 

    (μ)(δ_r) (x_n) 
        = δ_r(x_n) + y_n 
    + ∑_m=1^n-1δ_r(x_m) y_n-m
        = x_n - y_n +∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1(((u_n_1 y_n_2)  y_n_3)⋯) y_n_ℓ+1 
    + y_n 
    
       
    + ∑_m=1^n-1 u_m y_n-m 
    + ∑_m=1^n-1∑_λ=1^m-1 (-1)^λ∑_∈_m^λ+1(((u_m_1 y_m_2)⋯) y_m_λ+1) y_n-m

where we set u_n:=x_n-y_n and therefore we have 

    ∑_m=1^n-1 u_m y_n-m    = ∑_∈_n^2 u_n_1 y_n_2 .

Setting ℓ=λ+1 in the last sum, we have 2≤ℓ≤ n-1 
and ℓ≤ m≤ n-1 with 
⋃_m=λ^n-1_m^ℓ×_n-m^1 = _n^ℓ+1, 

therefore

    ∑_m=1^n-1∑_λ=1^m-1 (-1)^λ∑_∈_m^λ+1(((u_m_1 y_m_2)⋯) y_m_λ+1) y_n-m    = 
    ∑_ℓ=2^n-1 (-1)^ℓ∑_∈_n^ℓ+1(((u_n_1 y_n_2)  y_n_3)⋯) y_n_ℓ+1.

Thus, we finally obtain 

    (δ_r)(x_n)     = x_n .


For the second identity, we rewrite the comultiplication as 

    (x_n)    = x_n + y_n + ∑_∈_n^2 x_n_1 y_n_2

and using the fact that 
_n^ℓ+1= ⋃_m=1^n-1_m^1 ×_n-m^ℓ, 

and setting λ=ℓ, we rewrite the right codivision as  

    δ_r(x_n) 
       = u_n+ ∑_m=1^n-1∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ(((u_m y_k_1)  y_k_2)⋯) y_k_λ.

We then have 

    ()δ_r(x_n) 
        = (x_n)-z_n 
    + ∑_m=1^n-1∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ((((x_m)-z_m) z_k_1)⋯) z_k_λ
        = x_n + y_n + ∑_∈_n^2 x_n_1 y_n_2 
    + ∑_m=1^n-1∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ((x_m z_k_1)⋯) z_k_λ -z_n 
    
       
    + ∑_m=1^n-1∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ((y_m z_k_1)⋯) z_k_λ
       
    + ∑_m=1^n-1∑_∈_m^2∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ(((x_m_1 y_m_2) z_k_1)⋯) z_k_λ
       
    - ∑_m=1^n-1∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ((z_m z_k_1)⋯) z_k_λ .

When we then apply μ, we identify z_m=y_m and 
z_k_i=y_k_i for i=1,...,λ, and therefore we have  

    (μ)()δ_r(x_n) 
        = x_n + ∑_∈_n^2 x_n_1 y_n_2 
    + ∑_m=1^n-1∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ((x_m y_k_1)⋯) y_k_λ
       
    + ∑_m=1^n-1∑_∈_m^2∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ(((x_m_1 y_m_2) y_k_1)⋯) y_k_λ

where 

    ∑_m=1^n-1∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ((x_m y_k_1)⋯) y_k_λ    = ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1((x_n_1 y_n_2)⋯) y_n_ℓ+1 ,

and 

    ∑_∈_n^2 x_n_1 y_n_2 + ∑_m=1^n-1∑_∈_m^2∑_λ=1^n-m (-1)^λ∑_∈_n-m^λ(((x_m_1 y_m_2) y_k_1)⋯) y_k_λ
       
    = - ∑_∈_n^1+1 (-1)^1 x_n_1 y_n_1+1 
    - ∑_ℓ=2^n-1 (-1)^ℓ∑_∈_n^ℓ+1((x_n_1 y_n_2)⋯) y_n_ℓ+1 .

This we finally have

    (μ)()δ_r(x_n) 
        = x_n .


 §.§ Properties of the loop of invertible series


Loops satisfying weak versions of associativity have many applications,
for instance in Blaschke's Web Geometry through nets <cit.>.
It is therefore interesting to ask what kind of identities are satisfied
by the loops(A)of invertible series. 


Given an algebra A, the loop (A) satisfies an identity 

(*)     u(a,b,...,c) = v(a,b,...,c)

for any series a,b,...,c ∈(A) if and only if the identity (*) 
is satisfied in A, that is, for any elements a,b,...,c∈ A.

Roughly speaking, this result follows from the fact that the comultiplication 
 is linear on both sides on generators. 
More precisely, if in the identity (*) the operators u and v are 
multilinear, the implication 
“(*) on A  ⇒  (*) on (A)” 
is proved by direct inspection, and the opposite implication 
is proved by considering series of the form 
a=1+a_1 λ, b=1+b_1 λ,..., c=1+c_1 λ. 

If in the identity (*) the operators u and v are not multilinear, 
for instance the element a appears k times, it suffices to linearize
them, by considering the sum a=a^1+⋯ +a^k of k different elements. 


In particular, this result implies that(A)is a Moufang loop if and
only ifAis alternative, that(A)is a group if and only ifAis associative, and that(A)is an abelian group if and only ifAis commutative and associative.

We give below counterexamples to some interesting properties of loops which
fail on the loops(A)for associative algebrasA, which can be
deduced by the coloop bialgebra. 


The left and the right inverses of any a∈(A) do not coincide,
that is

a\ e ≠ e  a. 

In fact, the left and right inverses in (A) coincide if and only if
the left antipode S_l and the right antipode S_r of  coincide. 
Applying equations (<ref>), we find

    S_r(x_n)     := (ε)  δ_r(x_n) 
    
        = -x_n - ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1(((x_n_1 x_n_2)  x_n_3)⋯) x_n_ℓ+1

and 

    S_l(x_n)     := (ε)  δ_l(x_n)
    
        = -x_n - ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1
    x_n_1(⋯(x_n_2 (x_n_ℓ x_n_ℓ+1)) ),

therefore the two antipodes do not coincide. 
For instance, for a series a=1+a_1 λ, we have 

    e a     := a   S_r 
    = 1 - a_1 λ + a_1 a_1 λ^2 - (a_1 a_1)a_1 λ^3 
    + ((a_1 a_1)a_1)a_1 λ^4 + ⋯
     
    a\ e     := a   S_l 
    = 1 - a_1 λ + a_1 a_1 λ^2 - a_1(a_1 a_1) λ^3 
    + a_1(a_1(a_1 a_1)) λ^4 + ⋯

To have a counterexample to the equality e a=a\ e,
take for A the algebra of 2× 2 matrices over the sedenions,
spanned by 1 and by the imaginary units e_i for i=1,...,15. 
If a=1+a_1 λ is the series with coefficient 

a_1 = ([ e_1+e_10 e_5+e_14;        0        1 ]),

we have 

e a - a\ e 
= ([            0 -2(e_5+e_14);            0            0 ])  λ^3 + O(λ^4). 

The left and right inversions in (A) do not allow us to construct
the divisions, that is, 

a  b ≠ a   (e b)          
a\ b ≠ (a\ e)   b 

for any a,b∈(A). 

In fact, to show that a  b ≠ a  (e b) and 
a \ b ≠ (a\ e) b in the loop (A) is 
equivalent to show that 
δ_r ≠ ( S_r)         δ_l ≠ (S_l) 
in the coloop bialgebra . Let us show it for the right codivision. 
For any generator x_n, we have 

    ( S_r) (x_n) 
        = x_n + S_r(y_n) + ∑_m=1^n-1 x_m S_r(y_n-m) 
    
        = x_n - y_n - ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1((y_n_1 y_n_2)⋯) y_n_ℓ+1
       
    - ∑_m=1^n-1 x_m y_n-m 
    - ∑_m=1^n-1∑_λ=1^n-m-1 (-1)^λ∑_∈_n-m^λ+1
    x_m (((y_k_1 y_k_2)⋯) y_k_λ+1).

Writing the last two sums in terms of compositions of n yields 

    ( S_r) (x_n) 
        = u_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1( x_n_1 (((y_n_2 y_n_3)⋯) y_n_ℓ+1)- 
    ((y_n_1 y_n_2)⋯) y_n_ℓ+1),

which is clearly different from the expression of δ_r(x_n). 

A loop Q is  left alternative if a(ab)=(aa)b for any a,b∈ Q,
and it is  right alternative if (ab)b=a(bb). 
The proalgebraic loop  on the category  is not left nor right
alternative.

For this, it suffices to show that the coloop bialgebra  is not
right coalternative, that is (μ) K≠ 0, where 
K=() () is the
coassociator.  
The first deviation from right alternativity appears on the generator
x_3, since we have

    K(x_3)     = (x_1y_1)z_1-x_1(y_1z_1) 
     
    (μ) K(x_3)    = (x_1y_1)y_1-x_1(y_1 y_1) ≠ 0.

For instance, if A is the algebra of sedenions, the deviation from
right alternativity can be seen comparing (a b) b and a(b b)
for the two series 

a=1+(e_1+e_10)λ         
b=1+(e_5+e_14)λ
because (e_1+e_10)(e_5+e_14) = 0 and therefore 

(a b) b - a(b b) 
= -(e_1+e_10)(e_5+e_14)^2 λ^3 = 2(e_1+e_10) λ^3. 

A loop Q is  power associative if every element of the loop
generates an abelian subgroup. 
The proalgebraic loop  on the category  is not power
associative. In particular, power associativity requires the associativity
(aa)a=a(aa) for any element. Therefore, it suffices to show that 
μ  (μ) K(x_3) = (x_1x_1)x_1-x_1(x_1x_1) ≠ 0. 


For instance, if we take A to be the algebra of 2× 2 matrices 
with coefficients in the sedenion algebra, for the series 
a=1+a_1λ of Example <ref> with 

a_1 = ([ e_1+e_10 e_5+e_14;        0        1 ]) 

we have 

a_1^2a_1 = ([ -2(e_1+e_10)  -(e_5+e_14);            0            1 ]) 
         
a_1 a_1^2= ([ -2(e_1+e_10)     e_5+e_14;            0            1 ])  

and therefore 

(a a) a - a(aa) = ([            0 -2(e_5+e_14);            0            0 ])  λ^3. 


§ COLOOP OF FORMAL DIFFEOMORPHISMS


The group of formal diffeomorphisms (tangent to the identity) 
is the set of series(A)={ a=∑_n≥ 0 a_n λ^n+1 | a_0=1, a_n∈ A }with coefficientsa_ntaken in a commutative algebraA, 
endowed with the composition law(a∘ b)(λ)=a( b(λ)), 
unite(λ)=λ, and where the inverse of a seriesa(λ)is given by the Lagrange inversion formula <cit.>. 
It is a proalgebraic group on, represented by the Faà di Bruno Hopf 
algebra <cit.>, <cit.>
    =[x_n, n≥ 1]      (x_0=1) 
    (x_n)    =∑_m=0^n x_m ⊗∑_(p)(m+1)!p_0! p_1!⋯ p_n! x_1^p_1⋯ x_n^p_n

where the sum is done over the set of tuples(p_0,p_1,p_2,...,p_n)of non-negative integers such thatp_0+p_1+p_2+⋯ +p_n=m+1andp_1+2p_2+⋯ +np_n=n-m. 
In this section we show that this group can be extended as a proalgebraic loop 
to the category. 



 §.§ Loop of formal diffeomorphisms

Let A be a unital associative algebra, non necessarily commutative, and 
let λ be a formal variable. 
We call formal diffeomorphisms in λ with coefficients in A 
the formal series in the set 
(A)={ a=∑_n≥ 0 a_n λ^n+1 | 
a_0=1, a_n∈ A }, 
 
endowed with the composition law 

    a∘  b 
        = ∑_n≥ 0 ∑_m=0^n k_0,...,k_m≥ 0∑_k_0+⋯ +k_m=n-m 
    a_m b_k_0⋯ b_k_m λ^n+1
        = ∑_n≥ 0 (a_n + b_n + ∑_m=1^n-1 a_m ∑_l=1^n 
    m+1lk_1,...,k_m≥ 1∑_k_1+⋯ +k_m=n-m 
    b_k_1⋯ b_k_m) λ^n+1

and the unit e given by e_0=1 and e_n=0 for all n>1. 
For instance, 

    (a∘ b)_1     = a_1+b_1, 
     
    (a∘ b)_2     = a_2+2a_1b_1+b_2, 
    
    (a∘ b)_3     = a_3+3a_2b_1+a_1(2b_2+b_1^2)+b_3.

The indeterminate λ is not necessary to define the loop law, 
but helps to keep track of the degree of the terms in the sum. 

For any unital associative algebra A, the set (A) is a loop.  

It is clear that the composition is a well-defined operation, and that e 
is a unit. Let us show that the left and right divisions exist.  
 
i) Let us prove that there exists a right division  
satisfying the two equations (<ref>). 
Given two series a=∑ a_n λ^n+1 and b=∑ b_n λ^n+1, 
let us define the series a b=∑ (a b)_n λ^n+1 
so that (a b)∘ b = a, that is
∑_m=0^n k_0,...,k_m≥ 0∑_k_0+⋯ +k_m=n-m
(a b)_m b_k_0⋯ b_k_m = a_n      .  
 
From now on, in the sum over the integers k_0,...,k_p we omit to write 
that all integers can be zero. 
These equations are solved recursively, starting from (a b)_0=a_0=1. 
The nth term is given by 

    (a b)_n     = a_n - ∑_m=0^n-1 ∑_k_0+⋯ +k_m=n-m  
    (a b)_m b_k_0⋯ b_k_m.

