The Betti side of the double shuffle theory. I. The harmonic coproducts
Benjamin Enriquez Hidekazu Furusho


16pt 

The Betti side of the double shuffle theory. I. The harmonic coproducts]The Betti side of the 
double shuffle theory. 
  I. The harmonic coproducts


This paper is the first in a series which aims at: (a) giving a proof that the
associator relations between multizeta values imply the double shuffle and regularization
(DSR) ones, alternative to that of the second-named author's 2010 paper; 
(b) enhancing Racinet's construction of 
a torsor structure over the ℚ-scheme of DSR relations to an explicit bitorsor structure.

In this paper, we revisit Racinet's original DSR formalism, whose main character is an
algebra coproduct, called the harmonic coproduct, and we introduce a variant which is a
module coproduct; we explain the `de Rham' nature of this formalism and construct a `Betti'
counterpart of it; we show how both formalisms can be interpreted in terms of geometry,
following the ideas of Deligne and Terasoma's unfinished 2005 preprint; 
we use Bar-Natan's interpretation of associators as functors from
the category of parenthesized braids to that of chord diagrams to show that any associator
relates the Betti and de Rham geometric objects, both in the `algebraic' and in the `module'
setups; we derive that any associator relates the Betti and de Rham algebra coproducts, as
well as their module counterparts. These results will be used in the next parts of the series. 

amsalpha+
[
    Thibaud Arnoux  –  Lionel Tabourier    –   Matthieu Latapy 

	Sorbonne Université, CNRS, Laboratoire d'Informatique de Paris 6, LIP6, F-75005 	

	4 place Jussieu
 75005 
    Paris, France

    http://www.complexnetworks.fr 

        

     
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§ INTRODUCTION




 §.§ The context




  §.§.§ The bitorsor of mixed Tate motives over ℤ


<cit.> contains the construction of a ℚ-linear neutral Tannakian category 𝒯:=
MT(ℤ)_ℚ of mixed Tate motives over ℤ, equipped with `Betti' and `de Rham' fiber 
functors ω_,ω_:𝒯→Vec_ℚ to the category of finite dimensional 
ℚ-vector spaces. It gives rise to a ℚ-scheme 𝖨𝗌𝗈𝗆^⊗_𝒯
(ω_,ω_) : 𝐤↦Isom^⊗_𝒯(ω_⊗𝐤,
ω_⊗𝐤), and to `motivic Galois' ℚ-group schemes 𝖠𝗎𝗍^⊗_𝒯
(ω_?):𝐤↦Aut^⊗_𝒯(ω_X⊗𝐤), for 
?∈{,} (denoted G^? in <cit.>), 𝐤 running over ℚ-algebras. Then 
𝖨𝗌𝗈𝗆^⊗_𝒯(ω_,ω_)(𝐤) is equipped with commuting free and transitive 
actions of 𝖠𝗎𝗍^⊗_𝒯(ω_)(𝐤) from the left, and of 
𝖠𝗎𝗍^⊗_𝒯(ω_)(𝐤) from the right.

The triple formed by this set and these two groups is a bitorsor,
meaning that the set is equipped with commuting left and right, free and transitive actions of
the groups, and the assignment of the triple to 𝐤 is then a bitorsor ℚ-scheme.

The category 𝒯 is closely connected with the family of real 
numbers known as multiple zeta values (MZVs), which are given by 

    ∀ s≥ 1,k_1≥ 1,…,k_s-1≥1,k_s≥ 2,   ζ(k_1,…,k_s)=∑_0<n_1<⋯<n_s1 n_1^k_1⋯ n_s^k_s;
    zeta(k_1,…,k_s)@ζ(k_1,…,k_s)

it follows from their integral expressions that these numbers, as well as[We set i:=√(-1)i@i.]  
2πi, are periods of mixed Tate motives. 



  §.§.§ The bitorsor of associators


Combining <cit.> and <cit.>, one can exhibit a family of algebraic relations satisfied by the MZVs and 2πi, the 
`associator relations'. The corresponding ℚ-scheme is the scheme[An affine ℚ-scheme (resp., 
ℚ-group scheme) 𝖷 can be identified with a representable functor from the category of ℚ-algebras 
to that of sets (resp. groups), denoted 𝐤↦𝖷(𝐤), for 𝐤 running over 
ℚ-algebras.
] of associators 
𝖬; 

it is equipped with commuting free and transitive actions of affine ℚ-group schemes 
𝖦𝖳 and 𝖦𝖱𝖳 (the Grothendieck-Teichmüller group 
scheme and its graded version) from the left and from the right, so that 𝖬 is a bitorsor ℚ-scheme. 

In <cit.>, it is explained that there is a bitorsor ℚ-scheme morphism 
𝖨𝗌𝗈𝗆_𝒯^⊗(ω_,ω_)→𝖬, i.e., a morphism
between these ℚ-schemes, together with compatible ℚ-group scheme morphisms 
𝖠𝗎𝗍_𝒯^⊗(ω_)→𝖦𝖳 and  
𝖠𝗎𝗍_𝒯^⊗(ω_)→𝖦𝖱𝖳. 



  §.§.§ The double shuffle torsor


In <cit.> (see also <cit.>), another family of algebraic relations satisfied by MZVs and 2πi was exhibited, the 
`double shuffle and regularization' (DSR) relations. The corresponding ℚ-scheme was constructed in <cit.>: it is a 
functor[𝖣𝖬𝖱 stands for the French `double mélange et régularisation'.] 
𝖣𝖬𝖱^,:=(𝐤↦⊔_μ∈𝐤^×{μ}×𝖣𝖬𝖱_μ(𝐤)), which is equipped with a free and transitive left action of a ℚ-group scheme
𝖣𝖬𝖱^:=(𝐤↦𝐤^×⋉(𝖣𝖬𝖱_0(𝐤),⊛)), therefore 
with a torsor ℚ-scheme structure. An interpretation of 𝖣𝖬𝖱^ in terms of a stabilizer group was
obtained in <cit.>. 



  §.§.§ Relations between the associator and double shuffle schemes


The unfinished preprint <cit.>, based on the ideas of Deligne's letter to Racinet (April 21, 2001), contains an attempt 
at constructing a ℚ-scheme inclusion 𝖬↪𝖣𝖬𝖱^,, using constructions involving 
the categories of perverse sheaves over the moduli space 𝔐_0,4 and 𝔐_0,5. 

In <cit.>, a ℚ-scheme inclusion 𝖬↪𝖣𝖬𝖱^, was constructed; a compatible 
ℚ-group scheme inclusion 𝖦𝖱𝖳^op↪𝖣𝖬𝖱^ can 
be constructed, so that 
this ℚ-scheme inclusion can be upgraded to a ℚ-torsor inclusion. This work is based on the construction, 
based on bar-complex techniques and the combinatorics of multiple polylogarithms, of explicit linear forms on the algebra 
U(𝔭_5), in which the associator relations take place, and on study of the interaction of these linear forms
with the associator relations. 



 §.§ The motivation and objectives of the series of papers
 

The main motivation of this series of papers are (1) to gain a better understanding of the inclusion 
𝖬⊂𝖣𝖬𝖱^,, in particular whether it is an equality (2) making explicit the bitorsor structure of 
𝖣𝖬𝖱^,. In order to reach (1), it seems useful to obtain a conceptual interpretation of this inclusion. 

The objectives of the series of papers are therefore: (a) to extend the interpretation of the group 𝖣𝖬𝖱^ in 
terms of a stabilizer (<cit.>) to the torsor 𝖣𝖬𝖱^,, for which the ideas of <cit.> will prove to be useful; 
(b) to give a proof of 
the inclusion 𝖬↪𝖣𝖬𝖱^, based on this interpretation and on the interpretation of 
𝖬 obtained in <cit.>; (c) to enhance the torsor structure of 𝖣𝖬𝖱^, into a bitorsor structure.


The material of the series is distributed as follows: the present part I is preparational; part II (<cit.>) will be devoted to objectives (a) and (b), and part III (<cit.>) to objective (c). 



 §.§ The contents of the present paper (part I of the series)


After treating categorical and topological preliminaries (Part 1), we revisit the DSR formalism and construct its Betti counterpart 
(Part 2); interpret both formalisms in terms of braids and moduli spaces (Part 3); and derive the compatibility of their main 
constituents with associators (Part 4). The corresponding compatibility statements are Theorem 10.9 (compatibility of harmonic 
algebra coproducts with associators) and Theorem 11.13 (compatibility of harmonic module coproducts with associators). These 
statements will play a key role in the next papers of the series and should be viewed as the main results of the present paper. 
The necessary material for understanding their formulation is to be found in: <ref> and 
<ref> (associators and their Γ-functions), <ref> and 
<ref> (definitions of the algebra 𝒲̂^ and of the module 
ℳ̂^), <ref> and <ref> (definitions of the de Rham harmonic algebra 
coproduct Δ̂^𝒲, and of its module version Δ̂^ℳ,), <ref> 
and <ref> (definitions of the algebra 𝒲̂^ and of the module ℳ̂^), 
<ref> and <ref> (definitions of the Betti harmonic coproduct Δ̂^𝒲, 
and of its module version Δ̂^ℳ,), and <ref> (definitions of the algebra isomorphism 
comp^𝒲,(1)_(μ,Φ) and of the module isomorphism 
comp^ℳ,(10)_(μ,Φ)).



  §.§.§ Conventions

 
Throughout the paper, 𝐤k@𝐤 is a commutative and associative ℚ-algebra.
By a 𝐤-algebra we mean a 𝐤-module, equipped with a 𝐤-bilinear associative product; 
a  𝐤-algebra is called unital if it contains a (necessarily unique) unity element.  
 We denote by A^× the group of invertible elements of an associative unital 𝐤-algebra A. For 
u∈ A^×, we denote by Ad(u) the inner automorphism of A induced by u, so 
Ad(u)(a)=uau^-1 for a∈ A. If f:A→ B is a morphism of 𝐤-algebras, if M,N are respectively an 
A-module and a B-module, then a 𝐤-module morphism g:M→ N is said to be compatible with f 
iff g(am)=f(a)g(m) for any a∈ A, m∈ M. 



  §.§.§ Acknowledgements


The collaboration of both authors has been supported by grants JSPS KAKENHI JP15KK0159 and JP18H01110. 



PART: 
Categorical preliminaries

In this section, we introduce various symmetric monoidal categories of 𝐤-modules and functors relating them. 
This material will be used through the rest of the paper. 


(1) 𝐤-mod_k-mod_gr@𝐤-mod_ is the category where objects are ℤ-graded 𝐤-modules and where morphisms are 𝐤-module morphisms of degree 0; 

(2) 𝐤-mod_k-mod_fil@𝐤-mod_ is the category where objects are pairs (M,i↦ F^iM), where M is a 𝐤-module
and i↦ F^iM is a map ℤ→{𝐤-submodules of M}, which is decreasing (i.e. F^iM⊃ F^i+1M for 
i∈ℤ), and where the set of morphisms from (M,i↦ F^iM) to (N,i↦ F^iN) is the set of 𝐤-module morphisms 
f:M→ N, which are compatible with the filtrations on both sides. We denote by F^if:F^iM→ F^iN the induced 𝐤-module map
corresponding to i∈ℤ; 

(3) 𝐤-mod_,+k-mod_gr+@𝐤-mod_,+ is the full subcategory of 𝐤-mod_ of bounded below ℤ-graded 𝐤-modules, i.e. of modules M=⊕_i∈ℤM_i,  such that there exists i(M)∈ℤ, such that M_i=0 for i<i(M); 

(4) 𝐤-mod_,+k-mod_fil+@𝐤-mod_,+ 
is the full subcategory of 𝐤-mod_ of pairs (M,i↦ F^iM), such that 
there exists i(M)∈ℤ, such that F^i(M)M=M;

(5) 𝐤-mod_k-mod_top@𝐤-mod_ is the full subcategory of 𝐤-mod_,+ of pairs (M,i↦ F^iM), such that 
the map M→lim_← iM/F^iM is a 𝐤-module isomorphism, i.e. M is complete 
for the topology defined by i↦ F^iM. 




  §.§.§ Functors


There are tautological functors to each category from each of its full subcategories. We will denote some of these functors
as follows taut_^,+:𝐤-mod_→𝐤-mod_,+tautfiltop@taut_^,+ and 
taut_,+^,+:𝐤-mod_,+→𝐤-mod_,+tautgrfil@taut_,+^,+. 

There is also an `associated graded' functor :𝐤-mod_→𝐤-mod_gr@, which takes 
(M,i↦ F^iM) to gr(M):=⊕_i∈ℤgr_i(M), where gr_i(M):=F^iM/F^i+1M for 
i∈ℤ. It induces a functor gr:𝐤-mod_,+→𝐤-mod_,+. 

For each α∈ℤ, there are functors F^α:𝐤-mod_→𝐤-mod_,+ and 
(-)_≥α:𝐤-mod_→𝐤-mod_,+, defined by F^α(M,i↦ F^iM):=(F^α M,i↦ F^iM∩ F^α M), and M_≥α:=⊕_i≥αM_i for M=⊕_i∈ℤM_i. 

Finally, there is a `completion' functor (-)^∧:𝐤-mod_,+→𝐤-mod_, given by 
(M,i↦ F^iM)↦(j
M/F^jM,i↦jF^iM/F^jM); it gives rise to a functor 
(-)^∧∘taut_,+^,+:𝐤-mod_,+→𝐤-mod_. 

A diagram of all these functors and categories is as follows

    𝐤-mod_^(-)_≥0[r]   𝐤-mod_,+@<2pt>^taut_,+^,+[d]   
    𝐤-mod__F^0[r]^[u]   𝐤-mod_,+@<2pt>^[u] 
    @<2pt>^(-)^∧[r]   𝐤-mod_@<2pt>^taut_^,+[l]


 
1) The functor (-)^∧∘taut_,+^,+(f) : 𝐤-mod_,+→𝐤-mod_ is exact. 

2) If f is a morphism in 𝐤-mod_,+ such that the morphism (f) in 𝐤-mod_,+ is 
injective, then the morphism f̂ in 𝐤-mod_ is injective. 


1) Let 0→ A→ B→ C→ 0 be an exact sequence in 𝐤-mod_,+. Then for each k∈ℤ, 
its degree k component 0→ A_k→ B_k→ C_k→ 0  is exact. The image of 0→ A→ B→ C→ 0 by 
(-)^∧∘taut_,+^,+ is the sequence 0→ (⊕_k<0A_k)⊕(∏_k≥0A_k)→
(⊕_k<0B_k)⊕(∏_k≥0B_k)→(⊕_k<0C_k)⊕ (∏_k≥0C_k)→ 0, which is then exact. 

2) Let f:M→ N be a morphism in 𝐤-mod_,+ with (f) injective. Let i_0:=min(i(M),i(N)). 
For any k≥ i_0, the map f_≤ k:M/F^kM→ N/F^kN induced by f is compatible with the filtrations on both sides induced by 
the images of F^iM and F^iN, i≤ k. The associated graded map is ⊕_i=i_0^k-1gr_i(f), which is injective. 
Therefore f_≤ k is injective. So  f̂ is injective as well.  



  §.§.§ Symmetric monoidal structures


Each of the categories from Definition <ref>, (1)-(4), is equipped with a symmetric monoidal structure, induced at the level of objects 
by the tensor product of 𝐤-modules. The category 𝐤-mod_ is also equipped with a symmetric monoidal structure
arising from the completed tensor product (M,i↦ F^M)⊗̂(N,i↦ F^N):=(j(M⊗ N)/F^j(M⊗ N),
i↦jF^i(M⊗ N)/F^j(M⊗ N)). For X,Y objects in 𝐤-mod_, one has therefore 
X⊗̂Y=(taut_^,+(X)⊗taut_^,+(Y))^∧. The functors in the previous paragraphs are then all monoidal. 



PART: 
The de Rham and Betti frameworks of double shuffle theory

In this part, we recall from <cit.> the DSR formalism and complement it by the introduction of a harmonic module coproduct 
Δ^ℳ, (<ref>). We construct a variant of this formalism, where its basic ingredient, the free algebra 
in two generators, is replaced by the group algebra of the free group in the same number of generators (<ref>). 
We then construct families of isomorphisms relating the various ingredients of the formalism of <ref> with their 
counterparts of <ref> (<ref>). The formalism of <ref> (resp. <ref>) 
is called `de Rham' (resp. `Betti') for reasons explained in <ref>. 




§ THE DE RHAM FRAMEWORK OF DOUBLE SHUFFLE THEORY


We recall from <cit.> the basic formalism of double shuffle theory. Its main ingredients are: 𝐤-bialgebras 
(𝒱^,Δ^𝒱,) and (𝒲^,Δ^𝒲,), a 𝐤-coalgebra 
(ℳ^,Δ^ℳ,), related by an algebra inclusion 𝒲^↪𝒱^ 
and a 𝒱^-module structure on ℳ^, inducing a free rank one 𝒲^-module structure 
on ℳ^, compatible with the coproducts; completions of all these structures; and a “Γ-function” map 
𝒢(𝒱̂^)→𝐤[[t]], where the source is the set of group-like elements of the completion of 
𝒱̂^. 

We introduce the algebra and module structures in <ref>, the coproducts in <ref>, the completions 
in <ref>, the Γ-function map in <ref>. We explain the relation with the formalism 
of <cit.> in <ref>. 



 §.§ The algebras 𝒱^, 𝒲^ and the module ℳ^



Let 𝔣_2 be the free ℤ-graded 𝐤-Lie algebra over generators e_0 and e_1e_0, e_1@ e_0, e_1 of degree 1 and let 

    𝒱^:=U𝔣_2V^DR@𝒱^

be its enveloping algebra; it is a ℤ-graded algebra whose components of degree <0 vanish, therefore an algebra in 
𝐤-mod_gr,+. 

Then

    𝒲^:=𝐤⊕ (U𝔣_2)e_1⊂ U𝔣_2=𝒱^W^DR@𝒲^

is a ℤ-graded subalgebra of 𝒱^, hence a subalgebra of it in 𝐤-mod_,+. 

The quotient 𝐤-module 

    ℳ^:=𝒱^/𝒱^e_0 
    M^DR@ℳ^

is an object in 𝐤-mod_,+; it is a left module over 𝒱^, and by restriction also 
over 𝒲^. Denote by 1_1DR@1_ the class 
of 1∈𝒱^ in ℳ^. The map

    (-)· 1_
    : 𝒱^→ℳ^,   a↦ a· 1_-1_DR@(-)· 1_
 
is a surjective morphism in  𝐤-mod_,+, and its kernel is equal to (U𝔣_2)e_0. 
It follows from the direct sum decomposition U𝔣_2=𝐤⊕ (U𝔣_2)e_0⊕ (U𝔣_2)e_1 that  
ℳ^ is free of rank one as a 𝒲^-module, generated by 1_. We denote by 

    𝒲^→ℳ^,   a↦ a· 1_

the corresponding isomorphism of 𝒲^-modules. 



 §.§ The coproducts Δ^𝒱,,Δ^𝒲, and Δ^ℳ,



We denote by Δ^𝒱,:𝒱^→(𝒱^)^⊗2DeltaVDR@Δ^𝒱, the enveloping algebra 
coproduct; it is determined by the conditions that e_0,e_1 are primitive. 

One shows that 𝒲^ is freely generated by the elements
y_n:=-e_0^n-1e_1y_n@y_n, where n≥ 1. One denotes by 

    Δ^𝒲,:𝒲^→(𝒲^)^⊗2DeltaWDR@Δ^𝒲,

the algebra morphism determined by 

    ∀ n≥ 1,  Δ^𝒲,(y_n)=y_n⊗1+1⊗ y_n+∑_k=1^n-1y_k⊗ y_n-k;

it equips 𝒲^ with a Hopf algebra structure. 

We denote by 

    Δ^ℳ,:ℳ^→(ℳ^)^⊗2DeltaMDR@Δ^ℳ,

the 𝐤-module morphism determined by the conditions that Δ^ℳ,(1_)=(1_)^⊗2, and that it 
is compatible with the module structures on both sides and with the algebra morphism Δ^𝒲,.  Then 
(ℳ^,Δ^ℳ,) is a cocommutative and coassociative coalgebra. 


As a Hopf algebra,  (𝒲^,Δ^𝒲,) is isomorphic to the universal enveloping algebra U(𝔤^) of the 
free Lie algebra 𝔤^ with generators (u_k)_k≥1, the isomorphism being given by 𝔤^[[t]]∋∑_k≥1t^ku_k↦log(1+∑_k≥1t^ky_k)∈𝒲^[[t]]. 




 §.§ Completions


The pairs (𝒱^,Δ^𝒱,), (𝒲^,Δ^𝒲,) 
(resp. (ℳ^,Δ^ℳ,)) are Hopf algebras (resp. is a coalgebra) in 𝐤-mod_,+
(see Definition <ref>). Their images by the functor (-)^∧∘taut_,+^,+:𝐤-mod_,+→𝐤-mod_ (see <ref>) are topological Hopf algebras 
(𝒱̂^V^DR^@𝒱̂^,Δ̂^𝒱,DeltaVDR^@Δ̂^𝒱,) and 
(𝒲̂^W^DR^@𝒲̂^,Δ̂^𝒲,DeltaWDR^@Δ̂^𝒲,), and a topological coalgebra 
(ℳ̂^M^DR^@ℳ̂^,Δ̂^ℳ,DeltaMDR^@Δ̂^ℳ,); the topological 𝐤-module ℳ̂^ is then a free
rank 1 module over the topological 𝐤-algebra 𝒲̂^. 



 §.§ Γ-functions


Define a map 

    𝒱̂^→𝐤[[t]]^×,    g↦Γ_g,

where 

    Γ_g(t):=exp(∑_n≥ 1(-1)^n+1 n(g|e_0^n-1e_1)t^n)∈𝐤[[t]]^×Gamma@Γ_g

and where g↦ ((g|w))_w
is the map 𝒱̂^→𝐤^{words in e_0,e_1} such that the identity g=∑_w(g|w)w holds. 



 §.§ Relation with double shuffle theory


In order to explain how the above material is used in the theory of double shuffle relations between the MZVs given by 
(<ref>), we introduce the notation 𝒱̂_ℂ^,  
ℳ̂_ℂ^ (resp. 𝒱_ℚ^) for the spaces 𝒱̂^,  
ℳ̂^ (resp. 𝒱^) when 𝐤=ℂ (resp. 𝐤=ℚ). 

The series φ_KZ∈ℂ⟨⟨ A,B⟩⟩^×
from 2 in <cit.> can be viewed as an element of (𝒱̂_ℂ^)^× 
via the isomorphism given by A↦ e_0, B↦ e_1. It is a generating 
series for the MZVs, namely 

    φ_=1+∑_m,k_1,…,k_m>0,k_1>1
    (-1)^mζ(k_1,…,k_m)w(k_1,…,k_m),
    phiKZ@φ_

where w(k_1,…,k_m) is an explicit element in e_0^k_1-1e_1⋯ e_0^k_m-1e_1+e_1𝒱^_ℚ+𝒱^_ℚ e_0
(see <cit.> and <cit.>, Proposition 3.2.3). 

The double shuffle relations between MZVs can be formulated as follows (see <cit.>): 


  (1) φ_ is group-like for the coproduct Δ̂^𝒱,; 


  (2) (Γ_φ_^-1(-e_1)·φ_)· 1_∈ℳ̂^_ℂ
is group-like for the coproduct Δ̂^ℳ,.  

The correspondence between the present formalism and that of <cit.> is as follows. 



this paper     e_0,e_1     𝒱^, 𝒱̂^     Δ^𝒱,, Δ̂^𝒱,     𝒲^, 𝒲̂^     Δ^𝒲,, Δ̂^𝒲, 
 <cit.>     x_0,-x_1     𝐤⟨ x_0,x_1⟩,      Δ, Δ̂   𝐤⊕𝐤⟨ x_0,x_1⟩ x_1,      Δ_⋆, Δ̂_⋆ 
 
         𝐤⟨⟨ x_0,x_1⟩⟩          degree completion      
 





ℳ^, ℳ̂^    
   Δ^ℳ,, Δ̂^ℳ,    
    (-)· 1_:𝒱^→ℳ^, (-)· 1_:𝒱̂^→ℳ̂^  
    
 𝐤⟨ x_0,x_1⟩ /𝐤⟨ x_0,x_1⟩ x_0,     
  Δ_⋆, Δ̂_⋆    
    π_Y :𝐤⟨ x_0,x_1⟩→𝐤⟨ x_0,x_1⟩ /𝐤⟨ x_0,x_1⟩ x_0, 

 
degree completion          degree completion  
 





§ THE BETTI FRAMEWORK OF DOUBLE SHUFFLE THEORY


We construct a Betti version of the double shuffle formalism from <ref>. The analogues of 𝒱^, 𝒲^ and 
ℳ^ are a pair of filtered 𝐤-algebras 𝒱^, 𝒲^, and a filtered 𝐤-module 
ℳ^ (<ref>). An algebra presentation of 𝒲^ is established in <ref>, and used in 
<ref> to define a Hopf algebra coproduct Δ^𝒲, on 𝒲^ as well as a coassociative 
coproduct Δ^ℳ, on ℳ^; a Hopf algebra coproduct Δ^𝒱, on 𝒱^ is also 
defined. In <ref>, we show the compatibility of the coproducts with the filtrations, and we relate 
the resulting associated graded objects with the material of <ref>. We define the corresponding topological 
Hopf algebras and coassociative coalgebra in <ref>. 



 §.§ The algebras 𝒱^, 𝒲^ and the module ℳ^


Let F_2F_2@F_2 be the free group with generators X_0, X_1X_0, X_1@X_0, X_1. Let 

    𝒱^:=𝐤F_2
    V^B@𝒱^

be its group algebra. It is equipped with a decreasing algebra filtration 𝒱^=F^0𝒱^⊃ F^1𝒱^⊃⋯, where for i≥ 0, F^i𝒱^F^iVB@F^i𝒱^ is the i-th power of the augmentation ideal 𝒱^_+ of 𝒱^=𝐤F_2.
Then 

    𝒲^:=𝐤⊕𝒱^(X_1-1)⊂𝒱^W^B@𝒲^

is a subalgebra of 𝒱^; it is equipped with the induced filtration given by 

    ∀ i≥0,   F^i𝒲^:=F^i𝒱^∩𝒲^.
    F^iWB@F^i𝒲^
 
Then 𝒱^ is an algebra in 𝐤-mod_,+ and 𝒲^ is a subalgebra. 

The quotient 𝐤-module 

    ℳ^:=𝒱^/𝒱^(X_0-1)
    M^B@ℳ^

is a left module over 𝒱^, and by restriction over 𝒲^. Let 1_1B@1_ be the 
class of 1∈𝒱^ in ℳ^. The map  

    (-)· 1_
    :𝒱^→ℳ^,    a↦ a· 1_-1_B@(-)· 1_

is surjective, with kernel (𝐤F_2)(X_0-1). The 𝐤-module ℳ^ is equipped with a decreasing filtration 
given by F^iℳ^:=(F^i𝒱^)· 1_F^iMB@F^iℳ^  for i≥ 0, compatible with the filtrations of 𝒱^ and 𝒲^. 
Therefore ℳ^ is a 𝒱^-module and a 𝒲^-module in 𝐤-mod_,+. 

 
Let G be the free group over generators g_1,…,g_n, let ℤG be its group 
algebra with coefficients in ℤ and let (ℤG)_0 be the augmentation ideal of
this algebra. Then (ℤG)_0 is freely generated, as a left ℤG-module, by the family (g_1-1,…,g_n-1). 

The proof in <cit.> relies on the following statement (not explicitly stated there)

    ∀ N≥ 1,  #{g∈ G| g≠1,ℓ(g)<N}=#{(g,i)∈ G×[[1,n]]|ℓ(g)<N,ℓ(gg_i)<N}

which may be proved by counting arguments. Here is an alternative proof of this lemma. 

Alternative proof of Lemma <ref>. There is a unique morphism of left ℤG-modules

    a:(ℤG)^⊕ n→(ℤG)_0,    (f_1,…,f_n)↦∑_i=1^n f_i(g_i-1).

It follows from <cit.>, equation (2.3) (see also the argument in Proposition 6.2.6 from <cit.>) that the map a is surjective. 

The injectivity of a may then be established in two ways. 

(First proof of injectivity of a.) In <cit.>, endomorphisms ∂∂ g_i (i∈[[1,n]]) of the 
ℤ-module ℤG are constructed. These endomorphisms 
are characterized by the following properties: ∂∂ g_i(1)=0,  ∂∂ g_i(g_j)=δ_ij
(j∈[[1,n]]), and ∂∂ g_i(uv)=u∂∂ g_i(v)+∂∂ g_i(u)ϵ(v) for u,v
in ℤG, where ϵ:ℤG→ℤ is the counit map (augmentation). These properties 
imply the identities ∂∂ g_i(f(g_j-1))=δ_ijf for f∈ℤG, i∈[[1,n]]. As a consequence, 
the ℤ-module map 

    b:(ℤG)_0→(ℤG)^⊕ n,    f↦ (∂∂ g_1f,…,
    ∂∂ g_nf)

is such that b∘ a=id. It follows that a is injective and therefore an isomorphism of ℤG-modules.  

(Second proof of injectivity of a.) Let ℤ⟨⟨ x_1,…,x_n⟩⟩ be the ring of formal series in 
noncommutative variables x_1,…,x_n, with coefficients in ℤ. The group morphism 
G→ℤ⟨⟨ x_1,…,x_n⟩⟩^× given by g_i↦ 1+x_i  for i∈[[1,n]]
gives rise to an algebra morphism r:ℤG→ℤ⟨⟨ x_1,…,x_n⟩⟩,  
which is injective (see <cit.>, Ch. 2, Sec. 5, no. 3, Thm. 1). 
The map r∘ a:ℤG^⊕ n→ℤ⟨⟨ x_1,…,x_n⟩⟩ is  
given by (f_1,…,f_n)↦∑_i=1^n r(f_i)x_i and is therefore equal to the composed map 
ℤG^⊕ nr^⊕ n→ℤ⟨⟨ x_1,…,x_n⟩⟩^⊕ nφ→ℤ⟨⟨ x_1,…,x_n⟩⟩, where φ is given by (r_1,…,r_n)↦∑_i=1^n r_ix_i. 
This map is injective, as is r^⊕ n, which implies that r∘ a is injective, showing the injectivity of a. 


Lemma <ref> implies the direct sum decomposition 

    𝒱^=𝐤 1⊕𝒱^(X_1-1)⊕𝒱^(X_0-1)
    =𝒲^⊕𝒱^(X_0-1),

which implies that ℳ^ is free of rank one as a 𝒲^-module, generated by the class 
1_ of 1∈𝐤F_2. We denote by 

    𝒲^→ℳ^,   a↦ a· 1_

the corresponding isomorphism of 𝒲^-modules. 



 §.§ Presentation of the algebra 𝒲^


Set for k∈ℤ, 

    ξ_k^+:=X_0^k(X_1-1),   and  ξ_k^-:=X_0^k(X_1^-1-1)xi @ ξ_k^+, ξ_k^-.
 These are elements of 𝒲^⊂𝐤 F_2. 


The algebra 𝒲^ is the algebra  whose
generators are (ξ_k^+)_k∈ℤ, (ξ_k^-)_k∈ℤ, and relations are 

    ∀ k∈ℤ,   ξ_k^+ξ_0^-=ξ_k^-ξ_0^+=-ξ_k^+-ξ_k^-.



Let 𝒜 be the associative non-commutative algebra generated by 
(ξ_k^+)_k∈ℤ, (ξ_k^-)_k∈ℤ
divided by the two-sided ideal generated by the relation given in (<ref>).

One checks that there is an algebra morphism

    f:𝒜→𝐤⊕𝐤 F_2(X_1-1)=𝒲^,
 
uniquely determined by 

    ξ_k^+↦ X_0^k(X_1-1)  and ξ_k^-↦ X_0^k(X_1^-1-1).
 
We will prove that f is an algebra isomorphism.

Let us first prove that f is surjective. For this, it will suffice to prove that for any word w in X_0^±1,X_1^±1, there exists a 
polynomial without constant term P_w over a set of noncommutative variables {(ξ̃_k^+)_k∈ℤ,ξ̃_0^-} 
indexed by ℤ⊔{0}, such that 

    w· (X_1-1)=P_w((ξ_k^+)_k∈ℤ,ξ_0^-).

We argue by induction on the length |w| of w. If w=1, then P_w=ξ̃_0^+. Assume the statement for |w|<n 
and let w be a word of length n. Then w=gw', where g∈{X_0^±1,X_1^±1} and w' has length n-1. 
If g=X_1^±1, then 

    w·(X_1-1)=X_1^±1w'(X_1-1)=P_w((ξ_k^+)_k∈ℤ,ξ_0^-),
 
where P_w
:=(1+ξ̃_0^±)P_w'. 
If g=X_0^±1, then 

    w·(X_1-1)=X_0^±1w'(X_1-1)=P_w((ξ_k^+)_k∈ℤ,ξ_0^-),

where we set 

    P_w
    :=∑_k∈ℤξ̃_k±1^+(P_w')_k^+
    -ξ̃_±1^+(1+ξ̃_0^-)(P_w')_0^-
    ,

in which the expressions (P_w')_k^+ and (P_w')_0^- have the following meaning: for P a polynomial 
in the noncommutative variables (ξ̃_k^+)_k∈ℤ,ξ̃_0^- without constant term, 
(P)_k^+ and (P)_0^- are the polynomials in the same variables such that

    P=∑_k∈ℤξ̃_k^+(P)_k^++ξ̃_0^-(P)_0^-.
 
This proves that f:𝒜→𝐤⊕𝐤 F_2(X_1-1) is surjective. 

To prove that f is injective, we will: 


  a) equip the vector space 𝒜[t^±1] with an algebra structure *, such that 𝒜→𝒜[t^±1], a↦ a⊗1 is an algebra morphism; 

  b) construct an algebra isomorphism F:𝒜[t^±1]→𝐤 F_2; 

  c) check that F extends f, in the sense that the following diagram commutes

    𝒜^f[r] @^(->_-⊗1[d]    𝐤⊕𝐤 F_2(X_1-1)
    @^(->[d]
    𝒜[t^±1]_F[r]    𝐤 F_2


 The above diagram implies that f is injective, which finally implies that it is an isomorphism. 

(a) Construction of an algebra structure on 𝒜[t^±1]. 
Let 𝐤⟨(ξ̃_k^±)_k∈ℤ⟩_+ 
be the ideal of 𝐤⟨(ξ̃_k^±)_k∈ℤ⟩ generated by the generators 
(ξ̃_k^±)_k∈ℤ. We have a direct sum decomposition 
𝐤⟨(ξ̃_k^±)_k∈ℤ⟩=𝐤⊕𝐤⟨(ξ̃_k^±)_k∈ℤ⟩_+. 
Left multiplication induces a linear isomorphism 

    (⊕_k∈ℤ(𝐤ξ̃_k^+⊕𝐤ξ̃_k^-))⊗𝐤⟨(ξ̃_k^±)_k∈ℤ⟩∼→𝐤⟨(ξ̃_k^±)_k∈ℤ⟩_+,

therefore there is a linear automorphism T of 𝐤⟨(ξ̃_k^±)_k∈ℤ⟩_+, 
uniquely determined by 

    ∀ k∈ℤ, ∀ a∈𝐤⟨(ξ̃_k^±)_k∈ℤ⟩,    
    T(ξ̃_k^± a)=ξ̃_k+1^± a.

Let I be the two-sided ideal of 𝐤⟨(ξ̃_k^±)_k∈ℤ⟩ generated by (<ref>). 
 Then I⊂𝐤⟨(ξ̃_k^±)_k∈ℤ⟩_+. Set 𝒜^+:=
𝐤⟨(ξ̃_k^±)_k∈ℤ⟩_+/I, then 

    𝒜=𝐤⊕𝒜^+.

Let V⊂𝐤⟨(ξ̃_k^±)_k∈ℤ⟩_+ be the linear span of all 
ξ̃_k^+ξ̃_0^-+ξ̃_k^++ξ̃_k^- 
and ξ̃_k^-ξ̃_0^++ξ̃_k^++ξ̃_k^-, for k∈ℤ. Then 

    I=V·𝐤⟨(ξ̃_k^±)_k∈ℤ⟩+
    𝐤⟨(ξ̃_k^±)_k∈ℤ⟩_+· V·𝐤⟨(ξ̃_k^±)_k∈ℤ⟩.

It follows from the definition of T that T^±1 maps the subspace 𝐤⟨(ξ̃_k^±)_k∈ℤ⟩_+· V
·𝐤⟨(ξ̃_k^±)_k∈ℤ⟩ to itself. Moreover, for any 
a∈𝐤⟨(ξ̃_k^±)_k∈ℤ⟩, one has 

    T^±1((ξ̃_k^+ξ̃_0^-+ξ̃_k^++ξ̃_k^-)a)=(ξ̃_k±1^+ξ̃_0^-+ξ̃_k±1^++ξ̃_k±1^-)a,  
    T^±1((ξ̃_k^-ξ̃_0^++ξ̃_k^++ξ̃_k^-)a)=(ξ̃_k±1^-ξ̃_0^++ξ̃_k±1^++ξ̃_k±1^-)a,

which implies that T^±1 maps V·𝐤⟨(ξ̃_k^±)_k∈ℤ⟩ to itself. All this implies that 
T^±1(I)⊂ I, so T(I)=I. It follows that T induces a linear automorphism of 𝒜^+. Relation 
(<ref>) then implies: 

    for P,P'∈𝒜^+ and a∈ℤ, we have T^a(PP')=T^a(P)P'.


We now define a bilinear map 

    *:𝒜[t^±1]^⊗2→𝒜[t^±1]

as follows: 


  1) for a,b∈ℤ, (1⊗ t^a)*(1⊗ t^b)=1⊗ t^a+b; 

  2) for a∈ℤ, P∈𝒜^+, 

    (1⊗ t^a)*(P⊗ t^b):=T^a(P)⊗ t^b,   
    (P⊗ t^a)*(1⊗ t^b):=P⊗ t^a+b,


  3) for a,b∈ℤ, and P,Q∈𝒜^+, 

    (P⊗ t^a)*(Q⊗ t^b):=(P· T^a(Q))⊗ t^b,

where · is the product in 𝒜^+. 


It then follows from a case analysis, using (<ref>), that 

    the product * defines an associative algebra structure on 𝒜[t^±1].



(b) Construction of an algebra isomorphism (𝒜[t^±1],*)≃𝐤 F_2. The elements 
1⊗ t and 1⊗ t^-1 of (𝒜[t^±1],*) are mutually inverse; so are the elements 
1+ξ_0^+ and 1+ξ_0^- of 𝒜, and therefore the elements (1+ξ_0^+)⊗1 and (1+ξ_0^-)⊗1 of 
𝒜[t^±1]. All this implies that there is an algebra morphism 

    G:𝐤 F_2→(𝒜[t^±1],*)

uniquely determined by 

    X_0^±1↦ 1⊗ t^±1,   X_1^±1↦ (1+ξ_0^±)⊗1.