To prove that (a∘ b) b = a, i.e. that 
((a∘ b) b)_n = a_n for any n≥ 0, 
we proceed by induction. 
We have ((a∘ b) b)_0=(a∘ b)_0=1, therefore 
((a∘ b) b)_1 
= (a∘ b)_1 - ((a∘ b) b)_0 b_1 
= a_1+b_1-b_1 = a_1

and 

    ((a∘ b) b)_n     = (a∘ b)_n - ∑_m=0^n-1 ∑_k_0+⋯ +k_m=n-m((a∘ b) b)_m b_k_0⋯ b_k_m
        = ∑_m=0^n ∑_k_0+⋯ +k_m=n-m 
    a_p b_k_0⋯ b_k_m 
    - ∑_m=0^n-1 ∑_k_0+⋯ +k_m=n-m((a∘ b) b)_m b_k_0⋯ b_k_m
        = a_n +  ∑_m=0^n-1 ∑_k_0+⋯ +k_m=n-m(a_m-((a∘ b) b)_m) b_k_0⋯ b_k_m,

so, if we suppose that ((a∘ b) b)_m=a_m for any 
m≤ n-1, we have ((a∘ b) b)_n=a_n. 
 
ii) To prove the existence of the left division we proceed in the same way: 
the series a\ b that satisfies the identity 
a∘ (a\ b)=b of equations (<ref>), that is, 
∑_m=0^n ∑_k_0+⋯ +k_m=n-m 
a_m (a\ b)_k_0⋯ (a\ b)_k_m = b_n 
    , 

is given recursively by (a\ b)_0=1 and 

    (a\ b)_n     = b_n - ∑_m=1^n ∑_k_0+⋯ +k_m=n-m 
    a_m (a\ b)_k_0⋯ (a\ b)_k_m.

The identity a\ (a∘ b)=b means that, for any n≥ 0, we have 
(a\ (a∘ b))_n = b_n. This is proved by induction. 
We have (a\ (a∘ b))_0=1, therefore 
(a\ (a∘ b))_1 
= (a∘ b)_1 
- a_1 (a\ (a∘ b))_0 (a\ (a∘ b))_0 
= a_1+b_1-a_1 = b_1

and 

    (a\ (a∘ b))_n 
        = (a∘ b)_n - ∑_m=1^n ∑_k_0+⋯ +k_m=n-m 
    a_m (a\ (a∘ b))_k_0⋯(a\ (a∘ b))_k_m
        = ∑_m=0^n ∑_k_0+⋯ +k_m=n-m
    a_m b_k_0⋯ b_k_m 
    - ∑_m=1^n ∑_k_0+⋯ +k_m=n-m 
    a_m (a\ (a∘ b))_k_0⋯(a\ (a∘ b))_k_m
        = a_0b_n +  ∑_m=1^n ∑_k_0+⋯ +k_m=n-m 
    a_m (b_k_0⋯ b_k_m - (a\ (a∘ b))_k_0⋯(a\ (a∘ b))_k_m) 
        = b_n +  ∑_m=1^n-1 ∑_k_0+⋯ +k_m=n-m 
    a_m (b_k_0⋯ b_k_m - (a\ (a∘ b))_k_0⋯(a\ (a∘ b))_k_m) 
        
    + a_n (b_0⋯ b_0 - (a\ (a∘ b))_0⋯(a\ (a∘ b))_0)
        = b_n +  ∑_m=1^n-1 ∑_k_0+⋯ +k_m=n-m 
    a_m (b_k_0⋯ b_k_m - (a\ (a∘ b))_k_0⋯(a\ (a∘ b))_k_m),

so, if we suppose that (a\ (a∘ b))_m=b_m for any 
m≤ n-1, we have (a\ (a∘ b))_n=b_n. 


For instance, the first terms of the right division are

    (a b)_1     = a_1-b_1, 
     
    (a b)_2     = a_2-[b_2+2(a b)_1 b_1] 
    = a_2-2a_1b_1-(b_2-2b_1^2), 
     
    (a b)_3     = a_3-[b_3+(a b)_1(2b_2+b_1^2)+3(a b)_2b_1 ]
        = a_3-(2a_1b_2+3a_2b_1)+5a_1b_1^2-[b_3-(2b_1b_2+3b_2b_1)+5b_1^3],

and the first terms of the left division are 

    (a\ b)_1     =b_1-a_1, 
     
    (a\ b)_2     =b_2-[2a_1 (a\ b)_1+a_2] 
    = b_2-2a_1b_1-(a_2-2a_1^2), 
     
    (a\ b)_3     =b_3-[a_1(2(a\ b)_2+(a\ b)_1^2)
    +a_2(3(a\ b)_1)+a_3] 
        = b_3-(2a_1b_2+3a_2b_1)+(5a_1^2b_1+a_1b_1a_1-a_1b_1^2)
    -[a_3-(2a_1a_2+3a_2a_1)+5a_1^3].


We now prove that the loop of formal diffeomorphism is proalgebraic over 
associative algebras, and give its representative coloop bialgebra. 



 §.§ Faà di Bruno coloop bialgebra
 

As in Section <ref>, letX=_{x_n, n≥ 1}be the set of graded variablesx_nof degreen, and identify the tensor algebraT(X)=⟨ x_n, n≥ 1⟩with the set of non-commutative polynomials⟨ x_n, n≥ 1⟩. 
We endow this algebra with the structure of a coloop biagebra which represents
the loop. 

As before, to simplify the notations, we denote byx_n=x_n^(1)andy_n=x_n^(2)the generators taken in the two copies ofXinT(X) T(X)≅ T(X^(1)⊕ X^(2)). 

To describe the codivisions we need to introduce some sets of sequences 
and two types of related integer coefficients. 


For any ℓ≥ 1, let _ℓ denote the set of 
sequences = (m_1,...,m_ℓ) such that  

    m_1+⋯ +m_ℓ=ℓ,          
    m_1+⋯ +m_j≥ j      j=1,...,ℓ-1 .
 
For instance, for ℓ=1,2,3, we have 
_1 = { (1) },   _2 = { (2,0),(1,1) },   _3 = { (3,0,0),(2,1,0),(2,0,1),(1,2,0),(1,1,1) }. 


For any ℓ≥ 1 and any sequence (n_1,...,n_ℓ+1) of positive 
integers, we call  Lagrange coefficient[
These coefficients appear in the Lagrange inversion formula <cit.>, 
cf. <cit.>. 
]
the number
 
d_ℓ(n_1,...,n_ℓ) 
= ∑_∈_ℓn_1+1m_1⋯n_ℓ+1m_ℓ. 

For ℓ=0, _0 is empty and we set d_0 = 1.

For instance, for ℓ=1,2,3, we have

    d_1(n_1)     = n_1+11 ,
    
     
    d_2(n_1,n_2)     = n_1+12 + n_1+11n_2+11 ,
    
     
    d_3(n_1,n_2,n_3)
        = n_1+13 
    + n_1+12n_2+11 + n_1+12n_3+11
       
    + n_1+11n_2+12 
    + n_1+11n_2+11n_3+11 .

For any ℓ≥ 1, let _ℓ be the set of sequences
= (e_1,...,e_ℓ) of bits e_i∈{1,2}. 
For any ∈_ℓ, let _ℓ^ be the set of sequences 
= (m_1,...,m_ℓ)∈_ℓ such that  

    m_i=0      e_i=2,      i=2,...,ℓ .

The bits e_i will be used in Eq. (<ref>) to label
the generators x_i of the coloop bialgebra in order to determine to which
copy of the coproduct algebra the variables x_i^(e_i) belong.
To simplify the final formulas for the codivisions, we chose for e_i
the bits 1 and 2, even if, for the present discussion, the bits 1 and 0
would be more appropriate. 

In particular, if = (1,1,...,1) then _ℓ^ = _ℓ.
If  starts with the bit e_1=2, then _ℓ^ is empty,
because the condition (<ref>) implies that m_1≥ 1. 
If  starts with the bit e_1=1 and contains at least a bit value 2,
then the set _ℓ^ is a proper subset of _ℓ obtained
by keeping only those sequences  which have the value 0 in all
the positions where the bit value of  is 2. 
For instance, _1^(1) = _1 = { (1) } and for ℓ=2 we have

    _2^(1,1)    = _2 = { (2,0),(1,1) }, 
    
    _2^(1,2)    = { (2,0) }.

For ℓ=3, we have 

    _3^(1,1,1)    = _3 
    = { (3,0,0),(2,1,0),(2,0,1),(1,2,0),(1,1,1) }
    _3^(1,1,2)    = { (3,0,0),(2,1,0),(1,2,0) }
    _3^(1,2,1)    = { (3,0,0),(2,0,1) }
    _3^(1,2,2)    = { (3,0,0) } .


For any ℓ≥ 1, any sequence ∈_ℓ 
and any sequence (n_1,...,n_ℓ+1) of positive integers, we call 
 labeled Lagrange coefficient the number 

    d_ℓ^(n_1,...,n_ℓ)     = ∑_∈_ℓ^n_1+1m_1⋯n_ℓ+1m_ℓ.

For ℓ=0, _0 and _0 are empty and we set d_0^=d_0 1. 
Of course, if = (1,1,...,1) then 
d_ℓ^(n_1,...,n_ℓ) = d_ℓ(n_1,...,n_ℓ),
if  starts by 2 then d_ℓ^=0, 
and if  starts by 1 and contains some bit values equal to 2, then 
d_ℓ^(n_1,...,n_ℓ) < d_ℓ(n_1,...,n_ℓ). 

Here are the values of d_ℓ^ for ℓ=1,2,3, obtained by summing
the patterns according to the labeled sequences given in
(<ref>) and (<ref>): 

    d_1^(1)(n_1)     = d_1(n_1) = n_1+11 , 
    
     
    d_2^(1,1)(n_1,n_2)     = d_2(n_1,n_2) 
    = n_1+12 + n_1+11n_2+11,
    
     
    d_2^(1,2)(n_1,n_2)     = n_1+12 ,

where the term n_1+11n_2+11 disappears because it 
corresponds to the sequence = (1,1) which does not have the value 0
in the same position as the bit 2 in = (1,2), 

    d_3^(1,1,1)(n_1,n_2,n_3)     = d_3(n_1,n_2,n_3) 
    
        = n_1+13 
    + n_1+12n_2+11 + n_1+12n_3+11
       
    + n_1+11n_2+12 
    + n_1+11n_2+11n_3+11 ,
    
     
    d_3^(1,1,2)(n_1,n_2,n_3) 
        = n_1+13 
    + n_1+12n_2+11 
    + n_1+11n_2+12 , 
    
     
    d_3^(1,2,1)(n_1,n_2,n_3) 
        = n_1+13 + n_1+12n_3+11 ,
    
    
    d_3^(1,2,2)(n_1,n_2,n_3) 
        = n_1+13 .

We call  Faà di Bruno coloop bialgebra the free unital associative 
algebra 

    =⟨ x_n, n≥ 1⟩,      x_0=1
 
of non-commutative polynomials in the graded variables x_n, 
with the following graded co-operations: 

  *  
comultiplication :⟶ 
given by 

    (x_n) 
        = x_n + y_n + ∑_ℓ=1^n-1∑_∈_n^ℓ+1n_1+1ℓ 
    x_n_1 y_n_2⋯ y_n_ℓ+1 ,
 
where _n^ℓ+1 is the set of compositions of n of length ℓ+1, 
cf. (<ref>); 


  *  
counit ε:⟶ given by 
ε(x_n) = δ_n,0; 


  * 
right codivision 
δ_r:⟶ given by

    δ_r(x_n) 
        = ∑_ℓ=0^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    d_ℓ(n_1,...,n_ℓ) 
    (x_n_1-y_n_1)  y_n_2⋯ y_n_ℓ+1,

where the Lagrange coefficients d_ℓ are given in Def. <ref>;


  *  
left codivision 
δ_l:⟶ given by

    δ_l(x_n) 
        = 
    
    
    ∑_ℓ=0^n-1 (-1)^ℓ∑_∈_n^ℓ+1∑_∈_ℓ (-1)^ d_ℓ^(n_1,...,n_ℓ) 
    x^(e_1)_n_1 x^(e_2)_n_2⋯ x^(e_ℓ)_n_ℓ  (y_n_ℓ+1-x_n_ℓ+1)

where the set of sequences _ℓ and the labeled Lagrange coefficients 
d_ℓ^ are given in Def. <ref>, and where we set 

(-1)^ = (-1)^e_1+⋯ +e_ℓ-ℓ
and, according to the previous convention, we set 

    x^(e_i)_n     = {[ x_n    ; y_n     ]. .

In particular, since d_ℓ^=0 if e_1=2, the first variable is always
x^(e_1)_n_1=x_n_1.
 

For instance, on the first five generators, the comultiplication is 

    (x_1)    = x_1 + y_1 
    
    (x_2)    = x_2 + y_2 + 2x_1y_1 
    
    (x_3)    = x_3 + y_3 + ( 2x_1y_2 + 3x_2y_1 ) + x_1y_1^2
    
    (x_4)    = x_4 + y_4 + ( 2x_1y_3 + 3x_2y_2 + 4x_3y_1) 
    + ( x_1(y_1y_2+y_2y_1) + 3x_2y_1^2 ) 
    
    (x_5) 
       = x_5 + y_5 + ( 2x_1y_4 + 3x_2y_3 + 4x_3y_2 + 5x_4y_1 ) 
    
              
    + ( x_1(y_1y_3+y_2^2+y_3y_1) + 3x_2(y_1y_2+y_2y_1) +6x_3y_1^2 ) 
    + x_2y_1^3

the right codivision, withu_n=x_n-y_n, is 

    δ_r(x_1)    = u_1 
    
    δ_r(x_2)    = u_2 - 2u_1y_1 
    
    δ_r(x_3)    = u_3 - ( 2u_1y_2 + 3u_2y_1 ) + 5u_1y_1^2 
    
    
    
    
    δ_r(x_4)    = u_4 - ( 2u_1y_3 + 3u_2y_2 + 4u_3y_1 ) 
    + ( 5u_1y_1y_2 + 7u_1y_2y_1 + 9u_2y_1^2 ) - 14u_1y_1^3 
    
    
    
    
    
    
    
    
    δ_r(x_5)    = u_5 - ( 2u_1y_4 + 3u_2y_3 + 4u_3y_2 + 5u_4y_1 ) 
    
                
    + ( 5u_1y_1y_3 + 7u_1y_2^2 + 9u_1y_3y_1 + 9u_2y_1y_2 + 12u_2y_2y_1 
    + 14u_3y_1^2 ) 
    
                
    - ( 14u_1y_1^2y_2 + 19u_1y_1y_2y_1 + 23u_1y_2y_1^2 + 28u_2y_1^3 )   
    + 42u_1y_1^4

and the left codivision has additional terms which contain both 
variablesxandyin alternative order beside the first position 
which is alwaysx, and last position which is alwaysv_n=y_n-x_n:

    δ_l(x_1)    = v_1 
    
    δ_l(x_2)    = v_2 - 2x_1v_1  
    
    δ_l(x_3)    = v_3 - ( 2x_1v_2 + 3x_2v_1 ) + 5x_1^2v_1 - x_1y_1v_1 
    
    
    
    
    
    δ_l(x_4)    = v_4 - ( 2x_1v_3 + 3x_2v_2 + 4x_3v_1 ) 
    + ( 5x_1^2v_2 + 7x_1x_2v_1 + 9x_2x_1v_1 ) - 14x_1^3v_1 
    
            
    -( x_1y_1v_2 + x_1y_2v_1 + 3x_2y_1v_1 ) 
    + ( 4x_1^2y_1v_1 + 2x_1y_1x_1v_1) 
    
    
    
    
    
    
    
    
    
    
    
    
    δ_l(x_5)    = v_5 - ( 2x_1v_4 + 3x_2v_3 + 4x_3v_2 + 5x_4v_1 ) 
    
                
    + ( 5x_1^2v_3 + 7x_1x_2v_2 + 9x_1x_3v_1 + 9x_2x_1v_2 + 12x_2^2v_1 
    + 14x_3x_1v_1 ) 
    
                
    - ( 14x_1^3v_2 + 19x_1^2x_2v_1 + 23x_1x_2x_1v_1 + 28x_2x_1^2v_1 )   
    + 42x_1^4v_1 
    
            
    -( x_1y_1v_3 + x_1y_2v_2 + x_1y_3v_1 + 3x_2y_1v_2 + 3x_2y_2v_1 
    + 6x_3y_1v_1 ) 
    
                  
    + ( 4x_1^2y_1v_2 + 4x_1^2y_2v_1 + 9x_1x_2y_1v_1 + 10x_2x_1y_1v_1 
    
                         
    + 2x_1y_1x_1v_2 + 3x_1y_1x_2v_1 + 2x_1y_2x_1v_1 
    + 7x_2y_1x_1v_1 - x_2y_1^2v_1 ) 
    
                  
    - ( 14x_1^3y_1v_1 + 9x_1^2y_1x_1v_1 + 5x_1y_1x_1^2v_1 
    - x_1^2y_1^2v_1 - x_1y_1x_1y_1v_1 ) .
 

We now want to prove that the algebra given above is indeed a coloop bialgebra. 
The only difficulty is to prove that the codivisions satisfy the 
cocancellation properties (<ref>) and 
(<ref>), which are equivalent to some recurrence 
relations on the Lagrange coefficientsd_ℓandd_ℓ^. 

We prove in fact a stronger result, namely, that there exist some 
operatorsR_ℓandR_ℓ^defined on the tensor spaceT(A)over any positively graded algebraA, which produce the Lagrange coefficients 
and which satisfy the wished recurrence relations. 
These operators provide an alternative definition of the Faà di Bruno 
codivisions when applied to the non-unital associative coproduct 
algebraA= 
= T(X^(1)⊕ X^(2)). 



 §.§ Faà di Bruno co-operations in terms of recursive operators
 

LetA=⊕_n≥ 1 A_nbe a positively graded associative algebra over 
a field, and let us denote by|a|the degree of an elementa∈ A, 
that is, the integernsuch thata∈ A_n. 
The tensor algebraT(A) = ⊕_ℓ≥ 0 A^ℓis then bigraded, 
on one side by the tensor powerℓ, that we call  length, 
and on the other side by the grading induced by that ofA, that we call 
 degree,|a_1 a_2⋯ a_ℓ| = ∑_i=1^ℓ |a_i|.A multi-monomial is a homogeneous element ofT(A)with respect to the length,
that is, an element of the forma_1 a_2⋯ a_ℓfor someℓ≥ 1. 
ThenT(A)can be decomposed into the following direct sum with respect to 
the degree[
Note that if A had a null degree component A_0, then T(A) would contain 
an infinite sum of terms in each degree, namely
T(A)_0 = ⊕_p≥ 0 A_0^⊗ p and 
T(A)_n = ⊕_ℓ=1^n ⊕_∈_n^ℓ ⊕_p≥ 0ℓ+pℓ A_0^⊗ p 
A_n_1⋯ A_n_ℓ
for n≥ 1. 
]: 

    T(A)     = ⊕⊕_n≥ 1 ( ⊕_ℓ=1^n ⊕_∈_n^ℓ 
    A_n_1⋯ A_n_ℓ) ,

where the compositions∈_n^ℓare defined by 
eq. (<ref>). 


Let us define a graded linear operation 
 : T(A)  T(A) ⟶⊕ A

by setting 

    a  (b_1⋯ b_ℓ)
        = |a|+1ℓ  a  b_1⋯ b_ℓ
    
    (a_1⋯ a_ℓ')  (b_1⋯ b_ℓ”) 
        = a_1  (a_2⋯ a_ℓ'
    b_1⋯ b_ℓ”) 
    
        = |a_1|+1ℓ'+ℓ”-1  
    a_1⋯ a_ℓ'  b_1⋯ b_ℓ”

where the expressions on the right-hand side mean the product in the algebra 
A. 

In particular, if we apply these rules to 1∈ = A^ 0, we have 

    1  1     = 1 
    
     
    1  b     = b 
    
    
    1  (b_1⋯ b_ℓ)
        = 0     
    
    a  1     = a 
    
    
    (a_1⋯ a_ℓ)  1
        = |a_1|+1ℓ-1  a_1⋯ a_ℓ .

The restriction :A⊗ T(A) ⟶ A is a brace product 
on A which is symmetric if A is commutative and generalises the natural 
pre-Lie product of the Lie subalgebra of strictly positive generators in 
the Witt algebra (cf. <cit.>). 
Note however that  on T(A)  T(A) is not a multibrace product 
(cf. <cit.>), even excluding the scalar component, because the first 
non-trivial multibrace identity 
M_21(a b+b a;c)+M_11(M_11(a;b);c) = 
M_11(a;M_11(b;c))+M_12(a;b c+c b) is not satisfied. 
Moreover, a unit for  can not exist, because of length arguments, 
and  is not associative, since for any a,b,c∈ A we have 

(a b)  c - a (b c) = (|a|+1) |a|  abc ≠ 0. 

The algebraic structure described by the operator  in terms of generators 
and relations is an open question. 

We call  left recursive operatorL:T(A)⟶ T(A) 
the collection L={ L_ℓ, ℓ≥ 0 } of (non homogeneous) linear
operators L_0= :⟶ and 

    L_ℓ: 
    A^ℓ⟶⊕_λ=1^ℓ A^λ, 
        ℓ≥ 1

defined recursively by 

    L_ℓ(a_1,...,a_ℓ)    = ∑_i=0^ℓ-1 (-1)^ℓ-1-i( L_i(a_1,...,a_i) a_i+1)  
    a_i+2⋯ a_ℓ ,

where we denote 
L_ℓ(a_1,...,a_ℓ):= L_ℓ(a_1⋯ a_ℓ)
and L_0 is understood as acting on 1. 


The first left operators give

    L_1(a)     = L_0(1) a =1  a = a, 
    
    
    L_2(a,b)     = -( L_0(1) a)  b + L_1(a) b = 
    - a b + a b 
    
        = - a  b + |a|+11  ab , 
    
    
    L_3(a,b,c)     = (L_0(1) a) b c - (L_1(a) b)  c
    + L_2(a,b)  c 
    
        = a  b  c - (a b)  c 
    - (a b)  c + (a b)  c 
    
        = 
    a b c 
    - |a|+11  ab c 
    + (|a|+11|a|+|b|+11
    - |a|+12)  abc 
    .
 

The left operatorsL_ℓcan be easily described in a closed way. 


For any ℓ≥ 2 and any a_1,...,a_ℓ∈ A we have

    L_ℓ(a_1,...,a_ℓ) 
       = L_ℓ-1(a_1,...,a_ℓ-1)  a_ℓ 
    - L_ℓ-1(a_1,...,a_ℓ-1)  a_ℓ.


As a consequence, L_ℓ(a_1,...,a_ℓ) is the sum of the 2^ℓ-1 possible 
multi-monomials obtained by combining the operations  and  
with fixed parenthesizing on the left, namely

    L_ℓ(a_1,...,a_ℓ) 
       = ∑_σ_1,...,σ_ℓ-1∈{0,1} (-1)^σ_1+⋯ +σ_ℓ-1(⋯( (a_1 ∗_σ_1 a_2) ∗_σ_2 a_3 ) 
    ∗_σ_3⋯∗_σ_ℓ-2 a_ℓ-1) ∗_σ_ℓ-1 a_l

where we set 
∗_σ = {[  ;   ]. . 

By induction on ℓ. For ℓ=2, we have

    L_1(a_1)  a_2 - L_1(a_1)  a_2 
        = a_1 a_2 - a_1  a_2 = L_2(a_1,a_2).

Now suppose that for any i=2,...,ℓ-1 we have 

L_i(a_1,...,a_i) 
= L_i-1(a_1,...,a_i-1)  a_i - L_i-1(a_1,...,a_i-1)  a_i.

Let us expand the sum defining L_ℓ(a_1,...,a_ℓ). 
At each step, we separate the first two terms of the sum over i=0,...,ℓ-1:

    L_ℓ(a_1,...,a_ℓ) 
       = ∑_i=0^ℓ-1 (-1)^ℓ-1-i( L_i(a_1,...,a_i) a_i+1)  
    a_i+2⋯ a_ℓ
       = (-1)^ℓ-1 a_1 a_2⋯ a_ℓ 
    + (-1)^ℓ-2 (a_1 a_2) ⋯ a_ℓ
       
    + ∑_i=2^ℓ-1 (-1)^ℓ-1-i( L_i(a_1,...,a_i) a_i+1)  
    a_i+2⋯ a_ℓ
       = (-1)^ℓ-2 L_2(a_1,a_2) a_3⋯ a_ℓ 
    + (-1)^ℓ-3( L_2(a_1,a_2) a_3)⋯ a_ℓ
       
    + ∑_i=3^ℓ-1 (-1)^ℓ-1-i( L_i(a_1,...,a_i) a_i+1)  
    a_i+2⋯ a_ℓ = …

Iterating this expansion we obtain 

    L_ℓ(a_1,...,a_ℓ) 
       = (-1)^ℓ-(ℓ-1) L_ℓ-1(a_1,...,a_ℓ-1)  a_ℓ 
    + (-1)^ℓ-ℓ L_ℓ-1(a_1,...,a_ℓ-1)  a_ℓ
       = L_ℓ-1(a_1,...,a_ℓ-1)  a_ℓ 
    - L_ℓ-1(a_1,...,a_ℓ-1)  a_ℓ .

We call  right recursive operatorR:T(A)⟶ T(A) 
the collection R={ R_ℓ, ℓ≥ 0 } of (non homogeneous) linear
operators R_0= :⟶ and 

    R_ℓ: 
    A^ℓ⟶⊕_λ=1^ℓ A^λ, 
        ℓ≥ 1

defined recursively by 

    R_ℓ(a_1,...,a_ℓ)    = ∑_j=1^ℓ∑_∈_ℓ^j( a_1  R_p_1-1(a_2,...,a_p_1) ) 
    
       ( a_p_1+1 R_p_2-1(a_p_1+2,...,a_p_1+p_2) ) 
    ⋯
       ( a_p_1+⋯ +p_j-1+1 
    R_p_2-1(a_p_1+⋯ +p_j-1+2,...,a_p_1+⋯ +p_j) ) ,

where we denote 
R_ℓ(a_1,...,a_ℓ):= R_ℓ(a_1⋯ a_ℓ)
and where R_0 is understood as acting on 1. 


For instance, the first right operators are

    R_1(a)     = a  R_0(1) = a , 
    
    
    R_2(a,b)     = a R_1(b) + (a R_0(1))  (b R_0(1)) 
        = a b + a  b  
    = |a|+11  ab + a  b , 
    
    
    R_3(a,b,c)     = a R_2(b,c) + (a R_1(b))  c
    + a  (b  R_1(c)) + a  b c 
        = a (b c) 
    + a (b c) 
    + a  (b c)
    + (a b) c 
    + a b c 
    
        = 
    (|a|+11|b|+11 + |a|+12)  abc
    + |b|+11  a  bc 
       
    + |a|+11  ab c 
    + a b c .


Note that the right recursive operator is not just a flip of the left recursive
one, basically because the recursion defining the two operators takes place
on the left and on the right-hand side of, which is not a symmetric
operation. The precise relationship betweenR_ℓandL_ℓis given in
Cor. <ref>, after some preliminary results. 

The right operatorsR_ℓcan also be described in a closed way. 


Let _ℓ be the set of sequences satisfying (<ref>). 
For any ∈_ℓ, we define a length-homogeneous linear operator 
R^ℓ_:A^ℓ⟶ T(A) which nests the operation
 in a multi-monomial a_1⋯ a_ℓ according to
the sequence =(m_1,...,m_ℓ)∈_ℓ. The idea is the following: 

  * 
  The multi-monomial R^ℓ_(a_1,...,a_ℓ) is constructed by nesting
  tensor monomials of the form a_i Q_i+1(a_i+1,a_i+2,...)
  one into the other one, where Q_i+1(a_i+1,a_i+2,...) is a multi-monomial
  whose tensor factors can be single variables or monomials of the same form
  a_j Q'_j+1(a_j+1,a_j+2,...). 

  * 
  Every tensor monomial a_i Q_i+1(a_i+1,a_i+2,...) is   determined
  by the length of the multi-monomial Q_i+1(a_i+1,a_i+2,...) and that
  of the nested monomials of the same form. 
  The sequence  fixes the lengths of all the nested multi-monomials:
  
  * 
    The coefficient m_1 is the overall length of the multi-monomial
    R^ℓ_(a_1,...,a_ℓ) in the tensor algebra T(A), that is,
    we have R^ℓ_(a_1,...,a_ℓ)∈ A^ m_1.
  
  * 
    For i=2,...,ℓ-1, the coefficient m_i is the length of the
    multi-monomial Q_i(a_i,a_i+1,...) on which a_i-1 acts by :
    if m_i=0, then a_i-1 appears as an insolated tensor factor,
    if m_i≠ 0, then a_i-1 acts by  on a multimonomial of
    length m_i, which is determined by the values m_j for j>i. 
  
And now we give the algorithm to construct R^ℓ_(a_1,...,a_ℓ): 

  (1)
The coefficient m_1 tells us how many tensor factors we have to construct. 