One checks that the algebra homomorphism f: 𝒜→𝐤⊕𝐤 F_2(X_1-1) 
restricts to a morphism of algebras without unit 

    f^+:𝒜^+→𝐤 F_2(X_1-1),

and that the following diagram commutes

    𝒜^+^f^+[r]_T[d]   𝐤 F_2(X_1-1)^X_0·-[d]
    
    𝒜^+_f^+[r]   𝐤 F_2(X_1-1)

Define now a linear map 

    F:𝒜[t^±1]→𝐤 F_2

by 

    1⊗ t^a↦ X_0^a,    P⊗ t^b↦ f^+(P)X_0^b

for a,b∈ℤ, P∈𝒜^+. Then: 


  1) for a,b∈ℤ, F(1⊗ t^a)F(1⊗ t^b)=X_0^aX_0^b=X_0^a+b=F(1⊗ t^a+b)=F((1⊗ t^a)*(1⊗ t^b)); 

  2) for a∈ℤ, P∈𝒜^+, 

    F(1⊗ t^a)F(P⊗1)=X_0^a f^+(P)= f^+(T^a(P))=F(T^a(P)⊗1)=F((1⊗ t^a)*(P⊗1)),


    F(P⊗1)F(1⊗ t^a)=f^+(P)X_0^a=F(P⊗ t^a)=F((P⊗1)*(1⊗ t^a));


  3)for a,b∈ℤ, P,Q∈𝒜^+, 

    F(P⊗ t^a)F(Q⊗ t^b)=f^+(P)X_0^a f^+(Q)X_0^b= f^+(P) f^+(T^a(Q))X_0^b
    
        = f^+(P T^a(Q))X_0^b=F(PT^a(Q)⊗ t^b)=F((P⊗ t^a)*(Q⊗ t^b)).


All this implies that F is an algebra morphism 𝒜[t^±1]→𝐤 F_2.
Let us now show that the algebra morphisms F,G are mutually inverse. 

We have 

    F∘ G(X_0^±1)=F(1⊗ t^±1)=X_0^±1,


    F∘ G(X_1^±1)=F((1+ξ_0^±)⊗1)=1+F(ξ_0^±⊗1)=1+f^+(ξ_0^±)=1+(X_1^±1-1)=X_1^±1,

so F∘ G=id_𝐤 F_2. 
 
Similarly, G∘ F is an algebra endomorphism of (𝒜[t^±1],*). 

Given the structure of *, a collection of generators of this algebra is given by {g⊗ 1,1⊗ t^±1}, 
where g runs over a collection of algebraic generators of 𝒜, so such a collection of algebraic 
generators is {(ξ_k^±⊗1)_k∈ℤ,1⊗ t^±1}. One has 

    G∘ F(ξ_k^±⊗1)=G(f^+(ξ_k^±))=G(X_0^k(X_1^±1-1))=G(X_0)^*k*(G(X_1^±1)-1)
    
        =(1⊗ t^±1)^*k*(ξ_0^±⊗1) =ξ_k^±⊗1,

so G∘ F=id_𝒜[t^±1]. 

(c) Diagram involving f and F. The commutation of (<ref>) follows from the definition of F. 


Lemma <ref> leads to the following presentation:


The associative 𝐤-algebra 𝒲^ is generated by the unity element 1 and the elements

    Y_n^+:=(X_0-1)^n-1X_0(1-X_1)          (n>0), 
        Y_n^-:=(X_0^-1-1)^n-1X_0^-1(1-X_1^-1)      (n>0),  
        X_1    and  X_1^-1, Y_n @ Y_n^+, Y_n^-

with defining relations

    X_1X_1^-1=X_1^-1X_1=1.




By the presentations in the previous lemma, 
we have ξ_k^-=-ξ_k^+(1+ξ_0^-) 
and ξ_k^+=-ξ^-_k(1+ξ_0^+), 
from which we can deduce that 
𝒜 is generated by ξ_k^+ and ξ_-k^- for k⩾ 0
with defining equation (<ref>) only for k=0.

It is immediate to see that (Y_n^+)_n>0 (resp. (Y_n^-)_n>0)
can be expressed by 
linear combinations of  (ξ_n^+)_n>0 (resp. (ξ_n^-)_n>0) and vice versa
and that the equation (<ref>) for k=0 is equivalent to 
(<ref>),
from which our claim follows.




 §.§ The coproducts Δ^𝒱,,Δ^𝒲, and Δ^ℳ,


We denote by Δ^𝒱,:𝒱^→(𝒱^)^⊗2
DeltaVB@Δ^𝒱,
the group algebra coproduct; it is determined by the  
condition that the elements X_0^±1, X_1^±1 are group-like. 


There is a unique algebra morphism 

    Δ^𝒲,:𝒲^→(𝒲^)^⊗2DeltaWB@Δ^𝒲,

such that 

    Δ^𝒲,(X_1^±1)=X_1^±1⊗ X_1^±1,   Δ^𝒲,(Y_k^±)=Y_k^±⊗1+1⊗ Y_k^±+∑_k',k”>0,k'+k"=k 
    Y^±_k'⊗ Y^±_k”  for any k≥1.

 It equips 𝒲^ with a cocommutative Hopf algebra structure with antipode given by 
X_1^±1↦ X_1^∓1, Y_k^±↦∑_a≥1(-1)^a∑_k_1+⋯+k_a=k
Y^±_k_1⋯ Y^±_k_a.  


The relations between the generators (see Proposition <ref>) are obviously 
preserved, therefore Δ^𝒲, is well-defined. One directly checks the 
cocommutativity and coassociativity of Δ^𝒲,.  One also checks that for t a 
formal parameter, the series 1+∑_k≥1t^kY_k^± are group-like. It follows that 
Δ^𝒲, admits the announced antipode. 

We denote by 

    Δ^ℳ,:ℳ^→(ℳ^)^⊗2DeltaMB@Δ^ℳ,

the 𝐤-module morphism determined by the conditions that Δ^ℳ,(1_)=(1_)^⊗2, and that it 
is compatible with the module structures on both sides and with the algebra morphism Δ^𝒲,.  
Then (ℳ^,Δ^ℳ,) is a cocommutative and coassociative coalgebra. 


If 𝔤 is a Lie algebra and Γ is a group acting on 𝔤 by Lie algebra 
automorphisms, then the semidirect product U(𝔤)⋊Γ is 
equipped with a cocommutative Hopf algebra structure, defined by the conditions that the elements 
of 𝔤 are primitive and the elements of Γ are group-like. The Hopf algebra 
(𝒲^,Δ^𝒲,) is isomorphic to U(𝔤^)⋊ℤ, 
with 𝔤^ the free Lie algebra over generators 
(U_k^±,(ℓ))_k≥ 1,ℓ∈ℤ and the action of ℤ on 
𝔤^ being given by 1· U_k^±,(ℓ)=U_k^±,(ℓ+1). The isomorphism 
is given by 
𝔤^[[t]]∋∑_k≥1t^kU_k^±,(ℓ)↦ X_1^ℓ·log(1+∑_k≥1t^kY_k^±)· X_1^-ℓ∈𝒲^[[t]], ℤ∋± 1↦ 
X_1^±1∈𝒲^. 


 The automorphism of 𝒱^ induced by X_i↦ X_i^-1, i=0,1 restricts to a Hopf 
algebra automorphism θ of (𝒲^,Δ^𝒲,) such that X_1^±↦ X_1^∓ 
and Y_k^±↦ Y_k^∓ for k>0. 



 §.§ Filtrations and associated graded objects




  §.§.§ Associated graded object of (𝒱^,Δ^𝒱,)


Recall the decreasing algebra filtration of the algebra 𝒱^ (see <ref>). 
The associated graded algebra gr(𝒱^)=⊕_n≥0gr_n(𝒱^)
=⊕_n≥0F^n𝒱^/F^n+1𝒱^ is generated in degree 1; according to <cit.>, chap. 2, 5, no. 4, Theorem 2, 
there is an isomorphism of graded algebras

    gr(𝒱^)∼→𝒱^,   gr_1(𝒱^)∋ [X_i-1]↦ e_i∈ (degree 1 part of 𝒱^)
    ⊂𝒱^.

The algebra morphism Δ^𝒱,:𝒱^→(𝒱^)^⊗2 is then compatible with the filtrations on both sides, 
and the associated graded algebra morphism coincides with Δ^𝒱,:𝒱^→(𝒱^)^⊗2. 



  §.§.§ Filtration on (𝒲^,Δ^𝒲,) and associated graded object


Recall the decreasing algebra filtration of the algebra 𝒲^ (see <ref>). 


For n≥1, one has 

    F^n𝒲^=F^n-1𝒱^·(X_1-1)

(equality of subspaces of 𝒱^). 


Set e_i^±:=X_i^±1-1∈𝒱^ for i=0,1. 
By (<ref>), one has 

    𝒱^_+=𝒱^·e_0^+⊕𝒱^·e_1^+,
 which implies

    ∀ n≥ 0,   F^n𝒱^=(𝒱^_+)^n=∑_(a_1,…,a_n)∈{0,1}^n𝒱^e_a_1^+𝒱^⋯𝒱^e_a_n^+.

Since F^n𝒱^ is a right ideal in 𝒱^, one has F^n𝒱^=F^n𝒱^·𝒱^, which together with 
(<ref>) implies 

    ∀ n≥ 0,   F^n𝒱^=∑_(a_1,…,a_n)∈{0,1}^n𝒱^e_a_1^+𝒱^⋯𝒱^e_a_n^+𝒱^.

Let n≥ 1. (<ref>) implies 

    F^n𝒱^=∑_(a_1…,a_n-1)
      ∈{0,1}^n-1𝒱^e_a_1^+𝒱^⋯𝒱^e_a_n-1^+𝒱^e_0^+
    +
    ∑_(a_1…,a_n-1)
      ∈{0,1}^n-1𝒱^e_a_1^+𝒱^⋯𝒱^e_a_n-1^+𝒱^e_1^+,

which together with (<ref>) and  (<ref>) for n replaced by n-1 implies

    F^n𝒱^=0⊕ F^n-1𝒱^·e_0^+⊕ F^n-1𝒱^·e_1^+⊂𝐤 1⊕𝒱^·e_0^+⊕𝒱^·e_1^+=𝒱^.

As F^n𝒲^=F^n𝒱^∩ (𝐤 1⊕ 0⊕𝒱^·e_1^+), one obtains 
F^n𝒲^=F^n-1𝒱^·e_1^+ 



The morphism of graded algebras gr(𝒲^)→gr(𝒱^) induced by the compatibility of the inclusion 
𝒲^⊂𝒱^ with the filtrations is injective. 
The image of gr(𝒲^) under the isomorphism gr(𝒱^)≃𝒱^ is equal to 𝒲^⊂𝒱^.    


The injectivity of the map gr(𝒲^)→gr(𝒱^) follows from the fact that the filtration of 
𝒲^ is induced by  that of 𝒱^. We have 𝒲^=𝐤⊕ F^1𝒲^, which implies that 

    gr(𝒲^)=𝐤1⊕gr(F^1𝒲^)⊂𝒱^,

where 

    gr(F^1𝒲^)⊂⊕_n>0𝒱^_n.

The sequence of linear maps 𝒱^-·(X_1-1)⟶F^1𝒲^⊂𝒱^ is compatible with the  
filtrations whose nth steps are given by F^n𝒱^, F^n+1𝒲^,  and F^n+1𝒱^. The 
associated graded sequence of maps 𝒱^≃gr(𝒱^)→gr(F^1𝒲^)→gr(𝒱^)≃𝒱^ is given by 
𝒱^→𝒱^, x↦ xe_1. It follows from Lemma <ref> that the first map 
gr(𝒱^)→gr(F^1𝒲^) is surjective. All this implies the identification 
gr(F^1𝒲^)≃𝒱^ e_1, which together with (<ref>) implies the claim. 


We now describe explicitly the filtration of 𝒲^. Set e_i^±:=X_i^±1-1∈𝒱^ for i=0,1,  

    ∀ a_+,a_-≥ 0,  y_a_+,a_-^±:=(e_0^+)^a_+(e_0^-)^a_-e_1^±∈𝒲^,

and 

    ∀ a≥ 1,   V_a:=Span_𝐤( y_a_+,a_-^ϵ|a_++a_-≥ a-1,   ϵ∈{+,-})⊂𝒲^.

 
 
 1) For n≥1, F^n𝒲^ is equal to the sum ∑_k≥1∑_a_1,…,a_k|a_1+⋯+a_k≥ nV_a_1⋯ V_a_k of products (for the algebra structure of 𝒲^) of subspaces of 𝒲^. 
 
 2) F^0𝒲^=𝒲^=𝐤1⊕ F^1𝒲^. 
 

One has 𝒱^=𝐤1⊕ F^1𝒱^ and 𝒲^⊃𝐤1, which implies 2). Let n≥1. By Lemma <ref>, F^n𝒲^=F^n-1𝒱^·e_1^+. Since 
X_1 is invertible in 𝒱^, one has 𝒱^· X_1^-1=𝒱^, which together with (<ref>) implies F^n-1𝒱^· X_1^-1=F^n-1𝒱^. All this together with 
X_1^-1·e_1^+=- e_1^- implies 

    F^n𝒲^=F^n-1𝒱^·e_1^+ + F^n-1𝒱^·e_1^-.

One has 𝒱^_+=∑_s≥1∑_((a_1,ϵ_1),…,(a_s,ϵ_s))∈({0,1}×{+,-})^s𝐤·e_a_1^ϵ_1⋯e_a_s^ϵ_s, therefore 

    ∀ n≥ 0,   F^n𝒱^=∑_s≥ n∑_((a_1,ϵ_1),…,(a_s,ϵ_s))∈({0,1}×{+,-})^s𝐤·e_a_1^ϵ_1⋯e_a_s^ϵ_s,

which together with (<ref>) implies 

    ∀ n≥ 1,   F^n𝒲^=∑_s≥ n∑_((a_1,ϵ_1),…,(a_s-1,ϵ_s-1))∈({0,1}×{+,-})^s-1,
    ϵ_s∈{+,-}𝐤·e_a_1^ϵ_1⋯e_a_s-1^ϵ_s-1e_1^ϵ_s,

The fact that e_0^+ and e_0^- commute implies that the right-hand side of this equality can be 
identified with the sum of subspaces announced in 1). 


Define linear maps 

    ξ^±:𝐤[t,t^-1]→𝒲^,    f↦ξ^±(f):=f(X_0)(X_1^±1-1).


For a≥1, one has 

    V_a=ξ^+(((t-1)^a-1))+ξ^-(((t-1)^a-1))⊂𝒲^,

where ((t-1)^a-1):=(t-1)^a-1𝐤[t,t^-1]⊂𝐤[t,t^-1] is the ideal of 𝐤[t,t^-1]
generated by (t-1)^a-1. 


For a',a”≥0, one has y_a',a”^±=ξ^±((-t^-1)^a”(t-1)^a'+a”), therefore V_a=ξ^+(W_a)
+ξ^-(W_a), where 

    W_a:=Span{t^-a”(t-1)^a'+a”|a',a”≥0,a'+a”≥ a-1}⊂𝐤[t,t^-1].

One has 

    W_a=(t-1)^a-1·(Span(1,…,t^1-a)+(t-1)Span(1,…,t^-a)
    +(t-1)^2Span(1,…,t^-a-1)+⋯)
    
        =(t-1)^a-1·(Span(1,…,t^1-a)+Span(t,t^-a)
    +Span(t^2,…,t^-a-1)+⋯)=((t-1)^a-1).



 
For any k∈ℤ, one has 

    Δ^𝒲,(X_0^k(1-X_1))
    =X_0^k(1-X_1)⊗ 1+1⊗ X_0^k(1-X_1)
    +∑_i=1^k-1 X_0^i(1-X_1) ⊗ X_0^k-i(1-X_1),
 

    Δ^𝒲,(X_0^k(1-X_1^-1)) 
    =X_0^k(1-X_1^-1)⊗ 1+1⊗ X_0^k(1-X_1^-1)
    -∑_i=0^k X_0^i(1-X_1^-1) ⊗ X_0^k-i(1-X_1^-1),

where for A an abelian group, for f:ℤ→ A be a function and for a,b∈ℤ, we set 

    ∑_k=a^b f(k):= f(a)+⋯+f(b)     if b≥ a, 
     0     if b=a-1, 
     
    -f(a-1)-⋯-f(b+1)    if b<a-1



Let t be a formal parameter. As we have seen, the series 
s_±(t) :=1+∑_k≥1t^kY_k^± are group-like for Δ^𝒲,. 
One computes these series as follows 

    s_±(t)=1+tX_0^±1{1-t(X_0^±1-1)}^-1(1-X_1^±1)=(1+t-tX_0^±1)^-1
    (1+t-tX_0^±1X_1^±1)=s̃_±(u),

where u :=t/(1+t) and s̃_±(u ) :=(1-uX_0^±1)^-1(1-uX_0^±1X_1^±1). 
It follows that the series s̃_±(u) are group-like. Together with their expansions 
s̃_±(u)=1+∑_k≥1u^k X_0^± k(1-X_1^±1), this implies (<ref>) for 
k≥1 and (<ref>) for k≤-1. Since X_1^±1 are group-like, (<ref>) for k
is equivalent to (<ref>) for -k, which implies (<ref>) for k≤-1 and (<ref>) for 
k≥1 . Finally, (<ref>) and (<ref>) for k=0 are direct consequences of the fact that 
X_1^±1 are group-like.   


The assignments f↦t'f(t”)-t”f(t')t'-t”, f↦t'f(t')-t”f(t”)t'-t”  define linear maps 𝐤[t,t^-1]→𝐤[t',(t')^-1,t”,(t”)^-1].  We denote by 

    Op^±:𝐤[t,t^-1]↦𝐤[t,t^-1]^⊗ 2

the composition of this linear map with the isomorphism of their target algebra with 𝐤[t,t^-1]^⊗ 2 
given by t'↦ t⊗ 1, t”↦1⊗ t. 


For a∈ℤ, the image of t^a by the first map is -t't”(t^' a-2+⋯
+t^'' a-2) if a≥ 1, and (t't”)^a(t^' -a+⋯+t^'' -a) if a≤ 0, which implies the result
in this case. The result in the second case follows from the identity ∀ f∈𝐤[t,t^-1], 
Op^+(f)+Op^-(f)=f⊗1+1⊗ f.  


The map Δ^𝒲,:𝒲^→(𝒲^)^⊗2 is such that 

    ∀ f∈𝐤[t,t^-1],  ξ^±(f)↦ξ^±(f)⊗1+1⊗ξ^±(f)
    +(ξ^±)^⊗2(Op^±(f)).


Let k∈ℤ and f:=t^k. Then  

    ξ^+(f)=X_0^k(X_1-1)↦ξ^+(f)⊗1+1⊗ξ^+(f)-∑_i=1^k-1
    X_0^k-i(X_1-1)⊗ X_0^i(X_1-1)
    = ξ^+(f)⊗1
    
        +1⊗ξ^+(f)+(ξ^+)^⊗2(-∑_i=1^k-1t^' k-i t^'' i)
    =ξ^+(f)⊗1+1⊗ξ^+(f)+(ξ^+)^⊗2(t'f(t”)-t”f(t')t'-t”),

which implies the result in the case of ξ^+. The result for ξ^- follows from this, from 
ξ^-(f)=-ξ^+(f)X_1^-1, and from the relation between  Op^+ and Op^-
(see proof of Lemma <ref>). 



For n≥ 2, each of the maps Op^± defined in 
Lemma-Definition <ref>
takes ((t-1)^n-1) (see Lemma <ref>) to ∑_a=1^n-1((t-1)^a-1)⊗((t-1)^n-a-1). 


Assume that f∈((t-1)^n-1), that is f(t)=(t-1)^n-1a(t), with a(t)∈𝐤[t^±1]. Then 

    t'f(t”)-t”f(t')t'-t”=t'(t”-1)^n-1a(t”)-t”(t'-1)^n-1a(t')t'-t”
        =(t”-1)^n-1t'a(t”)-t”a(t') t'-t”+t”a(t')(t”-1)^n-1-(t'-1)^n-1 t'-t”

The second term of the last line belongs to the announced space ∑_a=1^n-1((t-1)^a-1)⊗((t-1)^n-a-1), 
while the first term belongs to 𝐤[t,t^-1]⊗ ((t-1)^n-1), which is contained in this space. This proves the result in the case 
of Op^+. This and the  relation between  Op^+ and Op^- (see proof of Lemma <ref>) imply 
the result in the case of Op^-. 


The morphism Δ^𝒲, is compatible with the filtration of 𝒲^, in other words, if 
n≥ 0, then Δ^𝒲,(F^n𝒲^)⊂∑_a,b≥0|a+b=nF^a𝒲^⊗ F^b𝒲^. 


Lemmas <ref>, <ref> and <ref> imply 
that ∀ a≥1, Δ^𝒲,(V_a)⊂ V_a⊗1+1⊗ V_a+∑_a',a”≥1,a'+a”=aV_a'⊗ V_a”. 
The result then follows from Lemma <ref>. 



The associated graded morphism of Δ^𝒲,:𝒲^→(𝒲^)^⊗2 
with respect to the filtrations of both sides identifies with Δ^𝒲,:𝒲^→(𝒲^)^⊗2 
under the isomorphisms gr(𝒲^)≃𝒲^. 


This follows from the fact that the elements y_n,n≥1 generate 𝒲^, from the second part of (<ref>) 
for ±=+, and from the fact that for n≥1, the class of Y_n^+=X_0(X_0-1)^n-1(1-X_1)∈ F^1𝒲^ in 
gr_1(𝒲^) is e_0^n-1·(-e_1)=y_n.  



  §.§.§ Associated graded object of (ℳ^,Δ^ℳ,)


Recall the decreasing filtration ℳ^=F^0ℳ^⊃ F^1ℳ^⊃⋯ on the 𝐤-module 
ℳ^, where for i≥ 0, F^iℳ^=F^i𝒱^· 1_ (see <ref>). 


For i≥0, one has F^iℳ^=F^i𝒲^· 1_. 


For i≥1, one has F^i𝒱^· 1_=(F^i-1𝒱^(X_0-1)+F^i-1𝒱^(X_1-1))· 1_
=F^i-1𝒱^(X_0-1)· 1_+F^i-1𝒱^(X_1-1)· 1_=F^i-1𝒱^(X_1-1)· 1_=F^i𝒲^· 1_, where the first (resp. last) equality follows from (<ref>) (resp. Lemma <ref>). 



(ℳ^,Δ^ℳ,) is a coalgebra in 𝐤-mod_fil,+; its image under gr
is the coalgebra (ℳ^,Δ^ℳ,) in 𝐤-mod_gr,+. 


The first statement follows from Proposition <ref> and from 
Δ^ℳ,(1_)=(1_)^⊗2. The morphism (-)· 1_:𝒱^→ℳ^ in 
𝐤-mod_fil,+ induces a morphism 𝒱^=gr(𝒱^)
gr((-)· 1_)→gr(ℳ^). If one equips 𝒱^ with the shifted 
filtration F^n(𝒱^[1]):=F^n-1(𝒱^), then (-)·(X_0-1):𝒱^[1]→𝒱^
is a morphism in 𝐤-mod_fil,+, such that the composed morphism 
𝒱^[1](-)·(X_0-1)→𝒱^(-)· 1_→ℳ^ is zero. 
Applying gr, one obtains that the composed morphism 𝒱^[1](-)· e_0→𝒱^→gr(ℳ^) is zero, therefore a morphism ℳ^→gr(ℳ^).  

By Lemma <ref>, the map 𝒲^→ℳ^, x↦ x· 1^ is an 
isomorphism in 𝐤-mod_fil,+, therefore it gives rise to an isomorphism 
gr(𝒲^)→gr(ℳ^). The following diagram 
commutes

    gr(𝒲^)^gr((-)· 1_)_≃[rr]      gr(ℳ^)
    𝒲^^≃_(-)· 1_[rr]^≃[u]      ℳ^[u]

which implies that ℳ^→gr(ℳ^) is an isomorphism. In the following diagram 

    gr(𝒲^)^⊗2^(gr((-)· 1_))^⊗2_≃[rrr]         gr(ℳ^)^⊗2
       gr(𝒲^)^gr((-)· 1_)_≃[r]^gr(Δ^𝒲,)[ul]   gr(ℳ^)_gr(Δ^ℳ,)[ur]   
       𝒲^_(-)· 1_^≃[r]^Δ^𝒲,[dl]^≃[u]   ℳ^_Δ^ℳ,[dr]_≃[u]   
     
    (𝒲^)^⊗2_((-)· 1_)^⊗2^≃[rrr]^≃[uuu]         (ℳ^)^⊗2_≃[uuu]

the inner square commutes by (<ref>), the external square commutes as it is the tensor square of 
(<ref>), the lower quadrangle commutes by the definition of Δ^ℳ,, the left quadrangle 
commutes by Proposition <ref>, the upper quadrangle commutes because (-)· 1_ induces an isomorphism 
of the coalgebras (𝒲^,Δ^𝒲,) and (ℳ^,Δ^ℳ,) in 
𝐤-mod_fil,+.  It follows that the right quadrangle commutes. 


One checks that ℳ^ is a module in 𝐤-mod_fil,+ over 𝒱^, therefore over 𝒲^, 
and that the image of this module structure under gr
is the module structure of ℳ^ over 𝒱^ and 𝒲^ in 
𝐤-mod_gr,+. 




 §.§ Completions


Extending the above definitions of F^k𝒳^ by F^k𝒳^:=𝒳^ for k<0 and 
𝒳∈{𝒱,𝒲,ℳ}, the pairs (𝒱^,Δ^𝒱,), 
(𝒲^,Δ^𝒲,) (resp. (ℳ^,Δ^ℳ,)) are Hopf algebras (resp. is a 
coassociative coalgebra) in 𝐤-mod_,+. Applying the functor (-)^∧:𝐤-mod_,+→𝐤-mod_ (see <ref>), one obtains topological Hopf algebras
(𝒱̂^V^B^@𝒱̂^,Δ̂^𝒱,DeltaVB^@Δ̂^𝒱,) and 
(𝒲̂^W^B^@𝒲̂^,Δ̂^𝒲,DeltaWB^@Δ̂^𝒲,), and a topological coalgebra 
(ℳ̂^M^B^@ℳ̂^,Δ̂^ℳ,DeltaMB^@Δ̂^ℳ,).
It follows from the compatibility of the decomposition (<ref>) 
with filtrations that ℳ̂^ is free of rank one as a 𝒲̂^-module. 

It follows from <ref>, Proposition <ref>, and Lemma <ref> that  
the image of the pair (𝒳^,Δ^𝒳,) by the functor gr:𝐤-mod_,+→𝐤-mod_,+ (see <ref>) is (𝒳^,Δ^𝒳,) 
for 𝒳∈{𝒱,𝒲,ℳ}. 



§ FAKE COMPARISON ISOMORPHISMS


In <ref> and <ref>, we introduced parallel `de Rham' and `Betti' structures, namely an inclusion 
of topological algebras 𝒲̂^↪𝒱̂^ (resp. 𝒲̂^↪𝒱̂^), and a surjective 𝒱̂^-module (resp. 𝒱̂^-module) morphism 
(-)· 1_:𝒱̂^↠ℳ̂^ (resp. (-)· 1_:𝒱̂^↠ℳ̂^), in which 𝒱̂^ (resp. 𝒱̂^) is the left regular module 
over itself. We introduce automorphisms of the de Rham algebra structures in <ref>, and of the de Rham module 
structures in <ref>. In <ref>, we define isomorphisms between the various Betti and 
de Rham structures. In <ref>, we relate these isomorphisms with the Betti-de Rham comparison isomorphisms, 
either of the fundamental group of 𝔐_0,4 with basepoint 1⃗, or of the fundamental groupoid of the same space 
with respect to the pair of basepoints (1⃗,0⃗). 



 §.§ The algebra automorphisms aut_(μ,g)^𝒱,(1), and 
aut_(μ,g)^𝒲,(1),


For (μ,g)∈𝐤^××𝒢(𝒱̂^), we denote by 
aut_(μ,g)^𝒱,(1), the automorphism of the topological 𝐤-algebra 
𝒱̂^ given by 

    aut_(μ,g)^𝒱,(1), : e_0↦ g·μ e_0· g^-1,   
    e_1↦μ e_1. autV1@aut_(μ,g)^𝒱,(1),

One checks that this automorphism restricts to an automorphism of the subalgebra 
𝒲̂^⊂𝒱̂^, which we denote 
aut_(μ,g)^𝒲,(1),autW1@aut_(μ,g)^𝒲,(1),. 



 §.§ The module automorphisms aut_(μ,g)^𝒱,(10), and 
aut_(μ,g)^ℳ,(10),


For (μ,g)∈𝐤^××𝒢(𝒱̂^). Denote by 
aut_(μ,g)^𝒱,(10), the automorphism of the topological 𝐤-module 
𝒱̂^ given by 

    ∀ v∈𝒱̂^,  aut_(μ,g)^𝒱,(10),(v):= 
    aut_(μ,g)^𝒱,(1),(v)· g. 
    autV10@aut_(μ,g)^𝒱,(10),

One checks that aut_(μ,g)^𝒱,(10), leaves the 𝐤-submodule 
𝒱̂^ e_0⊂𝒱̂^ invariant, and therefore induces an automorphism 
of the topological 𝐤-module ℳ̂^=𝒱̂^/𝒱̂^ e_0, 
which will be denoted aut_(μ,g)^ℳ,(10),autM10@aut_
(μ,g)^ℳ,(10),. 

One checks that for 𝒳∈{𝒱,ℳ} and 𝒜∈{𝒱,𝒲}, the 𝐤-module
automorphism aut_(μ,g)^𝒳,(10), is compatible (in the sense of <ref>) with the 
module structure of 𝒳̂^ over the algebra 𝒜̂^ and with the algebra automorphism 
aut_(μ,g)^𝒜,(1),. 



 §.§ The isomorphisms comp_(μ,Φ)^𝒱,(1), comp_(μ,Φ)^𝒲,(1), 
comp_(μ,Φ)^𝒱,(10) and comp_(μ,Φ)^ℳ,(10)


There is a unique isomorphism 

    iso^𝒱:𝒱̂^→𝒱̂^isoV@iso^𝒱

of topological 𝐤-algebras, defined by the condition X_i↦exp(e_i) for i=0,1. This isomorphism 
restricts to an isomorphism of topological 𝐤-algebras

    iso^𝒲:𝒲̂^→𝒲̂^.
    isoW@iso^𝒲

There is a commutative diagram 

    𝒱̂^^iso^𝒱[r]_-·(X_0-1)[d]   𝒱̂^^-·(exp(e_0)-1)[d]
    𝒱̂^^iso^𝒱[r]   𝒱̂^

The cokernels of the endomorphisms -· (exp(e_0)-1) and -· e_0 of 𝒱̂^ coincide as 
-· (exp(e_0)-1)=(-· e_0)∘(-·exp(e_0)-1e_0) and 
-·exp(e_0)-1e_0 is a linear automorphism. 
Taking vertical cokernels of the above diagram, we obtain a 𝐤-module isomorphism 

    iso^ℳ:ℳ̂^→ℳ̂^isoM@iso^ℳ

which is moreover compatible with the algebra isomorphism iso^𝒱. 

Let (μ,g)∈𝐤^××𝒢(𝒱̂^). 
Composing the algebra automorphism aut^𝒲,(1),_(μ,g) 
(resp. aut^𝒱,(1),_(μ,g)) of 
𝒲̂^ (resp. 𝒱̂^) with the algebra
isomorphism iso^𝒲 (resp. iso^𝒱), 
we obtain algebra isomorphisms 

    comp_(μ,g)^𝒲,(1)=aut^𝒲,(1),_(μ,g)∘iso^𝒲 : 
    𝒲̂^→𝒲̂^,
    compW1@comp_(μ,g)^𝒲,(1)  comp_(μ,g)^𝒱,(1)=aut^𝒱,(1),_(μ,g)∘iso^𝒱 : 
    𝒱̂^→𝒱̂^.
    compV1@comp_(μ,g)^𝒱,(1)

Composing the 𝐤-module automorphism aut_(μ,g)^𝒱,(10), (resp. 
aut_(μ,g)^ℳ,(10),) of 𝒱̂^ (resp. ℳ̂^)
with the 𝐤-module isomorphism iso^𝒱 (resp. iso^ℳ), one obtains 
𝐤-module isomorphisms  

    comp_(μ,g)^𝒱,(10):=aut_(μ,g)^𝒱,(10),∘iso^𝒱:𝒱̂^→𝒱̂^,
    compV10@comp_(μ,g)^𝒱,(10)  comp_(μ,g)^ℳ,(10):=aut_(μ,g)^ℳ,(10),∘iso^ℳ:ℳ̂^→ℳ̂^,
    compM10@comp_(μ,g)^ℳ,(10)
 
which are compatible with the previous algebra isomorphisms. 

 Let (μ,g)∈𝐤^××𝒢(𝒱̂^). Then 

    ∀ v^∈𝒱̂^,  comp_(μ,g)^𝒱,(10)(v^)=
    comp_(μ,g)^𝒱,(1)(v^)· g.
 


This follows from the definitions of comp_(μ,g)^𝒱,(1) (see second part of 
(<ref>)), of comp_(μ,g)^𝒱,(10) (see first part of 
(<ref>)), and from (<ref>), with v being replaced by 
iso_𝒱(v^). 


Let (μ,g)∈𝐤^××𝒢(𝒱̂^). Then the following diagram commutes

    𝒱̂^^comp_(μ,g)^𝒱,(10)[rr]_(-)· 1_[d]      𝒱̂^^(-)· 1_[d]
    ℳ̂^_comp_(μ,g)^ℳ,(10)[rr]      ℳ̂^



This follows from comp_(μ,g)^𝒳,(10)=aut_(μ,g)^𝒳,(10),∘iso^𝒳 for 𝒳∈{𝒱,ℳ}, from iso^ℳ∘((-)· 1_)=
((-)· 1_)∘iso^𝒱, and from aut^ℳ,(10),_(μ,g)∘((-)· 1_)
=((-)· 1_)∘aut^𝒱,(10),_(μ,g) (see <ref>). 



 §.§ Relation with comparison isomorphisms




  §.§.§ 
 Recall from 0.1.1 the notation 𝒯=MT(ℤ)_ℚ and the motivic Galois group 
schemes 𝖠𝗎𝗍_𝒯^⊗(ω_?)  for ?∈{,} (<cit.>). By <cit.>, there is an inclusion of 
ℚ-group schemes 𝖠𝗎𝗍_𝒯^⊗(ω_?)⊂𝖦^?, where 
𝖦^? is the ℚ-group scheme defined in <cit.>, 1.6 for ?=DR and <cit.>, 2.1 for ?=B.



  §.§.§ 
  For 𝖧 a ℚ-group scheme, denote by 𝖧-mod the tensor category of its finite-dimensional linear representations. An object of 𝖧-mod is then a pair (V,ρ) where V is a finite dimensional ℚ-vector 
space and ρ : 𝖧→𝖦𝖫(V) is a morphism of ℚ-group schemes. For ?∈{,}, there is a 
sequence of tensor functors 𝖦^?-modres_?→𝖠𝗎𝗍_𝒯^⊗(ω_?)-mod≃𝒯 where the first functor is induced by restriction and the second is the Tannaka equivalence (see <cit.>, 
2.1). Its composition with ω_? is the forgetful functor forget:𝖦^?-mod→Vec_ℚ.  



  §.§.§ 
  
For ?∈{,}, set 𝒱̂^?,(1)_ℚ:=(𝒱̂^?_ℚ,(μ,g)↦aut^𝒱,(1),?_(μ,g)) and 𝒱̂^?,(10)_ℚ:=
(𝒱̂^?_ℚ,(μ,g)↦aut^𝒱,(10),?_(μ,g)) (see <ref>
and <ref> for ?=DR and <cit.>, 2.2 for ?=B). By <cit.>, 1.6 (?=DR), and <cit.>, 2.2 (?=B), 
these are pro-objects in 𝖦^?-mod. Denote by m_? the product of 𝒱̂^?_ℚ and 
recall that Δ̂^𝒱,? denotes its 
coproduct. The category Pro(𝖦^?-mod) of pro-objects in 𝖦^?-mod is equipped with 
a tensor product, and (m_?,Δ̂^𝒱,?) defines a Hopf algebra structure on 
𝒱̂^?,(1)_ℚ in Pro(𝖦^?-mod), as well as a module-coalgebra 
structure over it on 𝒱̂^?,(10)_ℚ in the same category (this meaning that 
𝒱̂^?,(10)_ℚ is a left 𝒱̂^?,(1)_ℚ-module and 
a coassociative coalgebra, the left action morphism being a coalgebra morphism), both Hopf algebra and coalgebra structures
being cocommutative. 



  §.§.§ 
 Denote by M^∨ the dual of an object M of Pro(𝖦^?-mod); this is an object in 
the category Ind(𝖦^?-mod) of ind-objects in 𝖦^?-mod. Then 
(𝒱̂^?,(1)_ℚ)^∨ and (𝒱̂^?,(10)_ℚ)^∨ 
are objects in Ind(𝖦^?-mod); in this category, (𝒱̂^?,(1)_ℚ)^∨ is a commutative 
Hopf algebra and (𝒱̂^?,(10)_ℚ)^∨ is a commutative algebra-comodule over it. 



  §.§.§ 
 
In the rest of 3.4, we set X:=ℙ^1-{0,1,∞} and denote by 0,1 the vectors d/dx at points 0,1 of 
ℙ^1 (with coordinate z=x+iy). These are base points at infinity in X.  For x,y∈{0,1}, let P_y,x(X) be 
the motivic path space from x to y (see [DG], 4.3). 



  §.§.§ 
 
The images of (𝒱̂^?,(1)_ℚ)^∨ and (𝒱̂^?,(10)_ℚ)^∨ 
in Ind(𝒯) by Ind(res_?) are respectively isomorphic to 𝒪(P_1,1(X)) and 
𝒪(P_1,0(X)) in the notation of [DG], Rem. 4.23, the former being equipped with its commutative Hopf algebra 
structure in Ind(𝒯) and the latter  with its commutative algebra-module structure over it, both structures
corresponding to the 𝒯-group scheme structure of P_1,1(X) and to the 𝒯-torsor structure over it of 
P_1,0(X) (see [DG], 5). 



  §.§.§ 
 
For x,y∈{0,1}, prounipotent ℚ-group schemes π_1^(X,x) and 
π_1^alg,unip(X,x) are defined in [De], 10.5, as well as 

torsors π_1^(X,y,x) and π_1^alg,unip(X,y,x);  

we use the notation π_1^ instead of π_1^alg,unip. If ?∈{,}, then the functor 
ω_? :𝒯→Vec_ℚ takes the  𝒯-group scheme (resp. 𝒯-torsor) 
P_x,x(X) (resp. P_y,x(X)) to π_1^?(X,x) (resp. π_1^?(X,y,x)), so that forget takes the Hopf algebra and 
comodule algebra 𝒱̂^?,(1) and 𝒱̂^?,(10) in Pro(𝖦^?-mod) to 
the objects with the similar structures 𝒪(π_1^?(X,1))^∨ and 𝒪(π_1^?(X,1,0))^∨ in Pro(Vec_ℚ), 𝒪(-) being the ring of regular functions over a scheme. Then 
comp^𝒱,(1)_(2πi,φ_KZ) and 
comp^𝒱,(10)_(2πi,φ_KZ) can respectively be identified with the Betti-de Rham 
comparison algebra and module isomorphisms  ULieπ_1^(X,1)=𝒪(π_1^(X,1))^∨∼→𝒪(π_1^(X,1))^∨=ULieπ_1^(X,1)
and 𝒪(π_1^(X,1,0))^∨∼→𝒪(π_1^(X,1,0))^∨ given by [De], 12.16
(here ULie(-) is the cocommutative topological Hopf algebra attached to a ℚ-group scheme). The 
isomorphisms comp^𝒱,(1)_(μ,g) and comp^𝒱,(10)_(μ,g), for
(μ,g)∈𝐤^××𝒢(𝒱̂^) can therefore be viewed as `fake comparison isomorphisms'.

