  (2) 
  Start with a_1 and read the coefficient m_2: if m_2=0 write
  a_1 ⊗ a_2 ⋯, if m_2≠ 0 write a_1 (a_2⋯)
  and expect to close the parenthesis after a multi-monomial of length m_2. 

  (3)
  Then read the next coefficient of  and repeat the procedure of (2).
  For any i=2,...,ℓ, if m_i=0 write a_i-1⊗ a_i ⋯,
  if m_i≠ 0 write a_i-1 (a_i ⋯) and expect to close this
  parenthesis after a multi-monomial of length m_i. 

  (4)
  The procedure stops with the coefficient m_ℓ which, by definition of
  , can be only 0 or 1, and tells if the last pattern is
  a_ℓ-1⊗ a_ℓ or a_ℓ-1 a_ℓ. 

Let us give some examples of this algorithm, for ℓ=5. 
Fix a_1,...,a_5∈ A and set n_i=|a_i| for i=1,...,5. 
For = (2,1,0,2,0), the multi-monomial R^5_(2,1,0,2,0)(a_1,...,a_5) 
is composed of two tensor factors (because m_1=2). 
The variable a_1 acts by  on a multi-monomial of length 1 
(because m_2=1) which starts necessarily by a_2, 
and since a_2 does not act by  (because m_3=0), 
the first tensor factor is necessarily of the form a_1 a_2. 
Then the second tensor factor starts with a_3 acting by  on a 
multi-monomial of length 2 (because m_4=2), which starts 
necessarily by a_4. Since a_4 does not act by  (because m_5=0), 
the second tensor factor is necessarily of the form a_3 (a_4 a_5).
Therefore we finally have

    R^5_(2,1,0,2,0)(a_1,...,a_5) 
        = (a_1 a_2)(a_3 (a_4 a_5)) 
    
        = n_1+11n_3+12  a_1 a_2  a_3 a_4 a_5.


For =(2,1,2,0,0), the variable a_1 still acts by  
on a multi-monomial of length 1 which starts necessarily by a_2, 
but this time a_2 itself acts by  on a multi-monomial of length 2, 
and this exhausts the possible  operations. Finally, this time we have

    R^5_(2,1,2,0,0)(a_1,...,a_5) 
        = [a_1 (a_2 (a_3 a_4)]  a_5 
    
        = n_1+11n_2+12  a_1 a_2 a_3 a_4  a_5.


Note that the binomial coefficients given by a sequence ∈_ℓ
can be determined directly from the last ℓ-1 digits, plus an 
extra null value. For (2,1,0,2,0) we have exactly 
n_1+11n_2+10n_3+12n_4+10n_5+10 . 

and for (2,1,2,0,0) we have
n_1+11n_2+12n_3+10n_4+10n_5+10 . 


Two more examples of the algorithm: for =(3,0,2,0,0), 

    R^5_(3,0,2,0,0)(a_1,...,a_5) 
        = a_1 (a_2 (a_3 a_4))  a_5
    
        = n_2+12  a_1  a_2 a_3a_4  a_5 ,

and for =(4,0,1,0,0), 

    R^5_(4,0,1,0,0)(a_1,...,a_5) 
        = a_1  (a_2 a_3)  a_4 a_5 
    
        = n_2+11  a_1  a_2 a_3 a_4 a_5 .

For any ℓ≥ 1 and any (a_1,...,a_ℓ), we have 

    R_ℓ(a_1,...,a_ℓ) 
        = ∑_∈_ℓ R^ℓ_(a_1,...,a_ℓ).


For instance, forℓ=1,2,3we have

    R^1_(1)(a)     = a, 
    
    
    R^2_(1,1)(a,b)     = a b 
    = n_1+11  a b
    
    
    R^2_(2,0)(a,b)     = a b
    
    
    R^3_(1,1,1)(a,b,c)     = a (b c)  
    = n_1+11n_2+11  a b c .
    
     
    R^3_(1,2,0)(a,b,c)     = a (b c) 
    = n_1+12  a b c 
    
    
    R^3_(2,0,1)(a,b,c)     = a  (b c) 
    = n_2+11  a b c 
    
     
    R^3_(2,1,0)(a,b,c)     = (a b) c 
    = n_1+11  a b c 
    
     
    R^3_(3,0,0)(a,b,c)     = a b c

Comparing with the value ofR_1,R_2andR_3given above, the assertion 
is easily verified. 
 
Let us call R̃_ℓ(a_1,...,a_ℓ) the sum over 
∈_ℓ of Lemma <ref>, and prove that it solves 
equation (<ref>) by induction on ℓ. 

For ℓ=1,2,3 the assertion was proved in the examples. 
For any ℓ≥ 1, we then suppose that on the 
right-hand side of eq. (<ref>) we have 
R_p_i-1 = R̃_p_i-1 for any 1≤ i≤ j, and we set 

P_i=p_1+p_2+⋯ +p_i , 

so that 

    R_ℓ(a_1,...,a_ℓ) 
       = ∑_j=1^ℓ ∑_∈_ℓ^j ∑_i=1^j ∑_^(i)∈_p_i-1( a_1 R̃^p_1-1_^(1)(a_2,...,a_p_1) ) 
       ( a_p_1+1R̃^p_2-1_^(2)(a_p_1+2,...,a_P_2) ) 
    ⋯( a_P_j-1+1R̃^p_j-1_^(j)(a_P_j-1+2,...,a_P_j) ).

In this sum, we can note the following things: 

  * 
The running value j gives the length of the corresponding multi-monomial. 

  * 
In the first tensor factor, the value q^(1)_1 represents the length of 
R̃^p_1-1_^(1)(a_2,...,a_p_1), that is, a sequence number 
associated to a_1, and more generally ^(1) rules the nested 
operations up to the variable a_p_1-1. 
The last variable a_p_1 does not act on further variables and so it 
should be associated to a missing value 0. 
Therefore, the nested operations in the whole first tensor factor are ruled 
by the sequence (^(1),0). 

  * 
Similarly, for any i≤ j, the nested operations in the ith tensor factor 
are ruled by the sequence (^(i),0). 

Let us then associate to this expression the sequence
 = (j,^(1),0,^(2),0,...,^(j)), 

that is, 

    m_1 = j 
        m_2 = q^(1)_1, … , m_p_1 = q^(1)_p_1-1 , m_p_1+1 = 0 , 
        m_P_i-1+2 = q^(i)_1, … , m_P_i = q^(i)_p_i-1 , m_P_i+1 = 0 , 
      
        m_P_j-1+2 = q^(j)_1, … , m_P_j = q^(j)_p_j-1,

which has precisely length 

1+(p_1-1)+1+(p_2-1)+⋯ +1+(p_j-1) = p_1+⋯ +p_j = P_j= ℓ.   

Note that in the sum over the sequences =(p_1,...,p_j)∈^j_ℓ,  
where p_i≥ 1 for i=1,...,j, there occur the terms with p_i-1=0. 
In this case the multipolynomial R̃_p_i-1=R_0=1 has no variables, and 
the set _p_1-1=_0 is empty. The corresponding sequence ^(i) 
is then absent in , but its associated null value must be present, 
for any i=1,...,j-1, to preserve the total length ℓ. 
Following the rules of the algorithm given in Def. <ref>, 
we can therefore write 

    ( a_1 R̃^p_1-1_^(1)(a_2,...,a_p_1) ) ( a_p_1+1R̃^p_2-1_^(2)(a_p_1+2,...,a_P_2) ) 
    ⋯       
    ⋯( a_P_j-1+1R̃^p_j-1_^(j)(a_P_j-1+2,...,a_P_j) )       =    R_(a_1,...,a_ℓ) .


Let us call 
_ℓ = { = (j,^(1),0,^(2),0,...,^(j)) | 
1≤ j≤ℓ, ∈_ℓ^j, ^(i)∈_p_i-1  1≤ i≤ j }
the set of sequences obtained in this way. 
Then the equality 

    R_ℓ(a_1,a_2,...,a_ℓ) 
        = ∑_∈_ℓ R^ℓ_(a_1,...,a_ℓ)
    = R̃_ℓ(a_1,a_2,...,a_ℓ)

holds if we show that _ℓ = _ℓ. 

Let us first show that _ℓ⊂_ℓ. 
For fixed j,  and ^(1),...,^(j), we have 

    m_1+⋯ +m_ℓ    = j+∑_i=1^j (q^(i)_1+⋯ + q^(i)_p_i-1) 
        = j+(p_1-1)+(p_2-1)+⋯ +(p_j-1) 
        = j+ p_1+⋯ +p_j -j= ℓ.

For any h=1,...,ℓ, suppose that h belongs to the rth block, for 
some r≤ j, that is, 

h = P_r-1+1+k = p_1+⋯ +p_r-1+1+k, 

with 1≤ k≤ p_r-1. Then we have

    m_1+⋯ +m_h 
        = j+∑_i=1^r-1(q^(i)_1+⋯ + q^(i)_p_i-1) 
    + (q^(r)_1+⋯ + q^(r)_k)
    
        = j+(p_1-1)+(p_2-1)+⋯ +(p_r-1-1) 
    + (q^(r)_1+⋯ + q^(r)_k) 
        = (p_1+⋯ +p_r-1)+(q^(r)_1+⋯ +q^(r)_k)+(j-r)+1 
       ≥ P_r-1+1+k = h

because q^(r)_1+⋯ +q^(r)_k ≥ k and j-r≥ 0. 

Finally, let us show that there is a bijection between _ℓ and 
_ℓ. 
The set _ℓ is well known to be in bijection with the set PBT_ℓ+1 
of planar binary trees with ℓ+1 leaves (and a root). 
An explicit bijection Φ:_ℓ⟶ PBT_ℓ+1 is described
in <cit.>, Definition 2.16, using the  over and  under 
grafting operations on trees, namely 

t s = -.1st         
t s = -.1ts. 

The first values of Φ, for the empty sequence in _0 
and for (1)∈_1 and (2,0),(1,1)∈_2, are 
Φ( ) = ,     Φ(1) = , 
    Φ(2,0) = ,     Φ(1,1) = . 

So, for our purpose, it is enough to show that _ℓ is in bijection 
with PBT_ℓ+1. 
For this, since _ℓ⊂_ℓ, consider the map Φ 
restricted to _ℓ and let us show that the image Φ(_ℓ) 
coincides with PBT_ℓ+1. 

For a given sequence 
= (j,^(1),0,^(2),0,...,^(j))∈_ℓ, we have: 

  * 
The sequence  is decomposable as = (',^(j)) 
into the two well-defined sequences 
'=(j,^(1),0,^(2),0,...,^(j-1),0) ∈_P_j-1+1 
and ^(j)∈_p_j-1. 
In fact, if we set ℓ'=P_j-1+1, we have 

    m'_1+⋯ m'_ℓ'    = 
    j+(p_1-1)+⋯ +(p_j-1-1)= j+P_j-1-(j-1) = ℓ',

and for any h=1,...,ℓ' one can see that m'_1+⋯ m'_h≥ h 
with a computation similar to that used to show that ∈_ℓ. 
 
According to the definition of Φ, we then have 
Φ() = Φ(')Φ(^(j)). 
Graphically, if we denote the trees by t=Φ(), 
t'=Φ(') and t_j=Φ(^(j)), this means that 

t = -.1t't_j.  

  *  
The sequence ' is surely not decomposable because it is of the form 
'=(m”_1+1,m”_2,...,m”_ℓ”,0)
 
with 
”=(j-1,^(1),0,^(2),0,...,^(j-1))∈_ℓ”, 
    ℓ”=ℓ'-1. 
 
The sequence ” indeed belongs to _ℓ” for the same reason 
used to show that ∈_ℓ. 
Then, the sequence ' is not decomposable in position 1 because 
m'_1=m”_1+1 ≥ 2, and it is not decomposable in any position 
h=2,...,ℓ' because ”∈_ℓ” implies that 
m'_1+⋯ +m'_h = m”_1+1+m”_2+⋯ + m”_h ≥ h+1, and therefore 
surely m'_1+⋯ +m'_h ≠ h.  

Finally, according to the definition of Φ, we then have 
Φ(') = Φ(”). 
If we set t”=Φ(”), this means that 

t' = -.1t”
and therefore 

t = -.1t”t_j. 

  * 
The same arguments can be applied to the sequence ” and its new 
components, until we reach a full description of the tree t=Φ() 
in terms of the trees t_i = Φ(^(i)), for i=1,...,j, namely 

t = -.1t_1t_2t_j. 

Let us denote this tree by G^j(t_1,...,t_j). 

In conclusion, if we let j run from 1 to ℓ, we consider all possible 
sequences ∈^j_ℓ and for any i=1,...,j all trees 
t_i∈Φ(_p_i-1)= PBT_p_i, the result G^j(t_1,...,t_j) is 
any possible tree with number of leaves given by 
|G^j(t_1,...,t_j)| 
= 1+|t_1|+⋯ +|t_j| 
= 1+p_1+⋯ +p_j= ℓ+1.

In other words, we have 
Φ(_ℓ) = { t=G^j(t_1,...,t_j) | 
j=1,...,ℓ, ∈^j_ℓ, 
t_i ∈ PBT_p_i, 1≤ i≤ j }  = PBT_ℓ+1. 

For any ℓ≥ 0 and any a_1,...,a_ℓ+1∈ A, set n_i=|a_i|≥ 1 for 
i=1,...,ℓ+1.  
Then, for any sequence ∈_ℓ, we have 

    a_1 R_(a_2,...,a_ℓ+1) 
       = n_1+1m_1⋯n_ℓ+1m_ℓn_ℓ+1+10a_1 a_2⋯ a_ℓ+1 .

Therefore 

    a_1 R_ℓ(a_2,...,a_ℓ+1) 
       = d(n_1,...,n_ℓ+1) a_1 a_2⋯ a_ℓ+1 ,

where the Lagrange coefficients d(n_1,...,n_ℓ+1) are given in 
Def. <ref>. 


To describe the left codivision we introduce a last set of operators 
corresponding to the labeled Lagrange coefficients. 