PART: 
Geometric interpretation of double shuffle theory

The purpose of this part is to construct geometric interpretations of the `de Rham' and `Betti' DSR formalisms introduced in Part 
<ref>. These interpretations rely on general algebraic constructions which are explained in <ref>. 
Then <ref> and <ref> are devoted to the de Rham DSR formalism: we introduce geometric objects 
related with the de Rham fundamental groups in <ref>, namely infinitesimal braid Lie algebras and morphisms relating 
them; they are used in <ref> to give a geometric interpretation of the de Rham harmonic algebra coproduct following 
<cit.>, as well as of its module counterpart. <ref> and <ref> follow the same pattern on the Betti side: 
introduction of geometric objects related with the Betti fundamental groups in <ref>, namely braid groups and morphisms 
relating them, and geometric interpretations of the  Betti harmonic algebra and module coproducts in <ref>. 



§ ALGEBRAIC PRELIMINARIES FOR PART <REF>


In this section, we prove algebraic results that will be used in <ref> and <ref>. 
The first result allows for the construction of an algebra morphism starting from an algebra containing an ideal with 
freeness properties (<ref>, Lemma <ref>). The second result describes the behavior of 
this construction under isomorphisms (<ref>, Lemma <ref>). The third result 
shows how an algebra morphism may be constructed starting from an algebra morphism to a matrix algebra satisfying some conditions
(<ref>, Lemma <ref>). 
 


 §.§ Construction of algebra morphisms based on ideals with freeness properties




Let R be an associative algebra 
and let J⊂ R be a two-sided ideal. Assume that
(j_a)_a∈[[1,d]] is a family of elements of J, which constitutes a basis of J for its 
left R-module structure. For r∈ R, let (m_ab(r))_a,b∈[[1,d]] 
be the collection of elements of R defined by j_ar=∑_b=1^d m_ab(r)j_b. 

Then the map ϖ_R,J : R→ M_d(R)piRJ@ϖ_R,J, 
r↦ (m_ab(r))_a,b∈[[1,d]] is an algebra morphism.  


Let R^⊕ d→ J be the map (r_1,…,r_d)↦∑_a=1^d r_aj_a. This is an isomorphism of left 
R-modules. It sets up an isomorphism of algebras End_R-left(J)≃End_R-left(R^⊕ d), 
where the index "R-left" means endomorphisms of left R-modules. 
The map R→End_R-left(J), r↦(j↦ jr) is an algebra morphism when R is equipped 
with the opposite algebra structure. Similarly, the map M_d(R)→End_R-left(R^⊕ d) taking M∈ M_d(R) 
to the image under the canonical isomorphism R^⊕ d≃ M_1× d(R) of the endomorphism X↦ XM 
is an algebra isomorphism when M_d(R) is equipped with the opposite algebra structure. We then obtain a sequence of algebra 
morphisms 

    R^op→End_R-left(J)≃End_R-left(R^⊕ d)≃ M_d(R)^op,
 
which yields an algebra morphism R→ M_d(R). One checks that this morphism is given by the announced formula. 


Let R be an associative algebra, 
let J⊂ R be a two-sided ideal, free as a left 
R-module with basis (j_a)_a∈[[1,d]]. 
Let f:R→ R' be an algebra isomorphism, and let (j'_a)_a∈[[1,d]] be a basis of f(J)⊂ R' as a left R'-module. 

Then there exists a unique element P∈GL_d(R'), such that 
[ f(j_1);      ⋮; f(j_d) ]=P[ j'_1;    ⋮; j'_d ]
(equality in M_d× 1(R')). 
The morphisms ϖ_R,J:R→ M_d(R) 
and ϖ_R',J':R'→ M_d(R')
respectively attached to the data (J⊂ R,(j_a)_a∈[[1,d]]) and (J'⊂ R',(j'_a)_a∈[[1,d]])
as in Lemma <ref> are related by the following diagram 

    R^ϖ_R,J[rr]_f[d]       M_d(R)^M_d(f)[d]
     R'_ϖ_R',J'[r]    M_d(R')_Ad(P)[r]    M_d(R')



For r∈ R, one has [ j_1;   ⋮; j_d ]r
=_R,J(r)[ j_1;   ⋮; j_d ], therefore  
[ f(j_1);      ⋮; f(j_d) ]f(r)
=f(_R,J(r))[ f(j_1);      ⋮; f(j_d) ], that is 
P[ j'_1;    ⋮; j'_d ]f(r)
=f(_R,J(r))P[ j'_1;    ⋮; j'_d ]. Comparing this with the identity 
[ j'_1;    ⋮; j'_d ] r'
=_R',J'(r')[ j'_1;    ⋮; j'_d ] for r'∈ R', one obtains _R',J'(r')=P^-1· 
f(_R,J(f^-1(r')))· P. 

 §.§ Construction of algebra morphisms based on morphisms to matrix algebras



Let R be an associative algebra 
and let e∈ R. Define ·_e:R× R→ R
by r·_e r':=rer'. Then (R,·_e) is an associative algebra. 
The subspaces Re and eR of R are subalgebras. There is an algebra morphism 
mor_R,e:(R,·_e)→ RemorRe@mor_R,e, 
given by r↦ re.


Let R,S be associative algebras, 
let e∈ R, let n≥ 1, and let f:R→ M_n(S) be an algebra 
morphism. 
Assume that there exist elements row∈ M_1× n(S)row @ row and col∈ M_n× 1(S)col @ col, such that 
f(e)=col·row. 

 Then the map f̃:(R,·_e)→ S, defined by r↦row· f(r)·col
is an algebra morphism. One then has a commutative diagram

    R_≃[d]^f[r]   M_n(S)^row·(-)·col[d] 
     (R,·_e)_f̃[r]   S
 
where 

    row·(-)·col: M_n(S)→ S

is the linear map defined by m↦row· m·col; in diagram (<ref>), 
the horizontal arrows are algebra morphisms and the vertical arrows are vector space morphisms. 


For r,r'∈ R, one has 

    f̃(r·_e r')=f̃(rer')=row· f(rer')·col=row· f(r)f(e)f(r')·col
        = row· f(r)·col·row· f(r')·col=f̃(r)f̃(r').





§ INFINITESIMAL BRAID LIE ALGEBRAS


In <ref>, we recall the definition of the infinitesimal braid Lie algebras 
𝔱_n, 𝔭_n (n≥1), as well as the morphisms ℓ:𝔭_4→𝔭_5, 
pr_i:𝔭_5→𝔭_4 (i∈[[1,5]]) and pr_12:𝔭_5→(𝔭_4)^⊕2
relating them. 

In <ref>, we introduce an ideal J(pr_5) of the universal enveloping algebra U𝔭_5 arising from 
pr_5, and show its freeness as a left U𝔭_5UP_5@U(𝔭_5)-module (Lemma <ref>). 
Lemma <ref> then gives rise to an algebra morphism ϖ:U𝔭_5→ M_3(U𝔭_5). 
By composing ϖ with the morphisms ℓ and pr_12, we construct a morphism 
ρ:𝒱^→ M_3((𝒱^)^⊗2). In Lemma <ref>, by introducing suitable 
elements row_1∈ M_1×3((𝒱^)^⊗2) and 
col_1∈ M_3×1((𝒱^)^⊗2), we show that 
the morphism ρ satisfies the hypothesis of Lemma <ref> (the index 1 in 
the notation row_1 and col_1 is motivated by the fact that these elements should be thought of as associated to the tangential base point 1⃗ in 𝔐_0,4). Applying this lemma, we obtain in 
<ref> a morphism :(𝒱^,·_e_1)→ (𝒱^)^⊗2, which we
compute in Lemma <ref>. 



 §.§ Material on infinitesimal braid Lie algebras




  §.§.§ The Lie algebras 𝔱_n,𝔭_n



For n≥2, let 𝔱_nt_n @𝔱_n be the graded Lie 𝐤-algebra with generators t_ijt_ij @t_ij, i≠ j∈[[1,n]] of degree 1, 
and relations t_ji=t_ij, [t_ij,t_ik+t_jk]=0 for i,j,k all different in [[1,n]], and [t_ij,t_kl]=0 for i,j,k,l 
all different in [[1,n]]. The Lie algebra 𝔱_n is called the Drinfeld-Kohno, or infinitesimal braid, Lie algebra. 

For n≥ 4, we denote by 𝔭_np_nfrak@𝔭_n the graded Lie 𝐤-algebra with generators e_ije_ij @e_ij of degree 1, where i≠ j∈[[1,n]]
and relations e_ji=e_ij, ∑_j∈[[1,n]]-{i}e_ij=0, [e_ij,e_kl]=0 for i,j,k,l all distinct in [[1,n]]. 
The Lie algebra 𝔭_n is called the sphere infinitesimal braid Lie algebra. 

For n≥ 3, there are surjective morphisms of graded Lie algebras 𝔱_n↠𝔭_n+1 given by 
t_ij↦ e_ij for i≠ j∈[[1,n]],  𝔱_n+1↠𝔭_n+1 given by t_ij↦ e_ij 
for i≠ j∈[[1,n+1]], and an injective morphism 𝔱_n↪𝔱_n+1 given by t_ij↦ t_ij 
for i≠ j∈[[1,n]]; they fit in a commutative diagram 

    𝔱_n@->>[r]@^(->[d]   𝔭_n+1
    𝔱_n+1@->>[ur]

Moreover, the morphism 𝔱_n→𝔭_n+1 factorizes as 𝔱_n↠𝔱_n/Z(𝔱_n)≃𝔭_n+1, where Z(𝔱_n) is the 1-dimensional
center of 𝔱_n (it is concentrated in degree 1 and spanned by ∑_i<j∈[[1,n]]t_ij). 


Let P_nP_n@P_n (resp. K_nK_n @K_n) be the pure sphere (resp. Artin pure) braid group with n strands (see <ref> and 
<cit.>).  Its lower central series defines a descending group filtration. The associated graded ℤ-module is a 
ℤ-Lie algebra. Then gr(P_n)⊗𝐤≃𝔭_n, and 
gr(K_n)⊗𝐤≃𝔱_n. 




  §.§.§ The morphisms ℓ,pr_i and pr_12 between infinitesimal braid Lie algebras



There is a graded 
Lie algebra isomorphism
𝔭_4≃𝔣_2, where e_0=e_14=e_23, e_1=e_12=e_34. One also sets  
e_∞:=-e_0-e_1e_infty@e_∞, so e_∞=e_13=e_24. 
By abuse of notation, and following <cit.>, we set  

    ∀ i∈{0,1}, e_i:=(e_i,0)∈(𝔣_2)^2,      f_i:=(0,e_i)∈(𝔣_2)^2. f _0, f_1@ f_0, f_1
 
We then have e_i=e_i⊗1, f_i=1⊗ e_i in (𝒱^)^⊗2. 

One checks that there are Lie algebra morphisms pr_i:𝔭_5→𝔭_4pr 125@pr_1, pr_2, pr_5 for i=1,2,5, given by 



  elt x∈𝔭_5     e_12     e_13     e_14     e_15     e_23    e_24    e_25    e_34    e_35    e_45 

  pr_1(x)     0     0     0     0     e_0     e_∞     e_1     e_1     e_∞     e_0 
 
pr_2(x)     0     e_∞     e_0     e_1     0     0     0      e_1      e_0     e_∞ 

  pr_5(x)     e_1     e_∞     e_0     0     e_0     e_∞     0     e_1     0     0 
 



These morphisms give rise to the Lie algebra morphism 
pr_12:𝔭_5→𝔭_4^⊕2pr 12t@ pr_12 defined 
by pr_12(x):=(pr_1(x),pr_2(x)). 

There is a Lie algebra morphism ℓ:𝔭_4→𝔭_5 l @ℓ, given by 
e_0↦ e_23, e_1↦ e_12. It is such that pr_5∘ℓ
is the identity of 𝔭_4.

The isomorphism 𝔭_4≃𝔣_2 gives rise to an isomorphism 
U𝔭_4≃ U𝔣_2=𝒱^. 

The Lie algebra morphisms pr_12, pr_i and ℓ induce algebra morphisms between 
universal enveloping algebras, which will be denoted  
pr_12 :U𝔭_5
→ (𝒱^)^⊗2, 
 pr_i : U𝔭_5
 →𝒱^ and ℓ : 𝒱^→ U𝔭_5. 



 §.§ Constructions related to an ideal of U𝔭_5




  §.§.§ The structure of J(pr_5)


 We denote by J(pr_5)Jpr_5 @ J(pr_5) the kernel 
Ker(U𝔭_5
pr_5→𝒱^). 
This is a two-sided ideal of U𝔭_5.


In order to study the structure of J(pr_5), we prove the following Lemma <ref>. 


1) The Lie subalgebra of 𝔭_5 generated by the e_i5, i∈[[1,4]] is freely generated by the 
e_i5, i∈[[1,3]], and coincides with the ideal Ker(pr_5) of 𝔭_5; it will be denoted as
𝔣_3. 

2) Set 𝔭̃_4:=im(ℓ:𝔭_4→𝔭_5).
There is a direct sum decomposition 𝔭_5=𝔭̃_4⊕𝔣_3.  


  
1) follows from <cit.>, 1.1 with n=i=5 (the notation for 
𝔣_3 in loc. cit. is N_n). 
2) follows  from the facts that 𝔣_3=Ker(pr_5) and that 
pr_5∘ℓ=id_𝔭_4.  


Using <cit.>, one can prove that the centralizer of e_45 in 𝔭_5
decomposes as a direct sum 𝐤 e_45⊕𝔭̃_4.


 
The map (U𝔭_5)^⊕3→ J(pr_5), (p_i)_i∈[[1,3]]↦∑_i∈[[1,3]]p_i· e_i5 
is an isomorphism of left U𝔭_5-modules. 


The equality 𝔭_5=𝔭̃_4⊕𝔣_3 is a decomposition of the Lie algebra 𝔭_5 
as a direct sum of two Lie subalgebras. The tensor product of ℓ:U𝔭_4→ U𝔭_5 with the injection
U𝔣_3→ U𝔭_5, followed by the product in U𝔭_5, 
induces a linear map 

    codec : U𝔭_4⊗ U𝔣_3→ U𝔭_5.

This map is compatible with the PBW filtrations on both sides, and its associated graded map is the linear map 
S𝔭_4⊗ S𝔣_3→ S𝔭_5, which is an isomorphism of graded vector spaces, 
so that codec is an isomorphism of filtered vector spaces. 

The following diagram 

    U𝔭_4⊗ U𝔣_3^codec[rr]_id⊗ε[rrd]        
    U𝔭_5^pr_5[d]
    
           U𝔭_4

is commutative, where ε is the counit of U𝔣_3. Indeed, for p∈ U𝔭_4
and f∈ U𝔣_3, 

    pr_5∘codec(p⊗ f)=pr_5(ℓ(p)f)
    =(pr_5( ℓ(p)))·pr_5(f)=p·ε(f),

where the first equality follows from the definition of codec, 
the second equality follows from the algebra morphism property of 
pr_5, and the third equality follows from the facts that pr_5∘ℓ=id_𝔭_4 
and that 
𝔣_3=Ker(pr_5:𝔭_5→𝔭_4). 
This diagram implies that J(pr_5) is equal to the isomorphic image by codec
of the subspace U𝔭_4⊗ (U𝔣_3)_+ of U𝔭_4⊗ U𝔣_3, where 
(U𝔣_3)_+:=Ker(ε : U𝔣_3→𝐤) is the augmentation ideal of U𝔣_3, 
that is 

    J(pr_5)
    =im(U𝔭_4⊗ (U𝔣_3)_+codec→ U𝔭_5).


Since the e_i5, i∈[[1,3]], belong to 𝔣_3, the following diagram commutes

    (U𝔭_4⊗ U𝔣_3)^⊕ 3_codec^⊕3[d]^∼[r]
       U𝔭_4⊗ (U𝔣_3)^⊕ 3[r]    U𝔭_4⊗ U𝔣_3^codec[d]
     
    (U𝔭_5)^⊕ 3[rr]       U𝔭_5
 
where the lower horizontal map is given by (p_i)_i∈[[1,3]]↦∑_i∈[[1,3]]p_ie_i5, and the upper horizontal map
is given by the tensor product of the identity in U𝔭_4 with the map (U𝔣_3)^⊕ 3→ U𝔣_3, 
(φ_i)_i∈[1,3]↦∑_i∈[[1,3]]φ_ie_i5. 

Since U𝔣_3 is freely generated, as an associative algebra, by the e_i5, i∈[[1,3]], the latter map 
corestricts to a 𝐤-module isomorphism (U𝔣_3)^⊕ 3→ (U𝔣_3)_+. It follows
that the upper horizontal map of (<ref>) corestricts to an isomorphism 
U𝔭_4⊗ (U𝔣_3)^⊕ 3→ U𝔭_4⊗ (U𝔣_3)_+. 

Since the vertical maps of (<ref>) are isomorphisms, this implies that the lower 
horizontal map of (<ref>) corestricts to an isomorphism from (U𝔭_5)^⊕ 3
to the image by codec of U𝔭_4⊗ (U𝔣_3)_+, which is J(pr_5)
according to (<ref>). 



  §.§.§ A morphism :U𝔭_5→ M_3(U𝔭_5)


Lemma <ref> says that the hypothesis of Lemma <ref> is satisfied 
in the following situation: R=U𝔭_5, J=J(pr_5), d=3, (j_a)_a∈[[1,d]]=(e_i5)_i∈[[1,3]]. 
We denote by 

    :U𝔭_5→ M_3(U𝔭_5)
    pivar@

the algebra morphism given in this situation by Lemma <ref>. Then for p∈ U𝔭_5, 
(p)=(a_ij(p))_i,j∈[[1,3]], and 

    ∀ i∈[[1,3]],   e_i5p=∑_j∈[[1,3]]a_ij(p)e_j5

(equalities in U𝔭_5). 



  §.§.§ Construction and properties of a morphism :𝒱^→ M_3((𝒱^)^⊗2)



Define the algebra morphism 

    :𝒱^→ M_3((𝒱^)^⊗2)
    rhovar@

to be the composition 

    𝒱^≃ U𝔭_4ℓ→ U𝔭_5→
    M_3(U𝔭_5)M_3(pr_12)→
    M_3((U𝔭_4)^⊗2)≃ M_3((𝒱^)^⊗2),

where ℓ is as in <ref>,  is as in <ref>, 
and M_3(pr_12)M_3pr_12@M_3(pr_12) is the  
morphism induced by pr_12, i.e., taking (p_ij)_i,j∈[[1,3]] to (pr_12(p_ij))_i,j∈[[1,3]]. 


Set 

    row_1:=[  e_1 -f_1    0 ]∈ M_1× 3((𝒱^)^⊗2),  col_1:=[  1; -1;  0 ]∈ M_3× 1((𝒱^)^⊗2)
    row1 @ row_1col1 @ col_1

(recall that 
e_1,f_1∈ (𝒱^)^⊗2 are e_1⊗1,1⊗ e_1), then 

    (e_1)=col_1·row_1

(equality in M_3((𝒱^)^⊗2)). 


One has ℓ(e_1)=e_12. Let us compute (e_12). One has 

    e_15e_12=e_12e_15+[e_15,e_12]=e_12e_15+[e_25,e_15]=(e_12+e_25)e_15-e_15e_25, 
        e_25e_12=-e_25e_15+(e_12+e_15)e_25
    (applying the permutation of indices 1 and 2 to the previous equality),     e_35e_12=e_12e_35,

which implies that 

    (e_12)=[ e_12+e_25     -e_15         0;     -e_25 e_12+e_15         0;         0         0      e_12 ]∈ M_3(U𝔭_5).

The image of this matrix in M_3((𝒱^)^⊗2) is 

    (e_1)=[  e_1 -f_1    0; -e_1  f_1    0;    0    0    0 ]=[  1; -1;  0 ][  e_1 -f_1    0 ]=col_1·row_1.

Therefore (e_1)=col_1·row_1. 




  §.§.§ Construction and properties of a morphism :(𝒱^,·_e_1)→(𝒱^)^⊗2



Lemma <ref> shows that the hypothesis of Lemma <ref> is satisfied in the following situation: 
R=𝒱^, S=(𝒱^)^⊗2, e=e_1, n=3, f=, row_1 and col_1 are as in Lemma 
<ref>. We denote by

    :(𝒱^,·_e_1)→ (𝒱^)^⊗2rhovar^tilde@

the algebra morphism given in this situation by Lemma <ref>. 

Then for any f∈𝒱^, one has 

    (f)=row_1·(f)·col_1=row_1·{M_3(pr_12)∘∘ℓ(f)}·col_1
    ∈ (𝒱^)^⊗2.



For any n≥ 0, 

    (e_0^n)=e_1e_0^n+f_1f_0^n-∑_i=0^n-1(e_1e_0^i)·(f_1f_0^n-1-i)

(equality in (𝒱^)^⊗2=(U𝔣_2)^⊗2). 


According to (<ref>), (e_0^n)=row_1·(e_0^n)·col_1. 
As  is an algebra morphism, (e_0^n)=(e_0)^n. Then (e_0)=M_3(pr_12)∘∘ℓ(e_0)
=M_3(pr_12)((e_23)). 

Let us compute (e_23). One has 

    e_15e_23=e_23e_15, 
        e_25e_23=e_23e_25+[e_25,e_23]=e_23e_25+[e_35,e_25]=(e_23+e_35)e_25-e_25e_35, 
        e_35e_23=-e_35e_25+(e_23+e_25)e_35
    
    (applying the permutation of indices 2 and 3 to the previous equality),

which implies that 

    (e_23)=[      e_23         0         0;         0 e_23+e_35     -e_25;         0     -e_35 e_23+e_25 ]∈ M_3(U𝔭_5).

Then  

    (e_0)=M_3(pr_12)((e_23))=
    [         e_0           0           0;           0    -e_1+f_0        -e_1;           0 e_0+e_1-f_0     e_0+e_1 ]∈ M_3((𝒱^)^⊗2).

Set TT@T:=[    -e_1+f_0        -e_1; e_0+e_1-f_0     e_0+e_1 ]∈ M_2((𝒱^)^⊗2), 
then (e_0^n)=[ e_0^n     0;     0   T^n ], therefore 

    (e_0^n)=row_1·(e_0^n)·col_1=e_1e_0^n+[ -f_1    0 ] T^n
    [ -1;  0 ],

where the last equality follows from the form of row_1 and col_1. 

One checks that T=[  1  0; -1  1 ][  f_0 -e_1;    0  e_0 ][  1  0; -1  1 ]^-1, therefore 

    T^n=[  1  0; -1  1 ][  f_0 -e_1;    0  e_0 ]^n 
    [ 1 0; 1 1 ]=[  1  0; -1  1 ][                        f_0^n -∑_i=0^n-1
    f_0^ie_1e_0^n-1-i;                            0                        e_0^n ][ 1 0; 1 1 ],

so 

    (e_0^n)=e_1e_0^n+[ -f_1    0 ][                        f_0^n -∑_i=0^n-1
    f_0^ie_1e_0^n-1-i;                            0                        e_0^n ][ -1; -1 ]=e_1e_0^n+f_1f_0^n-∑_i=0^n-1
    f_1f_0^ie_1e_0^n-1-i;

the result then follows from the commutativity of e_s with f_t for s,t∈{0,1}.  



If 𝒜 is a unital associative algebra, then the following identity holds in M_2(𝒜)

    Ad([ 1 0; a 1 ])([ u v; 0 w ])
    =Ad([    1 a^-1;    0    1 ])([    a^-1wa         0; au-wa-ava    aua^-1 ])

provided 
u,v,w∈𝒜 and a∈𝒜^×. This implies 
the identity 

    T=[  1 -1;  0  1 ][         e_0           0; e_0+e_1-f_0         f_0 ][  1 -1;  0  1 ]^-1,
 allowing for an alternative computation of T^n.  




§ GEOMETRIC INTERPRETATION OF THE DE RHAM HARMONIC COPRODUCTS


In this section, we construct commutative diagrams relating the de Rham harmonic algebra and module coproducts
Δ^𝒲, and Δ^ℳ, with infinitesimal braid Lie algebras (diagrams 
(<ref>) and (<ref>)). 

These diagrams involve a localization 𝒱^[1 e_1] of the algebra 𝒱^, and 
a module ℳ^[1 e_1] over this algebra, which are introduced in 
<ref>. 

We prove the commutativity of the first diagram in <ref>: we first construct a
diagram relating Δ^𝒲, and the morphism ρ̃ from <ref>
(Lemma <ref>), and derive from there the commutativity of diagram (<ref>) relating 
Δ^𝒲, and ρ and row_1,col_1 (Proposition <ref>). 
The material in <ref> is inspired by <cit.>, more specifically by 6.3 and Proposition 6.2 in that paper. 

In <ref>, we introduce a column vector col_0 (the index 0 indicates that it should be thought of as 
associated to the tangential base point 0⃗ of 𝔐_0,4). We then prove the existence of a 
map δ:ℳ^→(ℳ^[1 e_1])^⊗2 fitting in a commutative 
diagram involving col_0; using diagram (<ref>), we then identify δ with 
Δ^ℳ,, which establishes diagram (<ref>) (Proposition <ref>).  

We finally construct completions of the diagrams (<ref>) and  (<ref>) 
in <ref>.  



 §.§ Localizations
 

Define 𝒱^[1/e_1]V^DR/e_1@𝒱^[1/e_1] to be the localization of 𝒱^ with respect to e_1, i.e. the unital 𝐤-algebra with generators 
e_0 and e_1^±1 and relations e_1e_1^-1=e_1^-1e_1=1; 
𝒱^[1/e_1]  is then equipped with a ℤ-grading given by deg(e_i)=1 for i=0,1. 
It follows that 𝒱^[1/e_1] in an algebra in 𝐤-mod_ (see Definition <ref>). 

Set also ℳ^[1/e_1]:=𝒱^[1/e_1]/𝒱^[1/e_1] e_0M^DR/e_1@ℳ^[1/e_1]. This is a 
ℤ-graded 𝒱^[1/e_1]-module, therefore ℳ^[1/e_1] is a 
𝒱^[1/e_1]-module in 𝐤-mod_. 

One checks that the natural ℤ-graded 𝐤-algebra morphism 𝒱^→𝒱^[1/e_1] 
and 𝐤-module morphism ℳ^→ℳ^[1/e_1] are injective. 



 §.§ Relationship between infinitesimal braid Lie algebras and Δ^𝒲,


Denote by 𝐤[e_0] the linear span in 𝒱^ of the elements e_0^n, n≥0. 


(𝒱^,·_e_1)
V^DR, e_1@(𝒱^,·_e_1)
is generated, as an associative (non-unital) algebra, by 
𝐤[e_0].  


For k≥1, the k-th power of the canonical injection, followed by the k-th fold product ·_e_1, sets up a linear
map 𝐤[e_0]^⊗ k→𝒱^.  One checks that this map is injective and that its image coincides with 
the part of 𝒱^ of e_1-degree equal to k-1. So the composition 
⊕_k≥1𝐤[e_0]^⊗ k→⊕_k≥1(𝒱^)^⊗ k→𝒱^, 
where the first map is the canonical injection and the second map is the iteration of the product ·_e_1, maps 
⊕_k≥1𝐤[e_0]^⊗ k injectively (in fact, bijectively) to 𝒱^. 

Recall that 𝒲^ is the subalgebra of 𝒱^ equal to 𝐤⊕𝒱^ e_1
(see <ref>). We set 

    𝒲^_+:=𝒱^ e_1.
    W^DR+@𝒲^_+

This is a (non-unital) subalgebra of 𝒲^. 

Since the right multiplication by e_1 is injective in 𝒱^, the algebra morphism 

    mor_𝒱^,e_1:(𝒱^,·_e_1)→𝒲^_+
    morVDRe1@ mor_𝒱^,e_1

(see <ref>) is an algebra isomorphism. 
One also checks that the algebra automorphism Ad(e_1^-1) of 𝒱^[1 e_1] restricts to an algebra morphism 
𝒱^→𝒱^[1 e_1]_≥0  . 


The following diagram is commutative

    (𝒱^,·_e_1) ^Ad(e_1f_1)^-1∘[rr]_mor_𝒱^,e_1^≃[d]       
    (𝒱^[1 e_1]_≥0)^⊗2
    𝒲^_+_Δ^𝒲,[rr]        (𝒲^)^⊗2@^(->[u]

where the top horizontal map is the composition of ρ̃ with the tensor square of the morphism
Ad(e_1^-1):𝒱^→𝒱^[1 e_1]_≥0  
and the right vertical map is the tensor square of the injection 𝒲^↪𝒱^↪𝒱^[1 e_1]_≥0. 


For n≥1, 

    Δ^𝒲,∘mor_𝒱^,·_e_1(e_0^n)
    =Δ^𝒲,(e_0^ne_1)=Δ^𝒲,(-y_n+1)
        =-y_n+1⊗1-1⊗ y_n+1-∑_i=1^n y_i⊗ y_n+1-i
        =e_0^ne_1⊗1+1⊗ e_0^ne_1-∑_i=1^n e_0^i-1e_1⊗ e_0^n-ie_1=Ad(e_1f_1)^-1∘(e_0^n),

where the third equality follows from (<ref>), 
and the last equality follows from Lemma <ref>.  

It follows that the two maps of the above diagram agree on e_0^n, n≥1. Since these maps are algebra morphisms, 
and since the family e_0^n, n≥0 generates (𝒱^,·_e_1) (see Lemma <ref>), 
this diagram commutes. 



The following diagram commutes

    𝒱^_≃^♢[d]^ρ[r]    
    M_3((𝒱^)^⊗2)^(e_1f_1)^-1row_1·(-)·col_1e_1f_1_♢♯[rrrr]            
    (𝒱^[1 e_1]_≥0)^⊗2
    
    (𝒱^,·_e_1)_mor_𝒱^,e_1^≃♯[d]
                   
    𝒲^_+_Δ^𝒲,[rrrrr]                (𝒲^)^⊗2@^(->[uu]

where ρ is as in (<ref>), 
row_1,col_1 are as in (<ref>), and Δ^𝒲, is as in <ref>; 
in this diagram, all the maps are algebra morphisms (resp. degree 0 maps), except for the maps marked with ♢ (resp. 
♯), which are only 𝐤-module morphisms (degree 1 maps).  


This follows from the combination the commutative diagram from Lemma <ref> with the commutative diagram

    𝒱^^ρ[rr]_≃[d]       M_3((𝒱^)^⊗2)
    ^(e_1f_1)^-1row_1·-·col_1e_1f_1[d]
     (𝒱^,·_e_1)
    _Ad(e_1f_1)^-1∘ρ̃[rr]
           (𝒱^[1 e_1]_≥0)^⊗2

which follows from the specialization, based on (<ref>), of the commutative diagram from Lemma <ref> 
to R=𝒱^, S=𝒱^[1 e_1]^⊗2, f=ρ, e=e_1, row=(e_1f_1)^-1row_1, 
col=col_1e_1f_1. 

The map ρ has degree 0 as it is a composition of maps of degree 0. The other degree statements follow from inspection of the degrees of 
e_1, row_1, col_1.  



 §.§ Relationship between infinitesimal braid Lie algebras and Δ^ℳ,
 


Set 

    col_0:=[           0; -e_1· 1_^⊗2;  e_1· 1_^⊗2 ]∈ 
    M_3×1((ℳ^)^⊗2).
    col_0@col_0




Denote by (a,x)↦ ax the action of M_3((𝒱^)^⊗2) on 
M_3× 1((ℳ^)^⊗2). Then 

    ρ(e_0)col_0=0.



Using (<ref>), one computes ρ(e_0)col_0=
[              0; -e_1f_0· 1_^⊗2;  e_1f_0· 1_^⊗2 ]
=0 in M_3× 1((𝒱^/𝒱^ e_0)^⊗2)
=M_3× 1((ℳ^)^⊗2). 



The map (e_1f_1)^-1row_1·(-)·col_0 : M_3((𝒱^)^⊗2)
→ℳ^[1/e_1]^⊗2 constructed out of matrix multiplication and of 
the tensor square of the map 𝒱^[1/e_1]⊗𝒱^⊗ℳ^→ℳ^[1/e_1] induced by the inclusions 𝒱^⊂𝒱^[1/e_1] and 
ℳ^⊂ℳ^[1/e_1], the product on 𝒱^[1/e_1], and the 
action of 𝒱^[1/e_1] on ℳ^[1/e_1], has image contained in 
(ℳ^[1/e_1]_≥-1)^⊗2. 


This map is given by (m_ij)_i,j∈[[1,3]]↦{f_1^-1(m_13-m_12)e_1+e_1^-1(m_22-m_23)e_1}· 1_^⊗2.  One has f_1^-1(m_13-m_12)e_1· 1_^⊗2∈ℳ^[1/e_1]_≥1⊗ℳ^[1/e_1]_≥-1 and 
e_1^-1(m_22-m_23)e_1· 1_^⊗2∈(ℳ^[1/e_1]_≥0)^⊗2, 
therefore the sum of these elements belongs to the announced space. 


There is a unique map δ:ℳ^→ (ℳ^[1/e_1]_≥-1)^⊗2delta@δ, 
such that the following diagram commutes 

    𝒱^^ρ[rr]_(-)· 1_[d]        M_3((𝒱^)^⊗2)
    ^(e_1f_1)^-1row_1·(-)·col_0[d]
    ℳ^_δ[rr]      
    (ℳ^[1/e_1]_≥-1)^⊗2

It is such that 

    δ(1_)=1_^⊗2.



If x∈𝒱^, then 

    (e_1f_1)^-1row_1·ρ(xe_0)·col_0=
    (e_1f_1)^-1row_1ρ(x)ρ(e_0)col_0=0

by (<ref>), therefore (
(e_1f_1)^-1row_1·(-)·col_0
)∘ρ(𝒱^ e_0)=0. The existence and uniqueness of δ follow. 
One then computes 

    δ(1_)=(e_1f_1)^-1row_1·ρ(1)·col_0=
    (e_1f_1)^-1row_1·col_0
    =(e_1f_1)^-1e_1f_1· 1_^⊗2=1_^⊗2.
 



The map δ satisfies the identity 

    ∀ x∈𝒲^,  ∀ m∈ℳ^,   δ(x· m)=Δ^𝒲,(x)·δ(m),

where the module structure in the left-hand side (resp. right-hand side) is that of ℳ^ over 
𝒲^ (resp. ℳ^[1/e_1]^⊗2 over (𝒲^)^⊗2).  


The identity is obvious if x=1. Assume now that x=ae_1 with a∈𝒱^ and that 
m∈ℳ^=𝒱^/𝒱^ e_0. Let m̃∈𝒱^ be a lift of m. 
Then 

    δ(x· m)=δ(ae_1· m)=(e_1f_1)^-1row_1·ρ(ae_1m̃)·col_0
    
        =(e_1f_1)^-1row_1·ρ(a)ρ(e_1)ρ(m̃)·col_0
    =(e_1f_1)^-1row_1·ρ(a)·col_1·row_1·ρ(m̃)·col_0
    
        = (e_1f_1)^-1row_1·ρ(a)·col_1· e_1f_1 · 
    (e_1f_1)^-1row_1·ρ(m̃)·col_0
    
        = Δ^𝒲,(x)·δ(m),
 
where row_1 is as in (<ref>), 
the second equality follows from (<ref>), the third equality follows from the fact that ρ
is an algebra morphism, the fourth equality follows from (<ref>) and the last equality follows from the 
combination of 
(<ref>) and the equality Δ^𝒲,(ae_1)=(e_1f_1)^-1row_1·ρ(a)
·col_1·(e_1f_1) for a∈𝒱^, which follows from equation (<ref>). 



The following diagram commutes 

    𝒱^^ρ[rrr]_(-)· 1_[d]           M_3((𝒱^)^⊗2)
    ^(e_1f_1)^-1row_1·(-)·col_0[d]
    ℳ^_Δ^ℳ,[rr]      (ℳ^)^⊗2@^(->[r]   (ℳ^[1/e_1]_≥-1)^⊗2

In this diagram, all the maps have degree 0. 


Combining (<ref>), (<ref>) and the fact that ℳ^ is a free 𝒲^-module of 
rank one with generator 1_, one obtains δ=Δ^ℳ,, which one injects in (<ref>) to get 
the result. The statement on degrees follows from inspection of the degrees of (e_1f_1)^-1row_1 and col_0. 




 §.§ Completions 
(commutativities of (A7) in (<ref>) and (M5) in (<ref>))



The following lemma will be used to prove the commutativities 
mentioned in the title of this subsection. 

The commutative diagram (<ref>) (resp. (<ref>)) gives rise to a commutative diagram between the
degree completions of its constituents, in which the completion of the map (𝒲^)^⊗2↪(𝒱^[1/e_1]_≥0)^⊗2 (resp. (ℳ^)^⊗2↪(ℳ^[1/e_1]_≥-1)^⊗2) is injective. 


By Proposition <ref>, if one equips 𝒱^ and M_3((𝒱^)^⊗2) with the 
shifted gradings 𝒱^[1]_n:=𝒱^_n-1 and M_3((𝒱^)^⊗2)[1]_n:=
M_3(((𝒱^)^⊗2)_n-1), then (<ref>) is a diagram in the category 𝐤-mod_,+. 
By Proposition <ref>, (<ref>) is similarly a diagram in the same category. Applying the functor
(-)^∧∘taut_,+^fil,+ (see <ref>), one obtains commutative diagrams in 
𝐤-mod_. The announced injectivities follows from Lemma <ref>, 1). 




§ BRAID GROUPS


In <ref>, we recall the definition of various families of braid groups (the Artin braid group, the sphere 
braid group and the modular group of the sphere with marked points), of their pure subgroups, and of a diagram of morphisms 
relating them (see (<ref>)). We recall the presentation of these groups and relate various generators by morphisms 
(Lemma <ref>). We then give a presentation of the modular group P_5^* which exhibits an order
5 cyclic symmetry, and may be viewed as an analogue of the presentation of 𝔭_5 in <cit.>, Proposition 4
(this presentation is not used is the sequel of the paper). In <ref>, we define morphisms 
ℓ:F_2→ P_5^*, pr_i:P_5^*→ F_2 (i∈[[1,5]]),  
pr_12:P_5^*→(F_2)^2 relating P_5^* with the free group with two generators F_2 or its square. 

In <ref>, we introduce an ideal J(pr_5) of the group algebra 𝐤P_5^* arising from pr_5, and show its freeness as a left 𝐤P_5^*-module 
(Lemma <ref>). Lemma <ref> then gives rise to an algebra morphism 
ϖ:𝐤P_5^*→ M_3(𝐤P_5^*). By composing ϖ with the morphisms 
ℓ and pr_12, we construct a morphism ρ:𝒱^→ 
M_3((𝒱^)^⊗2) (see (<ref>)). In Lemma <ref>, 
by introducing suitable elements row_1∈ M_1×3((𝒱^)^⊗2) and 
col_1∈ M_3×1((𝒱^)^⊗2), we show that the morphism ρ 
satisfies the hypothesis of Lemma <ref>. Applying this lemma, we obtain in <ref> 
a morphism :(𝒱^,·_X_1-1)→ (𝒱^)^⊗2, which we compute in 
Lemma <ref>. 



 §.§ Material on braid groups




  §.§.§ Braid groups


For X a topological space, let C_n(X)C_nX@C_n(X) denote its configuration space of n distinct points. 
Let also 𝔐_0,n+1M_0,n+1@ 𝔐_0,n+1 be the moduli space of smooth complex projective curves of genus 
zero with n+1 marked points. Below, we give a list
of topological spaces and simply-connected  subspaces, together with standard names and notation for the 
corresponding  fundamental groups (see <cit.>). 



range of n     n≥1 
    n≥1     n≥3
 
 space X     C_n(ℂ) 
    C_n(ℙ^1_ℂ)     C_n(ℙ^1_ℂ)/PGL_2(ℂ)≃𝔐_0,n  
 
subspace  b    𝒰_n     𝒰_n     ℬ_n 
 
 notation for π_1(X,b)     K_nK_n@K_n      P_nP_n@P_n 
    P^*_nP_n*@P^*_n  

  name of π_1(X,b)     
[  pure Artin; braid group ]
    [ pure sphere (Hurwitz);           braid group ]
    [   pure modular group of the; sphere with n marked points ]
  
 





range of n     n≥1 
    n≥1     n≥3

space X     C_n(ℂ)/S_n 
    C_n(ℙ^1_ℂ)/S_n     
[ C_n(ℙ^1_ℂ)/(S_n×PGL_2(ℂ));                ≃𝔐_0,n/S_n ] 
 
 subspace b     S_n·𝒰_n     S_n·𝒰_n     S_n·ℬ_n 
 
 notation for π_1(X,b)     B_nB_n@b_n      H_nH_n@H_n     B^*_n B_n*@B^*_n 
  name of π_1(X,b)     Artin braid group     
[ sphere (Hurwitz);      braid group ]
    
[        modular group of the; sphere with n marked points ]  
 


Here we set 𝒰_n:={(x_1,…,x_n)∈ℝ^n|x_1<⋯<x_n}⊂ C_n(ℂ)U_n @𝒰_n; 
we define  
ℬ̃_n⊂ C_n(ℙ^1)B_ncal^tilde@ℬ̃_n to be the set of n-tuples (x_1,…,x_n) in (ℙ^1_ℝ)^n, which lie on ℙ^1_ℝ in the counterclockwise order; one has 
ℬ̃_n=PGL_2^+(ℝ)·𝒰_n; we define 
ℬ_nB_ncal @ℬ_n as the quotient ℬ̃_n/PGL_2^+(ℝ); 
it is a simply-connected subspace of C_n(ℙ^1_ℂ)/PGL_2(ℂ). 
 
By homotopy exact sequence, the pure groups appear as the kernels of the natural morphisms of their non-pure counterparts to S_n, 
so 

    K_n=Ker(B_n→ S_n),    P_n=Ker(H_n→ S_n),    P_n^*=Ker(B_n^*→ S_n).


According to <cit.>, Appendix, B_n^* is isomorphic to the quotient of 
the Artin braid group B_n by the normal subgroup 
generated by η_neta_n @ η_n if n≥3, which corresponds to the winding of x_n around (x_1,…,x_n-1) (see 
(<ref>) for an expression in terms of standard generators), and by its center Z(B_n), 
which is isomorphic to ℤ. One has Z(B_n)⊂ K_n, and P_n+1^*≃ K_n/Z(B_n) for n≥2.
One also has P_n≃ P_n^*× C_2, where C_2 is the cyclic group of order 2
(see <cit.>, Proposition A4 iii) and also <cit.>, Corollary 2.1.2). 



The isomorphism P_n+1^*≃ K_n/Z(B_n) can be interpreted as follows. There is an isomorphism 
𝔐_0,n+1≃ C_n(ℂ)/Aff, with Aff=
{x↦ ax+b | a∈ℂ^×,b∈ℂ}; it gives rise to a homotopy exact sequence 
π_2(𝔐_0,n+1)→π_1(Aff)→π_1(C_n(ℂ))→π_1(𝔐_0,n+1) → 1. 
The spaces 𝔐_0,n are K(π,1)-spaces, as can be seen inductively from the homotopy exact sequences
of the fibrations 𝔐_0,n+1→𝔐_0,n, therefore π_2(𝔐_0,n+1)=1. One has 
K_n=π_1(C_n(ℂ)), P^*_n+1=π_1(𝔐_0,n+1) and π_1(Aff)=ℤ, so the above exact sequence 
implies P^*_n+1≃ K_n/ℤ. 



The isomorphism P_n≃ P_n^*× C_2 for n≥ 2 implies, in the notation of Remark <ref>, 
the isomorphism gr(P_n)≃gr(P_n^*)× C_2, 
therefore gr(P_n^*)⊗𝐤≃gr(P_n)⊗𝐤≃𝔭_n as 
ℚ⊂𝐤. 


A diagram of pure braid groups.
The canonical projection C_n+1(ℂ)→ C_n+1(ℙ^1_ℂ)/PGL_2(ℂ)
defines a morphism of topological spaces; this map takes 𝒰_n+1 to ℬ_n+1, therefore induces a group 
morphism K_n+1→ P_n+1^*. 

Define a morphism C_n(ℂ)→ C_n+1(ℙ^1_ℂ)/PGL_2(ℂ) to be the map taking 
(x_1,…,x_n) to the class of (x_1,…,x_n,∞). This map takes 𝒰_n to ℬ_n+1, therefore induces a group morphism K_n→ P_n+1^*. 

Let D be the open unit disc in ℂ, let C_n^D(ℂ) (resp. 𝒰_n^D) be the intersection of 
C_n(ℂ) (resp. 𝒰_n) with D^n. Then (C_n^D(ℂ),𝒰_n^D) is a deformation 
retract of (C_n(ℂ),𝒰_n), therefore K_n≃π_1(C_n^D(ℂ),𝒰_n^D). 
The morphism C_n^D(ℂ)→ C_n+1(ℂ) given by (x_1,…,x_n)↦ (x_1,…,x_n,2)
takes 𝒰_n^D to 𝒰_n+1, and therefore induces a morphism K_n→ K_n+1. 

Then the diagram 

    K_n@->>[r]@^(->[d]    P_n+1^*
     K_n+1@->>[ur]

commutes. 

Presentations of braid groups. 
According to <cit.>, the group B_n is presented for n≥1 
by generators σ_1,…,σ_n-1sigma @σ_1,…,σ_n-1, 
subject to relations σ_iσ_j=σ_jσ_i for |i-j|>1, and σ_iσ_i+1σ_i
=σ_i+1σ_iσ_i+1 for i∈[[1,n-1]]. The morphism B_n→ S_n is then given by 
σ_i↦(i,i+1). 

For i<j∈[[1,n]], set 

    x̃_ij:=
    (σ_j-2⋯σ_i)^-1σ_j-1^2(σ_j-2⋯σ_i)=
    (σ_j-1⋯σ_i+1)σ_i^2(σ_j-1⋯σ_i+1)^-1∈ B_n.
    x_ij^tilde @ x̃_ij

(when j=i+1, the products σ_j-2⋯σ_i and σ_j-1⋯σ_i+1 
are equal to 1 by convention, see (<ref>)). 
Then one checks that x̃_ij∈ K_n. 
According to <cit.>, a presentation of K_n is given by 
generators x̃_ij, i<j∈[[1,n]], subject to relations 

    (a_ijk,x̃_ij)=(a_ijk,x̃_ik)=(a_ijk,x̃_jk)=1

for i<j<k and i,j,k∈[[1,n]], where a_ijk=x̃_ijx̃_ikx̃_jka_ijk @ a_ijk, together with 

    (x̃_ij,x̃_kl)=(x̃_ik,x̃_ij^-1x̃_jlx̃_ij)=(x̃_il,x̃_jk)=1.
 
for i<j<k<l and i,j,k,l∈[[1,n]]. 

Set 

    ω_n:=x̃_12· (x̃_13x̃_23)⋯ (x̃_1n⋯x̃_n-1,n)∈ K_n.
    omega_n @ ω_n
 
Then ω_n is a generator of Z(B_n), therefore 

    P_n+1^*=K_n/⟨ω_n⟩.
 

For n≥ 2 and i∈[[1,n]], define x̃_i,n+1∈ K_n by 

    x̃_i,n+1:=(x̃_1i⋯x̃_i-1,ix̃_i,i+1⋯x̃_in)^-1.
    x_in+1 @ x̃_i,n+1

For i<j∈[[1,n+1]], we denote by x_ij∈ P_n+1^* x_ij @ x_ij the image of x̃_ij∈ K_n 
by the projection (<ref>). 


If i<j∈[[1,n+1]], then x_ij∈ P^*_n+1 is the image of x̃_ij∈ K_n+1 under the 
morphism K_n+1→ P^*_n+1.  


If j≤ n, then the image of x̃_ij∈ K_n under K_n→ P^*_n+1 is 
x_ij∈ P^*_n+1, and the image of the same element under K_n→ K_n+1
is x̃_ij∈ K_n, so the result follows from the commutativity of (<ref>). 

Assume now that j=n+1. According to <cit.>, three lines after (A1), one has 

    η_n+1=σ_n⋯σ_1^2⋯σ_n

(equality in B_n+1). Moreover, one checks that for i∈[[1,n]], 

    σ_i⋯σ_n·η_n+1^-1· (σ_i⋯σ_n)^-1
    =x̃_1i⋯x̃_i-1,ix̃_i,i+1⋯x̃_i,n+1

(equality in B_n+1). The left-hand side belongs to the kernel of the morphism B_n+1→ B_n+1^*, and the right-hand 
side belongs to the subgroup K_n+1⊂ B_n+1, therefore the latter side belongs to the kernel of the 
composed morphism K_n+1⊂ B_n+1→ B_n+1^*. As this morphism factors as K_n+1↠ P_n+1^*
↪ B_n+1^*, the right-hand side belongs to Ker(K_n+1↠ P_n+1^*). 


Diagrammatic representation.
The generators of B_n are depicted as follows when n=4,  
[xscale=0.5,yscale=0.3]
(3,0) nodeσ_1= ; 
[number of strands=4]  (braid) at (4,1)  s_1;
 , 
[xscale=0.5,yscale=0.3]
(3,0) nodeσ_2= ; 
[number of strands=4]   (braid) at (4,1)  s_2;
, 
[xscale=0.5,yscale=0.3]
(3,0) nodeσ_3= ; 
[number of strands=4]    (braid) at (4,1)  s_3;

and the convention for the product is 
[xscale=0.5,yscale=0.3]
(2.5,-0.7) nodeσ_2σ_1= ; 
[number of strands=4]    (braid) at (4,1)  s_1s_2;


The element x̃_ij∈ K_n is then depicted as follows 


           
                              (-1.2,0.5) nodex̃_ij=;
                 [-] (-0.5,0)–(-0.5,1) (-0.2,0)–(-0.2,1)
(0.2,0)–(0.2,1) (0.3,0)–(0.3,1) (0.6,0)–(0.6,1) (1.2,0)–(1.2,1)
(1.5,0)–(1.5,1);
                 [dotted] (-0.5,0.5)–(-0.2,0.5)
(0.3,0.5)–(0.6,0.5) (1.2,0.5)–(1.5,0.5);
                 [-] (0.8,1)–(0.8,0.5);
                 [color=white, line width=7pt](0,0)
..controls(0.1,0.5) and (0.9,0)        ..(1,0.5);
                 [-] (0,0) ..controls(0.1,0.5) and (0.9,0)
..(1,0.5);
                 [color=white, line width=7pt](1,0.5)
..controls(0.9,1.0) and (0.1,0.5)       ..(0,1);
                 [-] (1,0.5) ..controls(0.9,1.0) and (0.1,0.5)
   ..(0,1);
                 [color=white, line width=5pt] (0.8,0.5)–(0.8,0);
                 [-] (0.8,0.5)–(0.8,0);
[decorate,decoration=brace,mirror] (-0.6,-0.1) – (-0.1,-0.1)
node[midway,below]i-1;
[decorate,decoration=brace] (-0.6,1.1) – (0.6,1.1)
node[midway,above]j-1;
            


According to these conventions, the diagrammatic representatives of the elements x̃_i,5∈ B_4 given by 
(<ref>) are the following 





For a,b≥ 1, we define 
σ_a,b∈ B_a+bsigma_ab@σ_a,b to be the element represented by 



    [baseline=(current  bounding  box.center),xscale=0.8, yscale=1]
             (-0.5,0.5) nodeσ_a,b= ; 
            [-] (0,1) –(1.2, 0) (0.2,1)–(1.4,0)  (0.8,1)–(2,0);
            [decorate,decoration=brace] (-0.1,1.1) – (0.8,1.1) node[midway,above]a;
            [dotted] (0.3,0.9) –(0.8, 0.9)  ;
    
             [color=white, line width=5pt] (0,0) –(1.2, 1) (0.2,0)–(1.4,1)  (0.8,0)–(2,1);
             [-] (0,0) –(1.2, 1) (0.2,0)–(1.4,1)  (0.8,0)–(2,1);
             [decorate,decoration=brace] (1.1,1.1) – (2.1,1.1) node[midway,above]b;
             [dotted] (0.8,0.45) –(1.3, 0.45)  ;
              (2.7,0.5) node∈ B_a+b ; 
            .

Note the equality σ_1=σ_1,1^-1∈ B_2. 

The following result may be viewed as an analogue of Proposition 4 in <cit.>. 

For i∈  C_5≃ [[1,5]], define g_i∈ P_5^* by g_i:=x_i,i+1 (with the convention x_0,1:=x_1,5). 
The group P_5^* is presented by generators g_igi@ g_i (i∈ C_5) (i∈ C_5), subject to the relations

    (g_i,g_j)=1  if  i,j∈ C_5  and i-j≠±1,    
    (g_0,g_1)(g_1,g_2)(g_2,g_3)(g_3,g_4)
    (g_4,g_0)=1.



It follows from (<ref>) that for i∈[[1,4]], one has 

    x_i,5=(x_1i⋯ x_i-1,ix_i,i+1⋯ x_i4)^-1∈ P_5^*.

Relation ω_5=1, namely 

    x_12x_13x_23x_14x_24x_34=1,

together with relation (<ref>) for i=4, implies g_4=x_45=x_12x_13x_23, 
therefore 

    x_13=g_1^-1g_4g_2^-1.
 
Relation (<ref>)
together with (x_14,x_23)=1 yields 
x_12x_13x_14x_23x_24x_34=1,
which together with 
 (<ref>) for i=1, yields x_15=x_23x_24x_34. This relation yields

    x_24=g_2^-1g_0g_3^-1.

By the commutation of 
x_34 with x_23x_24x_34, this relation also yields g_0=x_15=x_24x_34x_23=x_24x_34g_2, therefore x_34^-1x_24^-1=g_2g_0^-1. 
(<ref>) implies x_14=x_23^-1x_13^-1x_12^-1x_34^-1x_24^-1, 
which after combination with the previous equality yields x_14=x_23^-1x_13^-1x_12^-1g_2
g_0^-1=
g_2^-1x_13^-1g_1^-1g_2g_0^-1. 
Combining with (<ref>), we obtain 

    x_14=g_4^-1g_2g_0^-1.
 
As P_5^* is a quotient of K_4 it is generated by {x_ij|i<j∈[[1,4]]]}, therefore also by 
the union of this set with g_0, which is equal to {g_i|i∈ C_5}∪{x_13,x_24,x_14}. 
Relations (<ref>), (<ref>) and (<ref>) then imply that a generating set is 
{g_i|i∈ C_5}. 

The group P_5^* may be viewed as generated by {g_i|i∈ C_5}∪{x_13,x_24,x_14}, subject to 
relations (<ref>), (<ref>), ω_5=1, and (<ref>) for i=1,4. 
These relations imply expressions (<ref>), (<ref>) and (<ref>) of 
x_13,x_24,x_14 in terms of {g_i|i∈ C_5}. Substituting these expressions in relations 
(<ref>), (<ref>), ω_5=1, and (<ref>) for i=1,4, we obtain
a presentation of P_5^* in terms of the generators {g_i|i∈ C_5}. The relations obtained in this way are the following. 
Relation ω_5=1 yields the commutation of g_0 with g_2. 
Relation (<ref>) for (i,j,k)=(1,2,3) yields the commutation of g_4 with g_1 and g_2.
Relation (<ref>) for (i,j,k)=(2,3,4) yields the commutation of g_0 with g_2 and g_3.
The first part of relation (<ref>), namely (x_12,x_34)=1, yields the commutation of 
g_1 with g_3. The last part of relation (<ref>), namely (x_14,x_23)=1, 
yields a consequence of the already obtained commutations of g_2 with g_0 and g_4.
The middle part of relation (<ref>), namely (x_13,x_12^-1x_24x_12)=1, together with the 
commutations (g_1,g_3)=(g_0,g_2)=1, yields the relation 
g_0g_2^-1g_1g_3^-1g_2g_4^-1g_3g_0^-1g_4g_1^-1=1, 
which by again using the commutation relations yields g_0g_4^-1g_1g_0^-1g_2g_1^-1g_3g_2^-1g_4g_3^-1=1, 
which is equivalent to the cyclic relation (g_0,g_1)(g_1,g_2)(g_2,g_3)(g_3,g_4)(g_4,g_0)=1. 

Relation (<ref>) for (i,j,k)=(1,2,4) splits as the conjunction of 
x_12x_14x_24=x_14x_24x_12 and x_14x_24x_12=x_24x_12x_14. 
The first relation yields a consequence of already obtained relations, namely 
(g_0,g_2)=(g_1,g_3)=(g_1,g_4)=1. 
After using (g_0,g_2)=(g_1,g_3)=1, the second relation yields 
g_0g_3^-1g_1g_4^-1g_2g_0^-1g_3g_1^-1g_4g_2^-1=1 which as above is equivalent to the already obtained cyclic relation.  

Relation (<ref>) for (i,j,k)=(1,3,4) splits as the conjunction of 
x_13x_14x_34=x_34x_13x_14 and x_34x_13x_14=x_14x_34x_13. 
The first relation yields a consequence of the already obtained relations 
(g_2,g_4)=(g_3,g_1)=(g_3,g_0)=1. After using 
(g_2,g_4)=1 and the commutation of g_1 with g_3^-1, the second relation yields
the already obtained relation g_0g_3^-1g_1g_4^-1g_2g_0^-1g_3g_1^-1g_4g_2^-1=1. 



Under the commutation relations, the cyclic relation ∏_i∈ C_5(g_i,g_i+1)=1 (using the notation 
∏_i∈ C_5a_i:=a_0a_1⋯ a_4) is equivalent to any of the relations 
∏_i∈ C_5g_ig_i+j^-1=1, where j∈ C_5-{0,1}. For j=2, this relation 
expresses as ∏_i∈ C_5(g_i^-1,g_i+1^-1)=1. The cyclic relation is also equivalent to the relation
 ∏_i∈ C_5(g_-i,g_-i-1)=1. All this proves that the group D_5× C_2 acts by automorphisms of 
P_5^* as follows: the dihedral group D_5D_5 @ D_5 acts by permutation of indices of the generators (g_i)_i∈ C_5
and the cyclic group C_2 acts by g_i↦ g_i^-1. 





  §.§.§ The morphisms ℓ,pr_i and pr_12
between braid groups


Let F_2F_2@F_2 be the free group with generators X_0, X_1
X_0, X_1@X_0, X_1.
Similarly to <ref>, we will abuse notation by setting for i=0,1

    X_i:=(X_i,1)∈(F_2)^2,    Y_i:=(1,X_i)∈(F_2)^2.
    Y_0, Y_1@Y_0, Y_1

We then have X_i=X_i⊗1, Y_i=1⊗ X_i in (𝒱^)^⊗2. 


1) There are group morphisms pr_i:P_5^*→ F_2pr_125_underline@ pr_1, pr_2, pr_5 for i=1,2,5, given by 


  x∈ P_5^*     x_12     x_13     x_14     x_15     x_23    x_24    x_25    x_34    x_35    x_45 

  pr_1(x)     1     1     1     1     X_0     (X_1X_0)^-1     X_1     X_1     (X_0X_1)^-1     X_0 
 
pr_2(x)     1     (X_0X_1)^-1     X_0     X_1     1     1     1      X_1      X_1^-1X_0X_1     
(X_0X_1)^-1 

  pr_5(x)     X_1     (X_0X_1)^-1     X_0     1     X_0     (X_1X_0)^-1     1     X_1     1     1 

 

2) There is a group morphism ℓ:F_2→ P_5^*l_underline@ℓ, given by X_0↦ x_23, X_1↦ x_12.  
We have pr_5∘ℓ=id_F_2. 


Direct computation. 

Define pr_12:P_5^*→ (F_2)^2pr_12t_underline@pr_12 as the morphism p↦ (pr_1(p),
pr_2(p)). We will denote by ℓ,pr_i and pr_12 
the Hopf algebra morphisms relating the group algebras 𝐤 F_2, (𝐤 F_2)^⊗2 and 𝐤 P_5^*
induced by ℓ,pr_i and pr_12. 

  The pure modular group of the sphere with 4 marked points P_4^* is freely generated by x_12,x_23, 
and therefore isomorphic to F_2. Composing with this isomorphism 
the morphisms pr_i (resp. ℓ), one obtains the morphisms P_5^*→ P_4^* (resp. P_4^*→ P_5^*) 
induced by the morphisms between moduli spaces consisting in forgetting the i-th marked point (resp. doubling the fourth marked
point). 




 §.§ Algebraic constructions related to an ideal of 𝐤 P_5^*




  §.§.§ The structure of J(pr_5)



We denote by J(pr_5)Jpr_5_underline @J(pr_5) the kernel 
Ker(𝐤 P_5^*pr_5→𝐤 F_2).
This is a two-sided ideal of 𝐤 P_5^*. 


Let F_3F_3 @ F_3 be the free group with generators a_i,
i∈[[1,3]]; there is a unique group morphism F_3→ P_5^*, given by 
a_i↦ x_i5 for i∈[[1,3]]. 


1) The morphisms pr_5 and F_3→ P_5^* fit in an exact sequence 1→ F_3→ P_5^*→ F_2→1.
As F_3→ P_5^* is injective, we will identify F_3 with its image in P_5^*.  

2) The map F_2× F_3→ P_5^*, (f,f')↦ℓ(f)·f'
is a bijection. 


1) follows from <cit.>, Proposition 2.1.3, based on <cit.>. 2) then follows from the fact that ℓ is a 
section of pr_5. 

 
The map (𝐤 P_5^*)^⊕3→ J(pr_5), (p_i)_i∈[[1,3]]↦∑_i∈[[1,3]]p_i· (x_i5-1) 
is an isomorphism of left 𝐤 P_5^*-modules. 


The bijection from Lemma <ref>, 2) induces a linear isomorphism 𝐤 F_2⊗𝐤 F_3≃→𝐤 P_5^*. For (f,f')∈ F_2× F_3, 
pr_5(ℓ(f)·f')=f as pr_5∘ℓ=id_F_2 and F_3=Ker(pr_5). It follows that the
following diagram is commutative 

    𝐤 F_2⊗𝐤 F_3^≃[r]_id⊗ε[dr]    𝐤 P_5^* ^pr_5[d]
       𝐤 F_2

It follows that J(pr_5) is the image of 𝐤 F_2⊗
(𝐤 F_3)_+ under the linear isomorphism 𝐤 F_2⊗𝐤 F_3≃→𝐤 P_5^*. 

Since x_i5∈ F_3 for i∈[[1,3]], the diagram 

    (𝐤 F_2⊗𝐤 F_3)^⊕3^≃[r][d]     (𝐤 P_5^*)^⊕3[d]
    𝐤 F_2⊗𝐤 F_3^≃[r]    𝐤 P_5^*

commutes, where the vertical maps are, on the left-hand side, the tensor product of the id_𝐤 F_2
with 𝐤 F_3^⊕3→𝐤 F_3, (f_i)_i∈[[1,3]]↦∑_i∈[[1,3]]f_i
·(x_i5-1), and on the right-hand side, the map given by the same formula.  
 
It follows from Lemma <ref> that 𝐤 F_2⊗
(𝐤 F_3)_+ is the isomorphic image of the left vertical map, therefore J(pr_5)
is the isomorphic image of (𝐤 P_5^*)^⊕3 by the right vertical map, which proves the claimed statement. 




  §.§.§ A morphism :𝐤 P_5^*→ M_3(𝐤 P_5^*)



Lemma <ref> says that the hypothesis of Lemma <ref> is satisfied 
in the following situation: R=𝐤 P_5^*, J=J(pr_5), d=3, (j_a)_a∈[[1,d]]
=(x_i5-1)_i∈[[1,3]]. We denote by 

    :𝐤 P_5^*→ M_3(𝐤 P_5^*)
    pivar_underline@

the algebra morphism given in this situation by Lemma <ref>. Then for p∈𝐤 P_5^*, 
(p)=(a_ij(p))_i,j∈[[1,3]], and 

    ∀ i∈[[1,3]],   (x_i5-1)p=∑_j∈[[1,3]]a_ij(p)(x_j5-1)

(equalities in 𝐤 P_5^*). 



  §.§.§ Construction and properties of a morphism 𝒱^→ M_3((𝒱^)^⊗2)


Define the algebra morphism 

    :𝒱^→ M_3((𝒱^)^⊗2)
    rhovar_underline@

to be the composition 

    𝒱^ℓ→𝐤 P_5^*→
    M_3(𝐤 P_5^*)M_3(pr_12)→
    M_3((𝒱^)^⊗2),

where ℓ is as in 2) of Lemma <ref>, 
 is as in <ref>, 
and 
M_3(pr_12)M_3pr_12 @M_3(pr_12) is the  
morphism induced by pr_12, i.e., taking (p_ij)_i,j∈[[1,3]] to (pr_12(p_ij))_i,j∈[[1,3]]. 


Set 

    row_1:=[ X_1-1 1-Y_1     0 ]∈ M_1× 3((𝒱^)^⊗2),   col_1:=[ Y_1;  -1;   0 ]∈ M_3× 1((𝒱^)^⊗2)
    row1_underline @ row_1col1_underline @ col_1
 
(where X_1,Y_1∈ F_2^2⊂ (𝒱^)^⊗2 are defined by (<ref>)), then 

    (X_1-1)=col_1·row_1

(equality in M_3((𝒱^)^⊗2)). 


One has ℓ(X_1)=x_12. Let us compute (x_12). One has 

    (x_15-1)x_12    =   (x_15-1)x_12x_15x_25x_25^-1x_15^-1
    =x_12x_15x_25 (x_15-1)x_25^-1x_15^-1
        
    =x_12x_15x_25(-x_15x_25^-1x_15^-1+1+x_25^-1x_15^-1)(x_15-1)
    +x_12x_15x_25(1-x_15)x_25^-1(x_25-1), 
    
    (x_25-1)x_12    =(x_25-1)x_12x_15x_25x_25^-1x_15^-1
    =x_12x_15x_25(x_25-1)x_25^-1x_15^-1
       =x_12x_15x_25(x_25^-1-1)x_15^-1(x_15-1)+x_12x_15(x_25-1) , 
    
    (x_35-1)x_12    =x_12(x_35-1) ,

which implies that 

    (x_12)
    =[ x_12x_15x_25(-x_15x_25^-1x_15^-1+1+x_25^-1x_15^-1)                        x_12x_15x_25(1-x_15)x_25^-1                                                  0;                     x_12x_15x_25(x_25^-1-1)x_15^-1                                           x_12x_15                                                  0;                                                  0                                                  0                                               x_12 ]∈ M_3(𝐤 P_5^*).

The image of (x_12-1) in M_3((𝒱^)^⊗2) by M_3(pr_12) is therefore 

    [ (X_1-1)Y_1 Y_1(1-Y_1)          0;      1-X_1      Y_1-1          0;          0          0          0 ]
    =[ Y_1;  -1;   0 ][ X_1-1 1-Y_1     0 ]
    =col_1·row_1.

Therefore (X_1-1)=col_1·row_1. 




  §.§.§ Construction and properties of a morphism :(𝒱^,·_X_1-1)→ 
(𝒱^)^⊗2


Lemma <ref> shows that the hypothesis of Lemma <ref> is satisfied in the 
following situation: R=𝒱^, S=(𝒱^)^⊗2, e=X_1-1, n=3, f=, row 
and col are row_1 and col_1 from Lemma 
<ref>. We denote by

    :(𝒱^,·_X_1-1)→ (𝒱^)^⊗2rhovar_underline^tilde@

the algebra morphism given in this situation by Lemma <ref>. 

Then for any f∈𝒱^, one has 

    (f)=row_1·(f)·col_1
    =row_1·{M_3(pr_12)∘∘ℓ(f)}·col_1
    ∈ (𝒱^)^⊗2.



For any k∈ℤ, 

    (X_0^k) 
    =(X_1-1)X_0^k⊗ 1+1⊗ (1-X_1^-1)X_0^kX_1
    -∑_i=1^k-1(X_1-1)X_0^i⊗ (1-X_1^-1)X_0^k-iX_1,


    (X_0^kX_1^-1)=(X_0^k) (X_1^-1⊗ X_1^-1)

(equalities in (𝒱^)^⊗2). 


According to (<ref>), (X_0^k)
=row_1·(X_0^k)·col_1. 
As  is an algebra morphism, (X_0^k)=(X_0)^k. Then 
(X_0)=M_3(pr_12)∘∘ℓ(X_0)
=M_3(pr_12)((x_23)). 

Let us compute (x_23).

As x_15 commutes with x_23, one has 

    (x_15-1)· x_23=x_23· (x_15-1).


Since x_25 commutes with x_23x_25x_35, one has 

    x_23^-1x_25x_23=x_25x_35x_25x_35^-1x_25^-1

which implies the first equality in the following chain of equalities 

    x_23^-1(x_25-1)x_23=x_25x_35x_25x_35^-1x_25^-1-1=x_25(x_35x_25x_35^-1x_25^-1-1)+x_25-1
    
        = x_25{x_35(x_25x_35^-1x_25^-1-1)+x_35-1}+x_25-1
    
        =  x_25[x_35{x_25(x_35^-1x_25^-1-1)+x_25-1}+x_35-1]+x_25-1
    
        = x_25(x_35[x_25{x_35^-1(x_25^-1-1)+x_35^-1-1}+x_25-1]+x_35-1)+x_25-1
    
        =x_25x_35x_25x_35^-1(x_25^-1-1)+x_25x_35x_25(x_35^-1-1)+x_25x_35(x_25-1)+x_25(x_35-1)+x_25-1
    
        = (-x_25x_35x_25x_35^-1x_25^-1+x_25x_35+1)(x_25-1)+
    (-x_25x_35x_25x_35^-1+x_25)(x_35-1),

the following equalities being immediate. Therefore 

    (x_25-1)· x_23=x_23(-x_25x_35x_25x_35^-1x_25^-1+x_25x_35+1)· (x_25-1)
    +x_23(-x_25x_35x_25x_35^-1+x_25)· (x_35-1).


Since x_23x_25x_35=x_35x_23x_25, one has x_23^-1x_35x_23=x_25x_35x_25^-1, 
which implies the first equality in the following chain of equalities 

    x_23^-1(x_35-1)x_23=x_25x_35x_25^-1-1=x_25(x_35x_25^-1-1)+x_25-1 
    
        =x_25{x_35(x_25^-1-1)+x_35-1}+x_25-1=x_25x_35(x_25^-1-1)+x_25-1+x_25(x_35-1)
    
        =(-x_25x_35x_25^-1+1)(x_25-1)+x_25(x_35-1),

the following equalities being immediate. Therefore  

    (x_35-1)· x_23=x_23(-x_25x_35x_25^-1+1)· (x_25-1)+x_23x_25· (x_35-1).

Equalities (<ref>), (<ref>) and (<ref>) imply that 

    (x_23)=[                                         x_23                                            0                                            0;                                            0 x_23(-x_25x_35x_25x_35^-1x_25^-1+x_25x_35+1)              x_23(-x_25x_35x_25x_35^-1+x_25);                                            0                     x_23(-x_25x_35x_25^-1+1)                                     x_23x_25 ]∈ 
    M_3(𝐤 P_5^*).

Then  

    (X_0)=M_3(pr_12)((x_23))=
    [                     X_0                       0                       0;                       0 (1-X_1)X_0+Y_1^-1Y_0Y_1           (1-X_1)X_0X_1;                       0  X_0-Y_1^-1Y_0Y_1X_1^-1                  X_0X_1 ]∈ M_3((𝒱^)^⊗2).

Set TT_@T:=[ (1-X_1)X_0+Y_1^-1Y_0Y_1           (1-X_1)X_0X_1;  X_0-Y_1^-1Y_0Y_1X_1^-1                  X_0X_1 ]∈ M_2((𝒱^)^⊗2), 
then (X_0^k)=[ X_0^k     0;     0   T^k ], therefore 

    (X_0^k)=row_1·(X_0^k)·col_1
    =(X_1-1)X_0^kY_1+[ 1-Y_1     0 ]T^k
    [ -1;  0 ]
    =(X_1-1)X_0^kY_1-(1-Y_1)(T^k)_11,

where the second equality follows from the form of row_1 and col_1, and where 
(T^k)_11 means the (1,1)-entry of T^k.  

One checks that 

    T=[       1       0; -X_1^-1       1 ][ a b; 0 c ][       1       0; -X_1^-1       1 ]^-1

where 

    [       1       0; -X_1^-1       1 ]^-1=[      1      0; X_1^-1      1 ]

and 

    [ a b; 0 c ]=
    [  Y_1^-1Y_0Y_1 (1-X_1)X_0X_1;             0  X_1^-1X_0X_1 ].


For k∈ℤ, one has 

    [ a b; 0 c ]^k=[                  a^k ∑_i=0^k-1a^ibc^k-i-1;                    0                  c^k ]

using the notation (<ref>)
so that 

    T^k=[       1       0; -X_1^-1       1 ][                  a^k ∑_i=0^k-1a^ibc^k-i-1;                    0                  c^k ][      1      0; X_1^-1      1 ]=[ a^k+
    ∑_i=0^k-1a^ibc^k-i-1· X_1^-1                                 *;                                                                   * ]

therefore 

    (T^k)_11=a^k+
    ∑_i=0^k-1a^ibc^k-i-1· X_1^-1
        = Y_1^-1Y_0^kY_1+
    ∑_i=0^k-1Y_1^-1Y_0^iY_1  · (1-X_1)X_0X_1·    X_1^-1X_0^k-i-1X_1 · X_1^-1
        = Y_1^-1Y_0^kY_1+
    ∑_i=0^k-1Y_1^-1Y_0^iY_1  · (1-X_1)X_0^k-i
        = Y_1^-1Y_0^kY_1+(1-X_1)X_0^k
    +∑_i=1^k-1Y_1^-1Y_0^iY_1· (1-X_1)X_0^k-i.


It follows that 

    (X_0^k)=(X_1-1)Y_1X_0^k-(1-Y_1)(T^k)_11
        =
    (X_1-1)Y_1X_0^k-(1-Y_1)
    {
    (1-X_1)X_0^k+Y_1^-1Y_0^kY_1
    +∑_i=1^k-1Y_1^-1Y_0^iY_1· (1-X_1)X_0^k-i}
       
    = (X_1-1)Y_1X_0^k-(1-Y_1)
    (1-X_1)X_0^k-(1-Y_1)Y_1^-1Y_0^kY_1
    -(1-Y_1)∑_i=1^k-1Y_1^-1Y_0^iY_1· (1-X_1)X_0^k-i
        = 
    (X_1-1)X_0^k+(Y_1-1)Y_1^-1Y_0^kY_1
    -∑_i=1^k-1(1-X_1)X_0^k-i· Y_1^-1(1-Y_1)Y_0^iY_1,
 
which implies the first identity. 

By Lemma <ref>, one has (X_1)=1+col_1·row_1. 
As 1+row_1·col_1 is equal to X_1Y_1 and is therefore invertible, one checks that the inverse of 1+col_1·row_1 is 1-col_1·(1+row_1·col_1)^-1·row_1, therefore 

    (X_1^-1)=1-col_1·(1+row_1
    ·col_1)^-1·row_1=1-col_1·(X_1Y_1)^-1·row_1.

Then 

    (X_0^kX_1^-1)=row_1·(X_0^k)·(
    1-col_1·(X_1Y_1)^-1·row_1)·col_1
    
        
    =row_1·(X_0^k)·col_1·(
    1-(X_1Y_1)^-1·row_1·col_1)=(X_0^k)·(X_1Y_1)^-1,

which proves the second statement. 




Identity (<ref>) from Remark <ref> implies another decomposition of T, 
namely 

    T=
    [    1 -X_1;    0    1 ][                     X_0                       0; X_0-Y_1^-1Y_0Y_1 X_1^-1            Y_1^-1Y_0Y_1 ][    1 -X_1;    0    1 ]^-1,

which as in this remark allows for an alternative computation of T^k. 




§ GEOMETRIC INTERPRETATION OF THE BETTI HARMONIC COPRODUCTS


The purpose of this section is to construct a commutative diagram relating the  Betti algebra and module coproducts 
Δ^𝒲, and Δ^ℳ, with braid groups (diagrams (<ref>) and (<ref>)), 
analogously to the de Rham diagrams (<ref>) and (<ref>). 

This construction involves a localization 𝒱^[1 X_1-1] of the algebra 𝒱^, and a module 
ℳ^[1 X_1-1] over this algebra, which are introduced and studied in <ref>. 
We prove the commutativity of diagram (<ref>) in <ref> (Proposition <ref>); 
this diagram relates Δ^𝒲, with ρ and row_1, col_1, 
and is a Betti analogue of diagram (<ref>). We prove the commutativity of the diagram (<ref>) in 
<ref> (Proposition <ref>); this statement is based on Proposition <ref> and relates 
Δ^ℳ, with ρ, row_1 and the column vector col_0 
(see Definition <ref>). We construct completions of the diagrams (<ref>) and (<ref>) in 
<ref>. 



 §.§ Localizations
 

Define 𝒱^[1 X_1-1]V^B/X_1-1@𝒱^[1 X_1-1] to be the localization of 𝒱^ with respect to X_1-1, i.e. the 𝐤-algebra
with generators X_0^±1,X_1^±1,(X_1-1)^-1 and relations expressing that u^-1 is a left and right inverse of u for 
u∈{X_0,X_1,X_1-1}. It is equipped with a collection of subspaces indexed by i∈ℤ, namely 

    F^i(𝒱^[1X_1-1])
    :=∑_
    n,i_0,…,i_n≥ 0,
    
    i_0+⋯+i_n-n=iF^i_0𝒱^
    (X_1-1)^-1⋯(X_1-1)^-1F^i_n𝒱^.
    F^iVB/X_1-1@F^i(𝒱^[1X_1-1])

This collection is decreasing and compatible with the product, and therefore equips 𝒱^[1X_1-1] with the structure of 
an algebra in 𝐤-mod_fil (see Definition <ref>). 