For any ℓ≥ 1, let _ℓ be the set of sequences of bits
e_i∈{1,2}, as in Def. <ref>. 
We call  labeled right recursive operatorR^:T(A)⟶ T(A) the collection 
R^ = { R^_ℓ, ℓ≥ 0, ∈_ℓ}  
of (non homogeneous) linear operators R^_0=:⟶ and 

    R^_ℓ: 
    A^ℓ⟶⊕_λ=1^ℓ A^λ,
        ℓ≥ 1

defined recursively by

    R_0^(1)(a) = a          R_0^(2)(a)= 0,

and, for ℓ≥ 2 and for any = (e_1,...,e_ℓ)∈_ℓ, by

    R^_ℓ(a_1,...,a_ℓ)    = 
    ∑_j=1^ℓ∑_∈_ℓ^j( R_1^(e_1)(a_1)  R_p_1-1^(e_2,...,e_p_1)(a_2,...,a_p_1) ) 
    
       ( a_p_1+1
    R_p_2-1^(e_p_1+2,...,e_p_1+p_2)(a_p_1+2,...,a_p_1+p_2) ) 
    ⋯
       ⋯( a_p_1+⋯ +p_j-1+1 
    R_p_j-1^(e_p_1+⋯ +p_j-1+2,...,e_ℓ)
    (a_p_1+⋯ +p_j-1+2,...,a_ℓ) ) .

It turns out that R_ℓ^=R_ℓ if =(1,1,...,1). 
If  starts by 2, then R_ℓ^=0. 
If  starts by 1 and contains a bit value e_i=2 (in position i),
then R_ℓ^ is obtained from R_ℓ by removing the term which
contains the factor a_i-1 Q_i.
 

For instance, for ℓ=2, we have _1 = { (1,1),(1,2),(2,1),(2,2) }
and therefore 

    R_2^(1,1)(a,b)
        = R_1^(1)(a) R_1^(1)(b) + R_1^(1)(a)  (b  R_0(1)) 
    
        = a b + a  b  
    
     
    R_2^(1,2)(a,b)
        = R_1^(1)(a) R_1^(2)(b) + R_1^(1)(a)  (b  R_0(1))  
    
        = a  b   
    
     
    R_2^(2,1)(a,b)
        = R_1^(2)(a) R_1^(1)(b) + R_1^(2)(a)  (b  R_0(1)) = 0
    
     
    R_2^(2,2)(a,b) 
        = R_1^(2)(a) R_1^(2)(b) + R_1^(2)(a)  (b  R_0(1)) = 0.

For ℓ=3, the set _2 contains 8 sequences, which give

    R_3^(1,1,1)(a,b,c) 
        = a R_2^(1,1)(b,c) 
    + (a R_1^(1)(b)) c 
    + a (b R_1^(1)(c))
    + a b c 
    
        = a (b c) 
    + a (b c) 
    + (a b) c 
    + a  (b c)
    + a b c 
    
    
    R_3^(1,1,2)(a,b,c) 
        = a R_2^(1,2)(b,c) 
    + (a R_1^(1)(b)) c 
    + a (b R_1^(2)(c))
    + a b c 
    
        = a (b c) 
    + (a b) c 
    + a b c 
    
    
    R_3^(1,2,1)(a,b,c) 
        = a R_2^(2,1)(b,c) 
    + (a R_1^(2)(b)) c 
    + a (b R_1^(1)(c))
    + a b c 
    
        = a  (b c)
    + a b c 
    
    
    R_3^(1,2,2)(a,b,c) 
        = a R_2^(2,2)(b,c) 
    + (a R_1^(2)(b)) c 
    + a (b R_1^(2)(c))
    + a b c 
    
        = a b c

and finally  R_3^(2,1,1) = R_3^(2,1,2) = R_3^(2,2,1) = R_3^(2,2,2) =0.  


































































The labeled right operations can also be given by a closed formula. 


For any ℓ≥ 2, for any sequence ∈_ℓ and for any 
a_1,...,a_ℓ∈ A, we have

    R_ℓ^(a_1,...,a_ℓ) 
        = ∑_∈_ℓ^ R^ℓ_(a_1,...,a_ℓ).


As a consequence, if for a_1,...,a_ℓ+1∈ A we denote n_i=|a_i| 
for i=1,...,ℓ+1, we then have 

    a_1 R_ℓ^(a_2,...,a_ℓ+1)     = 
    d_ℓ^(n_1,...,n_ℓ) a_1⋯ a_ℓ+1∈ A,

where the labeled Lagrange coefficients d_ℓ^(n_1,...,n_ℓ) 
are given in Def. <ref>. 

If =(1,1,...,1), and if  starts by 2, there is nothing to prove.
Otherwise, for any value e_i=2 in , we obtain
R_ℓ^(a_1,...,a_ℓ) from R_ℓ(a_1,...,a_ℓ) by removing the term
containing the factor a_i-1 Q_i. 
By Lemma <ref>, such a term is associated to a sequence 
∈_ℓ, and by Def. <ref> the factor 
a_i-1 Q_i corresponds to a non-zero value m_i. 
Therefore, in order to remove such terms, it suffices to consider sequences 
 with m_i=0 whenever e_i=2. 

The co-operations of the Faà di Bruno coloop bialgebra can be equivalently 
defined in terms of the recursive operators as follows: 

    (x_n) 
        = x_n + y_n + ∑_ℓ=1^n-1∑_∈_n^ℓ+1x_n_1(y_n_2⋯ y_n_ℓ+1)  , 
    
    δ_r(x_n) 
        = u_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    L_ℓ(u_n_1,y_n_2,...,y_n_ℓ) y_n_ℓ+1
        = u_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    u_n_1 R_ℓ(y_n_2,...,y_n_ℓ+1) ,  
    
    δ_l(x_n) 
        = v_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1∑_∈_ℓ (-1)^ x_n_1^(e_1) 
    R_ℓ^(x^(e_2)_n_2,...,x^(e_ℓ)_n_ℓ,v_n_ℓ+1)

where we recall that u_n=x_n-y_n, v_n=y_n-x_n, and also that
(-1)^=(-1)^e_1+⋯ +e_ℓ-ℓ and that the bit value in x^(e)_n
tells us in which copy of  falls the generator x_n,
cf. Def. <ref>.

It follows from the definition of  given in Def. <ref>, 
the expression of R_ℓ given in Cor. <ref>, and that of 
R_ℓ^ given in Lemma <ref>. The equivalence of the 
presentations of the right codivision in terms of R_ℓ and L_ℓ 
is proved in Cor. <ref> in next section. 


Note that in the termx_n_1^(e_1) 
R_ℓ^(x^(e_2)_n_2,...,x^(e_ℓ)_n_ℓ,v_n_ℓ+1)of the left
codivision (<ref>),
the labeled operatorR_ℓ^=R_ℓ^(e_1,...,e_ℓ)is applied to variables
which are also labeled, but only by the lastℓ-1bits of. 
For instance, no labels affect the variables in 

    δ_l(x_2)
        = v_2 - x_1^(1) R_1^(1)(v_1) + x_1^(2) R_1^(2)(v_1)
    
        = v_2 - x_1  v_1 + y_1  0 
    
        = v_2 - 2x_1 v_1

but labels do affect the variables in 

    δ_l(x_3)
        = v_3 - ( x_1^(1) R_1^(1)(v_2) + x_2^(1) R_1^(1)(v_1) ) 
    - ( -x_1^(2) R_1^(2)(v_2) - x_2^(1) R_1^(2)(v_1) )
    
       
    + x_1^(1) R_2^(1,1)(x_1^(1),v_1) - x_1^(1) R_2^(1,2)(x_1^(2),v_1)
    - x_1^(2) R_2^(2,1)(x_1^(1),v_1) 
    
       
    + x_1^(2) R_2^(2,2)(x_1^(2),v_1) 
    
        = v_3 - (x_1  v_2 + x_2  v_1) + x_1  (x_1  v_1)
    + x_1  (x_1  v_1) - x_1  (y_1  v_1)
    
        = v_3 - ( 21 x_1 v_2 + 31 x_2 v_1 )
    + ( 2121+22) x_1^2 v_1 
    - 22 x_1 y_1 v_1 
    
        = v_3 - (2x_1 v_2 + 3 x_2 v_1) + 5 x_1^2 v_1 - x_1 y_1 v_1.


 §.§ Functoriality of the diffeomorphisms loop


To prove the main theorem of this section we need some preliminary recurrence 
relations for the recursive operators, and consequently for the Lagrange 
coefficients.
 
For any ℓ≥ 1 and any sequence (n_1,...,n_ℓ+1) of positive 
integers, the coefficients d_ℓ+1(a_1,...,a_ℓ) satisfy the 
following recursive equation: 

    d_ℓ(n_1,...,n_ℓ)    = ∑_j=1^ℓ∑_∈_ℓ^jn_1+1j d_p_1-1(n_2,...,n_p_1) d_p_2-1(n_p_1+2,...,n_p_1+p_2)⋯
       ⋯ d_p_j-1(n_p_1+⋯ +p_j-1+2,...,n_ℓ).

Applying a_1 ( ) to the recursive expression
(<ref>) of R_ℓ(a_2,a_3,...,a_ℓ+1),
and using Cor. <ref>, immediately gives the result. 

For any ℓ≥ 0 and any a_1,...,a_ℓ+1∈ A, the following recursive 
equation holds:  

    a_1  R_ℓ(a_2,...,a_ℓ+1)
       = ∑_i=0^ℓ-1 (-1)^ℓ-1-i( a_1  R_i(a_2,...,a_i+1) ) 
     (a_i+2⋯ a_ℓ+1 ).

Modulo the factor a_1 a_2⋯ a_ℓ+1, this means that 

    d_ℓ(n_1,...,n_ℓ) 
        = ∑_i=0^ℓ-1 (-1)^ℓ-1-in_1+⋯ + n_i+1+1ℓ-i d_i(n_1,...,n_i).

The two assertions are equivalent, and the second one appears as a recursion
for the coefficients in the non-commutative Lagrange inversion formula. 
It is essentially based on the Chu-Vandermonde identity and can be 
proved[
We warmly thank Jiang Zeng for pointing out this method to us.] 
using the hypergeometric function _2F_1 or using some trick as in 
<cit.>, Lemma 2.15. 

For any ℓ≥ 0 and any a_1,...a_ℓ+1∈ A we have 

    a_1  R_ℓ(a_2,...,a_ℓ+1)    = L_ℓ(a_1,...,a_ℓ)  a_ℓ+1

By induction on ℓ. For ℓ=0,1 the identity is easily verified, 
because 

    a_1 1     = a_1 = 1  a_1 
    
     
    a_1  R_1(a_2)     = a_1  a_2 = L_1(a_1)  a_2.

Now suppose that for i=1,...,ℓ-1 we have 
a_1  R_i(a_2,...,a_i+1) = L_i(a_1,...,a_i)  a_i+1. 
Then by Lemma <ref> and Def. <ref> we have

    a_1  R_ℓ(a_2,...,a_ℓ+1)
       = ∑_i=0^ℓ-1 (-1)^ℓ-1-i( a_1  R_i(a_2,...,a_i+1) ) 
     (a_i+2⋯ a_ℓ+1 ) 
    
        = ∑_i=0^ℓ-1 (-1)^ℓ-1-i( L_i(a_1,...,a_i) a_i+1) 
     (a_i+2⋯ a_ℓ+1 ) 
    
        = ∑_i=0^ℓ-1 (-1)^ℓ-1-i( ( L_i(a_1,...,a_i) a_i+1) 
     a_i+2⋯ a_ℓ)  a_ℓ+1
        = L_ℓ(a_1,...,a_ℓ)  a_ℓ+1 .

In the case ≠ (1,1,...,1), whether there exists an operator 
L_ℓ^ :A^ℓ⟶ T(A) such that 

a_1 R_ℓ^(a_2,...,a_ℓ+1) = L_ℓ^(a_1,...,a_ℓ) a_ℓ+1
is an open question. 

For any ℓ≥ 1 and any a_1,...,a_ℓ+1∈ A, the following recursive 
equation holds:  

    a_1  R_ℓ(a_2,...,a_ℓ+1)
       = ∑_i=1^ℓ (-1)^i-1( a_1  (a_2⋯ a_i+1) ) 
     R_ℓ-i(a_i+2⋯ a_ℓ+1 ) .

Modulo the factor a_1 a_2⋯ a_ℓ+1, and if we call n_i=|a_i| for 
i=1,...,ℓ+1, this means that 

    d_ℓ(n_1,...,n_ℓ) 
        = ∑_i=1^ℓ (-1)^i-1n_1+1i d_ℓ-i(n_1+⋯+ n_i+1,n_i+2,...,n_ℓ).

The two assertions are equivalent. Let us prove the second one by induction 
on ℓ. 
Let us call d̃_ℓ(n_1,...,n_ℓ) the right-hand side of 
equation (<ref>). 
For ℓ=1, the sum in d̃_1(n) has only one term for i=1,
which gives

    d̃_1(n)     = (-1)^1-1n+11 d_0 = d_1(n).

Now suppose that eq. (<ref>) holds for any 
1≤ k≤ℓ-1, that is, we have

    d_k(n_1,...,n_k) 
        = ∑_i=1^k (-1)^i-1n_1+1i d_k-i(n_1+⋯+ n_i+1,n_i+2,...,n_k),

and prove it for ℓ. For this, we write d_ℓ(n_1,...,n_ℓ) 
using the recursion given in Lemma <ref> as a sum over 
0≤ k≤ℓ-1, and separate the term k=0 to which we can not apply 
the inductive hypothesis. 
Then we expand the factor d_k(n_1,...,n_k) using the inductive 
hypothesis and exchange the sums over k and i. We finally obtain 

    d_ℓ(n_1,...,n_ℓ) 
        = ∑_k=1^ℓ-1 (-1)^ℓ-1-kn_1+⋯+ n_k+1+1ℓ-k d_k(n_1,...,n_k) 
    + (-1)^ℓ-1n_1+1ℓ
        
    = ∑_i=1^ℓ-1 (-1)^i-1n_1+1i ∑_k=i^ℓ-1 (-1)^ℓ-1-kn_1+⋯+ n_k+1+1ℓ-k 
    d_k-i(n_1+⋯+ n_i+1,n_i+2,...,n_k)
    
        
    + (-1)^ℓ-1n_1+1ℓ .

Then, d_ℓ(n_1,...,n_ℓ) is equal to 

    d̃_ℓ(n_1,...,n_ℓ) 
        = ∑_i=1^ℓ-1 (-1)^i-1n_1+1i 
    d_ℓ-i(n_1+⋯+ n_i+1,n_i+2,...,n_ℓ)
    
       
    + (-1)^ℓ-1n_1+1ℓ ,

if and only if, for any 1≤ i≤ℓ-1, we have 

    ∑_k=i^ℓ-1 (-1)^ℓ-1-kn_1+⋯+ n_k+1+1ℓ-k 
    d_k-i(n_1+⋯+ n_i+1,n_i+2,...,n_k)
       
        
    = d_ℓ-i(n_1+⋯+ n_i+1,n_i+2,...,n_ℓ) .

This identity is easily verifyed by setting j=k-i, 
p_1 = n_1+⋯ + n_i+1 and p_j= n_i+j for 2≤ j≤ℓ-1-i, 
since it gives

    ∑_j=0^ℓ-i-1 (-1)^ℓ-i-1-jp_1+⋯+ p_j+1+1ℓ-i-j 
    d_j(p_1,p_2,...,p_j)
        = d_ℓ-i(p_1,p_2,...,p_ℓ-i)

which holds again by Lemma <ref>. 

For any ℓ≥ 1 and any a_1,...,a_ℓ∈ A, the following recursive 
equation holds:  

    L_ℓ(a_1,...,a_ℓ) 
       = ∑_i=1^ℓ-1 (-1)^i-1 
    L_ℓ-i(a_1  (a_2⋯ a_i+1),a_i+2,...,a_ℓ) 
    + (-1)^ℓ-1 a_1⋯ a_ℓ .

It suffices to write 

    a_1  R_ℓ(a_2,...,a_ℓ+1)
       = ∑_i=1^ℓ-1 (-1)^i-1( a_1 (a_2 ⋯ a_i+1) ) 
     R_ℓ-i(a_i+2⋯ a_ℓ+1 ) 
    
        
    + (-1)^ℓ a_1  (a_2⋯ a_ℓ+1)

after Lemma <ref>, and to apply the equality 
b_1 R_j(b_2,...,b_j+1) = L_j(b_1,...,b_j) b_j+1 
everywhere. 

For any ℓ≥ 2, any ∈_ℓ and any a_1,...,a_ℓ∈ A, 
we have

    R_ℓ^(a_1,...,a_ℓ) 
        = R_1^(e_1)(a_1)  R_ℓ-1^(e_2,...,e_ℓ)(a_2,...,a_ℓ)
    
       
    + ∑_i=1^ℓ-1 R_i^(e_1,...,e_i)(a_1,...,a_i) (a_i+1 R_ℓ-i-1^(e_i+2,...,e_ℓ)(a_i+2,...,a_ℓ)) .

The term j=1 in the defining recursion (<ref>) gives 
exactly 

R_1^(e_1)(a_1)  R_ℓ-1^(e_2,...,e_ℓ-1)(a_2,...,a_ℓ), 

se it remains to prove that 

    ∑_j=2^ℓ∑_∈_ℓ^j( R_1^(e_1)(a_1)  R_p_1-1^(e_2,...,e_p_1)(a_2,...,a_p_1) ) 
    
       ( a_p_1+1 R_p_2-1^(e_p_1+2,...,e_p_1+p_2)(a_p_1+2,...,a_p_1+p_2) ) 
    ⋯
       ⋯( a_p_1+⋯ +p_j-1+1 
    R_p_j-1^(e_p_1+⋯ +p_j-1+2,...,e_ℓ)
    (a_p_1+⋯ +p_j-1+2,...,a_ℓ) ) 
    = 
    
       
    = ∑_i=1^ℓ-1 R_i^(e_1,...,e_i)(a_1,...,a_i) (a_i+1 R_ℓ-i-1^(e_i+2,...,e_ℓ)(a_i+2,...,a_ℓ)) .

Let us prove this identity by induction. For ℓ=2 and 3, it is easy to
verify on the above examples that 

    R_2^(a,b) 
        = R_1^(e_1)(a)  R_1^(e_2)(b) 
    + R_1^(e_1)(a) (b R_0(1)), 
    
    
    R_3^(a,b,c) 
        = R_1^(e_1)(a)  R_2^(e_2,e_3)(b,c) 
    + R_1^(e_1)(a) (b R_1^(e_3)(c))
    + R_2^(e_1,e_2)(a,b) (c  R_0(1)).

Now suppose it holds up to order ℓ-1, and let us prove it at order ℓ. 

Consider the left-hand side of eq. (<ref>). Since j≥ 2, 
we can write 
^j_ℓ = ⋃_i=j-1^ℓ-1^j-1_i ×^1_ℓ-i
and decompose ∈^j_ℓ into (p_1,...,p_j-1)∈^j-1_i 
and (p_j)∈^1_ℓ-i for any value i=j-1,...,ℓ-1. 
We then have p_1+⋯ +p_j-1=i and p_j=ℓ-(p_1+⋯ +p_j-1)=ℓ-i. 
Therefore the left-hand side can be written as 

    ∑_j=2^ℓ∑_i=j-1^ℓ-1∑_(p_1,...,p_j-1)∈_i^j-1δ_e_1,1 ( a_1  R_p_1-1^(e_2,...,e_p_1)(a_2,...,a_p_1) ) 
    ⋯
       ⋯( a_P_j-2+1 R_p_j-1-1^(e_p_j-2+2,...,e_p_j-1)(a_P_j-2+2,...,a_i) ) 
       ( a_i+1 
    R_ℓ-i-1^(e_i+2,...,e_ℓ)(a_i+2,...,a_ℓ) ) = 
    
       
    = ∑_i=1^ℓ-1 ( ∑_k=1^i ∑_(p_1,...,p_k)∈_i^kδ_e_1,1 ( a_1  R_p_1-1^(e_2,...,e_p_1-1)(a_2,...,a_p_1) ) 
    ⋯. 
    
       . ⋯( a_P_k-1+1 R_p_k-1^(e_P_k-1+2,...,e_P_k)(a_P_k-1+2,...,a_i) ) ) 
       ( a_i+1 
    R_ℓ-i-1^(e_i+2,...,e_ℓ)(a_i+2,...,a_ℓ) ),

where P_k=p_1+⋯ +p_k. 
Applying the inductive hypothesis to the sum over k leads to the result. 

The associative algebra  is indeed a coloop bialgebra and 
represents the loop of formal diffeomorphisms  as a functor 
:⟶. 

As a consequence, given an associative algebra A, a series 
a=∑_n≥ 0 a_n λ^n+1∈(A) can be seen as an algebra 
homomorphism a:⟶ A defined on the generators 
of  by a(x_n)=a_n, and the right and left division a b and 
a\ b are given at any order n by the following closed formulas: 

    (a b)_n     = μ_A (a b) δ_r(x_n) 
    
        = a_n-b_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    d_ℓ(n_1,...,n_ℓ) 
    (a_n_1-b_n_1)  b_n_2⋯ b_n_ℓ+1 , 
    
     
    (a\ b)_n     = μ_A (a b) δ_l(x_n) 
    
        = b_n-a_n + ∑_ℓ=2^n-1 (-1)^ℓ∑_∈_n^ℓ+1∑_∈_ℓ (-1)^ d_ℓ^(n_1,...,n_ℓ) 
    a_n_1 c^(e_1)_n_2⋯ c^(e_ℓ-1)_n_ℓ  (b_n_ℓ+1-a_n_ℓ+1) ,

where c^(e_i)_n_i= a_n_i if e_i=1 and c^(e_i)_n_i= b_n_i if e_i=2.  

The free associative algebra  clearly represents the sets 
(A) over associative algebras A, and the comultiplication  
is just the Faà di Bruno comultiplication  seen with values in 
 instead of ⊗, therefore 
it clearly represents the loop law given in Definition 
<ref>. 
Thus, the theorem is proved if we show that  is indeed a coloop 
bialgebra. 

The comultiplication  satisfies the compatibility relation with 
the standard counit, because  does, and coassociativity is not required. 
So it remains to check that the codivisions δ_r and δ_l 
given in Def. <ref> satisfy the identities 
(<ref>) and (<ref>). Since these maps 
are algebra morphisms, it suffices to verify these identities on the 
generators x_n, for any n≥ 1. 
 
i) Let us start with the right codivision and show that it satisfies  
the first identity (<ref>), namely
 
(μ)   (δ_r) (x_n) = x_n,
 
which explicitely gives the recurrence (with u_n=x_n-y_n) 

    δ_r(x_n) 
       = u_n -∑_m=1^n-1∑_ℓ=1^n-m∑_∈_n-m^ℓδ_r(x_m)(y_k_1⋯ y_k_ℓ).

Expanding δ_r(x_n) in terms of the left recursive opeators, 
this equation becomes

    u_n + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    L_ℓ(u_n_1,y_n_2,...,y_n_ℓ)  y_n_ℓ+1
       
    = u_n -∑_m=1^n-1∑_j=1^n-m∑_∈_n-m^j∑_i=0^m-1 (-1)^i ∑_∈_m^i+1( L_i(u_p_1,y_p_2,...,y_p_i)  y_p_i+1) 
    (y_q_1⋯ y_q_j) 
    
       
    = u_n +∑_ℓ=i+j=1^n-1∑_i=0^ℓ-1 (-1)^i+1∑_m=i+1^n-1∈_n-m^ℓ-i∑_∈_m^i+1( L_i(u_p_1,y_p_2,...,y_p_i)   y_p_i+1) 
    (y_q_1⋯ y_q_ℓ-i) .

Now, since 
⋃_m=i+1^n-1_m^i+1×_n-m^ℓ-i≅_n^ℓ+1, 

let us call =(,), that is,

(n_1,n_2,...,n_ℓ+1) = (p_1,...,p_i+1,q_1,...,q_j). 

Then, the recursion (<ref>)
is equivalent, for any n≥ 1, any 1≤ℓ≤ n-1 and any 
∈_n^ℓ+1, to the equation

    L_ℓ(u_n_1,y_n_2,...,y_n_ℓ)  y_n_ℓ+1
       
    = ∑_i=0^ℓ-1 (-1)^ℓ-1-i( L_i(u_n_1,y_n_2,...,y_n_i)   y_n_i+1) 
     (y_n_i+2⋯ y_n_ℓ+1 ) 
    
       
    = ∑_i=0^ℓ-1 (-1)^ℓ-1-i( ( L_i(u_n_1,y_n_2,...,y_n_i)   y_n_i+1) 
     y_n_i+2⋯ y_n_ℓ) 
     y_n_ℓ+1 ,

which holds by definition of L_ℓ. 

The second identity (<ref>), namely 

(μ)   () δ_r(x_n) = x_n,   
 
is better developed using the expansion over the right recursive operators, 
and explicitely gives the recurrence 

    ∑_ℓ=1^n-1∑_∈^ℓ+1_n (-1)^ℓ 
    x_n_1 R_ℓ(y_n_2,...,y_n_ℓ+1) 
    
        
    = ∑_m=1^n-1∑_p=1^n-m∑_i=1^p∑_∈^i_p∑_j=0^n-m-p∑_∈^j_n-m-p (-1)^j+1(x_m (y_p_1⋯ y_p_i)) 
     R_j(y_q_1,...,y_q_j) .

Rewriting the sums in terms of m=1,...,n-1, ℓ=i+j=1,...,n-m, 
i=1,...,ℓ and j=ℓ-i, this gives a sum over 
∈_p^i and ∈_n-m-p^ℓ-i for 
p=i,...,n-m. That is, we get a sum over =(,)∈_n-m^ℓ
and consequently a sum over =(m,)∈_n^ℓ+1: 

    ∑_ℓ=1^n-1∑_∈^ℓ+1_n (-1)^ℓ 
    x_n_1 R_ℓ(y_n_2,...,y_n_ℓ+1) 
    
        
    = ∑_m=1^n-1∑_ℓ=1^n-m∑_i=1^ℓ∑_p=i^n-m∑_∈^i_p∑_∈^ℓ-i_n-m-p 
    (-1)^ℓ-i+1(x_m (y_p_1⋯ y_p_i)) 
     R_ℓ-i(y_q_1,...,y_q_ℓ-i) 
    
        
    = ∑_m=1^n-1∑_ℓ=1^n-m∑_∈^ℓ_n-m∑_i=1^ℓ (-1)^ℓ-i+1(x_m (y_k_1⋯ y_k_i)) 
     R_ℓ-i(y_k_i+1,...,y_k_ℓ) 
    
        
    = ∑_ℓ=1^n-1∑_∈^ℓ+1_n (-1)^ℓ∑_i=1^ℓ (-1)^i-1(x_n_1 (y_n_2⋯ y_n_i+1)) 
     R_ℓ-i(y_n_i+2,...,y_n_ℓ+1) .

Therefore, for any n≥ 2, any ℓ=1,..., n-1 and any sequence 
∈^ℓ+1_n, eq. (<ref>)
is equivalent to the recurrence equation 

    x_n_1 R_ℓ(y_n_2,...,y_n_ℓ+1) 
        = ∑_i=1^ℓ (-1)^i-1(x_n_1 (y_n_2⋯ y_n_i+1)) 
     R_ℓ-i(y_n_i+2,...,y_n_ℓ+1),

which is proved in Lemma <ref>. 
 
ii) Let us show now that the left codivision given in 
Def. (<ref>) satisfies the identities 
(<ref>). 
The first identity (<ref>), namely 
 
(μ)   (δ_l) (x_n) = y_n,
 
explicitely gives the recurrence (with v_n=y_n-x_n)

    δ_l(x_n)  
       = v_n - ∑_m=1^n-1 ∑_λ=1^n-m ∑_∈_n-m^λ 
    x_m(δ_l(y_k_1)⋯δ_l(y_k_λ)),

where y_k is just the kth generator x_k in the second copy of the free 
product algebra , therefore the formula for 
δ_l(y_k) is just the same as for δ_l(x_k). 

To show this, we consider the expansion
(<ref>) of δ_l(x_n) given in
Thm. <ref>. Since R_ℓ^=0 when e_1=2,
we have x_n_1^(e_1)=x_n_1 and we can rewrite
(<ref>) as 

    δ_l(x_n)  
        = v_n + ∑_m=1^n-1 x_m∑_ℓ=1^n-m (-1)^ℓ∈_ℓ∑_∈_n-m^ℓ (-1)^  
    R_ℓ^(x^(e_2)_n_1,...,x^(e_ℓ)_n_ℓ-1,v_n_ℓ).