Set also ℳ^[1 X_1-1]:=𝒱^[1 X_1-1]/𝒱^[1 X_1-1](X_0-1)M^B/X_1-1@ℳ^[1 X_1-1]. This is a module over 
𝒱^[1X_1-1].
For i∈ℤ, let 
F^i(ℳ^[1X_1-1])
F^iMB/X_1-1@F^i(ℳ^[1X_1-1])
 be the image of 
F^i(𝒱^[1X_1-1]) under the canonical projection (-)· 1_:𝒱^[1X_1-1]
→ℳ^[1X_1-1]. This collection is decreasing and compatible with the collection of subspaces of 
𝒱^[1X_1-1] and the action of this algebra, therefore equips ℳ^[1 X_1-1] with the 
structure of a module over 𝒱^[1X_1-1] in 𝐤-mod_fil. 

The natural maps 𝒱^→𝒱^[1 X_1-1] and ℳ^→ℳ^[1 X_1-1]
are compatible algebra and module morphisms in 𝐤-mod_fil. 


1) The canonical morphism 𝒱^→𝒱^[1 X_1-1] is injective. 

2) The image of 𝒱^[1 X_1-1] under the functor gr:𝐤-mod_fil→𝐤-mod_gr 
(see <ref>) is isomorphic to 𝒱^[1 e_1].  


1) Denote by 𝒱^[1 e_0]^∧ the completion of 𝒱^[1 e_1] with respect to large 
degrees (see <ref>). It may be viewed as an algebra in 𝐤-mod_fil by setting 
F^α(𝒱^[1 e_1]^∧):=∏_β≥α𝒱^[1 e_1]_β. There is a unique 
algebra morphism φ:𝒱^[1 X_1-1]→𝒱^[1 e_1]^∧ in 𝐤-mod_fil
given by X_i^±1↦(1+e_i)^±1 for i=0,1 and (X_1-1)^-1↦ e_1^-1. It induces an algebra morphism 
grφ:gr𝒱^[1 X_1-1]→gr𝒱^[1 e_1]^∧≃𝒱^[1 e_1] in 𝐤-mod_gr. Its composition with 𝒱^≃gr𝒱^gr(can)→gr𝒱^[1 X_1-1], can 
being the morphism 𝒱^→𝒱^[1 X_1-1], is the canonical map 
𝒱^→𝒱^[1 e_1], which is injective. It follows that gr(can) is injective. 
Since ∩_k≥0F^k𝒱^=0, this implies that can is injective. 

2) There is a unique graded algebra morphism ψ:𝒱^[1 e_1]→gr𝒱^[1 X_1-1], induced by 
e_i↦ [X_i-1]∈gr_1𝒱^[1 X_1-1] for i=0,1 and e_1^-1↦[(X_1-1)^-1]∈gr_-1𝒱^[1 X_1-1]. One has grφ∘ψ=id, which implies that ψ
is injective. For α∈ℤ,  for n,α_0,…,α_n as in the right-hand side of (<ref>), and for 
v_i∈ F^α_i𝒱^ for i=1,…,n, the degree α component ψ_α of ψ maps 
[v_0]e_1^-1⋯ e_1^-1[v_n]∈𝒱^[1 e_1]_α, where [v_i]∈gr_α_i𝒱^≃𝒱^_α_i to [v_0(X_1-1)^-1⋯(X_1-1)^-1v_n]∈gr_α(𝒱^[1 X_1-1]), 
therefore ψ_α is surjective. It follows that ψ is surjective. 


1) The morphism ℳ^→ℳ^[1 X_1-1] is injective. 

2) The image of ℳ^[1 X_1-1] under the functor gr (see <ref>) is isomorphic 
to ℳ^[1 e_1].  


1) Let ℳ^[1 e_1]^∧ the completion of ℳ^[1 e_1] with respect to large degrees 
(see <ref>); it can be identified with 𝒱^[1 e_1]^∧/𝒱^[1 e_1]^∧ e_0. 
The map φ in the proof of Lemma <ref> maps the left ideal generated by X_0-1 
to the left ideal generated by e_0, therefore induces a morphism φ:ℳ^[1 X_1-1]→ℳ^[1 e_1]^∧ in 𝐤-mod_fil, which gives rise to a morphism grφ:
grℳ^[1 X_1-1]→grℳ^[1 e_1]^∧. Its composition with 
ℳ^≃grℳ^gr(can_ℳ)→grℳ^
[1 X_1-1], where can_ℳ is the morphism ℳ^→ℳ^[1 X_1-1], is the canonical 
map ℳ^→ℳ^[1 e_1], which is injective. This implies the injectivity of gr(can_ℳ),
then of can_ℳ as in the proof of Lemma <ref>.  

2) The composition of ψ from the proof of Lemma <ref> with gr((-)· 1_
) takes e_0 to 0, therefore takes 
𝒱^[1 e_1]e_0 to 0, therefore induces a map ψ:ℳ^[1 e_1]→grℳ^[1 X_1-1]. One has grφ∘ψ=id, which implies 
that ψ is injective. The equality gr((-)· 1_)
∘ψ=ψ∘((-)· 1_), where 
(-)· 1_:
𝒱^[1 e_1] →ℳ^[1 e_1] is the canonical projection, the surjectivity of ψ
and that of gr((-)· 1_) (which follows from the construction of the filtration in ℳ^[1 X_1-1])
imply the surjectivity of ψ. 



 §.§ Relationship between braid groups and Δ^𝒲,
 


Denote by 𝐤[X_0^±1] the linear span in 𝐤 F_2 of the elements X_0^k, k∈ℤ, 
by 𝐤[X_0^±1]X_1^-1 the linear span of the elements X_0^kX_1^-1, k∈ℤ. The sum of 
these submodules of 𝒱^=𝐤 F_2 is direct.  


(𝒱^,·_X_1-1)V^B, X_1-1@(𝒱^,·_X_1-1) is generated, as an 
associative algebra, by 𝐤[X_0^±1]⊕𝐤[X_0^±1]X_1^-1.  


For s≥ 0 and for (k_0,…,k_s)∈ℤ^s+1, (ϵ_1,…,ϵ_s)∈{±1}^s, set 

    w(k_0,…,k_s|ϵ_1,…,ϵ_s):=X_0^k_0X_1^ϵ_1X_0^k_1⋯ X_0^k_s-1X_1^ϵ_s
    X_0^k_s∈ F_2.

If s≥ 0, let (F_2)_s be the subset of F_2 of all the elements (<ref>), where (k_0,…,k_s)∈ℤ^s+1, (ϵ_1,…,ϵ_s)∈{±1}^s; so when s=0, (F_2)_0 is the set of all X_0^k, k∈ℤ. 

One has for s≥ 0, (k_0,…,k_s+1)∈ℤ^s+1, (ϵ_2,…,ϵ_s+1)∈{±1}^s, 

    [ w(k_0,…,k_s+1|1,ϵ_2,…,ϵ_s+1)=X_0^k_0·_X_1-1
    w(k_1,…,k_s+1|ϵ_2,…,ϵ_s+1);                                   +w(k_0+k_1,k_2,…,k_s+1|ϵ_2,…,ϵ_s+1), ]


    [ w(k_0,…,k_s+1|-1,ϵ_2,…,ϵ_s+1)=-X_0^k_0X_1^-1·_X_1-1
    w(k_1,…,k_s+1|ϵ_2,…,ϵ_s+1);                                           +w(k_0+k_1,k_2,…,k_s+1|ϵ_2,…,ϵ_s+1). ]

These identities enable one to prove by induction on s≥0 that (F_2)_s is contained in the associative 
subalgebra of (𝒱^,·_X_1-1) generated by 𝐤[X_0^±1]⊕𝐤[X_0^±1]X_1^-1.
The statement then follows from the fact that the union for s≥0 of all (F_2)_s is equal to F_2.   


Recall that 𝒲^ is the subalgebra of 𝒱^ equal to 𝐤⊕𝒱^(X_1-1)
(see <ref>). We set 

    𝒲^_+:= 𝒱^(X_1-1).
    W^B+@𝒲^_+

This is a (non-unital) subalgebra of 𝒲^. It is equipped with the filtration induced by 𝒲^. 

Since the right multiplication by X_1-1 is injective in 𝒱^, the algebra morphism 

    mor_𝒱^,X_1-1morVBX_1-1@mor_𝒱^,X_1-1
    :(𝒱^,·_X_1-1)→𝒲^_+

(see <ref>) is an algebra isomorphism. 

One checks that the automorphisms 
Ad(X_1-1)^-1 and Ad(1-X_1^-1)^-1 of 𝒱^[1 X_1-1] map 𝒱^
to F^0(𝒱^[1 X_1-1]). 


The following diagram is commutative

    (𝒱^,·_X_1-1) ^Ad(X_1-1)^-1(1-Y_1^-1)^-1∘[rrrr]
    _mor_𝒱_,X_1-1^≃[d]             F^0(𝒱^[1X_1-1])^⊗2
    𝒲^_+_Δ^𝒲,[rrrr]              (𝒲^)^⊗2@^(->[u]

where the top horizontal map is the composition with ρ̃ of tensor product of the maps 
𝒱^→ F^0(𝒱^[1 X_1-1]) given by Ad(X_1-1)^-1 and Ad(1-X_1^-1)^-1
and the right vertical map is the tensor square of 𝒲^↪𝒱^↪
F^0(𝒱^[1X_1-1]). 


For k∈ℤ, 

    Δ^𝒲,∘mor_𝒱^,X_1-1(X_0^k)
    =Δ^𝒲,(X_0^k(X_1-1))= -X_0^k(1-X_1)⊗ 1-1⊗ X_0^k(1-X_1)
    
        -∑_i=1^k-1 X_0^i(1-X_1) ⊗ X_0^k-i(1-X_1)
    =Ad(X_1-1)^-1(Y_1-1)^-1(
    (X_1-1)X_0^k⊗1+1⊗ (X_1-1)X_0^k
        -∑_i=1^k-1 (X_1-1)X_0^i⊗ (X_1-1)X_0^k-i)
    =Ad((X_1-1)^-1(Y_1-1)^-1Y_1)((X_0^k)),

and similarly 

    Δ^𝒲,∘mor_𝒱^,X_1-1(X_0^kX_1^-1)
    =Δ^𝒲,(X_0^k(1-X_1^-1))
    =X_0^k(1-X_1^-1)⊗ 1+1⊗ X_0^k(1-X_1^-1)
    
        -∑_i=0^k X_0^i(1-X_1^-1) ⊗ X_0^k-i(1-X_1^-1)
    =Ad(X_1-1)^-1(Y_1-1)^-1(
    (X_1-1)X_0^kX_1^-1⊗1 
        +1⊗ (X_1-1)X_0^kX_1^-1
     -∑_i=0^k (X_1-1)X_0^iX_1^-1⊗ (X_1-1)X_0^k-iX_1^-1) 
    
        =Ad(X_1-1)^-1(Y_1-1)^-1((X_1-1)X_0^kX_1^-1⊗ X_1^-1+X_1^-1⊗ (X_1-1)X_0^kX_1^-1
        -∑_i=1^k-1 (X_1-1)X_0^iX_1^-1⊗ (X_1-1)X_0^k-iX_1^-1)
    =Ad((X_1-1)^-1(Y_1-1)^-1Y_1)((X_0^k))(X_1^-1⊗ X_1^-1)
    
        
    = Ad((X_1-1)^-1(Y_1-1)^-1Y_1)((X_0^kX_1^-1));

in each of these sequence of equalities, the second equality follows from 
(<ref>) (resp. (<ref>)), the fourth 
equality follows from Lemma <ref>. 

It follows that the two maps of the announced diagram agree on the family of elements X_0^k, X_0^kX_1^-1, k∈ℤ. 
Since these maps are algebra morphisms, and since this family generates (𝒱^,·_X_1-1) (see Lemma 
<ref>), this diagram commutes. 


The following diagram commutes

    𝒱^_≃^♢[d]^ρ[r]
        
    M_3((𝒱^)^⊗2)^(X_1-1)^-1(1-Y_1^-1)^-1row_1
    ·(-)·col_1(X_1-1)(1-Y_1^-1)_♢♯[rrrrrr]
                      
    F^0(𝒱^[1/X_1-1])^⊗2
     (𝒱^,·_X_1-1)_mor_𝒱^,X_1-1^≃♯[d]                  
    𝒲^_+_Δ^𝒲,[rrrrrrr]
                          
    (𝒲^)^⊗2@^(->[uu]

where ρ is as in (<ref>), row_1, col_1, are as in (<ref>), and Δ^𝒲, is as in <ref>; in this diagram, all the maps are 
𝐤-algebra morphisms (resp. compatible with the filtrations), except for the maps marked with ♢ (resp. ♯), which 
are only 𝐤-module morphisms (resp. increase the filtration degrees by 1).  



The map marked ♢♯ is well-defined as it is the composition of the map row_1·(-)·col_1:M_3((𝒱^)^⊗2)→(𝒱^)^⊗2 and of the tensor product of the maps 
𝒱^→ F^0(𝒱^[1 X_1-1]) given by Ad(X_1-1)^-1 and Ad(1-X_1^-1)^-1. 

The commutation of (<ref>) follows from the combination the commutative diagram from Lemma 
<ref> with the specialization, 
based on using (<ref>), of the commutative diagram from Lemma <ref> to R=𝒱^, 
S=𝒱^[1X_1-1]^⊗2, e=X_1-1, row=(X_1-1)^-1(1-Y_1^-1)^-1row_1, 
col=col_1(X_1-1)(1-Y_1^-1), which yields the commutative diagram 

    𝒱^^ρ[rrr]_≃[d]          M_3((𝒱^)^⊗2)
    ^(X_1-1)^-1(1-Y_1^-1)^-1row_1·(-)·col_1(X_1-1)(1-Y_1^-1)[d]
     
    (𝒱^,·_X_1-1)
    _Ad(X_1-1)^-1(1-Y_1^-1)^-1∘ρ̃[rrr]          F^0(𝒱^[1 X_1-1])^⊗2
 
The filtration statement about mor_𝒱^,X_1-1 follows from: X_1-1∈ F^1𝒱^, the filtration of 
𝒱^ is compatible with its algebra structure. The filtration statement about the map marked with ♢♯
follows from: same statement regarding 𝒱^[1 X_1-1]^⊗2, the components of the row (resp. column) element 
in this map belong to F^-1(𝒱^[1 X_1-1]^⊗2) (resp. F^2(𝒱^[1 X_1-1]^⊗2)). The 
filtration statement about Δ^𝒲, is Proposition <ref>. The filtration statement on 
ρ will be proved in Lemma <ref>. 


Recall that F_3 may be viewed as a normal subgroup of P_5^* (see Lemma <ref>). 
For any g∈ P_5^*, for any i∈[[1,3]], there exists an element w(g,i)∈ F_3, such that 

    g^-1x_i5g=w(g,i)x_i5w(g,i)^-1.



As F_3 is normal in P_5^*, for any g∈ P_5^*, the inner automorphism of P_5^* given by 
conjugation by g^-1 restricts to an automorphism θ_g of F_3. Then g↦θ_g defines a 
group anti-homomorphism θ:P_5^*→Aut(F_3). Let Aut^*(F_3)aut^*F_3@Aut^*(F_3) be the subgroup of Aut(F_3), consisting of all the automorphisms taking each x_i5, i∈[[1,3]], 
to a conjugate of this element. When g∈ F_3, θ_gtheta_g@θ_g is an inner automorphism of F_3, therefore belongs to Aut^*(F_3). One computes 

    θ_x_12:x_15↦Ad(x_15x_25)(x_15),  x_25↦Ad(x_15)(x_25),  x_35↦ x_35,


    θ_x_23:x_15↦ x_15,  x_25↦Ad(x_25x_35)(x_25),  x_35↦Ad(x_25)(x_35),

which implies that the images by θ of x_12^-1 and x_23^-1, and therefore also of F_2, lie in Aut^*(F_3). 
Together with Lemma <ref>, this implies that the image of P_5^* is contained in Aut^*(F_3),
and therefore the announced statement. 



Equip 𝐤 P_5^* with the adic filtration of its augmentation ideal. The algebra morphism :𝐤 P_5^*→ 
M_3(𝐤 P_5^*) is compatible with the filtrations on both sides. 


Let J:=(𝐤 P_5^*)_+. The announced statement means that for any n≥0, 
(J^n)⊂ M_3(J^n). This is obvious for n=0, let us prove it for n=1. 

As  is linear, and as J is spanned by the g-1, where g∈ P_5^*, it suffices to show that 
(g)-id∈ M_3(J) for any g∈ P_5^*, where id is the unit matrix in M_3(𝐤 P_5^*). 
Let i∈[[1,3]]; then 

    (x_i5-1)g=g· w(g,i)(x_i5-1)w(g,i)^-1=g· w(g,i)(x_i5-1)+g· w(g,i)(x_i5-1)(w(g,i)^-1-1)
 
where the first equality follows from (<ref>). 

By Lemma <ref>, the map (𝐤 F_3)^⊕ 3→(𝐤 F_3)_+, (f_i)_i∈[[1,3]]↦∑_i∈[[1,3]]f_i· (x_i5-1) is bijective. For w∈ F_3, we denote by 
(f_i(w))_i∈[[1,3]] the preimage of w-1 in (𝐤 F_3)^⊕ 3. Then 
∑_i∈[[1,3]]f_i(w)· (x_i5-1)=w-1. Substituting w by w(g,i)^-1 in 
this identity, one obtains  

    (x_i5-1)g=g· w(g,i)(x_i5-1)+∑_j∈[[1,3]]g· w(g,i)(x_i5-1)f_j(w(g,i)^-1)· (x_j5-1).

It follows that 

    (g)_ij=g· w(g,i)δ_ij+g· w(g,i)(x_i5-1)f_j(w(g,i)^-1).

As g· w(g,i)∈ P_5^*, g· w(g,i)≡ 1 mod J. Moreover, as J is a two-sided ideal, 
g· w(g,i)(x_i5-1)f_j(w(g,i)^-1)∈ J. It follows that (g)_ij≡δ_ij
mod J, and therefore that (g)-id∈ M_3(J). 
Therefore (J)⊂ M_3(J). 

Then for n≥1, (J^n)⊂(J)^n⊂ M_3(J)^n⊂ M_3(J^n). 



The algebra morphism ρ is compatible with the filtrations of its source and target algebras.  


The morphisms ℓ and pr_12 are compatible with the filtrations as they can be identified with 
algebra morphisms between group algebras induced by group morphisms, the group algebras being equipped with the adic filtrations of their augmentation ideals. Together with Lemma <ref>, this implies the result. 




 §.§ Relationship between braid groups and Δ^ℳ,
  


Set 

    col_0:=[                       0;    (1-X_1)Y_1^-1· 1_^⊗2; (1-X_1^-1)Y_1^-1· 1_^⊗2 ]∈ 
    M_3×1((ℳ^)^⊗2).
    col_0_underline@col_0




Denote by (a,x)↦ ax the action of M_3((𝒱^)^⊗2) on 
M_3× 1((ℳ^)^⊗2). Then 

    ρ(X_0-1)col_0=0.



(<ref>) implies the equalities

    ρ(X_0)col_0=
    [                                       0;        Y_1^-1Y_0Y_1(1-X_1)Y_1^-1· 1_^⊗2; -Y_1^-1Y_0Y_1X_1^-1(1-X_1)Y_1^-1· 1_^⊗2 ]
    =[                          0;    (1-X_1)Y_1^-1Y_0· 1_^⊗2; (1-X_1^-1)Y_1^-1Y_0· 1_^⊗2 ]
    =col_0
 
in M_3× 1((𝒱^/𝒱^(X_0-1))^⊗2)
=M_3× 1((ℳ^)^⊗2). 




The map (X_1-1)^-1(1-Y_1^-1)^-1row_1·(-)·col_0 : M_3((𝒱^)^⊗2)
→ℳ^[1/X_1-1]^⊗2 
defined similarly to the map in Lemma <ref>, replacing exponents DR by B and e_1 by X_1-1 
in localizations, 
has image contained in F^-1(ℳ^[1/X_1-1])^⊗2. 


This map is given by (m_ij)_i,j∈[[1,3]]↦{(Y_1-1)^-1Y_1(m_13X_1^-1-m_12)Y_1^-1(X_1-1)
}· 1_^⊗2
+{(X_1-1)^-1Y_1(m_22-m_23X_1^-1)Y_1^-1(X_1-1)}· 1_^⊗2.  The first (resp. second) summand belongs to 
F^1(ℳ^[1/X_1-1])⊗ F^-1(ℳ^[1/X_1-1]) (resp. 
F^0(ℳ^[1/X_1-1])^⊗2), 
therefore their sum belongs to the announced space. 


There is a unique map δ:ℳ^→ F^-1(ℳ^[1/X_1-1])^⊗2delta_underline@δ, 
such that the following diagram commutes 

    𝒱^^ρ[rr]_(-)· 1_[d]        M_3((𝒱^)^⊗2)
    ^
    (X_1-1)^-1(1-Y_1^-1)^-1row_1
    ·(-)·col_0[d]
    ℳ^_δ[rr]      
    F^-1(ℳ^[1/X_1-1])^⊗2

It is such that 

    δ(1_)=1_^⊗2.



If x∈𝒱^, then 

    (X_1-1)^-1(1-Y_1^-1)^-1row_1·ρ(x(X_0-1))·col_0=
    (X_1-1)^-1(1-Y_1^-1)^-1row_1·ρ(x)
    ρ(X_0-1)col_0=0

by (<ref>), therefore (
(X_1-1)^-1(1-Y_1^-1)^-1row_1·(-)·col_0
)∘ρ(𝒱^(X_0-1))=0. The existence and uniqueness of δ follow. 
One then computes 

    δ(1_)=(X_1-1)^-1(1-Y_1^-1)^-1row_1
    ·ρ(1)·col_0
    =(X_1-1)^-1(1-Y_1^-1)^-1row_1·col_0
    
        =
    (X_1-1)^-1(1-Y_1^-1)^-1·
    (1-Y_1)(1-X_1)Y_1^-1· 1_^⊗2=1_^⊗2.




The map δ satisfies the identity 

    ∀ x∈𝒲^,  ∀ m∈ℳ^,   δ(x· m)=Δ^𝒲,(x)·δ(m),

where the module structure in the left-hand side (resp. right-hand side) is that of ℳ^ over 
𝒲^ (resp. ℳ^[1/X_1-1]^⊗2 over 
(𝒲^)^⊗2).  


The identity is obvious if x=1. Assume now that x=a(X_1-1) with a∈𝒱^ and that 
m∈ℳ^=𝒱^/𝒱^(X_0-1). Let m̃∈𝒱^ be a lift of m. 
Then 

    δ(x· m)=δ(a(X_1-1)· m)
    =
    (X_1-1)^-1(1-Y_1^-1)^-1row_1
    ·ρ(a(X_1-1)m̃)·col_0
    
        =
    (X_1-1)^-1(1-Y_1^-1)^-1row_1
    ·ρ(a)ρ(X_1-1)ρ(m̃)·col_0
    
        =
    (X_1-1)^-1(1-Y_1^-1)^-1row_1
    ·ρ(a)·col_1·row_1·ρ(m̃)·col_0
    
       = 
    (X_1-1)^-1(1-Y_1^-1)^-1row_1
    ·ρ(a)·col_1· (X_1-1)(1-Y_1^-1)· 
    (X_1-1)^-1(1-Y_1^-1)^-1row_1·ρ(m̃)·col_0
        = Δ^𝒲,(x)·δ(m),
 
where row_1 is as in (<ref>), 
the second equality follows from (<ref>), the third equality follows from the fact that
ρ is an algebra morphism, the fourth equality follows from (<ref>), 
and the last equality follows from the combination of 
(<ref>) and the equality 

    Δ^𝒲,(a(X_1-1))=
    (X_1-1)^-1(1-Y_1^-1)^-1row_1·ρ(a)
    ·col_1·(X_1-1)(1-Y_1^-1)

for a∈𝒱^, which follows from (<ref>). 



The following diagram commutes 

    𝒱^^ρ[rrr]_(-)· 1_[d]           M_3((𝒱^)^⊗2)
    ^(X_1-1)^-1(1-Y_1^-1)^-1row_1·(-)·col_0[d]
    ℳ^_Δ^ℳ,[rr]      (ℳ^)^⊗2@^(->[r]   F^-1(ℳ^[1/X_1-1])^⊗2

where the right vertical map is as in Proposition <ref>.  All the maps in this diagram are compatible with the filtrations.  



Combining (<ref>), (<ref>) and the fact that ℳ^ is a free 𝒲^-module 
of rank one with generator 1_, one obtains δ=Δ^𝒲,, which we inject in 
(<ref>) to get the result. 

The filtration statement on ρ is Lemma <ref>. The filtration statement on 
Δ^ℳ, follows from Lemma <ref>. The filtration statement on the right vertical map 
follows from: the components of the row vector belong to F^-1(𝒱^[1X_1-1]^⊗2), the components 
of the column vector belong to F^1(ℳ^[1X_1-1]^⊗2), the action of 
𝒱^[1X_1-1] on ℳ^[1X_1-1] is compatible with the filtrations. 



 §.§ Completions 
(commutativities of (A1) in (<ref>) and (M1) in (<ref>))


The following lemma will be used to prove the commutativities 
mentioned in the title of this subsection. 

The commutative diagram (<ref>) (resp. (<ref>)) gives rise to a commutative diagram between the
completions of its constituents with respect to the filtrations, in which the completion of the map (𝒲^)^⊗2↪ F^0(𝒱^[1/X_1-1])^⊗2 (resp. (ℳ^)^⊗2↪ F^-1(ℳ^[1/X_1-1])^⊗2) is injective.  


By Proposition <ref>, if one equips 𝒱^ and M_3((𝒱^)^⊗2) with the 
shifted filtration F^n(𝒱^[1]):=F^n-1(𝒱^) and F^n(M_3((𝒱^)^⊗2)[1]):=
M_3(F^n-1(𝒱^)^⊗2), then (<ref>) is a diagram in the category 𝐤-mod_,+. 
By Proposition <ref>, (<ref>) is similarly a diagram in the same category. Applying the functor
(-)^∧ (see <ref>), one obtains commutative diagrams in 𝐤-mod_. The injectivity of the
completions of the said maps follows from Lemma <ref>, 2), together with the identification of the associated 
graded maps with the maps (𝒲^)^⊗2→(𝒱^[1/e_1]_≥0)^⊗2 and 
(ℳ^)^⊗2→ℳ^[1/e_1]_≥-1^⊗2 (see Proposition <ref>, 
Lemma <ref>, 2), Lemma <ref>, Lemma <ref>, 2)), which are known to be injective. 




PART: 
Associators and harmonic coproducts

In this part, we recall the formalism of associators and its interpretation in terms of categories related to braids, and do some 
related computations of matrices (<ref>). We combine these computations with the geometric 
interpretations of the Betti and de Rham algebra coproducts obtained in Part <ref> to show Theorem 
<ref>, according to which any associator relates these two algebra coproducts 
(<ref>). We follow a similar approach in <ref> to show Theorem 
<ref>, according to which any associator relates the Betti and de Rham module coproducts.  



§ ASSOCIATORS AND COMPARISON ISOMORPHISMS


The purpose of this section is to recall some facts on associators, and in particular how these objects
relate braid groups to infinitesimal braid Lie algebras.

In <ref>, we recall the definition of the set of associators over 𝐤 with a given parameter 
μ∈𝐤^× (Definition <ref>). In <ref>, we recall the construction and properties of Γ-functions of associators (Lemma <ref>). In 
<ref>, we recall from <cit.> that the choice of an associator gives rise to a functor 
from a category 𝐏𝐚𝐁 of parenthesized braids to a category 𝐏𝐚𝐂𝐃 of parenthesized 
chord diagrams; we compute images of particular morphisms of 𝐏𝐚𝐁. This leads, for each associator (μ,Φ), to a collection comp_(μ,Φ)^P of morphisms from algebras associated to braid groups to algebras associated to 
infinitesimal braid Lie algebras, indexed by parenthesized words P in one letter ∙. In <ref>, we express particular elements of the pure braid group K_4  in terms of elements 
σ_a,b∈ B_4. We apply these results to explicitly compute the images of these elements in the modular group 
P_5^* in the group of invertible elements of the completion (U𝔭_5)^∧ of the enveloping algebra of 
the infinitesimal braid Lie algebra 𝔭_5, first under the functor 
comp^(∙(∙∙))∙_(μ,Φ) (<ref>, Proposition 
<ref>), then under the functor comp^((∙∙)∙)∙_(μ,Φ) 
(<ref>, Proposition <ref>). This leads to the definition of 
matrices P_(μ,Φ), R_(μ,Φ) and P_(μ,Φ), R_(μ,Φ) (<ref>) 
and to their computation in <ref> and <ref>. 



 §.§ The set 𝖬(𝐤) of associators




Let 𝒜=F^0𝒜⊃ F^1𝒜⊃⋯ be a filtered algebra over 𝐤, complete 
with respect to its filtration. Let a_0,a_1∈ F^1𝒜. Then there is a morphism 
ev_a_0,a_1ev_a@ev_a_0,a_1:𝒱̂^→𝒜, uniquely determined by the conditions e_i↦ a_i for i=0,1. 


For Φ∈𝒱̂^, we set Φ(a_0,a_1):=ev_a_0,a_1(Φ)Phi(a_0,a_1) @ Φ(a_0,a_1)
(this is an element of 𝒜). 



Assume that 𝒜=𝒱̂^, (a_0,a_1):=(e_0,e_1), then 
ev_a_0,a_1=id_𝒱̂^, therefore Φ=Φ(e_0,e_1)Phi(e_0,e_1) @ Φ(e_0,e_1). 


Recall the graded Lie algebra 𝔱_4 from <ref>. 

  (<cit.>)
If μ∈𝐤, one defines the set 𝖬_μ(𝐤)M_muk @ 𝖬_μ(𝐤) 

to be the set of elements Φ∈𝒱̂^, which are group-like for Δ̂^𝒱, and such that  

    Φ(b,a)=Φ(a,b)^-1,


    e^(μ/2)bΦ(c,b)e^(μ/2)cΦ(a,c)e^(μ/2)aΦ(b,a)=1

(equalities in 𝐤⟨⟨ a,b⟩⟩, called the 2-cycle and hexagon conditions; here c:=-a-b), 

    Φ(t_12,t_23+t_24)·Φ(t_13+t_23,t_34)=Φ(t_23,t_34)·Φ(t_12+t_13,t_24+t_34)·Φ(t_12,t_23)

(equality in (U𝔱_4)^∧, called the pentagon condition).

We call 𝖬(𝐤):={(μ,Φ)|μ∈𝐤^×,Φ∈𝖬_μ(𝐤)}M_k @ 𝖬(𝐤) the set of associators over 𝐤. 
 

In <cit.>, it was proved  that the first two equalities are consequences of the latter one, 
where μ is obtained from the expansion Φ=1+μ^2 24[e_0,e_1]+ terms of degree ≥3.   
These two equalities also imply the relation 

    e^-(μ/2)bΦ(c,b)e^-(μ/2)cΦ(a,c)e^-(μ/2)aΦ(b,a)=1

in 𝐤⟨⟨ a,b⟩⟩. 


In <cit.>, 2, it is proved that φ_KZ∈(𝒱̂_ℂ^)^×
(see <ref>) belongs to 𝖬_1(ℂ), and that 𝖬_μ(ℚ) is nonempty, from 
which one derives that 𝖬_μ(𝐤) is nonempty for 𝐤 any ℚ-algebra and any 
μ∈𝐤^×. 




 §.§ Γ-functions


Let Φ∈𝖬_μ(𝐤). Let φ_0,φ_1 be the elements of 𝒱̂^
defined by the equality Φ=1+φ_0 e_0+φ_1 e_1 (equality in 𝒱̂^). 
Let e_0,e_1e_0, e_1^bar @ e_0,e_1 be free commutative formal variables; there is a unique continuous 𝐤-algebra morphism 
𝒱̂^→𝐤[[e_0,e_1]]
k[[e_0,e_1]]@𝐤[[e_0,e_1]], denoted f↦ f^abf^ab@f^ab, such that e_i↦e_i for i=0,1.  

 
Let μ∈𝐤 and Φ∈𝖬_μ(𝐤). Let Γ_Φ(t)∈𝐤[[t]]^× be as in 
(<ref>). 


  * One has the identity 

    (1+φ_1 e_1)^ab=Γ_Φ(-e_0)Γ_Φ(-e_1)Γ_Φ(-e_0-e_1)

in 𝐤[[e_0,e_1]]. 


  * Γ_Φ satisfies the identity

    Γ_Φ(t)Γ_Φ(-t)=μ t e^μ t/2-e^-μ t/2

in 1+t^2𝐤[[t]].  



In <cit.>, one attaches to Φ a collection (ζ_Φ(n))_n≥2zeta_Φ@ζ_Φ(n) of elements of 𝐤
with the following properties: (a) for n even ≥ 2, one has ζ_Φ(n)=μ^n·ζ(n)/(2πi)^n; 
(b) the series Γ̃_Φ(t) (denoted Γ_Φ(t) in <cit.>) defined by 
Γ̃_Φ(t):=exp(-∑_n≥2ζ_Φ(n)t^n/n) is such that 

    (1+φ_1 e_1)^ab=
    Γ̃_Φ(e_0+e_1)Γ̃_Φ(e_0)Γ̃_Φ(e_1)

(identity in 𝐤[[e_0,e_1]]). 
Then 

    (r.h.s. of (<ref>))=exp(-∑_n≥ 2ζ_Φ(n) n{(e_0+e_1)^n-e_0^n-e_1^n})
    
        =exp(-∑_n≥ 2ζ_Φ(n) n{ne_0^n-1e_1+O(e_1^2)})  
    = 1-∑_n≥ 2ζ_Φ(n)e_0^n-1e_1+O(e_1^2),

and 

    (l.h.s. of (<ref>))=1+∑_n≥ 2(Φ|e_0^n-1e_1)e_0^n-1e_1+O(e_1^2).

Then (<ref>) implies ζ_Φ(n)=-(Φ|e_0^n-1e_1) for n≥2. It follows that 
Γ_Φ(t)=1/Γ̃_Φ(-t), 
where Γ_Φ is as in (<ref>). Plugging this equality in (<ref>), we obtain 
(<ref>). This proves 1). 

Let us prove 2). Since logΓ(1-t)=γ t+∑_n≥ 2ζ(n) t^n/n, 
the series exp(2∑_n even,n≥ 2ζ(n)t^n n) is equal to 
Γ(1+t)Γ(1-t). The identities Γ(t+1)=tΓ(t) and Γ(t)Γ(-t)
=tsin(π t) imply that the latter series equals (2πi t) / (e^(2πi t)/2-
e^-(2πi t)/2), therefore 

    exp(2∑_n even,n≥ 2ζ(n)t^n n)=2πi te^(2πi t)/2-
    e^-(2πi t)/2.

Then 

    Γ_Φ(t)Γ_Φ(-t)=(Γ̃_Φ(t)Γ̃_Φ(-t))^-1
    =exp(2∑_n even,n≥ 2ζ_Φ(n)t^n n)
    =exp(2∑_n even,n≥ 2ζ(n) (μ t/(2πi))^n n)
    
        =μ t e^μ t/2-e^-μ t/2,

which proves 2). 


Lemma <ref> can also be derived by combining two results from <cit.>, namely 
the inclusion result 𝖬_μ(𝐤)⊂𝖣𝖬𝖱_μ(𝐤) 
and the result on Γ-functions for elements of 𝖣𝖬𝖱_μ(𝐤).  



The following identities 

    φ_1(α,β)=1β(Γ_Φ(-α)Γ_Φ(-β)Γ_Φ(-α-β)-1), 
      φ_0(α,β)=-1α(Γ_Φ(-α)Γ_Φ(-β)Γ_Φ(-α-β)-1),

hold in the commutative formal series ring 𝐤[[α,β]]. 


The first identity follows from the image of (<ref>) by the isomorphism of formal series rings taking 
(e_0,e_1) to (α,β). One has Φ(e_0,e_1)∈𝒢(𝒱̂^), which implies that 
logΦ(e_0,e_1)∈𝔣̂_2. One also knows that the degree 1 component of the series 
logΦ(e_0,e_1) in e_0,e_1 is zero. This implies (logΦ(e_0,e_1))^ab=0, therefore 
Φ(e_0,e_1)^ab=1. This implies the equality 1+αφ_0(α,β)+βφ_1(α,β)=1. 
Together with the first identity of (<ref>), this implies the second identity of (<ref>). 



 §.§ Functors arising from associators
 



  §.§.§ The category 𝐏𝐚𝐁


For n≥0, let Par_nParn@Par_n be the set of parenthesizations of the word ∙∙⋯∙_n letters. 
Set Par:=⊔_n≥ 0Par_nPar@Par. The set Par is a monoid with product denoted (P,Q)↦ PQ. 
For P∈Par, we denote by |P| the integer n such that P∈Par_n. 

We denote by 𝐏𝐚𝐁PaB@𝐏𝐚𝐁 the category with set of objects given by Ob(𝐏𝐚𝐁):=Par
and set of morphisms given by 𝐏𝐚𝐁(P,Q):=∅ if |P|≠|Q|, 𝐏𝐚𝐁(P,Q):=B_|P| if |P|=|Q|, where 
B_|P| is the Artin braid group with |P| strands. 

The composition 𝐏𝐚𝐁(P,Q)×𝐏𝐚𝐁(Q,R)→𝐏𝐚𝐁(P,R) is given by the product in B_|P|
if |P|=|Q|=|R|, more precisely ((Pf→Q),(Qg→R))↦
(Pgf→R) for f,g∈ B_|P|. 

We now introduce particular morphisms of 𝐏𝐚𝐁: 

(a) for P,Q,R∈Par, we denote by 

    a_P,Q,R∈𝐏𝐚𝐁((PQ)R,P(QR))
    a_PQR@a_P,Q,R

the image of 1∈ B_|P|+|Q|+|R|≃𝐏𝐚𝐁((PQ)R,P(QR));   

(b) for P∈Par and b∈ B_|P|, we denote by 

    b_P∈𝐏𝐚𝐁(P)
    b_p@b_P
 
the image of b∈ B_|P|≃𝐏𝐚𝐁(P);  

(c) for P,Q∈Par, we denote by 

    σ_P,Q∈𝐏𝐚𝐁(PQ,QP)
    sigma_PQ@σ_P,Q

the image of σ_|P|,|Q|∈ B_|P|+|Q|≃𝐏𝐚𝐁(PQ,QP), where σ_|P|,|Q| is as in (<ref>).  



  §.§.§ The category 𝐏𝐚𝐂𝐃


For n≥1, the permutation group S_nS_n@S_n acts by automorphisms of 𝔱_n by permutation of indices via σ· t_ij
=t_σ(i)σ(j). 
This action gives rise to an algebra structure on the tensor product 
(U𝔱_n)^∧⊗𝐤 S_n, and to a topological Hopf algebra on it, defined by the conditions that 
the elements of S_n be group-like and the elements of 𝔱_n be primitive; the resulting topological Hopf algebra structure
is denoted by (U𝔱_n)^∧⋊ S_nUT_nkS_n@ U(𝔱_n)^∧⋊𝐤 S_n. 


Following <cit.>, we denote by 𝐏𝐚𝐂𝐃PaCD^@𝐏𝐚𝐂𝐃 the category with set of objects Par and sets of morphisms given by 
𝐏𝐚𝐂𝐃(P,Q):=(U𝔱_|P|)^∧⋊ S_|P| if |P|=|Q| and 𝐏𝐚𝐂𝐃(P,Q):=0 
otherwise, and with composition induced by the product in (U𝔱_|P|)^∧⋊ S_|P| in the same way as the composition 
in 𝐏𝐚𝐁 is induced by the product in braid groups. 

We also denote by 𝐏𝐚𝐂𝐃^*PaCD^ast@𝐏𝐚𝐂𝐃^* the analogue of 𝐏𝐚𝐂𝐃, where the Lie algebra 𝔱_|P|
is replaced by its quotient 𝔭_|P|+1. One then has a natural functor 𝐏𝐚𝐂𝐃→𝐏𝐚𝐂𝐃^*. 



  §.§.§ The functor comp_(μ,Φ):𝐏𝐚𝐁→𝐏𝐚𝐂𝐃


It follows from <cit.>, Theorem 1, that each 
pair (μ,Φ), with μ∈𝐤^× and Φ∈𝖬_μ(𝐤), gives rise to a functor 

    comp_(μ,Φ):𝐏𝐚𝐁→𝐏𝐚𝐂𝐃comp@comp_(μ,Φ)

determined by 

    comp_(μ,Φ)(σ_∙,∙)=exp(-μ 2t_12)·([ 1 2; 2 1 ])∈𝐏𝐚𝐂𝐃(∙∙),


    comp_(μ,Φ)(a_∙,∙,∙)=Φ(t_12,t_23)
    ∈𝐏𝐚𝐂𝐃((∙∙)∙,∙(∙∙)),

and its compatibility with operations of extensions, cabling and strand removal from <cit.> 
in both the source and target categories.  

Let us again denote by 

    comp_(μ,Φ):𝐏𝐚𝐁→𝐏𝐚𝐂𝐃^*

the composition of functors 𝐏𝐚𝐁comp_(μ,Φ)→𝐏𝐚𝐂𝐃→𝐏𝐚𝐂𝐃^*. 


For P,Q∈Par with |P|=|Q|=n, we will denote by 

    comp_(μ,Φ)^P,Q:B_n→(U𝔭_n+1)^∧⋊ S_n
    comp^PQ@comp_(μ,Φ)^P,Q

the composed map 

    B_n≃𝐏𝐚𝐁(P,Q)
    comp^P,Q_(μ,Φ)→𝐏𝐚𝐂𝐃^*(P,Q)
    ≃(U𝔭_n+1)^∧⋊ S_n.

It corestricts to a map B_n→((U𝔭_n+1)^∧)^×⋊ S_n, which is a group morphism 
denoted comp_(μ,Φ)^P:=comp_(μ,Φ)^P,P
comp^P@comp_(μ,Φ)^P
 when Q=P.  

The map comp_(μ,Φ)^P,Q restricts and corestricts to a map K_n→
(U𝔭_n+1)^∧, which factors through a map 
P_n+1^*→(U𝔭_n+1)^∧. The latter map induces an isomorphism 
𝒱̂^(𝔐_0,n+1)∼→𝒱̂^(𝔐_0,n+1), 
where 𝒱̂^(𝔐_0,n+1):=(𝐤P_n+1^*)^∧, 
𝒱̂^(𝔐_0,n+1):=(U𝔭_n+1)^∧. 

It follows from the categorical formalism that the algebra isomorphisms 
comp_(μ,Φ)^(∙(∙∙))∙ and 
comp_(μ,Φ)^((∙∙)∙)∙ : 
𝒱̂^(𝔐_0,5)→𝒱̂^(𝔐_0,5) are related by 

    comp_(μ,Φ)^(∙(∙∙))∙=Ad(Φ(e_12,e_23))∘comp_(μ,Φ)^((∙∙)∙)∙



There are natural identifications comp_(μ,Φ)^𝒱,(1)≃comp_(μ,Φ)^(∙∙)∙ and comp_(μ,Φ)^𝒱,(10)≃comp_(μ,Φ)^∙(∙∙),(∙∙)∙.
 



  §.§.§ Images by comp_(μ,Φ) of particular morphisms




Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). 

1) If P,Q are in Par, with |P|=a, |Q|=b, then 

    comp_(μ,Φ)(σ_P,Q) =
     ([   1   ⋯   a a+1   ⋯ a+b; b+1   ⋯ a+b   1   ⋯   b ]) 
     ·exp(μ 2∑_β∈ a+[[1,b]]e_β,a+b+1
     +μ∑_α<β∈ a+[[1,b]]e_α,β)
      
       
      = ([   1   ⋯   a a+1   ⋯ a+b; b+1   ⋯ a+b   1   ⋯   b ]) ·exp(μ 2∑_α∈ [[1,a]]e_α,a+b+1
     +μ∑_α<β∈ [[1,a]]e_α,β)
       ∈𝐏𝐚𝐂𝐃^*(PQ,QP).


2) If P,Q,R are in Par, with |P|=a, |Q|=b, |R|=c, then 

    comp_(μ,Φ)(a_P,Q,R)=Φ(∑_γ∈ a+b+[[1,c]]e_γ,a+b+1,
    ∑_α∈[[1,a]]e_α,a+b+1)
    ∈𝐏𝐚𝐂𝐃^*((PQ)R,P(QR)).



1) It follows from (<ref>) that if P,Q are in Par, with |P|=a, |Q|=b, then 

    comp_(μ,Φ)(σ_P,Q)=
    ([   1   ⋯   a a+1   ⋯ a+b; b+1   ⋯ a+b   1   ⋯   b ]) 
     ·exp(-μ 2∑_α∈ A,β∈ Bt_α,β)
    ∈𝐏𝐚𝐂𝐃(PQ,QP),

where A:=[[1,a]], B:=a+[[1,b]]. 
The image of -1 2∑_α∈ A,β∈ Bt_α,β under 𝔱_a+b→𝔭_a+b+1 is 

    -12∑_α∈ A,β∈ Be_α,β 
    =1 2∑_β∈ B(-∑_α∈ Ae_α,β-∑_β'∈ Be_β,β')
     +∑_α<β∈ Be_α,β= 1 2∑_β∈ Be_β,a+b+1
     +∑_α<β∈ Be_α,β.

By symmetry, one also has 

    -12∑_α∈ A,β∈ Be_α,β 
    =1 2∑_α∈ Ae_α,a+b+1+∑_α<β∈ Be_α,β.

This implies that the image of comp_(μ,Φ)(σ_P,Q) in 𝐏𝐚𝐂𝐃^*(PQ,QP)
is the announced value. 

2) 
It follows from (<ref>) that if P,Q,R are in Par, with |P|=a, |Q|=b, |R|=c, then 

    comp_(μ,Φ)(a_P,Q,R)=Φ(∑_α∈ A,β∈ Bt_α,β,
    ∑_β∈ B,γ∈ Ct_β,γ)
    ∈𝐏𝐚𝐂𝐃((PQ)R,P(QR)),

where A:=[[1,a]], B:=a+[[1,b]], C:=a+b+[[1,c]]. 

For I⊂[[1,a+b+c]], set e_I:=∑_i,i'∈ Ie_i,i', and for I,J⊂[[1,a+b+c]] disjoint, set e_IJ:=∑_i∈ I,j∈ Je_i,j. 
Then 

    ∑_γ∈ Ce_γ,a+b+c+1=-∑_γ∈ C,α∈ Ae_α,γ
    -∑_γ∈ C,β∈ Be_β,γ-∑_γ,γ'∈ Ce_γ,γ'
    =z+e_AB,

where 

    z=-e_AB-e_AC-e_BC-e_C.

Similarly,

    ∑_α∈ Ae_α,a+b+c+1=z'+e_BC,

where 

    z'=-e_AB-e_AC-e_BC-e_A,

One immediately has [e_AB,e_C]=[e_BC,e_A]=0. Moreover, the relations [e_α,β+e_α',β,e_α,α']=0
for α,α'∈ A and β∈ B,  
and [e_α”,β,e_α,α']=0 for β∈ B and α,α',α”∈ A all distinct, 
imply that [e_AB,e_A]=0. Similarly, [e_AC,e_A]=0 and [e_AC,e_C]=[e_BC,e_C]=0. Finally, 

    ∑_α,α'∈ A,β∈ B,γ∈ C[e_α,γ,e_α',β]
    +∑_α∈ A,β,β'∈ B,γ∈ C[e_β,γ,e_α,β']
    =∑_α∈ A,β∈ B,γ∈ C[e_α,γ,e_α,β]
    +∑_α∈ A,β∈ B,γ∈ C[e_β,γ,e_α,β]
    
       =∑_α∈ A,β∈ B,γ∈ C[e_α,γ+e_β,γ,e_α,β]=0
 
therefore [e_AC+e_BC,e_AB]=0. 

All this implies

    [z,z']=[z,e_AB]=[z',e_BC]=[z,e_BC]=[z',e_AB]=0,

which, together with the fact that Φ is the exponential of a Lie series without linear terms, implies that 
comp_(μ,Φ)(a_P,Q,R)∈𝐏𝐚𝐂𝐃^*((PQ)R,P(QR)) is the announced value. 




 §.§ Relations in B_4



One has the following equalities in B_4:  

    x̃_1,5=σ_3,1σ_1,3,


    x̃_2,5=σ_2,2·x̃_4,5·σ_2,2^-1.


    x̃_3,5=σ_1,3·x̃_4,5·σ_1,3^-1.


    x̃_4,5=σ_1,3σ_3,1,


    x̃_2,5=σ_1,3σ_2,2^-1·x̃_1,5·σ_2,2σ_1,3^-1,



These equalities follow from the fact their left-hand sides can be represented as indicated in Figure <ref>.









 §.§ Computations of images of x̃_i5 under comp_(μ,Φ)^((∙∙)∙)∙
 



  §.§.§ Relations in 𝐏𝐚𝐁

 

1) The automorphism (x̃_1,5)_((∙∙)∙)∙ of the object ((∙∙)∙)∙ of the category 𝐏𝐚𝐁 is equal to the composition

    ((∙∙)∙)∙a_∙∙,∙,∙→
    (∙∙)(∙∙)a_∙,∙,∙∙→∙(∙(∙∙))(x̃_15)_∙(∙(∙∙))→∙(∙(∙∙))a_∙,∙,∙∙^-1→
    (∙∙)(∙∙)a_∙∙,∙,∙^-1→
    ((∙∙)∙)∙

of isomorphisms of this category. 

2) The automorphism (x̃_2,5)_((∙∙)∙)∙ of the object ((∙∙)∙)∙ of the category 𝐏𝐚𝐁 is equal to the composition

    ((∙∙)∙)∙a_∙∙,∙,∙→
    (∙∙)(∙∙)σ^-1_∙∙,∙∙→
    (∙∙)(∙∙)a^-1_∙∙,∙,∙→
    ((∙∙)∙)∙(x̃_45)_((∙∙)∙)∙→
        
    ((∙∙)∙)∙a_∙∙,∙,∙→
    (∙∙)(∙∙)σ_∙∙,∙∙→
    (∙∙)(∙∙)a_∙∙,∙,∙^-1→
    ((∙∙)∙)∙

of isomorphisms of this category. 

3) The automorphism (x̃_3,5)_((∙∙)∙)∙ of the object ((∙∙)∙)∙ of the category 𝐏𝐚𝐁 is equal to the composition

    ((∙∙)∙)∙σ^-1_∙,(∙∙)∙→∙((∙∙)∙)a^-1_∙,∙∙,∙→
    (∙(∙∙))∙(x̃_45)_(∙(∙∙))∙→
    (∙(∙∙))∙a_∙,∙∙,∙→∙((∙∙)∙)σ_∙,(∙∙)∙→
    ((∙∙)∙)∙

of isomorphisms of this category. 

4) The automorphism (x̃_4,5)_((∙∙)∙)∙ of the object ((∙∙)∙)∙ of the category 𝐏𝐚𝐁 is equal to the composition

    ((∙∙)∙)∙σ_(∙∙)∙,∙→∙((∙∙)∙)σ_∙,(∙∙)∙→
    ((∙∙)∙)∙

of isomorphisms of this category. 


Let 𝐁 be the groupoid with set of objets equal to ℤ_≥ 0 and sets of morphisms given by 
𝐁(n):=B_n if n≥0 (with the convention B_0=1) and 𝐁(n,m):=∅ if m≠ n. 

There is a functor
forget:𝐏𝐚𝐁→𝐁 given by forget(P):=|P| at the level of objects and defined at the level of 
morphisms by the condition that the map forget:𝐏𝐚𝐁(P,Q)→𝐁(|P|,|Q|) is given by the composition 
𝐏𝐚𝐁(P,Q)≃ B_|P|≃𝐁(|P|,|Q|) when |Q|=|P|, and is the only self-map of the empty set if |Q|≠|P|. 

Then 

    ∀ P,Q∈Ob(𝐏𝐚𝐁),   𝐏𝐚𝐁(P,Q)→𝐁(|P|,|Q|)  is an isomorphism.


Each of the items of the lemma states the equality of two elements of 𝐏𝐚𝐁((∙(∙∙))∙). In view 
of (<ref>), this equality is a consequence of the equality of their images in 𝐁(4)=B_4. The latter equality is itself a consequence of Lemma <ref>: (<ref>) (resp. 
(<ref>), (<ref>)) implies the equality of elements of B_4 relevant to 2) (resp. 3), 4)).  



The equalities from Lemma <ref> can be translated in the graphical language of <cit.> as
the statement that for i∈[[1,4]], the isomorphism (x̃_i5)_((∙∙)∙)∙ of the object 
((∙∙)∙)∙ of the category 𝐏𝐚𝐁 is equal to the composition of morphisms 
indicated in Figure <ref>. 









  §.§.§ Computation of images of morphisms in 𝐏𝐚𝐁



Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has

    comp_(μ,Φ)^((∙∙)∙)∙(x̃_1,5)=
    Ad(Φ(e_4,5,e_12,5)^-1Φ(e_34,5,e_1,5)^-1
    )(exp(μ e_1,5)),


    comp_(μ,Φ)^((∙∙)∙)∙(x̃_2,5)
    =Ad(
    Φ(e_4,5,e_12,5)^-1exp(μ 2e_34,5)Φ(e_34,5,e_2,5)^-1
    )
    (exp(μ e_2,5)),


    comp_(μ,Φ)^((∙∙)∙)∙(x̃_3,5)
    = Ad( exp(μ 2e_4,5)Φ(e_3,5,e_4,5))( exp(μ e_3,5)),


    comp_(μ,Φ)^((∙∙)∙)∙(x̃_4,5)
    =exp(μ e_4,5).



For i∈[[1,4]], comp_(μ,Φ)^((∙∙)∙)∙(x̃_i,5) is equal to comp_(μ,Φ)((x̃_i,5)_((∙∙)∙)∙).  One computes

    comp_(μ,Φ)((x̃_1,5)_∙(∙(∙∙)))=
    comp_(μ,Φ)(σ_∙(∙∙),∙)∘comp_(μ,Φ)(σ_∙,∙(∙∙)) 
     
        
    =([ 1 2 3 4; 2 3 4 1 ]) 
     ·exp(μ 2e_45)·([ 1 2 3 4; 4 1 2 3 ]) 
     ·exp(μ 2e_15) =exp(μ e_15),

where the second equality follows from the first equality in (<ref>) for 
(P,Q)=(∙(∙∙),∙) and from the second equality in (<ref>) for 
(P,Q)=(∙,∙(∙∙)), and the last equality is a computation in (U𝔭_5)^∧⋊ S_4. 

Then 

    comp_(μ,Φ)((x̃_1,5)_((∙∙)∙)∙)=
    comp_(μ,Φ)(a_∙∙,∙,∙)^-1∘comp_(μ,Φ)(a_∙,∙,∙∙)^-1
       ∘comp_(μ,Φ)((x̃_1,5)_∙(∙(∙∙)))
    ∘comp_(μ,Φ)(a_∙,∙,∙∙)∘comp_(μ,Φ)(a_∙∙,∙,∙)
    
       =Φ(e_4,5,e_12,5)^-1·Φ(e_34,5,e_1,5)^-1·exp(μ e_15)·Φ(e_34,5,e_1,5)·Φ(e_4,5,e_12,5),

where the first equality follows from Lemma <ref>, 1), and the second equality follows from (<ref>) for 
(P,Q,R)=(∙∙,∙,∙) and (∙,∙,∙∙) and from (<ref>). 
This proves (<ref>). 

One computes

    comp_(μ,Φ)((x̃_4,5)_((∙∙)∙)∙)=
    comp_(μ,Φ)(σ_∙,(∙∙)∙)∘comp_(μ,Φ)(σ_(∙∙)∙,∙)
     
        
    =([ 1 2 3 4; 4 1 2 3 ]) 
     ·exp(μ 2e_15) ·([ 1 2 3 4; 2 3 4 1 ]) 
     ·exp(μ 2e_4,5)=exp(μ e_4,5),

where the first equality follows from Lemma <ref>, 2), and the second 
equality follows from the second equality in (<ref>) for (P,Q)=(∙,(∙∙)∙)
and from the first equality in (<ref>) for (P,Q)=(∙,(∙∙)∙). This proves (<ref>). 

Then 

    comp_(μ,Φ)((x̃_2,5)_((∙∙)∙)∙)=
    comp_(μ,Φ)(a_∙∙,∙,∙)^-1∘comp_(μ,Φ)(σ_∙∙,∙∙)∘comp_(μ,Φ)(a_∙∙,∙,∙)
    
       ∘comp_(μ,Φ)((x̃_4,5)_((∙∙)∙)∙)∘comp_(μ,Φ)(a_∙∙,∙,∙)^-1∘comp_(μ,Φ)(σ_∙∙,∙∙)^-1∘comp_(μ,Φ)(a_∙∙,∙,∙)
    
        
    =Φ(e_4,5,e_12,5)^-1
    ·exp(μ 2e_34,5+μ e_3,4)·([ 1 2 3 4; 3 4 1 2 ]) 
    ·Φ(e_4,5,e_12,5)
    
       · e^μ e_4,5·Φ(e_4,5,e_12,5)^-1·([ 1 2 3 4; 3 4 1 2 ]) 
     ·exp(-μ 2e_34,5-μ e_3,4)·Φ(e_4,5,e_12,5)
     
        = Φ(e_4,5,e_12,5)^-1·exp(μ 2e_34,5)·exp(μ e_3,4)·Φ(e_2,5,e_34,5)·
     e^μ e_2,5·Φ(e_2,5,e_34,5)^-1
       ·exp(-μ e_3,4)·exp(-μ 2e_34,5)·Φ(e_4,5,e_12,5)
      
        =Φ(e_4,5,e_12,5)^-1·exp(μ 2e_34,5)·Φ(e_2,5,e_34,5) · 
     e^μ e_2,5·Φ(e_2,5,e_34,5)^-1·exp(-μ 2e_34,5)·Φ(e_4,5,e_12,5) 
     
        =Φ(e_4,5,e_12,5)^-1·exp(μ 2e_34,5)·Φ(e_34,5,e_2,5)^-1· 
     e^μ e_2,5·Φ(e_34,5,e_2,5)
     ·exp(-μ 2e_34,5)·Φ(e_4,5,e_12,5),

where the first equality follows from Lemma <ref>, 3), the second equality follows from 
(<ref>) for (P,Q,R)=(∙∙,∙,∙), from the second equality in (<ref>) 
for (P,Q)=(∙∙,∙∙) and from (<ref>), the third equality follows from computation in  
(U𝔭_5)^∧⋊ S_4, in particular the fact that e_3,4 commutes with e_34,5, the fourth equality follows from the
fact the e_3,4 commutes with e_2,5 and e_34,5, and the last equality follows from the 2-cycle identity 
(<ref>). This proves (<ref>). 

Finally

    comp_(μ,Φ)((x̃_3,5)_((∙∙)∙)∙)=
    comp_(μ,Φ)(σ_∙,(∙∙)∙)∘comp_(μ,Φ)(a_∙,∙∙,∙)
    
       ∘comp_(μ,Φ)((x̃_4,5)_(∙(∙∙))∙)
    ∘comp_(μ,Φ)(a_∙,∙∙,∙)^-1∘comp_(μ,Φ)(σ_∙,(∙∙)∙)^-1
        
    =  ([ 1 2 3 4; 4 1 2 3 ]) 
     ·exp(μ 2e_1,5)  ·Φ(e_4,5,e_1,5)· e^μ e_4,5·Φ(e_4,5,e_1,5)^-1·exp(-μ 2e_1,5)·([ 1 2 3 4; 2 3 4 1 ]) 
    
        =  exp(μ 2e_4,5)  ·Φ(e_3,5,e_4,5)· e^μ e_3,5·Φ(e_3,5,e_4,5)^-1·exp(-μ 2e_4,5),

where the first equality follows from Lemma <ref>, 4), the second equality follows from (<ref>) for 
(P,Q,R)=(∙,∙∙,∙), from the second equality in (<ref>) for (P,Q)=(∙,(∙∙)∙)
and from (<ref>), and the third equality follows from computation in  
(U𝔭_5)^∧⋊ S_4. This proves (<ref>). 





 §.§ Computations of images of x̃_i5 under comp_(μ,Φ)^(∙(∙∙))∙

 



  §.§.§ Relations in 𝐏𝐚𝐁



1) The automorphism (x̃_1,5)_(∙(∙∙))∙ of the object (∙(∙∙))∙ of the category 𝐏𝐚𝐁 is equal to the composition

    (∙(∙∙))∙a_∙,∙∙,∙→∙((∙∙)∙)σ_∙,(∙∙)∙→
    ((∙∙)∙)∙σ_(∙∙)∙,∙→∙((∙∙)∙)a_∙,∙∙,∙^-1→
    (∙(∙∙))∙

of isomorphisms of this category. 

2) The automorphism (x̃_2,5)_(∙(∙∙))∙ of the object (∙(∙∙))∙ of the category 𝐏𝐚𝐁 is equal to the composition

    (∙(∙∙))∙σ_∙,∙(∙∙)^-1→∙(∙(∙∙))a^-1_∙,∙,∙∙→
    (∙∙)(∙∙)σ_∙∙,∙∙→
    (∙∙)(∙∙)a_∙,∙,∙∙→∙(∙(∙∙))(x̃_1,5)_∙(∙(∙∙))→∙(∙(∙∙))
    
       a_∙,∙,∙∙^-1→
    (∙∙)(∙∙)σ_∙∙,∙∙^-1→
    (∙∙)(∙∙)a_∙,∙,∙∙→∙(∙(∙∙))σ_∙,∙(∙∙)→
    (∙(∙∙))∙

of isomorphisms of this category. 

3) The automorphism (x̃_3,5)_(∙(∙∙))∙ of the object 
(∙(∙∙))∙ of the category 𝐏𝐚𝐁 is equal to the composition

    (∙(∙∙))∙σ_∙,∙(∙∙)^-1→∙(∙(∙∙))
    a^-1_∙,∙,∙∙→(∙∙)(∙∙)
    a^-1_∙∙,∙,∙→((∙∙)∙)∙(x̃_4,5)_((∙∙)∙)∙→((∙∙)∙)∙
       a_∙∙,∙,∙→(∙∙)(∙∙)
    a_∙,∙,∙∙→∙(∙(∙∙))
    σ_∙,∙(∙∙)→
    (∙(∙∙))∙

of isomorphisms of this category. 

4) The automorphism (x̃_4,5)_(∙(∙∙))∙ of the object (∙(∙∙))∙ of the category 𝐏𝐚𝐁 is equal to the composition

    (∙(∙∙))∙σ_∙(∙∙),∙→∙(∙(∙∙))σ_∙,∙(∙∙)→
    (∙(∙∙))∙

of isomorphisms of this category. 


Each of the items of the lemma states the equality of two elements of 𝐏𝐚𝐁((∙(∙∙))∙). In view 
of (<ref>), this equality is a consequence of the equality of their images in 𝐁(4)=B_4 under the 
morphism of groupoids forget:𝐏𝐚𝐁→𝐁 from the proof of Lemma <ref>. 
The latter equality is itself a consequence of Lemma <ref>: (<ref>) 
(resp., (<ref>), (<ref>), (<ref>)) implies the equality of elements of B_4 relevant to 1) 
(resp., 2), 3), 4)). 


The equalities from Lemma <ref> can be translated in the graphical language of <cit.> as
the statement that for i∈[[1,4]], the isomorphism (x̃_i5)_(∙(∙∙))∙ of the object 
(∙(∙∙))∙ of the category 𝐏𝐚𝐁 is equal to the composition of morphisms 
indicated in Figure <ref>.






  §.§.§ Computation of images of morphisms in 𝐏𝐚𝐁



Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has

    comp_(μ,Φ)^(∙(∙∙))∙(x̃_1,5)=
    Ad(Φ(e_1,5,e_4,5))(exp(μ e_15)),


    comp_(μ,Φ)^(∙(∙∙))∙(x̃_2,5)
    =Ad( exp(μ 2e_4,5)
      Φ(e_23,5,e_4,5) exp(μ 2e_23,5) Φ(e_2,5,e_14,5))(exp(μ e_2,5)),


    comp_(μ,Φ)^(∙(∙∙))∙(x̃_3,5)
    =Ad(  exp(μ 2e_4,5)
     ·Φ(e_23,5,e_4,5)·Φ(e_3,5,e_14,5)
    )(exp(μ e_3,5)),


    comp_(μ,Φ)^(∙(∙∙))∙(x̃_4,5)
    =exp(μ e_4,5).



For i∈[[1,4]], comp_(μ,Φ)^(∙(∙∙))∙(x̃_i,5) is equal to comp_(μ,Φ)((x̃_i,5)_(∙(∙∙))∙). One computes

    comp_(μ,Φ)((x̃_1,5)_(∙(∙∙))∙)
    =comp_(μ,Φ)(a_∙,∙∙,∙^-1∘σ_(∙∙)∙,∙∘σ_∙,(∙∙)∙∘
    a_∙,∙∙,∙)
    
        
    =comp_(μ,Φ)(a_∙,∙∙,∙)^-1∘comp_(μ,Φ)(σ_(∙∙)∙,∙)∘comp_(μ,Φ)(σ_∙,(∙∙)∙)∘comp_(μ,Φ)(a_∙,∙∙,∙)
    
        = Φ(e_4,5,e_1,5)^-1·([ 1 2 3 4; 2 3 4 1 ]) 
     ·exp(μ 2e_4,5)
     ·([ 1 2 3 4; 4 1 2 3 ]) 
     ·exp(μ 2e_1,5)
     ·Φ(e_4,5,e_1,5)
     
        =Φ(e_1,5,e_4,5)·exp(μ e_15)·Φ(e_1,5,e_4,5)^-1,

where the first equality follows from Lemma <ref>, 1), the third equality from the relation between composition and product, 
from (<ref>) for (P,Q,R)=(∙,∙∙,∙), from the first equality in (<ref>) for 
(P,Q)=((∙∙)∙,∙) and from the second equality in (<ref>) for 
(P,Q)=(∙,(∙∙)∙), and the last equality follows 
from the duality relation (<ref>). This proves (<ref>). 

Similarly, 

    comp_(μ,Φ)((x̃_2,5)_(∙(∙∙))∙)
    
        
    =comp_(μ,Φ)(σ_∙,∙(∙∙)∘
    a_∙,∙,∙∙∘σ_∙∙,∙∙^-1∘
    a_∙,∙,∙∙^-1∘
    (x̃_1,5)_∙(∙(∙∙))∘
    a_∙,∙,∙∙∘σ_∙∙,∙∙∘
    a_∙,∙∙,∙^-1∘σ_∙,∙(∙∙)^-1)
    
        
    =comp_(μ,Φ)(σ_∙,∙(∙∙))∘comp_(μ,Φ)(a_∙,∙,∙∙)∘comp_(μ,Φ)(σ_∙∙,∙∙)^-1∘comp_(μ,Φ)(a_∙,∙,∙∙)^-1
       ∘comp_(μ,Φ)(x̃_1,5)_∙(∙(∙∙))∘comp_(μ,Φ)(a_∙,∙,∙∙)∘comp_(μ,Φ)(σ_∙∙,∙∙)∘comp_(μ,Φ)(a_∙,∙∙,∙)^-1
    comp_(μ,Φ)(σ_∙,∙(∙∙))^-1
        = 
    Ad(
    ([ 1 2 3 4; 4 1 2 3 ]) 
     ·exp(μ 2e_1,5)
     ·Φ(e_34,5,e_1,5)·([ 1 2 3 4; 3 4 1 2 ]) 
     ·exp(-μ 2e_34,5-μ e_3,4)·Φ(e_34,5,e_1,5)^-1)(exp(μ e_1,5))
       = 
     Ad(
      exp(μ 2e_4,5)
     ·Φ(e_23,5,e_4,5)
     ·([ 1 2 3 4; 2 3 4 1 ]) 
     ·exp(-μ 2e_34,5)·Φ(e_34,5,e_1,5)^-1) 
    (exp(μ e_15))
     
        
     = Ad( exp(μ 2e_4,5)
     ·Φ(e_23,5,e_4,5)
     ·exp(-μ 2e_14,5)·Φ(e_14,5,e_2,5)^-1) (exp(μ e_2,5))
     
        
     = Ad( exp(μ 2e_4,5)
     ·Φ(e_23,5,e_4,5)
     ·exp(μ 2e_23,5)·Φ(e_2,5,e_14,5))(exp(μ e_2,5)) ;

the first equality follows from Lemma <ref>, 2), the third equality follows from the relation between composition 
and product, from (<ref>) for (P,Q,R)=(∙,∙,∙∙) and from the second equality in (<ref>) 
for (P,Q)=(∙,∙(∙∙)) and (∙∙,∙∙). The two next equalities are computations in 
U(𝔭_5)^∧⋊ S_4, more precisely both are based on the effect of conjugation by an element of S_4 on 
an element of 𝔭_5  is this algebra, and the fourth equality also uses that 𝐤S_4→ 
U(𝔭_5)^∧⋊ S_4 is an algebra morphism, 
and the fact that e_34 commutes with both e_34,5 and e_15. The sixth equality follows from 
e_1234,5=0 and from the duality identity (<ref>). This proves (<ref>). 

One computes 

    comp_(μ,Φ)((x̃_3,5)_(∙(∙∙))∙)
    
        
    =comp_(μ,Φ)(σ_∙,∙(∙∙)∘
    a_∙,∙,∙∙∘
    a_∙∙,∙,∙∘
    (x̃_4,5)_((∙∙)∙)∙∘
    a_∙∙,∙,∙^-1∘
    a_∙,∙,∙∙^-1∘σ_∙,∙(∙∙)^-1)
    
        
    =comp_(μ,Φ)(σ_∙,∙(∙∙))∘comp_(μ,Φ)(a_∙,∙,∙∙)∘comp_(μ,Φ)(a_∙∙,∙,∙)∘comp_(μ,Φ)((x̃_4,5)_((∙∙)∙)∙)∘
       ∘comp_(μ,Φ)(a_∙∙,∙,∙)^-1∘comp_(μ,Φ)(a_∙,∙,∙∙)^-1∘comp_(μ,Φ)(σ_∙,∙(∙∙))^-1
        =
    Ad(
     ([ 1 2 3 4; 4 1 2 3 ]) 
     ·exp(μ 2e_1,5)
     ·Φ(e_34,5,e_1,5)·Φ(e_4,5,e_12,5)) 
    (exp(μ e_45))
     
       =  Ad(exp(μ 2e_4,5)
     ·Φ(e_23,5,e_4,5)·Φ(e_3,5,e_14,5)) 
    (exp(μ e_3,5)),

where the first equality follows from Lemma <ref>, 3), the third equality follows from 
(<ref>) for (P,Q,R)=(∙∙,∙,∙) and (∙,∙,∙∙),  
from the second equality in (<ref>) for (P,Q)=(∙,(∙∙)∙), and the last equality is a  
computation in (U𝔭_5)^∧⋊ S_4. This proves (<ref>). 

One computes

    comp_(μ,Φ)((x̃_4,5)_(∙(∙∙))∙)
    =comp_(μ,Φ)(
    σ_∙,∙(∙∙)∘σ_∙(∙∙),∙)
    =comp_(μ,Φ)(
    σ_∙,∙(∙∙))∘comp_(μ,Φ)(σ_∙(∙∙),∙)
    
        
    =([ 1 2 3 4; 4 1 2 3 ]) 
     ·exp(μ 2e_15) ·([ 1 2 3 4; 2 3 4 1 ]) 
     ·exp(μ 2e_45)=exp(μ e_45),

where the first equality follows from Lemma <ref>, 4), and 
the third equality follows from the second equality in (<ref>) for (P,Q)=(∙,∙(∙∙))
and from the first equality in (<ref>) for (P,Q)=(∙,∙(∙∙)). This proves (<ref>). 




 §.§ Definition of the matrices P_(μ,Φ), Q_(μ,Φ) and P_(μ,Φ), 
Q_(μ,Φ)


Recall from Lemma <ref>, 1) the graded Lie subalgebra 𝔣_3 of 𝔭_5 generated 
by the 
e_i5, i∈[[1,4]]. It follows from Propositions <ref> and <ref> that 
for i∈[[1,4]], both comp^((∙∙)∙)∙_(μ,Φ)(x̃_i5-1) and 
comp^(∙(∙∙))∙_(μ,Φ)(x̃_i5-1)  belong to the augmentation ideal of the degree 
completion (U𝔣_3)^∧ of its universal enveloping algebra.  This augmentation ideal is freely generated, as a left 
(U𝔣_3)^∧-module, by the elements e_i5, i∈[[1,3]]. It follows that there are uniquely defined collections 
(p_ij)_i,j∈[[1,3]] and (q_ij)_i,j∈[[1,3]] of elements of (U𝔣_3)^∧, such that 

    ∀ i,j∈[[1,3]],  comp^((∙∙)∙)∙_(μ,Φ)(x̃_i5-1)
    =∑_i∈[[1,3]]p_ije_j5,  comp^(∙(∙∙))∙_(μ,Φ)(x̃_i5-1)
    =∑_i∈[[1,3]]q_ije_j5

(equalities in (U𝔣_3)^∧). 


Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One sets P_(μ,Φ):=(p_ij)_i,j∈[[1,3]]∈ M_3((U𝔣_3)^∧), 
Q_(μ,Φ):=(q_ij)_i,j∈[[1,3]]∈ 
M_3((U𝔣_3)^∧).
P_mu,Phi@P_(μ,Φ)
Q_mu,Phi@Q_(μ,Φ) 

One then has 

    M_3× 1(comp^((∙∙)∙)∙_(μ,Φ))
    ([ x̃_15-1; x̃_25-1; x̃_35-1 ])
    =P_(μ,Φ)[ e_15; e_25; e_35 ],    
    M_3× 1(comp^(∙(∙∙))∙_(μ,Φ))
    ([ x̃_15-1; x̃_25-1; x̃_35-1 ])
    =Q_(μ,Φ)[ e_15; e_25; e_35 ]

(equalities in M_3× 1((U𝔣_3)^∧)). 

 If μ∈𝐤^× and Φ∈𝖬_μ(𝐤), then the matrices P_(μ,Φ) and Q_(μ,Φ) belong to GL_3((U𝔣_3)^∧). 


This follows from the fact that that both p_ij and q_ij are equal to μδ_ij modulo the augmentation ideal of 
(U𝔣_3)^∧, which is equal to the direct product of the components of U𝔣_3 of degrees >0. 



For i,j∈[[1,3]], one sets p_ij:=pr_12^∧(p_ij)∈ (𝒱^)^⊗ 2,∧, 
q_ij:=pr_12^∧(q_ij)∈ (𝒱^)^⊗ 2,∧. 



Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One sets P_(μ,Φ):=pr_12^∧(P_(μ,Φ))
=(p_ij)_i,j∈[[1,3]]∈GL_3((𝒱^)^⊗ 2,∧), 
Q_(μ,Φ):=pr_12^∧(Q_(μ,Φ))
=(q_ij)_i,j∈[[1,3]]∈GL_3((𝒱^)^⊗ 2,∧).
P_mu,Phi^bar@P_(μ,Φ)
Q_mu,Phi^bar@Q_(μ,Φ)  




 §.§ Computation of P_(μ,Φ) and P_(μ,Φ)
 


Recall the elements φ_0,φ_1∈𝒱̂^ associated with Φ (<ref>). 

Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has

    P_(μ,Φ)=   
       diag(Φ(e_4,5,e_12,5)^-1Φ(e_34,5,e_1,5)^-1,  Φ(e_4,5,e_12,5)^-1e^μ 2e_34,5Φ(e_34,5,e_2,5)^-1
    ,   e^μ 2e_4,5Φ(e_4,5,e_3,5)^-1)·
       ·[
    [                                                              (e^μ e_1,5-1)Φ(e_34,5,e_1,5)
    ·(φ_1-φ_0)(e_4,5,e_12,5)
    +(e^μ e_1,5-1)(φ_1-φ_0)(e_34,5,e_1,5)
    +e^μ e_1,5-1  e_1,5                                                                                       (e^μ e_1,5-1)Φ(e_34,5,e_1,5)
    · (φ_1-φ_0)(e_4,5,e_12,5)
    -(e^μ e_1,5-1)φ_0(e_34,5,e_1,5)                                                                                                                             -(e^μ e_1,5-1)Φ(e_34,5,e_1,5)
    ·φ_0(e_4,5,e_12,5);                        (e^μ e_2,5-1)Φ(e_34,5,e_2,5)
    · e^-μ 2e_34,5(φ_1-φ_0)(e_4,5,e_12,5)
    -(e^μ e_2,5-1)Φ(e_34,5,e_2,5)e^-μ 2e_34,5-1e_34,5
    - (e^μ e_2,5-1)φ_0(e_34,5,e_2,5) (e^μ e_2,5-1)Φ(e_34,5,e_2,5)
    · e^-μ 2e_34,5(φ_1-φ_0)(e_4,5,e_12,5)
    -(e^μ e_2,5-1)Φ(e_34,5,e_2,5)e^-μ 2e_34,5-1e_34,5
    +(e^μ e_2,5-1)(φ_1-φ_0)(e_34,5,e_2,5)
    +e^μ e_2,5-1e_2,5                                                                                                               -(e^μ e_2,5-1)Φ(e_34,5,e_2,5) 
    · e^-μ 2e_34,5φ_0(e_4,5,e_12,5);                                                                                                -(e^μ e_3,5-1)Φ(e_4,5,e_3,5)e^-μ 2e_4,5-1e_4,5
    -(e^μ e_3,5-1)φ_0(e_4,5,e_3,5)                                                                                                -(e^μ e_3,5-1)Φ(e_4,5,e_3,5)e^-μ 2e_4,5-1e_4,5
    -(e^μ e_3,5-1)φ_0(e_4,5,e_3,5)                                                                        -(e^μ e_3,5-1)Φ(e_4,5,e_3,5)e^-μ 2e_4,5-1e_4,5
    +(e^μ e_3,5-1)(φ_1-φ_0)(e_4,5,e_3,5)
    +e^μ e_3,5-1e_3,5 ]]

where diag(d_1,d_2,d_3) denotes the diagonal matrix with diagonal elements d_1,d_2,d_3.   