Then eq. (<ref>) is clearly verified for n=1, 
because δ_l(x_1)=v_1, and for any n≥ 2 and any m=1,...,n-1, it is 
equivalent to the equation

    ∑_λ=1^μ∑_∈_μ^λδ_l(x_k_1)⋯δ_l(x_k_λ) 
        = 
    
       
    = v_μ - ∑_ℓ=2^μ (-1)^ℓ∑_∈_μ^ℓ ∑_∈_ℓ (-1)^ 
    R_ℓ^(x^(e_2)_n_1,...,x^(e_ℓ)_n_ℓ-1,v_n_ℓ),

for any μ=n-m=1,...,n-1.

Let us prove this equation by induction on μ. 
For μ=1 we again have δ_l(x_1)=v_1. 
So, suppose that eq. (<ref>) holds up to order 
μ-1 and prove it at order μ. 

On the left-hand side of eq. (<ref>), 
we separate the term λ=1 and observe that, for λ≥ 2, 
we can decompose =(k_1,...,k_λ-1,k_λ)∈_μ^λ 
into (,ν) ∈_μ-ν^λ-1×_ν^1 with 

    q_i    = k_i     
    ν   = k_λ .

Since 
_μ^λ 
= ⋃_ν=1^μ-λ+1_μ-ν^λ-1×_ν^1, 

the left-hand side of eq. (<ref>) 
can then be written as 

    δ_l(x_μ) + ∑_ν=1^μ-1( ∑_i=1^μ-ν∑_∈_μ-ν^iδ_l(x_q_1)⋯δ_l(x_q_i) ) 
    δ_l(x_ν) .

We then apply the inductive hypothesis (<ref>)
to the sum over i=1,...,μ-ν, and expand the single factors
δ_l(x_μ) and δ_l(x_ν) as in
(<ref>), thus obtaining

    ∑_λ=1^μ∑_∈_μ^λδ_l(x_k_1)⋯δ_l(x_k_λ) 
        = 
    
       
    = v_μ + ∑_λ=1^μ-1 (-1)^λ∑_∈_μ^λ+1∑_∈_λ (-1)^ 
    x^(e_1)_n_1 R_λ^(x^(e_2)_n_2,...,x^(e_λ)_n_λ,v_n_λ+1)
    
       
    + ∑_ν=1^μ-1( 
    v_μ-ν - ∑_i=2^μ-ν (-1)^i ∑_∈_μ-ν^i∑_'∈_i (-1)^' 
    R_i^'(x^(e_2')_p_1,...,x^(e_i')_p_i-1,v_p_i) 
    ) 
       ( 
    v_ν + ∑_j=1^ν-1 (-1)^j ∑_∈_ν^j+1∑_”∈_j (-1)^” 
    x^(e_1”)_q_1 R_j^”(x^(e_2”)_q_2,...,x^(e_j”)_q_j,v_q_j+1) 
    ) .

Finally, it remains to prove that the right-hand side of
eq. (<ref>) coincides with the right-hand side of
eq. (<ref>). The first term v_μ appears
in both formulas, let us compare the other terms.

The first term in eq. (<ref>) is

    A     = ∑_λ=1^μ-1 (-1)^λ∑_∈_μ^λ+1∑_”∈_λ (-1)^” x_n_1^(e_1”)
    R_λ^”(x^(e_2”)_n_2,...,x^(e_λ”)_n_λ,v_n_λ+1) .

We apply the trick

    a     = R_1^(1)(a) = ∑_(e')∈_1 R_1^(e')(a)

to the element x_n_1^(e_1”), then set ℓ = λ+1 and
= (e',”)=(e',e_1”,...,e_λ”), and get 

    A     = - ∑_ℓ=2^μ (-1)^ℓ∑_∈_μ^ℓ∑_∈_ℓ (-1)^ ( R_1^(e_1)(x^(e_2)_n_1) 
    R_ℓ-1^(e_2,...,e_ℓ)(x^(e_3)_n_2,...,x^(e_ℓ)_n_ℓ-1,v_n_ℓ) 
    ).

The second term in eq. (<ref>) is

    B     = ∑_ν=1^μ-1 v_μ-ν v_ν.

We apply the second trick

    v_n    = -(x^(1)_n-x^(2)_n) = -∑_(e')∈_1 (-1)^e'-1 x^(e')_n
    
        = -∑_=(e_1,e_2)∈_2 (-1)^ R_1^(e_1)(x^(e_2)_n)

to the element v_μ-ν, and get 

    B     = - ∑_ν=1^μ-1 ∑_∈_2 (-1)^ 
    R_1^(e_1)(x^(e_2)_n)  v_ν
        = - ∑_∈_μ^2 ∑_∈_2 (-1)^ 
    R_1^(e_1)(x^(e_1)_n_1)  v_n_2.

Using again (<ref>), the third term in
eq. (<ref>) becomes 

    C     = ∑_ν=1^μ-1∑_j=2^ν-1 (-1)^j ∑_∈_ν^j+1∑_”'∈_j (-1)^”' v_μ-ν(x^(e_1”')_q_1
    R_j^”'(x^(e_2”')_q_2,...,x^(e_j”')_q_j,v_q_j+1))
    
        = - ∑_ν=1^μ-1∑_j=2^ν-1 (-1)^j ∑_∈_ν^j+1”'∈_j”∈_1∑_'∈_1
    (-1)^' (-1)^” (-1)^”'R_1^(e')(x^(e”)_μ-ν)(x^(e_1”')_q_1
    R_j^”'(x^(e_2”')_q_2,...,x^(e_j”')_q_j,v_q_j+1)).

We set ℓ=1+1+j, =(μ-ν,)∈_μ^ℓ and
=(e',e”,”')∈_ℓ, and obtain 

    C     = - ∑_ℓ=3^μ-1 (-1)^ℓ∑_∈_μ^ℓ∑_∈_ℓ (-1)^  
    R_1^(e_1)(x^(e_2)_n_1)(x_n_2^(e_3) 
    R_ℓ-2^(e_3,...,e_ℓ)(x^(e_4)_n_3,...,x^(e_ℓ)_n_ℓ-1,v_n_ℓ) 
    ) .

The fourth term in eq. (<ref>) is

    D     = - ∑_ν=1^μ-1∑_i=2^μ-ν (-1)^i ∑_∈_μ-ν^i∑_'∈_i (-1)^' 
    R_i^'(x^(e_2')_p_1,...,x^(e_i')_p_i-1,v_p_i)  v_ν.

We write v_p_i=-∑_”∈_1 (-1)^” x^(e”)_n using
(<ref>), and set ℓ=i+1, =(,ν)∈_μ^ℓ
and =(',”)∈_ℓ. Then we have 

    D     = - ∑_ℓ=3^μ (-1)^ℓ∑_∈_μ^ℓ ∑_∈_ℓ (-1)^ ∑_i=2^ℓ-1
    R_i^(e_1,...,e_i)(x^(e_2)_n_1,...,x^(e_ℓ-1)_n_i)  v_ℓ.

With similar manipulations, setting ℓ=i+j+1 and
=(,)∈ C_μ^ℓ, the last term in
eq. (<ref>) is

    E     = - ∑_ν=1^μ-1∑_i=2^μ-ν∑_j=2^ν-1 (-1)^i+j∑_∈_μ-ν^i∑_∈_ν^j+1∑_'∈_i∑_”'∈_j  (-1)^' (-1)^”'
       
    R_i^'(x^(e_2')_p_1,...,x^(e_i')_p_i-1,v_p_i) (x_q_1 R_j^”'(x^(e_2”')_q_2,...,x^(e_j”')_q_j,v_q_j+1)) 
    
        = - ∑_ℓ=4^μ (-1)^ℓ∑_∈_μ^ℓ∑_∈_ℓ (-1)^∑_i=2^ℓ-2 
    R_i^(e_1,...,e_i)(x^(e_2)_n_1,...,x^(e_i+1)_n_i) 
       (x^(e_i+2)_n_i+1 
    R_ℓ-i-1^(e_i+2,...,e_ℓ)(x^(e_i+3)_n_i+2,...,x^(e_ℓ)_n_ℓ-1,
    v_n_ℓ)) .

We now observe that the sum B extends C to the value ℓ=2, and that
D extends E to the value ℓ=3. Alltogether, we have  

    B+C+D+E 
        = - ∑_ℓ=3^μ (-1)^ℓ∑_∈_μ^ℓ∑_∈_ℓ (-1)^∑_i=1^ℓ-1 
    R_i^(e_1,...,e_i)(x^(e_2)_n_1,...,x^(e_i+1)_n_i) 
       (x^(e_i+2)_n_i+1 R_ℓ-i-1^(e_i+2,...,e_ℓ)
    (x^(e_i+3)_n_i+2,...,x^(e_ℓ)_n_ℓ-1,v_n_ℓ)) .

Therefore, eq. (<ref>) is then equivalent, 
for any ℓ≥ 3, any ∈_ν^ℓ and any ∈_ℓ, 
to the following recursion 

    R_ℓ^(x^(e_2)_n_1,...,x^(e_ℓ)_n_ℓ-1,v_n_ℓ) 
        = R_1^(e_1)(x^(e_2)_n_1) 
    R_ℓ-1^(e_2,...,e_ℓ)(x^(e_3)_n_2,...,x^(e_ℓ)_n_ℓ-1,v_n_ℓ) 
    
       
    + ∑_i=1^ℓ-1 
    R_i^(e_1,...,e_i)(x^(e_2)_n_1,...,x^(e_i+1)_n_i) 
    (x^(e_i+2)_n_i+1 R_ℓ-i-1^(e_i+2,...,e_ℓ)
    (x^(e_i+3)_n_i+2,...,x^(e_ℓ)_n_ℓ-1,v_n_ℓ)) ,

which is proved in Lemma <ref>. 
 

The second identity (<ref>), namely 

(μ)   () δ_l(x_n) = y_n,   
 
can not be expressed as a recurrence on R_ℓ^, because these operators 
do not show up explicitely to which factor of  
the variables belong. 
Then, let us use the recursion (<ref>) to 
describe δ_l and prove the second identity by induction on n.  

The identity is verified for n=1 because we have 

    (μ)   () δ_l(x_1) 
       = x_1+y_1-x_1 = y_1.

Then, suppose it holds up to the degree n-1
Since  and μ are algebra homomorphisms, if we apply the 
operator D=(μ)   () to the expression 

    δ_l(x_n)  
       = v_n - ∑_m=1^n-1 ∑_λ=1^n-m ∑_∈_n-m^λ 
    x_m(δ_l(y_k_1)⋯δ_l(y_k_λ)),

we obtain, for D(δ_l(x_n)), the sum of 

    D(v_n) 
        = y_n + ∑_m=1^n-1 ∑_λ=1^n-m ∑_∈_n-m^λ 
    x_m (y_k_1⋯ y_k_ℓ)

and of 

    - ∑_m=1^n-1 ∑_λ=1^n-m ∑_∈_n-m^λ 
    x_m( D(δ_l(y_k_1)) ⋯ D(δ_l(y_k_λ))).

Therefore the second identity is satisfied if, for any m=1,...,n-1, any 
λ=1,...,n-m and any ∈_n-m^λ, we have 

    D(δ_l(y_k_1)) ⋯ D(δ_l(y_k_λ)
        = y_k_1⋯ y_k_ℓ,

which is true by inductive hypothesis. 


 §.§ Properties of the diffeomorphisms loop

The coloop bialgebra  has a two-sided antipode S such that 

    δ_r     = ( S) ,

while the identity δ_l = (S ) does not hold.
Moreover, the antipode in the Faà di Bruno coloop bialgebra coincides with 
that in the non-commutative Faà di Bruno Hopf algebra given in <cit.>, 
that is, 

    S(x_n) 
        = - ∑_ℓ=0^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    d_ℓ+1(n_1,...,n_ℓ+1) x_n_1  x_n_2⋯ x_n_ℓ+1 .

i) In a coloop bialgebra, the left and right antipodes are given respectively 
by 

S_l=(ε) δ_l
         
S_r=(ε) δ_r, 

cf. (<ref>). 
Let us show that for  these two operators coincide, and therefore 
the two-sided antipode is well defined by S:=S_l=S_r. 

Indeed, let us fix n≥ 1. For the right antipode we have

    S_r(x_n) 
        = ∑_ℓ=0^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    d_ℓ+1(n_1,...,n_ℓ+1) (ε(x_n_1)-y_n_1)  y_n_2⋯ y_n_ℓ+1
        = - ∑_ℓ=0^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    d_ℓ+1(n_1,...,n_ℓ+1) x_n_1  x_n_2⋯ x_n_ℓ+1 ,

where n_1>0 implies ε(x_n_1)=0, and where we renamed 
the variables y as x because S_r takes values in . 
For the left antipode we have 

    S_l(x_n) 
        = ε(y_n)-x_n 
    - ∑_∈_n^2 d_1(n_1) x_n_1(ε(y_n_2)-x_n_2)
    
       
    + ∑_ℓ=2^n-1 (-1)^ℓ∑_∈_n^ℓ+1∑_∈_ℓ-1 (-1)^ d_ℓ+1^(n_1,...,n_ℓ+1) 
    (ε) (x_n_1 x^(e_1)_n_2⋯ x^(e_ℓ-1)_n_ℓ  
    v_n_ℓ+1).

Since ε kills the terms where some y appears, in the sum over 
the sequences ∈_ℓ-1 there only remains the sequence 
=(1,1,...,1), for which d_ℓ+1^=d_ℓ+1 and (-1)^=+, 
and therefore we have 

    S_l(x_n)     = -x_n + ∑_∈_n^2 d_1(n_1) x_n_1 x_n_2
       
    - ∑_ℓ=2^n-1 (-1)^ℓ∑_∈_n^ℓ+1 d_ℓ+1(n_1,...,n_ℓ+1) x_n_1 x_n_2⋯ x_n_ℓ+1
        = S_r(x_n).


ii) Let us now prove the identity δ_r = ( S).  
For any generator x_n of , we have

    ( S) (x_n) 
        = x_n + S(y_n) 
    + ∑_j=1^n-1∑_∈_n^j+1n_1+1j 
    x_n_1 S(y_n_2)⋯ S(y_n_j+1) 
    
       
    = x_n - y_n - ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    d_ℓ+1(n_1,...,n_ℓ+1) y_n_1  y_n_2⋯ y_n_ℓ+1
       
    + ∑_j=1^n-1∑_∈_n^j+1∑_p_1=1^n_2⋯∑_p_j=1^n_j+1 
    (-1)^p_1+⋯ +p_j∑_^1∈_n_2^p_1⋯∑_^j∈_n_j+1^p_jn_1+1j d_p_1(q^1_1,...,q^1_p_1) ⋯
       ⋯ d_p_j(q^j_1,...,q^j_p_j)x_n_1 y_q^1_1⋯ y_q^1_p_1⋯ y_q^j_1⋯ y_q^j_p_j .