The computation of the first line of the matrix follows from (<ref>) and from

    (e^μ e_1,5-1)Φ(e_34,5,e_1,5)Φ(e_4,5,e_12,5)
    
        
    =(e^μ e_1,5-1)Φ(e_34,5,e_1,5)φ_0(e_4,5,e_12,5)e_4,5
    +(e^μ e_1,5-1)Φ(e_34,5,e_1,5)φ_1(e_4,5,e_12,5)e_12,5
        
    +(e^μ e_1,5-1)φ_0(e_34,5,e_1,5)e_34,5
    +(e^μ e_1,5-1)φ_1(e_34,5,e_1,5)e_1,5+(e^μ e_1,5-1)
    
       
    =((e^μ e_1,5-1)Φ(e_34,5,e_1,5)(φ_1-φ_0)(e_4,5,e_12,5)
    +(e^μ e_1,5-1)(φ_1-φ_0)(e_34,5,e_1,5)
    +e^μ e_1,5-1  e_1,5)e_1,5
        +((e^μ e_1,5-1)Φ(e_34,5,e_1,5)(φ_1-φ_0)(e_4,5,e_12,5)
    +(e^μ e_1,5-1)(-φ_0)(e_34,5,e_1,5)
    )e_2,5
        +((e^μ e_1,5-1)Φ(e_34,5,e_1,5)(-φ_0)(e_4,5,e_12,5)
    )e_3,5.

The computation of the second line follows from (<ref>) and from 

    (e^μ e_2,5-1)Φ(e_34,5,e_2,5)e^-μ 2e_34,5Φ(e_4,5,e_12,5) 
    =(e^μ e_2,5-1)Φ(e_34,5,e_2,5)e^-μ 2e_34,5φ_0(e_4,5,e_12,5)e_4,5
        
    +(e^μ e_2,5-1)Φ(e_34,5,e_2,5)e^-μ 2e_34,5φ_1(e_4,5,e_12,5)e_12,5
    +(e^μ e_2,5-1)Φ(e_34,5,e_2,5)(e^-μ 2e_34,5-1)
    
        +(e^μ e_2,5-1)φ_0(e_34,5,e_2,5)e_34,5+(e^μ e_2,5-1)φ_1(e_34,5,e_2,5)e_2,5+(e^μ e_2,5-1)
    
        
    = (
    (e^μ e_2,5-1)Φ(e_34,5,e_2,5)e^-μ 2e_34,5(φ_1-φ_0)(e_4,5,e_12,5)
    -(e^μ e_2,5-1)Φ(e_34,5,e_2,5)
    e^-μ 2e_34,5-1e_34,5
        
    +(e^μ e_2,5-1)(-φ_0)(e_34,5,e_2,5))e_1,5
    + (
    (e^μ e_2,5-1)Φ(e_34,5,e_2,5)e^-μ 2e_34,5(φ_1-φ_0)(e_4,5,e_12,5)
    
        -(e^μ e_2,5-1)Φ(e_34,5,e_2,5)e^-μ 2e_34,5-1e_34,5  
    +(e^μ e_2,5-1)(φ_1-φ_0)(e_34,5,e_2,5)+e^μ e_2,5-1e_2,5)e_2,5
        + (
    (e^μ e_2,5-1)Φ(e_34,5,e_2,5)e^-μ 2e_34,5(-φ_0)(e_4,5,e_12,5))e_3,5.

The computation of the third line follows from (<ref>), from the 2-cycle identity (<ref>), and from 

    (e^μ e_3,5-1)Φ(e_4,5,e_3,5)e^-μ 2e_4,5=
    (e^μ e_3,5-1)Φ(e_4,5,e_3,5)(e^-μ 2e_4,5-1)+(e^μ e_3,5-1)φ_0(e_4,5,e_3,5)e_4,5
        +(e^μ e_3,5-1)φ_1(e_4,5,e_3,5)e_3,5+(e^μ e_3,5-1)
    =(-(e^μ e_3,5-1)Φ(e_4,5,e_3,5)e^-μ 2e_4,5-1e_4,5
        -(e^μ e_3,5-1)φ_0(e_4,5,e_3,5))e_1,5
    + (-(e^μ e_3,5-1)Φ(e_4,5,e_3,5)e^-μ 2e_4,5-1e_4,5
        -(e^μ e_3,5-1)φ_0(e_4,5,e_3,5))e_2,5
    + (-(e^μ e_3,5-1)Φ(e_4,5,e_3,5)e^-μ 2e_4,5-1e_4,5
        +(e^μ e_3,5-1)(φ_1-φ_0)(e_4,5,e_3,5)
    +e^μ e_3,5-1e_3,5)e_3,5.




Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has 

    P_(μ,Φ)
    =     
       diag(Φ(e_0,e_1)^-1Φ(f_∞,f_1)^-1, 
    Φ(e_0,e_1)^-1e^-μ 2e_1Φ(f_∞,f_1)^-1e^-μ 2f_1, 
    e^μ 2e_0Φ^-1(e_0,e_∞)e^μ 2f_∞Φ^-1(f_∞,f_0) )
    
       ·[
    [                                                       (e^μ f_1-1)(φ_1-φ_0)(e_0+f_∞,e_1+f_1)
    +(e^μ f_1-1)(φ_1-φ_0)(-(e_1+f_1),f_1)
    +e^μ f_1-1  f_1                                                                          (e^μ f_1-1)·
    ·(φ_1-φ_0)(e_0+f_∞,e_1+f_1)
    -(e^μ f_1-1)φ_0(-(e_1+f_1),f_1)                                                                                                               -(e^μ f_1-1)·
    ·φ_0(e_0+f_∞,e_1+f_1);                   (e^μ e_1-1)e^μ 2(e_1+f_1)·
    · (φ_1-φ_0)(e_0+f_∞,e_1+f_1)
    -(e^μ e_1-1)e^μ 2(e_1+f_1)-1-(e_1+f_1)
    - (e^μ e_1-1)φ_0(-(e_1+f_1),e_1) (e^μ e_1-1)e^μ 2(e_1+f_1)·
    ·(φ_1-φ_0)(e_0+f_∞,e_1+f_1)
    -(e^μ e_1-1)e^μ 2(e_1+f_1)-1-(e_1+f_1)
    +(e^μ e_1-1)(φ_1-φ_0)(-(e_1+f_1),e_1)
    +e^μ e_1-1e_1                                                                                                 -(e^μ e_1-1)e^μ 2(e_1+f_1)·
    ·φ_0(e_0+f_∞,e_1+f_1);                                          -(e^μ (e_∞+f_0)-1)Φ(e_0,e_∞)·
    ·Φ(f_∞,f_0)e^-μ 2(e_0+f_∞)-1e_0+f_∞
    -(e^μ (e_∞+f_0)-1)φ_0(e_0+f_∞,e_∞+f_0)                                          -(e^μ (e_∞+f_0)-1)Φ(e_0,e_∞)·
    ·Φ(f_∞,f_0)e^-μ 2(e_0+f_∞)-1e_0+f_∞
    -(e^μ (e_∞+f_0)-1)φ_0(e_0+f_∞,e_∞+f_0)          -(e^μ (e_∞+f_0)-1)Φ(e_0,e_∞)·
    ·Φ(f_∞,f_0)e^-μ 2(e_0+f_∞)-1e_0+f_∞
    +(e^μ(e_∞+f_0)-1)·
    ·(φ_1-φ_0)(e_0+f_∞,e_∞+f_0)
    +e^μ (e_∞+f_0)-1e_∞+f_0 ]]



One transforms the expression obtained from Proposition <ref> using identities Φ(e_0+f_∞,e_1+f_1)=Φ(e_0,e_1)Φ(f_∞,f_1), Φ(e_0+f_∞,e_∞+f_0)=Φ(e_0,e_∞)Φ(f_∞,f_0), 
Φ(-(e_1+f_1),f_1)=Φ(-(e_1+f_1),e_1)=1, which follow from the fact Φ is a group-like element with vanishing linear terms 
in e_0,e_1. 



 §.§ Computation of Q_(μ,Φ) and Q_(μ,Φ)



Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has

    Q_(μ,Φ)
    =     
       diag(Φ(e_15,e_45), 
    e^μ 2e_45Φ(e_23,5,e_45) e^μ 2e_23,5Φ(e_25,e_14,5), 
    e^μ 2e_45Φ(e_23,5,e_45) Φ(e_35,e_14,5) )·
       ·[
    [                                                                                                                                                                                                 e^μ e_15-1e_15
    +(e^μ e_15-1)(φ_1-φ_0)(e_45,e_15)                                                                                                                                                                                                                      -(e^μ e_15-1)φ_0(e_45,e_15)                                                                                                                                                                                                                      -(e^μ e_15-1)φ_0(e_45,e_15);                                                                                                            -(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5·
    ·Φ(e_45,e_23,5)
    e^-μ 2e_45-1e_45
    -(e^μ e_25-1)Φ(e_14,5,e_25)·
    · e^-μ 2e_23,5φ_0(e_45,e_23,5) -(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5·
    ·Φ(e_45,e_23,5)e^-μ 2e_45-1e_45
    +(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5·
    · (φ_1-φ_0)(e_45,e_23,5)
    + (e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5-1e_23,5
     +(e^μ e_25-1)(φ_1-φ_0)(e_14,5,e_25)
    +e^μ e_25-1e_25                         -(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5·
    ·Φ(e_45,e_23,5)e^-μ 2e_45-1e_45
    +(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5·
    ·(φ_1-φ_0)(e_45,e_23,5)
    + (e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5-1e_23,5
    -(e^μ e_25-1)φ_0(e_14,5,e_25);                                                                                                                                        -(e^μ e_35-1)Φ(e_14,5,e_35)·
    ·Φ(e_45,e_23,5)e^-μ 2e_45-1e_45
    - (e^μ e_35-1)Φ(e_14,5,e_35)φ_0(e_45,e_23,5)                                                                                            -(e^μ e_35-1)Φ(e_14,5,e_35)
    ·
    ·Φ(e_45,e_23,5) e^-μ 2e_45-1e_45
    + (e^μ e_35-1)Φ(e_14,5,e_35)
    ·
    ·
    (φ_1-φ_0)(e_45,e_23,5)
    - (e^μ e_35-1)φ_0(e_14,5,e_35)                                                                         -(e^μ e_35-1)Φ(e_14,5,e_35)·
    ·Φ(e_45,e_23,5) e^-μ 2e_45-1e_45
    + (e^μ e_35-1)Φ(e_14,5,e_35)·
    ·(φ_1-φ_0)(e_45,e_23,5)
    +(e^μ e_35-1)(φ_1-φ_0)(e_14,5,e_35)
    + e^μ e_35-1e_35 ]]



The computation of the first line of the matrix follows from (<ref>), from the 2-cycle identity (<ref>), and from 

    (e^μ e_15-1)Φ(e_45,e_15)=(e^μ e_15-1)(1+φ_0(e_45,e_15)e_45+φ_1(e_45,e_15)e_15)
    
        =(e^μ e_15-1e_15
    +(e^μ e_15-1)φ_1(e_45,e_15))e_15+(e^μ e_15-1)φ_0(e_45,e_15)e_45
       =(e^μ e_15-1e_15+(e^μ e_15-1)(φ_1-φ_0)(e_45,e_15))e_15
    -(e^μ e_15-1)φ_0(e_45,e_15)e_25-(e^μ e_15-1)φ_0(e_45,e_15)e_35.


The computation of the second line of the matrix follows from (<ref>), from the 2-cycle identity (<ref>), and from

    (e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5Φ(e_45,e_23,5)e^-μ 2e_45
    =(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5Φ(e_45,e_23,5)·
       ·(e^-μ 2e_45-1)
    +(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5 
    (φ_0(e_45,e_23,5)e_45
    +φ_1(e_45,e_23,5)e_23,5) 
    
       + (e^μ e_25-1)Φ(e_14,5,e_25)(e^-μ 2e_23,5-1)
    + (e^μ e_25-1)(φ_0(e_14,5,e_25)e_14,5+φ_1(e_14,5,e_25)e_25)
     +(e^μ e_25-1)
       
    =(
    -(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5Φ(e_45,e_23,5)
    e^-μ 2e_45-1e_45
    -(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5φ_0(e_45,e_23,5)) e_15
       +(
    -(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5Φ(e_45,e_23,5)
    e^-μ 2e_45-1e_45
    +(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5 
    (φ_1-φ_0)(e_45,e_23,5)
       
    + (e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5-1e_23,5
     +(e^μ e_25-1)(φ_1-φ_0)(e_14,5,e_25)
    +e^μ e_25-1e_25) e_25
       +(
    -(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5Φ(e_45,e_23,5)
    e^-μ 2e_45-1e_45
    +(e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5 
    (φ_1-φ_0)(e_45,e_23,5)
       
    + (e^μ e_25-1)Φ(e_14,5,e_25)e^-μ 2e_23,5-1e_23,5
     -(e^μ e_25-1)φ_0(e_14,5,e_25))e_35


The computation of the third line of the matrix follows from (<ref>), from the 2-cycle identity (<ref>), 
and from

    (e^μ e_35-1)Φ(e_14,5,e_35)Φ(e_45,e_23,5) e^-μ 2e_45 
    =(e^μ e_35-1)Φ(e_14,5,e_35)Φ(e_45,e_23,5) (e^-μ 2e_45-1)
    
       + (e^μ e_35-1)Φ(e_14,5,e_35)(φ_0(e_45,e_23,5)e_45+φ_1(e_45,e_23,5)e_23,5)
    
       + (e^μ e_35-1)(φ_0(e_14,5,e_35)e_14,5+φ_1(e_14,5,e_35)e_35)+ (e^μ e_35-1)
    
       
    =( -(e^μ e_35-1)Φ(e_14,5,e_35)Φ(e_45,e_23,5) e^-μ 2e_45-1e_45
    - (e^μ e_35-1)Φ(e_14,5,e_35)φ_0(e_45,e_23,5)) e_15
       
    +( -(e^μ e_35-1)Φ(e_14,5,e_35)Φ(e_45,e_23,5) e^-μ 2e_45-1e_45
    +(e^μ e_35-1)Φ(e_14,5,e_35)(φ_1-φ_0)(e_45,e_23,5)
    
       
    - (e^μ e_35-1)φ_0(e_14,5,e_35))e_25
    +( -(e^μ e_35-1)Φ(e_14,5,e_35)Φ(e_45,e_23,5) e^-μ 2e_45-1e_45
       +(e^μ e_35-1)Φ(e_14,5,e_35)(φ_1-φ_0)(e_45,e_23,5)
    + (e^μ e_35-1)(φ_1-φ_0)(e_14,5,e_35)
    + e^μ e_35-1e_35)e_35.




Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has

    Q_(μ,Φ)= 
       diag(Φ(f_1,f_∞), e^μ 2f_∞Φ(f_0,f_∞)e^μ 2f_0Φ(e_1,e_0), 
     e^μ 2(e_0+f_∞)Φ(f_0,f_∞)Φ(e_∞,e_0))·
       ·[
    [                                                                                                                                                                                                       e^μ f_1-1f_1
    +(e^μ f_1-1)(φ_1-φ_0)(e_0+f_∞,f_1)                                                                                                                                                                                                                          -(e^μ f_1-1)φ_0(e_0+f_∞,f_1)                                                                                                                                                                                                                          -(e^μ f_1-1)φ_0(e_0+f_∞,f_1);                                                                                                          -(e^μ e_1-1)Φ(e_0,e_1)e^-μ 2(-e_0+f_0)·
    ·Φ(f_∞,f_0)
    e^-μ 2(e_0+f_∞)-1e_0+f_∞
    -(e^μ e_1-1)Φ(e_0,e_1) e^-μ 2(-e_0+f_0)·
    ·φ_0(e_0+f_∞,-e_0+f_0) -(e^μ e_1-1)Φ(e_0,e_1)e^-μ 2(-e_0+f_0)·
    ·Φ(f_∞,f_0)e^-μ 2(e_0+f_∞)-1e_0+f_∞
    +(e^μ e_1-1)Φ(e_0,e_1)e^-μ 2(-e_0+f_0)·
    · (φ_1-φ_0)(e_0+f_∞,-e_0+f_0)
    + (e^μ e_1-1)Φ(e_0,e_1)e^-μ 2(-e_0+f_0)-1-e_0+f_0
     +(e^μ e_1-1)(φ_1-φ_0)(e_0-f_0,e_1)
    +e^μ e_1-1e_1                       -(e^μ e_1-1)Φ(e_0,e_1)e^-μ 2(-e_0+f_0)·
    ·Φ(f_∞,f_0)e^-μ 2(e_0+f_∞)-1e_0+f_∞
    +(e^μ e_1-1)Φ(e_0,e_1)e^-μ 2(-e_0+f_0)·
    ·(φ_1-φ_0)(e_0+f_∞,-e_0+f_0)
    + (e^μ e_1-1)Φ(e_0,e_1)e^-μ 2(-e_0+f_0)-1-e_0+f_0
    -(e^μ e_1-1)φ_0(e_0-f_0,e_1);                                                                                                                               -(e^μ (e_∞+f_0)-1)Φ(e_0,e_∞)·
    ·Φ(f_∞,f_0)e^-μ 2(e_0+f_∞)-1e_0+f_∞
    - (e^μ (e_∞+f_0)-1)Φ(e_0,e_∞)·
    ·φ_0(e_0+f_∞,-e_0+f_0)                                                                             -(e^μ (e_∞+f_0)-1)Φ(e_0,e_∞)
    ·
    ·Φ(f_∞,f_0) e^-μ 2(e_0+f_∞)-1e_0+f_∞
    + (e^μ (e_∞+f_0)-1)Φ(e_0,e_∞)
    ·
    ·
    (φ_1-φ_0)(e_0+f_∞,-e_0+f_0)
    - (e^μ (e_∞+f_0)-1)φ_0(e_0-f_0,e_∞+f_0)                                                  -(e^μ (e_∞+f_0)-1)Φ(e_0,e_∞)·
    ·Φ(f_∞,f_0) e^-μ 2(e_0+f_∞)-1e_0+f_∞
    + (e^μ (e_∞+f_0)-1)Φ(e_0,e_∞)·
    ·(φ_1-φ_0)(e_0+f_∞,-e_0+f_0)
    +(e^μ (e_∞+f_0)-1)(φ_1-φ_0)(e_0-f_0,e_∞+f_0)
    + e^μ (e_∞+f_0)-1e_∞+f_0 ]]



One transforms the expression obtained from Proposition <ref> using the commutativity of e_0 with f_0,f_∞, 
as well as the identities Φ(e_0+f_∞,e_1+f_1)=Φ(e_0,e_1)Φ(f_∞,f_1), 
Φ(e_0+f_∞,e_∞+f_0)=Φ(e_0,e_∞)Φ(f_∞,f_0), 
Φ(-(e_1+f_1),f_1)=Φ(-(e_1+f_1),e_1)=1, which follow from the fact Φ is a group-like element with vanishing linear terms 
in e_0,e_1. 



§ ASSOCIATORS AND HARMONIC ALGEBRA COPRODUCTS


The purpose of this section is the proof of the first main result of this paper, namely the compatibility of associators with 
harmonic algebra coproducts (Theorem <ref>). This proof relies on the commutativity of 
diagram (<ref>), which is divided into subdiagrams (A1) to (A8). Among them, subdiagram (A1) (resp. (A7)) 
expresses the interpretation of the Betti (resp. de Rham) algebra coproduct and was proved in Lemma <ref> 
(resp. Lemma <ref>). The main remaining diagrams are (A3) and (A4), both of which relate, by means of the 
matrix P_(μ,Φ) from <ref> and the algebra comparison isomorphisms, 
`de Rham' and `Betti' objects: namely, the morphisms ρ̂ and 
ρ̂ in the case of (A3), and the morphisms row_1·(-)·col_1 and 
row_1·(-)·col_1 in the case of (A4). <ref> to 
<ref> are devoted to the commutativity of (A3); more precisely, <ref> to 
<ref> are devoted to the commutativity of diagrams relating constituents of ρ̂ and 
ρ̂, and these results are gathered in <ref> for proving the commutativity of (A3). 
<ref> to <ref> are devoted to the commutativity of (A4); 
more precisely, <ref> (resp. <ref>) is devoted to the proof of 
equalities relating P_(μ,Φ), col_1 and col_1 (resp. P_(μ,Φ), 
row_1 and row_1), based on the explicit computation of P_(μ,Φ) in 
<ref>, and these results are combined in <ref> for proving the commutativity of (A4).  
Theorem <ref> is formulated and proved in <ref>. 



 §.§ Commutative diagram relating ℓ̂ and ℓ̂



One checks that for (μ,Φ)∈𝐤^××𝒢(𝒱̂^), the morphism  

    comp_(μ,Φ)^𝒱,(1):𝒱̂^→𝒱̂^

defined in <ref> is an isomorphism of topological Hopf algebras, where the two sides are equipped with the coproducts
Δ^𝒱, and Δ^𝒱,, and is given by  

    X_0↦Φ(e_0,e_1)e^μ e_0Φ(e_0,e_1)^-1,    X_1↦ e^μ e_1.

Then: 

Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). The following diagram of topological Hopf algebras 

    𝒱̂^^ℓ̂[r]_comp_(μ,Φ)^𝒱,(1)[d]    
    (𝐤P_5^*)^∧^comp_(μ,Φ)^((∙∙)∙)∙[d]
    
    𝒱̂^_ℓ̂[r]
        (U𝔭_5)^∧

is commutative, where comp_(μ,Φ)^𝒱,(1) is as in (<ref>), 
ℓ̂l_underline^hat@ℓ̂ is the completed version of ℓ from <ref>, 
comp_(μ,Φ)^((∙∙)∙)∙ is as in <ref>, and 
ℓ̂l^hat@ℓ̂ is the completed version of ℓ from 
<ref>.


The following equalities x_12=σ_1^2=σ_1,1^-2 hold in B_2 (see 
<ref>) and in view of <ref>, (c), 
imply the equality (x_12)_∙∙=σ_∙,∙^-2 in 
𝐏𝐚𝐁(∙∙). The following equalities 
(x_12)_((∙∙)∙)∙=(σ_∙,∙^-2⊗id_∙)⊗id_∙
and (x_23)_(∙(∙∙))∙=(id_∙⊗σ_∙,∙^-2)⊗id_∙ similarly hold in 𝐏𝐚𝐁(((∙∙)∙)∙) and 
𝐏𝐚𝐁((∙(∙∙))∙). <ref> then implies the equalitites  

    comp_(μ,Φ)^((∙∙)∙)∙(x_12)
    =e^μ e_12,   comp_(μ,Φ)^(∙(∙∙))∙(x_23)
    =e^μ e_23.

Then  

    comp_(μ,Φ)^((∙∙)∙)∙(x_23)=
    comp_(μ,Φ)((x_23)_((∙∙)∙)∙)
    
       =comp_(μ,Φ)((a_∙,∙,∙⊗id_∙)^-1∘(x_23)_(∙(∙∙))∙∘(a_∙,∙,∙⊗id_∙))
    =Φ(e_12,e_23)^-1e^μ e_23Φ(e_12,e_23),

where the last identity uses the second part of (<ref>). 

The first part of (<ref>) implies that comp_(μ,Φ)^((∙∙)∙)∙∘ℓ̂(X_1)=e^μ e_12, which is equal to ℓ̂∘comp_(μ,Φ)^𝒱,(1)(X_1). 
(<ref>) implies comp_(μ,Φ)^((∙∙)∙)∙∘ℓ̂(X_0)=
Φ(e_12,e_23)^-1e^μ e_23Φ(e_12,e_23). One has 
ℓ̂∘comp_(μ,Φ)^𝒱,(1)(X_0)=Φ(e_23,e_12)e^μ e_23Φ(e_23,e_12)^-1. 
The 2-cycle identity (<ref>) implies that these images are equal. Therefore the images of both X_0 and X_1
under comp_(μ,Φ)^((∙∙)∙)∙∘ℓ̂ and 
ℓ̂∘comp_(μ,Φ)^𝒱,(1) are the same. 





 §.§ Commutative diagram relating M_3(pr_12^∧) and M_3(pr^∧_12)




Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). The following 

    (𝐤 P_5^*)^∧_comp_(μ,Φ)^((∙∙)∙)∙[d]
    ^pr_1^∧[rrr]         𝒱̂^^comp_(μ,Φ)^𝒱,(1)[d]
     (U𝔭_5)^∧_pr_1^∧[r]   𝒱̂^      𝒱̂^^Ad(Φ(e_0,e_1)^-1)[ll]   (𝐤P_5^*)^∧_comp_(μ,Φ)^((∙∙)∙)∙[d]
    ^pr_2^∧[rrr]         𝒱̂^^comp_(μ,Φ)^𝒱,(1)[d]
     (U𝔭_5)^∧_pr_2^∧[r]   𝒱̂^      𝒱̂^^Ad(Φ(e_∞,e_1)^-1e^(μ/2)e_1)[ll]

are commutative diagrams of topological Hopf algebras.  


One computes 

    comp_(μ,Φ)^((∙∙)∙)∙(x_34)=
    comp_(μ,Φ)((x_34)_((∙∙)∙)∙)
    
        =comp_(μ,Φ)((a_∙∙,∙,∙)^-1∘(x_34)_(∙∙)(∙∙)∘ a_∙∙,∙,∙)
    =Φ(e_12,3,e_34)^-1e^μ e_34Φ(e_12,3,e_34).


It follows from Proposition <ref> that the elements x_i,i+1, i∈ C_5 generate P_5^*. So it suffices to check 
the commutativity of the diagrams on these elements. Using (<ref>) and (<ref>), 
(<ref>),  (<ref>) and (<ref>), the images of these generators 
under the two maps can be shown to be given by the following table. 

 

 generator x of  P_5^*     x_12      x_23      x_34      x_45      x_15

  [                              common value of; Ad(Φ(e_0,e_1)^-1)∘comp_(μ,Φ)^𝒱,(1)∘pr_1^∧(x);            and pr_1^∧∘comp_(μ,Φ)^((∙∙)∙)∙(x) ]
    1     e^μ e_0     Φ(e_0,e_1)^-1e^μ e_1Φ(e_0,e_1)     e^μ e_0     1

 

 
This implies that the first diagram commutes. 

The situation in the case of the second diagram is given by the following table  

 

 generator x of  P_5^*     x_12      x_23      x_34      x_45      x_15

  [ Ad(Φ(e_∞,e_1)^-1e^(μ/2)e_1); ∘comp_(μ,Φ)^𝒱,(1)∘pr_2^∧(x) ]
    1     1     [ Φ(e_∞,e_1)^-1e^μ e_1·;           ·Φ(e_∞,e_1) ]     ⋆     
[ Φ(e_∞,e_1)^-1e^μ e_1·;           ·Φ(e_∞,e_1) ]

 pr_2^∧∘comp_(μ,Φ)^((∙∙)∙)∙(x)
    1     1     [ Φ(e_∞,e_1)^-1e^μ e_1·;           ·Φ(e_∞,e_1) ]     e^μ e_∞     
[ Φ(e_∞,e_1)^-1e^μ e_1·;           ·Φ(e_∞,e_1) ]

 

 
where ⋆=Φ(e_∞,e_1)^-1e^-(μ/2)e_1Φ(e_0,e_1)e^-μ e_0Φ(e_0,e_1)^-1e^-(μ/2)e_1Φ(e_∞,e_1). 

By the hexagon identity (<ref>), one has 

    e^(μ/2)t_23Φ(t_12,t_23)e^(μ/2)t_12=Φ(t_13,t_23)e^(μ/2)(t_12+t_23)Φ(t_12,t_13)
 
and by the exchange of t_12 and t_23, also 

    e^(μ/2)t_12Φ(t_23,t_12)e^(μ/2)t_23=Φ(t_13,t_12)e^(μ/2)(t_12+t_23)Φ(t_23,t_13).

Taking the product of the last equality with the previous one and using the 2-cycle relation, we get 

    e^(μ/2)t_12Φ(t_23,t_12)e^μ t_23Φ(t_12,t_23)e^(μ/2)t_12
    =Φ(t_13,t_12)e^μ(t_12+t_23)Φ(t_12,t_13)

Inverting, using the 2-cycle identity and conjugating by Φ(t_13,t_12)^-1e^-(μ/2)t_12e^(μ/2)t_12, 
we get 

    Φ(t_13,t_12)^-1e^-(μ/2)t_12Φ(t_12,t_23)^-1e^-μ t_23Φ(t_23,t_12)^-1e^-μ t_12
    (Φ(t_13,t_12)^-1e^-(μ/2)t_12)^-1=e^-μ t_2,13.


Taking the image of the resulting identity by the 
morphism (U𝔱_3)^∧→𝒱̂^ given by t_12↦ e_1, t_23↦ e_0, 
t_13↦ e_∞ and using the 2-cycle identity, one obtains the equality 
⋆=e^μ e_∞. It follows that the second diagram commutes. 


 
Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). The following diagram commutes 

    M_3((𝐤 P_5^*)^∧)_M_3(comp_(μ,Φ)^((∙∙)∙)∙)[d]
    ^M_3(pr_12^∧)[rr]
           M_3((𝒱^)^⊗2∧)^M_3((comp_(μ,Φ)^𝒱,(1))^⊗2)[d]
     
    M_3((U𝔭_5)^∧)
    _M_3(pr_12^∧)[r]    M_3((𝒱^)^⊗2∧) _Ad(κ_(μ,Φ))[r]    
    M_3((𝒱^)^⊗2∧)

where 

    κ_(μ,Φ):=Φ(e_0,e_1)· e^-(μ/2)f_1Φ(f_∞,f_1)∈(𝒱^)^⊗2∧,  
    kappa_mu,Phi@κ_(μ,Φ)
 
and Ad(κ_(μ,Φ)) denotes the automorphism taking each entry of a matrix to its image by the automorphism 
Ad(κ_(μ,Φ)) of (𝒱^)^⊗2∧. 


Combining the tensor product of the two diagrams from Lemma <ref> with the diagram expressing the 
compatibility of comp_(μ,Φ)^((∙∙)∙)∙ with the coproducts of its source and 
target, one obtains the following commutative diagram

    (𝐤 P_5^*)^∧_comp_(μ,Φ)^((∙∙)∙)∙[d]
    ^pr_12^∧[rrrrr]                (𝒱^)^⊗2∧^(comp_(μ,Φ)^𝒱,(1))^⊗2[d]
     (U𝔭_5)^∧_pr_12^∧[r]    (𝒱^)^⊗2∧             (𝒱^)^⊗2∧^Ad(Φ(e_0,e_1)^-1Φ(f_∞,f_1)^-1e^(μ/2)f_1)[llll]

from where we derive the announced commutative diagram.  



 §.§ Commutative diagram relating  and 



Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). The following diagram is commutative 

    (𝐤 P_5^*)^∧^[rr]_comp_(μ,Φ)^((∙∙)∙)∙[d]       
    M_3((𝐤 P_5^*)^∧)^M_3(comp_(μ,Φ)^((∙∙)∙)∙)[d]
     
    (U𝔭_5)^∧_[r]    M_3((U𝔭_5)^∧)_Ad(P_(μ,Φ))[r]
       M_3((U𝔭_5)^∧)

where pivar^hat@,pivar_underline^hat@ are the 
completions of , (see <ref>, <ref>) and 
P_(μ,Φ)∈GL_3((U𝔣_3)^∧)⊂GL_3((U𝔭_5)^∧) is 
given by Definition <ref>. 


This follows from Lemma <ref>, with R=(𝐤P_5^*)^∧, 
J=J(pr_5)^∧, (j_a)_a∈[[1,d]]=(x_i5-1)_i∈[[1,3]], 
R'=(U𝔭_5)^∧, J'=J(pr_5)^∧, (j'_a)_a∈[[1,d]]=(e_i5)_i∈[[1,3]], 
f=comp_(μ,Φ)^((∙∙)∙)∙. 



 §.§ Commutative diagram relating  and  ((A3) in (<ref>))



Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤) and let κ_(μ,Φ)·P_(μ,Φ)∈GL_3((𝒱^)^⊗2∧) be the matrix obtained 
by the left multiplication by the scalar κ_(μ,Φ)∈((𝒱^)^⊗2∧)^× 
(see (<ref>)) of the entries of the matrix P_(μ,Φ)∈GL_3((𝒱^)^⊗2∧) 
(see Definition <ref>). The following diagram commutes 

    𝒱̂^^[rr]_comp_(μ,Φ)^𝒱,(1)[d]       
    M_3((𝒱^)^⊗2∧)^M_3((comp_(μ,Φ)^𝒱,(1))^⊗2)[d]
    𝒱̂^_[r]    M_3((𝒱^)^⊗2∧)
    _Ad(κ_(μ,Φ)·P_(μ,Φ))[r] 
        M_3((𝒱^)^⊗2∧)
 
where ρ̂, ρ̂ are as in  <ref>, <ref>. 


One checks that the following diagram commutes 

    M_3((U𝔭_5)^∧)^Ad(P_(μ,Φ))[r]_M_3(pr_12^∧)[rrrd]    
    M_3((U𝔭_5)^∧)
    ^M_3(pr_12^∧)[r]
        M_3((𝒱^)^⊗2∧)^Ad(κ_(μ,Φ))[r]    
    M_3((𝒱^)^⊗2∧)
              M_3((𝒱^)^⊗2∧)_Ad(κ_(μ,Φ)·P_(μ,Φ))[u].

The result follows from the juxtaposition of this diagram and of the diagrams from Lemmas <ref>, 
<ref> and <ref>. 



 §.§ Relationship between col_1, col_1 and P_(μ,Φ)




Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has 

    (comp_(μ,Φ)^𝒱,(1))^⊗2(col_1)=κ_(μ,Φ)·P_(μ,Φ)·col_1· v_(μ,Φ)

(equality in M_3×1((𝒱^)^⊗2∧)), where col_1, col_1 
are as in (<ref>), (<ref>), 
κ_(μ,Φ), P_(μ,Φ)
are as in (<ref>), Definition <ref> and where  

    v_(μ,Φ):=1μe^μ f_1Γ_Φ(e_1)Γ_Φ(f_1)Γ_Φ(e_1+f_1)∈(𝒱^)^⊗2∧. 
    v_mu,Phi@v_(μ,Φ)



Let d_(μ,Φ) and P̃_(μ,Φ) be the matrices of the right-hand side of equality of Corollary 
<ref>, so that P_(μ,Φ)=d_(μ,Φ)·P̃_(μ,Φ). One computes 

    P̃_(μ,Φ)·col_1=[  (e^μ f_1-1)(φ_1(-(e_1+f_1),f_1)+1 f_1); -(e^μ e_1-1)(φ_1(-(e_1+f_1),e_1)+1 e_1);                                       0 ]
    =[  e^μ f_1-1f_1Γ_Φ(e_1+f_1)Γ_Φ(-f_1)Γ_Φ(e_1); -e^μ e_1-1e_1Γ_Φ(e_1+f_1)Γ_Φ(-e_1)Γ_Φ(f_1);                                          0 ]
        
    =[  μ e^(μ/2)f_1; -μ e^(μ/2)e_1;             0 ]Γ_Φ(e_1+f_1)Γ_Φ(e_1)Γ_Φ(f_1),

where the second equality follows from φ_1(α,β)=1β(Γ_Φ(-α)Γ_Φ(-β)Γ_Φ(-α-β)-1) for α,β commutative formal variables (see (<ref>)), the third equality follows from the functional equation (<ref>) for Γ_Φ. Therefore 

    P̃_(μ,Φ)·col_1· v_(μ,Φ)=
    [        e^(3μ/2)f_1; - e^(μ/2)e_1+μ f_1;                  0 ].
 
On the other hand, κ_(μ,Φ)d_(μ,Φ) is a diagonal matrix of the form diag(e^-(μ/2)f_1,e^-(μ/2)e_1-μ f_1,*), 
with *∈(𝒱^)^⊗2∧. Then 

    κ_(μ,Φ)P_(μ,Φ)·col_1· v_(μ,Φ)=κ_(μ,Φ)d_(μ,Φ)·P̃_(μ,Φ)·col_1· v_(μ,Φ)=[ e^μ f_1;      -1;       0 ]
    =(comp_(μ,Φ)^𝒱,(1))^⊗2(col_1).





 §.§ Relationship between row_1, row_1 and P_(μ,Φ)




Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has 

    M_1×3((comp_(μ,Φ)^𝒱,(1))^⊗2)(row_1)·κ_(μ,Φ)·P_(μ,Φ)
    =u_(μ,Φ)·row_1

(equality in M_1×3((𝒱^)^⊗2∧)), where 
row_1, row_1 
are as in (<ref>), (<ref>), 
κ_(μ,Φ), P_(μ,Φ)
are as in (<ref>), Definition <ref>, and where

    u_(μ,Φ):=
    μ e^-μ f_1e^μ(e_1+f_1)-1  e_1+f_1Γ_Φ(e_1+f_1)Γ_Φ(e_1)Γ_Φ(f_1)
    ∈(𝒱^)^⊗2∧. 
    u_mu,Phi@u_(μ,Φ)



One has M_1×3((comp_(μ,Φ)^𝒱,(1))^⊗2)(row_1)=
[ e^μ e_1-1 1-e^μ f_1         0 ]. With d_(μ,Φ) as in the proof of 
Lemma <ref>, this implies  

    M_1×3((comp_(μ,Φ)^𝒱,(1))^⊗2)
    (row_1)·κ_(μ,Φ)d_(μ,Φ)=
    e^-(μ/2)f_1[                     e^μ e_1-1 -e^-(μ/2)(e_1+f_1)(e^μ f_1-1)                             0 ].

Then with P̃_(μ,Φ) as in the proof of Lemma <ref>, 

    M_1×3((comp_(μ,Φ)^𝒱,(1))^⊗2)
    (row_1)·κ_(μ,Φ)·P_(μ,Φ) 
    =M_1×3((comp_(μ,Φ)^𝒱,(1))^⊗2)
    (row_1)·κ_(μ,Φ)d_(μ,Φ)·P̃_(μ,Φ)
       
    =e^-(μ/2)f_1[                     e^μ e_1-1 -e^-(μ/2)(e_1+f_1)(e^μ f_1-1)                             0 ]·P̃_(μ,Φ). 
    