Set ℓ=p_1+⋯ + p_j, then 1≤ j ≤ℓ≤ n-1. 
Since 
⋃_n_1=1^n-ℓ ⋃_p_1=1^n_2⋯⋃_p_j=1^n_j+1 
C^1_n_1× C^p_1_n_2×⋯× C^p_j_n_j+1 = C^ℓ+1_n,

because n_1+n_2+⋯ + n_j+1=n, if we rename the sequence 
(n_1,q^1_1,...,q^1_p_1,...,q^j_1,...,q^j_p_j) as ∈ C^ℓ+1_n, 
the sum over j becomes

    ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1( 
    ∑_j=1^ℓ∑_∈_ℓ^jn_1+1j d_p_1(n_2,...,n_p_1+1) ⋯ 
    d_p_j(n_P_j-1+2,...,n_ℓ+1))  
    x_n_1 y_n_2⋯ y_n_ℓ+1,

where Pi=p_1+⋯ +p_i for i=1,...,j. 
Using the recurrence proved in Corollary <ref>, 
we finally obtain 

    ( S) (x_n) 
        = x_n - y_n - ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    d_ℓ+1(n_1,...,n_ℓ+1) y_n_1  y_n_2⋯ y_n_ℓ+1
        
    + ∑_ℓ=1^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    d_ℓ+1(n_1,...,n_ℓ+1) x_n_1  y_n_2⋯ y_n_ℓ+1
        = δ_r(x_n).


iii) The first counterexample to the analogue identity 
δ_l = (S ) is on the generator x_3, 
for which we have 

    (S ) (x_3)    = 
    v_3 - (2x_1v_2 + 3x_2v_1) + 5x_1^2v_1 - x_1v_1y_1 ,

where v_n=y_n-x_n, while 

    δ_l(x_3)    = 
    v_3 - (2x_1v_2 + 3x_2v_1) + 5x_1^2v_1 - x_1y_1v_1 .


This result allows us on one side to deduce some properties of the loop of
formal diffeomorphisms, and on the other side to compare the Faà di Bruno
coloop bialgebra with the non-commutative Faà di Bruno Hopf algebra. 


  * 
The proalgebraic loop  is not right alternative, nor power associative. 

  * 
Nevertheless,  has two-sided inverses and, for a given an associative 
algebra A and an element a∈(A), the inverse 
a^-1 = a\ e = e  a is given by the usual Lagrange 
coefficients, namely 

    (a^-1)_n 
        = - ∑_ℓ=0^n-1 (-1)^ℓ∑_∈_n^ℓ+1 
    d_ℓ+1(n_1,...,n_ℓ+1) a_n_1  a_n_2⋯ a_n_ℓ+1 .

  * 
The inversion allows us to construct the right division, that is, 
a  b = a∘ (e b) for any a,b∈(A), 
but it does not allow us to construct the left division, because
b\ a ≠ (b\ e)∘ a if a_n≠ 0 and b_m≠ 0 
for some n,m≥ 1.

1. 
The loop (A) is right alternative if and only if the coloop bialgebra 
 is right coalternative, that is (μ) K=0, where 
K=() -() is the 
coassociator. 

The first deviation from right alternativity appears on the generator x_5. 
If we temporarily denote by x_n=x_n^(1), y_n=x_n^(2) and z_n=x_n^(3) 
the three copies of the generators in H H H, we get 

    K(x_5) 
        = 6x_3(y_1z_1-z_1y_1) 
          + x_2[3(y_2z_1-z_2y_1)+3(y_1z_2-z_1y_2) 
      + (8y_1^2z_1-7y_1z_1y_1-z_1y_1^2) 
                + (8y_1z_1^2-7z_1y_1z_1-z_1^2y_1)] 
          +x_1[(y_3z_1-z_3y_1)+(y_2z_2-z_2y_2)+(y_1z_3-z_1y_3) 
    +3(y_2y_1z_1-y_2z_1y_1) 
            +2(y_1y_2z_1-y_1z_2y_1)+2(y_1^2z_2-y_1z_1y_2)+2(y_1z_1z_2-z_1y_1z_2) 
            +(3y_2z_1^2-2z_2y_1z_1)+(2y_1z_2z_1-3z_1y_2z_1) 
                + (5y_1^2z_1^2+y_1z_1^3 - 4y_1z_1y_1z_1-y_1z_1^2y_1-z_1y_1z_1^2)]

and therefore

(μ) K(x_5) = x_1(y_2y_1^2-y_1y_2y_1) ≠ 0. 


The generator x_5 corresponds to the power λ^6 of usual series
with substitution law, therefore the deviation from right alternativity
computed on the generator x_5 can be detected by comparing the values
(a∘ b)∘ b and a∘(b∘ b) for the two series 

a(λ)=λ+a_1λ^2          
b(λ)=λ+b_1λ^2+b_2λ^3 

with a_1=1 and b_2b_1^2≠ b_1b_2b_1. 
For instance[
The authors warmly thank J. M. Pérez-Izquierdo for comunicating 
this example.], by taking the 2× 2 elementary matrices b_1=E_11 and 
b_2=E_21, for which b_2b_1^2=b_2 and b_1b_2b_1=0. 
 

The same computation shows that (A) is not power associative, because 
μ  (μ) K(x_5) = x_1x_2x_1^2-x_1^2x_2x_1 ≠ 0. 

For a series c(λ)=λ+c_1λ^2+c_2λ^3, 
we then have (c∘ c)∘ c ≠ c∘ (c∘ c) if 
c_1c_2c_1^2 ≠ c_1^2c_2c_1. For instance, this is verified for the two 
2× 2 matrices 

c_1 = ([ 1 1; 0 1 ]) 
         
c_2= ([ 1 0; 1 0 ]) , 

for which we have 

c_1c_2c_1^2 = ([ 2 4; 1 2 ]) 
         
c_1^2c_2c_1= ([ 3 3; 1 1 ]) . 
 

2. 
The left and right inverses of a∈(A) can be found using 
respectively the left antipode S_l and right antipode S_r of , 
according to the standard rule

(e  a)_n = a(S_l(x_n)) 
         
(a\ e)_n = a(S_r(x_n)) . 

By Proposition <ref> we have S_r=S_l, therefore 
e  a=a\ e. 
 

3. 
The identity a  b = a∘ (e b) in the loop (A) is 
equivalent to the identity δ_r = ( S_r) 
in , proved in Proposition <ref>. 
The analogue identity for the left division does not hold. 


The commutative Faà di Bruno Hopf algebra which represents the classical 
proalgebraic group, mentioned at the beginning of 
section <ref>, admits a non-commutative lift 
<cit.>
    = ⟨ x_n, n≥ 1⟩,      (x_0=1) 
    
    (x_n)     = ∑_m=0^n x_m ⊗∑_(k) x_k_0⋯ x_k_m ,

where the sum is over the set of tuples(k_0,k_1,k_2,...,k_m)of non-negative integers such thatk_0+k_1+k_2+⋯ +k_m=n-m. 
Sinceis not a group over associative algebras, the existence 
of this Hopf algebra is not  a priori ensured by the extention 
of the functorfrom_to_. 


The image of the coloop bialgebra  under the canonical projection 
π given in Def. <ref> is the non-commutative 
Faà di Bruno Hopf algebra , that is, 

()^⊗ = . 

Indeed, we have ()^⊗ = as an algebra, and eventhough 
 is not coassociative, the comultiplication ()^⊗ coincides 
with  and therefore it is coassociative with respect to the 
component-wise multiplication in . 

The assertion is then proved because, by Prop. <ref>, 
the antipode in  is unique and coincides with that in  
on generators.  


§ APPENDIX: CATEGORICAL PROOFS WITH TANGLES


Tangle diagrams are an efficient tool to prove formal (categorical) properties. 
Tangles are drawings suitable to represent operations and co-operations in a 
monoidal category, cf. <cit.><cit.>, 
and therefore can be used to encode the structure of coloops in a category(,,I). In the context of non-associative algebras they have been 
used in <cit.> to code deformations of the enveloping 
algebra of a Malcev algebra, seen as the infinitesimal structure of a Moufang 
loop.

Tangles are drawings to be read from the top to the bottom as concatenation 
of operations acting on objects related by the monoidal product, and not by 
a cartesian (or tensor) product.
Here is the list of the tangles needed to represent all the operations and
the co-operations in coloops, with their defining identities. 
 18cm Categorical maps




τ    twist            
invertible
   =  𝕀𝕀
𝕀𝕀   
Sect. <ref>




μ    folding map            
associative 
   [0.5] 𝕀
[0.5]  =  𝕀[0.5]   
Sect. <ref>


          commutative 
   =  𝕀𝕀   
Sect. <ref>


           
unital 
   𝕀
 =  𝕀
 =  𝕀
𝕀   
Sect. <ref>




u    unit          folding morphism 
   =   
Sect. <ref>
 18cm Coloop structure maps




Δ    comultiplication
        
folding morphism 
   = lcr      
𝕀      𝕀   
Sect. <ref>


          unital 
   [.5] 
 =   
Sect. <ref>


          counital 
   𝕀
 =  
𝕀
 =  𝕀
𝕀   
Eq. (<ref>)





ε    counit          folding morphism 
   [.5] 
 =   
Sect. <ref>


           
unital 
   =  ∅   
Sect. <ref>






δ_r   2.6cmright codivision     * [-1] r    folding morphism 
   * 
 = lcr*       * 
𝕀      𝕀   
Sect. <ref>


          unital 
   [.5] 
* 
 =   
Sect. <ref>


          right cocancellation 
   [.5] 
* [.5] 
𝕀 =  [.5] * 
[.5] 
𝕀 =  𝕀
𝕀   
Eq. (<ref>)






δ_l    left codivision 
   * [-1] l    folding morphism 
   * 
 = lcr*       * 
𝕀      𝕀   
Sect. <ref>


          unital 
   [.5] 
* 
 =   
Sect. <ref>


          left cocancellation 
   [.5] 
[.5] * 
𝕀 =  [.5] * 
[.5] 
𝕀 =  𝕀
𝕀   
Eq. (<ref>)
 18cm Further coloop maps




S_r    right antipode 
   r 
 
:=   ^r
* 
𝕀    folding morphism 
   [.5] 
 r 
 = lcr r   r   
Sect. <ref>


          unital 
   r
 =   
Sect. <ref>




S_l    left antipode 
   l 
 
:=   ^l
* 
𝕀    folding morphism 
   [.5] 
 l 
 = lcr l  l   
Sect. <ref>


          unital 
   l 
 =   
Sect. <ref>18cm Properties of coloops




   μ δ_r = μ δ_l = u ε   ^r
* 
𝕀𝕀
 =   ^l
* 
𝕀𝕀
 =   
Eq. (<ref>)





partial counitality
   (ε) δ_r 
= (ε) δ_l =   ^r
* 
𝕀
 =   ^l
* 
𝕀
 =  𝕀
𝕀   
Eq. (<ref>)





3.5cmleft and right 
 5-terms identities   μ (S_r)  = μ ( S_l)  = u ε   r𝕀
 =  
𝕀 l
 =   
Eq. (<ref>)
 Proof of Eq. (<ref>): Forδ_r^r
    * 
    𝕀𝕀
     =  
    [.5] * 
    [.5] 
    
     =  
    𝕀
    𝕀
     =

and similarly forδ_l. 
  Proof of Eq. (<ref>): Forδ_r^r
    * 
    𝕀
     =  [.5] 
    * [.5] 
    𝕀
     =  [.5] 
    * [.5] 
    𝕀
    [.5] 𝕀
    [.5]  =  
    𝕀
    𝕀
     =  𝕀
    𝕀

and similarly forδ_l. 
  Proof of Eq. (<ref>): ForS_rr𝕀
     =  [.5] 
    * [.5] 
    𝕀𝕀
    
     =

and similarly forS_l. 
 18cm Properties of cogroups




coassociative
   ()  = ()   [0.5] 
[0.5] 𝕀 =  
𝕀[0.5]   
Eq. (<ref>)





unique antipode
   S_r=S_l=:S   ^r
* 
𝕀
 =  ^l
* 
𝕀
 =: 
 S   
Prop. <ref>




5-terms identity
   μ (S)  = μ ( S)  = u ε   S𝕀
 =  
𝕀 S
 =   
Eq. (<ref>)





3.5cmright coinverse 
 property   δ_r = ( S)   [.5] 𝕀[.5]  r 
* 
𝕀𝕀 =  
𝕀 S   
Eq. (<ref>)





3.5cmleft coinverse 
 property   δ_l = (S)   [.5] 𝕀[.5]  l 
* 
𝕀𝕀 =  
 S𝕀   
Eq. (<ref>)














If a coloop H is coassociative, then it has a two-sided antipode satisfying 
the 5-terms identity (<ref>) and the coinverse properties 
(<ref>). Therefore it is a cogroup. 

Assume that  is coassociative. then we have S_l=S_r because 
 l 
 =  
 l 
[2] 𝕀
 =  [1.5] 
[1.5] 𝕀
 r 𝕀[.5]  l 
[1.5] 𝕀
[1.5]  = 
𝕀[1.5] 
 r [.5] 𝕀 l 
𝕀[1.5] 
 = 
 r 
𝕀[2] 
 =  r 

and therefore S:= S_l=S_r satisfies the 5-terms identity becuase of 
(<ref>). 

For the the coinverse properties, let us show that the operator 
R:= ( S) satisfies the right cocancellations 
(<ref>), and therefore it coincides with δ_r. 
In fact, we have
[.5] 
[-.5] R[.5] * [.5] 
𝕀 =  [.5] 
[.5] 
𝕀 S 𝕀
𝕀 =  [.5] 
[.5] 
𝕀 S 𝕀
𝕀 =  
𝕀
𝕀 =  𝕀
𝕀
and
R[.5] * 
[.5] 
𝕀 =  [.5] 
[.5] 
𝕀𝕀 S 
𝕀 =  [.5] 
[.5] 
𝕀𝕀 S 
𝕀 =  
𝕀
𝕀 =  𝕀
𝕀
Similarly, the operator (S) satisfies the left-cocancellations 
(<ref>), and therefore it coincides with δ_l. 
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