       
    =e^-(μ/2)f_1(e^μ e_1-1)(e^μ f_1-1)·
       ·[                  (φ_1-φ_0)(-(e_1+f_1),f_1)+e^-(μ/2)(e_1+f_1)φ_0(-(e_1+f_1),e_1)
    +1 f_1+e^-(μ/2)(e_1+f_1)e^(μ/2)(e_1+f_1)-1-e_1-f_1 -φ_0(-(e_1+f_1),f_1)-e^-(μ/2)(e_1+f_1)(φ_1-φ_0)(-(e_1+f_1),e_1)
    -e^-(μ/2)(e_1+f_1) e_1+e^-(μ/2)(e_1+f_1)e^(μ/2)(e_1+f_1)-1-e_1-f_1                                                                                                                                  0 ]




















Then 

    (φ_1-φ_0)(-(e_1+f_1),f_1)+e^-(μ/2)(e_1+f_1)φ_0(-(e_1+f_1),e_1)
    +1 f_1+e^-(μ/2)(e_1+f_2)e^(μ/2)(e_1+f_2)-1-e_1-f_1
        =
    (1-(e_1+f_1)+1 f_1)(Γ_Φ(e_1+f_1)Γ_Φ(-f_1)Γ_Φ(e_1))
    +e^-(μ/2)(e_1+f_1)·1 e_1+f_1(Γ_Φ(e_1+f_1)Γ_Φ(-e_1)Γ_Φ(f_1))
    
       = 
    Γ_Φ(e_1+f_1)Γ_Φ(e_1)Γ_Φ(f_1)·(
    (1-(e_1+f_1)+1 f_1)Γ_Φ(f_1)Γ_Φ(-f_1)
    +e^-(μ/2)(e_1+f_1)·1 e_1+f_1Γ_Φ(e_1)Γ_Φ(-e_1)
    )
    
       = 
    Γ_Φ(e_1+f_1)Γ_Φ(e_1)Γ_Φ(f_1)·(
    (1-(e_1+f_1)+1 f_1)μ f_1 e^(μ/2)f_1e^μ f_1-1
    +e^-(μ/2)(e_1+f_1)·1 e_1+f_1μ e_1 e^(μ/2)e_1e^μ e_1-1)
    
        = 
    Γ_Φ(e_1+f_1)Γ_Φ(e_1)Γ_Φ(f_1)·e^-(μ/2)f_1(e^μ e_1-1)(e^μ f_1-1)·μ e_1 e_1+f_1(e^μ(e_1+f_1)-1).

where the first equality follows from (<ref>) and the third equality follows from (<ref>); 
similarly 
 
    -φ_0(-(e_1+f_1),f_1)-e^-(μ/2)(e_1+f_1)(φ_1-φ_0)(-(e_1+f_1),e_1)
    -e^-(μ/2)(e_1+f_1) e_1
        +e^-(μ/2)(e_1+f_1)e^(μ/2)(e_1+f_1)-1-e_1-f_1
        =1-(e_1+f_1)Γ_Φ(e_1+f_1)Γ_Φ(-f_1)Γ_Φ(e_1)
    -e^-(μ/2)(e_1+f_1)(1-(e_1+f_1)+1 e_1)Γ_Φ(e_1+f_1)Γ_Φ(-e_1)Γ_Φ(f_1)
       =
    Γ_Φ(e_1+f_1)Γ_Φ(e_1)Γ_Φ(f_1)·(
    1-(e_1+f_1)μ f_1e^(μ/2)f_1-e^-(μ/2)f_1
    -e^-(μ/2)(e_1+f_1)(1-(e_1+f_1)+1 e_1)μ e_1e^(μ/2)e_1-e^-(μ/2)e_1)
    
       =
    Γ_Φ(e_1+f_1)Γ_Φ(e_1)Γ_Φ(f_1)·e^-(μ/2)f_1(e^μ e_1-1)(e^μ f_1-1)·-μ f_1 e_1+f_1 (e^μ(e_1+f_1)-1),

where the first equality follows from (<ref>), the second equality follows from  (<ref>).  
It follows that the right-hand side of (<ref>) is equal to 

    e^-μ f_1·Γ_Φ(e_1+f_1)Γ_Φ(e_1)Γ_Φ(f_1)e^μ(e_1+f_1)-1  e_1+f_1[  μ e_1 -μ f_1      0 ]=u_(μ,Φ)·row_1.





 §.§ Commutative diagram relating row_1·(-)·col_1 and row_1·(-)·col_1 ((A4) in (<ref>))



Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). The following diagram of 𝐤-module morphisms is commutative, 
where u·(-)· v is the linear map x↦ uxv 

    M_3((𝒱^)^⊗2∧) ^row_1·(-)·col_1[rr]
    ^≃_M_3((comp_(μ,Φ)^𝒱,(1))^⊗2)[d]       (𝒱^)^⊗2∧_≃^(comp_(μ,Φ)^𝒱,(1))^⊗2[d]
     
    M_3((𝒱^)^⊗2∧)      (𝒱^)^⊗2∧
     
    M_3((𝒱^)^⊗2∧)_row_1·(-)·col_1[rr]
    ^Ad(κ_(μ,Φ)·P_(μ,Φ))_≃[u]       
    (𝒱^)^⊗2∧^≃_u_(μ,Φ)·(-)· v_(μ,Φ)[u]




This follows from the juxtaposition of the obviously commutative diagram  

    M_3((𝒱^)^⊗2∧) ^row_1·(-)·col_1[rrrrrrr]
    ^≃_M_3((comp_(μ,Φ)^𝒱,(1))^⊗2)[d]                      (𝒱^)^⊗2∧_≃^(comp_(μ,Φ)^𝒱,(1))^⊗2[d]
     
    M_3((𝒱^)^⊗2∧)_M_1×3((comp_(μ,Φ)^𝒱,(1))^⊗2)
    (row_1)·(-)·
    M_3×1((comp_(μ,Φ)^𝒱,(1))^⊗2)(col_1)[rrrrrrr]                     (𝒱^)^⊗2∧

with the following diagram

    M_3((𝒱^)^⊗2∧)^M_1×3((comp_(μ,Φ)^𝒱,(1))^⊗2)
    (row_1)·(-)·
    M_3×1((comp_(μ,Φ)^𝒱,(1))^⊗2)(col_1)[rrrrrrr]                     (𝒱^)^⊗2∧
     
    M_3((𝒱^)^⊗2∧)_row_1·(-)·col_1[rrrrrrr]
    ^Ad(κ_(μ,Φ)·P_(μ,Φ))_≃[u]                      (𝒱^)^⊗2∧^≃_u_(μ,Φ)·(-)· v_(μ,Φ)[u]

whose commutativity follows from (<ref>) and (<ref>). 






 §.§ Associators and harmonic algebra coproducts




Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). Set 

    B_Φ:=Γ_Φ(-e_1)Γ_Φ(-f_1)Γ_Φ(-e_1-f_1)∈((𝒲^)^⊗2∧)^×.
    B_Phi@B_Φ

The following diagram commutes

    𝒲̂^^Δ̂^𝒲,[rr]
    _comp_(μ,Φ)^𝒲,(1)[d]       (𝒲^)^⊗2∧^(comp_(μ,Φ)^𝒲,(1))^⊗2[d]
    𝒲̂^_Δ̂^𝒲,[r]    (𝒲^)^⊗2∧_Ad(B_Φ)[r]   
     (𝒲^)^⊗2∧
 




If (μ,g)∈𝐤^××(𝒱̂^)^×, there is a unique extension 
of the algebra automorphism aut^𝒱,(1),_(μ,g) of 𝒱̂^ 
(see <ref>) to an algebra automorphism aut^𝒱_loc,(1)_(μ,g)autVloc1@aut^𝒱_loc,(1)_(μ,g) of its localization 
𝒱̂^[1 e_1] (see <ref>). The algebra isomorphism iso^𝒱:
𝒱̂^→𝒱̂^ from <ref> also extends uniquely to an algebra isomorphism 
iso^𝒱_loc:𝒱̂^[1X_1-1]→𝒱̂^[1 e_1]isoVloc@iso^𝒱_loc.
We denote by 

    comp^𝒱_loc,(1)_(μ,g):𝒱̂^[1X_1-1]→𝒱̂^[1 e_1]
    compVloc1@comp^𝒱_loc,(1)_(μ,g)
 
the algebra isomorphism given by the composition aut^𝒱_loc,(1)_(μ,g)∘iso^𝒱_loc. 
It is compatible with the filtrations on both sides, and in particular it induces an algebra isomorphism between 
F^0𝒱̂^[1X_1-1] and 𝒱̂^[1 e_1]_≥0. 

Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). Consider the diagram 

    𝒲̂^_+
    ^Δ̂^𝒲,[rrrrrr]^*[@] to 15pt comp_(μ,Φ)^𝒲,(1)[ddddd]
    @^*[@] to -10ptmor_𝒱̂^,X_1-1[rd] 
    @[dddddr]|(A2)               @[dllll]|(A1)    
    (𝒲^)^⊗2∧^*[@] to 15pt (comp_(μ,Φ)^𝒲,(1))^⊗2[ddd]@^(->[ld] 
    
       𝒱̂^^ to 20ptρ̂[r]
    ^*[@] to 4pt to 4pt comp_(μ,Φ)^𝒱,(1)[dd]
    [lu]    M_3((𝒱^)^⊗2∧)^    row_1·(-)·col_1[rr]
    ^M_3((comp_(μ,Φ)^𝒱,(1))^⊗2)[d]        
    (𝒱^)^⊗2∧^Ad((X_1-1)^-1·
    ·(1-Y_1^-1)^-1)[r]
    ^(comp_(μ,Φ)^𝒱,(1))^⊗2[d]
         
    F^0𝒱^[1X_1-1]^⊗2∧^(comp_(μ,Φ)^𝒱_loc,(1))^⊗2[d]    
    
       @[r]|(A3)    M_3((𝒱^)^⊗2∧)^Ad((κ_(μ,Φ)·P_(μ,Φ))^-1)[d]
    @[rr]|(A4)       
    (𝒱^)^⊗2∧^u_(μ,Φ)^-1·(-)· v_(μ,Φ)^-1[d]
    @[r]|(A5)    
    (𝒱^[1e_1]_≥0)^⊗2∧^ B_Φ^-1·(-)· B_Φ·
    ·e_1+f_1e^μ(e_1+f_1)-1[d] 
       
    
       𝒱̂^_ to 20ptρ̂[r]^mor_𝒱̂^,e_1[d]     M_3((𝒱^)^⊗2∧)_row_1·(-)·col_1[rr]        
    (𝒱^)^⊗2∧_Ad((e_1f_1)^-1)[r] 
    @[dll]|(A7)    
    (𝒱^[1 e_1]_≥0)^⊗2∧@[dr]|(A6)   
    (𝒲^)^⊗2∧^*[@] to 10ptAd(B_Φ^-1)[dd] 
    
    
       𝒲̂^_+
    _Δ̂^𝒲,[rrrr]
    [ld] 
             @[dll]|(A8)    
    (𝒲^)^⊗2∧@^(->[u]
    @^*[@] to 23pt to 0pt (-)·e_1+f_1e^μ(e_1+f_1)-1[rd]  
       
    
    𝒲̂^_+
    _Δ̂^𝒲,[rrrrrr]
    @^*[@] to 5pt(-)·e^μ e_1-1e_1[ru] 
                      (𝒲^)^⊗ 2∧[ul]

It is divided into subdiagrams (A1) to (A8). 
In this diagram: 


  * the commutativity of (A1) (resp. (A3), (A4), (A7)) follows from Lemma <ref> (resp. 
Lemma <ref>, Lemma <ref>, Lemma <ref>); 


  * the commutativity of (A2) follows from the definition of mor_-,-, from the compatibilities of 
comp_(μ,Φ)^𝒱,(1) and comp_(μ,Φ)^𝒲,(1), and from 
the fact that comp_(μ,Φ)^𝒱,(1) is an algebra morphism taking
X_1-1 to e^μ e_1-1; 


  * the commutativity of (A5) follows from the equalities 

    B_Φ^-1· (e^μ e_1-1)^-1(1-e^-μ f_1)^-1
    =(e_1f_1)^-1u_(μ,Φ)^-1,


    (e^μ e_1-1)(1-e^-μ f_1)· B_Φ·e_1+f_1e^μ(e_1+f_1)-1
    =v_μ,Φ^-1e_1f_1


    (comp_(μ,Φ)^𝒱_loc,(1))^⊗2((X_1-1)^±1(1-Y_1^-1)^±1)=
    (e^μ e_1-1)^±1(1-e^-μ f_1)^±1
 
in (𝒱^[1 e_1])^⊗2∧, where the two first equalities follow from (<ref>), 
and the last equality is immediate;  


  * the commutativity of (A6) follows from the compatibility of 
comp_(μ,Φ)^𝒱_loc,(1) and comp_(μ,Φ)^𝒲,(1), 
and the commutativity of (A8) follows from the fact that Δ̂^𝒲, 
is an algebra morphism, such that e_1↦ e_1+f_1. 

Using successively the commutativities of (A6), (A1), (A5), (A4), (A3), (A7), (A8), (A2), and denoting by map_i (i=1,2) 
the two maps 𝒲̂^→(𝒲^)^⊗2∧ which can be derived from diagram (<ref>), 
one sees that the two maps 

    𝒱̂^^mor_𝒱^,X_1-1[rr]      𝒲̂^_+^map_i[r]   
    (𝒲^)^⊗2∧^B_Φ^-1·(-)· B_Φe_1+f_1e^μ(e_1+f_1)-1[rrr]         
    (𝒲^)^⊗2∧↪
    (𝒱^[1 e_1]_≥0)^⊗2∧

coincide (i=1,2). 
Since the map (𝒲^)^⊗2∧^B_Φ^-1·(-)
· B_Φe_1+f_1e^μ(e_1+f_1)-1[rrr]         
(𝒲^)^⊗2∧↪
(𝒱^[1 e_1]_≥0)^⊗2∧ is injective, and since mor_𝒱^,X_1-1:
𝒱̂^→𝒲̂^_+ is a bijection, it follows that the restrictions of map_1 and map_2 
to 𝒲̂^_+ coincide.  One has map_1(1)=1^⊗2=map_2(1), which implies that 
the restrictions of map_1 and map_2 to 𝐤1⊂𝒲̂^ also coincide. 
Since 𝒲̂^ is the direct sum of 𝐤1 and 𝒲̂^_+, it follows that the maps 
map_1 and map_2 coincide, therefore that (<ref>) commutes. 



§ ASSOCIATORS AND HARMONIC MODULE COPRODUCTS


The purpose of this section is to prove the second main result of this paper, namely the compatibility of associators with harmonic module
coproducts (Theorem <ref>). This proof relies on the commutativity of diagram (<ref>). The main new 
ingredients in this diagram are: a family of localized versions of the comparison isomorphisms from <ref>, 
which are constructed in <ref>; and a matrix R_(μ,Φ), which is defined in <ref>. 
Diagram (<ref>) is divided into subdiagrams (M1) to (M5). Among them, subdiagram (M1) 
(resp. (M5)) expresses the geometric interpretation of the Betti (resp. de Rham) module coproduct and was proved in Lemma 
<ref> (resp. Lemma <ref>). The main remaining diagrams are (M3) and (M4), which relate, 
by means of the module comparison isomorphisms and the matrices R_(μ,Φ) from <ref> and 
P_(μ,Φ) from 
<ref>, `de Rham' and `Betti' objects: namely, the morphisms ρ̂ and ρ̂ in the 
case of (M3), and the morphisms row_1·(-)·col_0 and row_1·(-)
·col_0 in the case of (M4). The commutativity of (M3) is established in <ref> using the module 
morphism properties of its constituents over the constituents of (A3) in (<ref>), the commutativity of (A3), and the 
definition of R_(μ,Φ). The commutativity of (M4) is established in <ref> to <ref>;  more precisely, 
<ref> is devoted to the expression of R_(μ,Φ) in terms of the matrix Q_(μ,Φ) from 
<ref>; <ref> is devoted to the proof of an equality relating R_(μ,Φ), col_0 and 
col_0, based in <ref> and the computation of Q_(μ,Φ) in <ref>; 
this result is combined in <ref> with the equality relating P_(μ,Φ), row_1 and 
row_1 from <ref> to prove the commutativity of (M4). Theorem <ref> 
is formulated and proved in <ref>. 



 §.§ Localized versions of comparison isomorphisms


For (μ,g)∈𝐤^××(𝒱̂^)^×, let 

    comp_(μ,g)^𝒱_loc,(10):𝒱^[1 X_1-1]^∧→𝒱^[1 e_1]^∧compVloc10@comp^𝒱_loc,(10)_(μ,g)

be the 𝐤-module isomorphism given by 

    ∀ v∈𝒱^[1 X_1-1]^∧,  comp_(μ,g)^𝒱_loc,(10)(v):=
    comp_(μ,g)^𝒱_loc,(1)(v)· g

(see (<ref>)). One checks that it restricts to an isomorphism 
𝒱^[1 X_1-1]^∧(X_0-1)→𝒱^[1 e_1]^∧ e_0. 


For (μ,g)∈𝐤^××(𝒱̂^)^×, 
 comp_(μ,g)^ℳ_loc,(10):ℳ^[1 X_1-1]^∧→ℳ^[1 e_1]^∧
 compM10loc@comp^ℳ_loc,(10)_(μ,g)
is the 𝐤-module isomorphism induced by comp_(μ,g)^𝒱_loc,(10). 



For (μ,g)∈𝐤^××(𝒱̂^)^×, the isomorphism 
comp_(μ,g)^ℳ_loc,(10)
is compatible with the filtrations on both sides. Moreover, the following diagram commutes

    ℳ̂^[d]^comp^ℳ,(10)_(μ,g)[rr]      ℳ̂^[d]
    
    ℳ^[1 X_1-1]^∧_comp^ℳ_loc,(10)_(μ,g)[rr]      ℳ^[1 e_1]^∧



The first statement follows from the fact that comp_(μ,g)^𝒱_loc,(10) is compatible 
with filtrations. It follows from its definition (see (<ref>)) that the algebra isomorphisms 
comp^𝒱,(1)_(μ,g) and comp^𝒱_loc,(1)_(μ,g) intertwine the algebra morphisms 
𝒱̂^→𝒱^[1X_1-1]^∧ and 𝒱̂^→𝒱^[1e_1]^∧. 
Expressing the resulting identity and multiplying it by g, one sees that the 𝐤-module morphisms 
comp^𝒱,(10)_(μ,g) and comp^𝒱_loc,(10)_(μ,g) intertwine the same 
algebra morphisms. The announced diagram follows by projection. 

The compatibility of the algebra morphism comp_(μ,g)^𝒱,(1) and the 𝐤-module morphism 
comp_(μ,g)^ℳ,(10) with the module structures (see <ref>) extends to the localized 
situation as follows. 


For (μ,g)∈𝐤^××(𝒱̂^)^×, the 𝐤-module morphism 
comp_(μ,g)^ℳ_loc,(10) is compatible (in the sense of <ref>)
with the 𝐤-algebra morphism comp_(μ,g)^𝒱_loc,(1) and with the module 
structure of ℳ^[1 X_1-1]^∧ (resp. ℳ^[1 e_1]^∧) over 
𝒱^[1 X_1-1]^∧ (resp. 𝒱^[1 e_1]^∧). 


It follows from (<ref>) and from the algebra morphism property of 
comp_(μ,g)^𝒱_loc,(1)
that 

    ∀ v,v'∈𝒱^[1 X_1-1]^∧,  comp_(μ,g)^𝒱_loc,(10)(vv')=
    comp_(μ,g)^𝒱_loc,(1)(v)
    comp_(μ,g)^𝒱_loc,(10)(v').

Applying to this identity the projection 𝒱^[1 e_1]^∧→ℳ^[1 e_1]^∧, one obtains: 

    ∀ v∈𝒱^[1 X_1-1]^∧,∀ m∈ℳ^[1 X_1-1]^∧,  comp_(μ,g)^ℳ_loc,(10)(vm)=
    comp_(μ,g)^𝒱_loc,(1)(v)
    comp_(μ,g)^ℳ_loc,(10)(m).





 §.§ Definition of the matrix R_(μ,Φ) and commutativity of (M3) in (<ref>)
 

Recall that Φ∈(𝒱̂^)^×, and that ρ̂:𝒱̂^→ M_3((𝒱^)^⊗2∧)
is an algebra morphism. It follows that ρ̂(Φ)∈GL_3((𝒱^)^⊗2∧). 


Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). We set

    R_(μ,Φ):=ρ̂(Φ)^-1P_(μ,Φ)^-1κ_(μ,Φ)^-1Φ(e_0,e_1)Φ(f_0,f_1)∈GL_3((𝒱^)^⊗2∧).
    R_mu,Phi@R_(μ,Φ)
 



Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). The diagram 

    𝒱̂^^ρ̂[r]_comp^𝒱,(10)_(μ,Φ)[dd]
        M_3((𝒱^)^⊗2∧)
    ^M_3((comp^𝒱,(10)_(μ,Φ))^⊗2)[d]
        M_3((𝒱^)^⊗2∧)
    𝒱̂^^ρ̂[r]   
    M_3((𝒱^)^⊗2∧)_κ_(μ,Φ)P_(μ,Φ)·(-)· R_(μ,Φ)[u]

((M3) in (<ref>)) commutes. 


Let v∈𝒱̂^. Then 

    M_3((comp^𝒱,(10)_(μ,Φ))^⊗2)(ρ̂(v))
    =M_3((comp^𝒱,(1)_(μ,Φ))^⊗2)(ρ̂(v))·Φ(e_0,e_1)Φ(f_0,f_1)
    
        
    =Ad(κ_(μ,Φ)P_(μ,Φ))(ρ̂(comp_(μ,Φ)^𝒱,(1)(v)))·Φ(e_0,e_1)Φ(f_0,f_1)
    
        
    =Ad(κ_(μ,Φ)P_(μ,Φ))(ρ̂(comp_(μ,Φ)^𝒱,(10)(v)·Φ^-1))·Φ(e_0,e_1)Φ(f_0,f_1)
    
        
    =Ad(κ_(μ,Φ)P_(μ,Φ))(ρ̂(comp_(μ,Φ)^𝒱,(10)(v))·ρ̂(Φ)^-1)·Φ(e_0,e_1)Φ(f_0,f_1)
    
        
    =κ_(μ,Φ)P_(μ,Φ)·ρ̂(comp_(μ,Φ)^𝒱,(10)(v))·ρ̂(Φ)^-1·
    (κ_(μ,Φ)P_(μ,Φ))^-1·Φ(e_0,e_1)Φ(f_0,f_1)
    
        
    =κ_(μ,Φ)P_(μ,Φ)·ρ̂(comp_(μ,Φ)^𝒱,(10)(v))·
    R_(μ,Φ),

where the first and third equalities follow from (<ref>), the second equality follows from
Lemma <ref>, the fourth equality follows from the fact that ρ̂ is an algebra morphism, 
and the last equality follows from (<ref>). 




 §.§ Computation of  R_(μ,Φ) 
in terms of Q_(μ,Φ)

 

Let us first relate the elements P_(μ,Φ) and Q_(μ,Φ) from Definition <ref>. 


Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has 

    Q_(μ,Φ)=ℓ̂(Φ)^-1· P_(μ,Φ)·ϖ̂(ℓ̂(Φ))

(equality in GL_3((U𝔭_5)^⊗2∧)). 


The following sequence of equalities holds in M_3×1((U𝔭_5)^⊗2∧)

    Q_(μ,Φ)·[ e_15; e_25; e_35 ]=
    M_3×1(comp_(μ,Φ)^(∙(∙∙))∙)
    ([ x_15-1; x_25-1; x_35-1 ])
    = ℓ̂(Φ)^-1·
    M_3×1(comp_(μ,Φ)^((∙∙)∙)∙)
    ([ x_15-1; x_25-1; x_35-1 ])·ℓ̂(Φ)
    
        = ℓ̂(Φ)^-1· P_(μ,Φ)·[ e_15; e_25; e_35 ]·ℓ̂(Φ)
    = ℓ̂(Φ)^-1· P_(μ,Φ)·ϖ̂(ℓ̂(Φ))·[ e_15; e_25; e_35 ],

since: the first (resp. third) equality follows from the second (resp. first) part of (<ref>);  
the second equality follows from (<ref>) together with Φ(e_12,e_23)=ℓ̂(Φ)^-1, which follows from 
the definition of ℓ̂ (see <ref>) and (<ref>); 
the last equality follows from (<ref>). 

The result then follows from the injectivity of the map ((U𝔭_5)^∧)^⊕3→ (U𝔭_5)^∧, 
(p_i)_i∈[[1,3]]↦∑_i∈[[1,3]]p_i· e_i5 (see Lemma <ref>). 



Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has 

    P_(μ,Φ)ρ̂(Φ)=Q_(μ,Φ)

(equality in GL_3((𝒱^)^⊗2,∧)). 


One has 

    Q_(μ,Φ)=pr_12^∧(Q_(μ,Φ))
    =pr_12^∧(ℓ̂(Φ)^-1· P_(μ,Φ)·ϖ̂(ℓ̂(Φ)))
    
       =pr_12^∧(ℓ̂(Φ))^-1·pr_12^∧(P_(μ,Φ))·pr_12^∧(ϖ̂(ℓ̂(Φ)))
    =1·P_(μ,Φ)·ρ̂(Φ)
    =P_(μ,Φ)ρ̂(Φ),

where the first equality follows from Definition <ref>, the second equality follows from Lemma <ref>, 
the third equality from the fact that pr_12^∧ is an algebra morphism. 

One has pr_12^∧∘ℓ̂(e_1)=0. As pr_12^∧ is an algebra morphism and as the 
logarithm of Φ is a Lie series in e_0,e_1 without degree 1 terms, this implies pr_12^∧∘ℓ̂(Φ)=1. 
Together with Definition <ref> and the definition of ρ̂ (see <ref>), this implies the fourth 
equality, hence the result.  


Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has 

    R_(μ,Φ)=Q_(μ,Φ)^-1κ_(μ,Φ)^-1Φ(e_0,e_1)Φ(f_0,f_1)

(equality in GL_3((𝒱^)^⊗2∧)).


This follows from Definition <ref> and Lemma <ref>.  




 §.§ Relationship between col_0, col_0 and R_(μ,Φ)
 


Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). The following equalities hold in (ℳ^)^⊗2∧: 

    (-(e^μ e_1-1)φ_1(e_0-f_0,e_1)-e^μ e_1-1e_1) e_1 1_^⊗2
    =(1-e^μ e_1)Φ(e_0,e_1)1_^⊗2


    ((e^μ(e_∞+f_0)-1)φ_1(e_0-f_0,e_∞+f_0)+e^μ(e_∞+f_0)-1e_∞+f_0) e_1 1_^⊗2
    = (1-e^μ e_∞)Φ(e_0,e_∞) 1_^⊗2


    Φ(e_0,e_1)e^(μ/2)e_0Φ(e_∞,e_0)(1-e^μ e_∞) Φ(e_0,e_∞) 1_^⊗2
    =
     (1-e^-μ e_1) Φ(e_0,e_1)1_^⊗2.



The following holds in (ℳ^)^⊗2∧: 

    (-(e^μ e_1-1)φ_1(e_0-f_0,e_1)-e^μ e_1-1e_1) e_1 1_^⊗2
    =-(e^μ e_1-1)(φ_1(e_0-f_0,e_1)e_1+1)  1_^⊗2
        =(1-e^μ e_1)(φ_0(e_0-f_0,e_1)(e_0-f_0)+φ_1(e_0-f_0,e_1)e_1+1)  1_^⊗2
    =(1-e^μ e_1)Φ(e_0-f_0,e_1)1_^⊗2
        =(1-e^μ e_1)Φ(e_0,e_1)1_^⊗2

as the second equality follows from e_0 1_^⊗2=f_0 1_^⊗2=0, the third equality from the relation 
between φ_0,φ_1 and Φ (see <ref>), and the last equality from the 
commutation of f_0 with e_0 and e_1. This proves (<ref>). 

The following holds in (ℳ^)^⊗2∧: 

    ((e^μ(e_∞+f_0)-1)φ_1(e_0-f_0,e_∞+f_0)+e^μ(e_∞+f_0)-1e_∞+f_0) e_1 1_^⊗2
        
    =-((e^μ(e_∞+f_0)-1)φ_1(e_0-f_0,e_∞+f_0)+e^μ(e_∞+f_0)-1e_∞+f_0) (e_∞+f_0)1_^⊗2
        =
    -(e^μ(e_∞+f_0)-1)(φ_1(e_0-f_0,e_∞+f_0)(e_∞+f_0)+
    1) 1_^⊗2
       
    =-(e^μ(e_∞+f_0)-1)(φ_0(e_0-f_0,e_∞+f_0)(e_0-f_0)+φ_1(e_0-f_0,e_∞+f_0)(e_∞+f_0)+
    1) 1_^⊗2
         
    = 
    -(e^μ(e_∞+f_0)-1)Φ(e_0-f_0,e_∞+f_0) 1_^⊗2
        
    =(1-e^μ(e_∞+f_0))Φ(e_0,e_∞) 1_^⊗2= 
    (1-e^μ e_∞)Φ(e_0,e_∞) 1_^⊗2

as the first and third equalities follow from e_0 1_^⊗2=f_0 1_^⊗2=0, the fourth from 
the relation between φ_0,φ_1 and Φ (see <ref>), the fifth from 
the commutation of f_0 with e_0 and e_∞, and the last from the same fact, together with f_0 1_^⊗2=0. This 
proves (<ref>). 

The following holds in (𝒱^)^⊗2∧: 

    Φ(e_0,e_1)e^(μ/2)e_0Φ(e_∞,e_0)(1-e^μ e_∞) Φ(e_0,e_∞) 
    
       
    =e^-(μ/2)e_1Φ(e_∞,e_1)(e^-(μ/2)e_∞- e^(μ/2)e_∞)Φ(e_0,e_∞)
    
       
    = e^-(μ/2)e_1·(e^(μ/2)e_1Φ(e_0,e_1) e^(μ/2)e_0- e^-(μ/2)e_1Φ(e_0,e_1) e^-(μ/2)e_0)
    
        
    =(1-e^-μ e_1)Φ(e_0,e_1)+Φ(e_0,e_1)(e^(μ/2)e_0-1)-e^-(μ/2)e_1Φ(e_0,e_1)(e^-(μ/2)e_0-1)

as the first equality follows from the 2-cycle identity (<ref>) and the hexagon identity (<ref>) for 
(a,b,c)=(e_0,e_∞,e_1) and the second equality follows from the hexagon identities (<ref>) and (<ref>) 
for (a,b,c)=(e_0,e_1,e_∞). Applying this equality to 1_^⊗2, and using e_0 1_^⊗2=0, one 
obtains (<ref>). 


Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has 

    Φ(e_0,e_1)^-1Φ(f_0,f_1)^-1M_3× 1((comp_(μ,Φ)^ℳ,(10))^⊗2)(col_0 ) 
    =R_(μ,Φ)^-1col_0

(equality in M_3× 1((ℳ^)^⊗2∧)), where col_0, col_0 
are as in Definitions <ref> and <ref>.


Using Lemmas <ref> and <ref>, one obtains


    M_3×1((comp_(μ,Φ)^ℳ,(10))^⊗2)(col_0)
    
    =
    
    e^-μ f_1Φ(f_0,f_1)[                            0;  (1-e^μ e_1)Φ(e_0,e_1) 1_^⊗2; (1-e^-μ e_1)Φ(e_0,e_1) 1_^⊗2 ].


On the other hand, 

    Φ(e_0,e_1) Φ(f_0,f_1)R_(μ,Φ)^-1·col_0
    =κ_(μ,Φ)·Q_(μ,Φ)·col_0 
    
       =κ_(μ,Φ)diag(Φ(f_1,f_∞), 
    e^μ 2f_∞Φ(f_0,f_∞)e^μ 2f_0Φ(e_1,e_0), 
    e^μ 2(e_0+f_∞)Φ(f_0,f_∞)Φ(e_∞,e_0))·
       ·[                                                                      0;                 (-(e^μ e_1-1)φ_1(e_0-f_0,e_1)-e^μ e_1-1e_1) e_1 
    1_^⊗2; ((e^μ(e_∞+f_0)-1)φ_1(e_0-f_0,e_∞+f_0)+e^μ(e_∞+f_0)-1e_∞+f_0) e_1 1_^⊗2 ]
       
    =[                                                                                                                                            0;                                        e^-(μ/2)f_1Φ(f_∞,f_1)
    e^μ 2f_∞Φ(f_0,f_∞)e^μ 2f_0(-(e^μ e_1-1)φ_1(e_0-f_0,e_1)-e^μ e_1-1e_1) e_1 1_^⊗2; Φ(e_0,e_1)· e^-(μ/2)f_1Φ(f_∞,f_1)
    e^μ 2(e_0+f_∞)Φ(f_0,f_∞)Φ(e_∞,e_0) 
    ((e^μ(e_∞+f_0)-1)φ_1(e_0-f_0,e_∞+f_0)+e^μ(e_∞+f_0)-1e_∞+f_0) e_1 1_^⊗2 ]
       
    =[                                                                                                 0;                      e^-(μ/2)f_1Φ(f_∞,f_1)
    e^μ 2f_∞Φ(f_0,f_∞)e^μ 2f_0 
    (1-e^μ e_1)Φ(e_0,e_1)1_^⊗2; Φ(e_0,e_1)· e^-(μ/2)f_1Φ(f_∞,f_1)
    e^μ 2(e_0+f_∞)Φ(f_0,f_∞)Φ(e_∞,e_0) 
    (1-e^μ e_∞)Φ(e_0,e_∞) 1_^⊗2 ]
       
    =e^-(μ/2)f_1Φ(f_∞,f_1)e^μ 2f_∞Φ(f_0,f_∞)e^μ 2f_0[                                                         0;                                (1-e^μ e_1)Φ(e_0,e_1)1_^⊗2; Φ(e_0,e_1)e^μ 2e_0Φ(e_∞,e_0) 
    (1-e^μ e_∞)Φ(e_0,e_∞) 1_^⊗2 ]
       
    =e^-μ f_1Φ(f_0,f_1)
    [                                                         0;                                (1-e^μ e_1)Φ(e_0,e_1)1_^⊗2; Φ(e_0,e_1)e^μ 2e_0Φ(e_∞,e_0) 
    (1-e^μ e_∞)Φ(e_0,e_∞) 1_^⊗2 ]
        =e^-μ f_1Φ(f_0,f_1)
    [                           0;  (1-e^μ e_1)Φ(e_0,e_1)1_^⊗2; (1-e^-μ e_1)Φ(e_0,e_1)1_^⊗2 ]

where the first (resp. second, third) equality follows from Corollary <ref> (resp. Corollary <ref>, 
(<ref>)), the fourth equality from the equalities (<ref>) and (<ref>), the fifth 
equality from the commutativity of e^μ 2f_0 with e_∞ and e_0, and from e^μ 2f_0
1_^⊗2=1_^⊗2, the sixth equality from the hexagon identity (<ref>) for (a,b,c)
=(f_0,f_1,f_∞), and the last equality follows from (<ref>). (<ref>) then follows from the 
combination of (<ref>) and (<ref>). 



 §.§ Commutative diagram relating row_1·(-)·col_0 and 
row_1·(-)·col_0 ((M4) in (<ref>))

 


Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). One has 

    M_1× 3((comp_(μ,Φ)^𝒱_loc,(1))^⊗2)((X_1-1)^-1(1-Y_1^-1)^-1row_1 )
    =B_Φ(e_1f_1)^-1row_1(κ_(μ,Φ)P_(μ,Φ))^-1

(equality in M_1× 3(𝒱^[1 e_1]^⊗2∧)). 


This follows from (<ref>), from the fact that 
(comp_(μ,Φ)^𝒱_loc,(1))^⊗2 is 
an algebra automorphism, and from the equality (comp_(μ,Φ)^𝒱,(1))^⊗2((X_1-1)(1-Y_1^-1))
=e_1f_1 u_(μ,Φ)B_Φ^-1, which follows from (<ref>) and (<ref>). 


 
Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). The diagram 

    M_3((𝒱^)^⊗ 2∧)^(X_1-1)^-1(1-Y_1^-1)^-1row_1·(-)·col_0[rrrr]
    _M_3((comp^𝒱,(10)_(μ,Φ))^⊗2)[d]
                ℳ^[1/X_1-1]^⊗2∧^(comp^ℳ_loc,(10)_(μ,Φ))^⊗2[dd] 
    
    M_3((𝒱^)^⊗2∧)
    _(κ_(μ,Φ)P_(μ,Φ))^-1·(-)· R_(μ,Φ)^-1[d]             
    
    M_3((𝒱^)^⊗2∧)_(e_1f_1)^-1row_1·(-)·col_0[rr] 
          ℳ^[1/e_1]^⊗2∧_B_Φ·(-)[rr] 
          ℳ^[1/e_1]^⊗2∧

((M4) in (<ref>)) commutes, where B_Φ is as in (<ref>). 


Let v∈ M_3((𝒱^)^⊗ 2∧). One has 

    (comp_(μ,Φ)^ℳ_loc,(10))^⊗2((X_1-1)^-1(1-Y_1^-1)^-1row_1· v·col_0 )
    
        =M_1× 3((comp_(μ,Φ)^𝒱_loc,(1))^⊗2)((X_1-1)^-1(1-Y_1^-1)^-1row_1 ) · 
    M_3((comp_(μ,Φ)^𝒱,(1))^⊗2(v))·
       · 
    M_3× 1((comp_(μ,Φ)^ℳ,(10))^⊗2)(col_0 ) 
    
         =M_1× 3((comp_(μ,Φ)^𝒱_loc,(1))^⊗2)((X_1-1)^-1(1-Y_1^-1)^-1row_1 ) · 
    M_3((comp_(μ,Φ)^𝒱,(10))^⊗2(v))·
       ·Φ(e_0,e_1)^-1Φ(f_0,f_1)^-1 
    M_3× 1((comp_(μ,Φ)^ℳ,(10))^⊗2)(col_0 ) 
    
        
    =
    B_Φ(e_1f_1)^-1row_1(κ_(μ,Φ)P_(μ,Φ))^-1· M_3((comp_(μ,Φ)^𝒱,(10))^⊗2)(v)· 
    R_(μ,Φ)^-1col_0,

where the first identity follows from Lemma <ref>, the second 
identity follows from (<ref>), and the last identity follows from 
(<ref>) and (<ref>). This implies
the commutation of the diagram.  



 §.§ Associators and harmonic module coproducts




Let μ∈𝐤^× and Φ∈𝖬_μ(𝐤). Then the
diagram 

    ℳ̂^^Δ̂^ℳ,[rrr]
    _comp^ℳ,(10)_(μ,Φ)[d]
             (ℳ^)^⊗ 2∧^(comp^ℳ,(10)_(μ,Φ))^⊗ 2[d]
    
    ℳ̂^_Δ̂^ℳ,[r]
       
    (ℳ^)^⊗ 2∧_B_Φ·(-)[rr]
          
    (ℳ^)^⊗ 2∧

commutes, where B_Φ is given by (<ref>).  


Consider the following diagram. It is divided into the subdiagrams (M1), ..., (M5).  

    ℳ̂^^Δ̂^ℳ,[rr]
    ^*[@] to 4pt to 4pt comp^ℳ,(10)_(μ,Φ)[dddd]
           (ℳ^)^⊗2∧@^(->[rr]       
    F^-1(ℳ^[1/X_1-1])^⊗ 2∧^(comp^ℳ_loc,(10)_(μ,Φ))^⊗2[dddd] 
    
    
       𝒱̂^@[ur]|(M1)@[dr]|(M3)@[dddl]|(M2)^[r]
    _(-)· 1_[ul]
    ^*[@] to 4pt to 4pt comp^𝒱,(10)_(μ,Φ)[dd]
        M_3((𝒱^)^⊗ 2∧)^*[@] to 4pt to 10pt (X_1-1)^-1(1-Y_1^-1)^-1row_1·(-)·col_0[urr]
    ^M_3((comp^𝒱,(10)_(μ,Φ))^⊗2)[d]
    @[ddrr]|(M4)      
    
          M_3((𝒱^)^⊗2∧)^(κ_(μ,Φ)P_(μ,Φ))^-1· (-)· R_(μ,Φ)^-1[d]
          
    
       𝒱̂^^[r]
    [dl]
    
    @[dr]|(M5)    M_3((𝒱^)^⊗2∧)
    
    ^*[@] to 4pt to 10pt (e_1f_1)^-1row_1·(-)·col_0[rd]
          
    
    ℳ̂^_Δ̂^ℳ,[rr]
    @^*[@] to 0pt to 10pt(-)· 1_[ur] 
    
          (ℳ^)^⊗2∧@^(->[r]   
    (ℳ^[1/e_1]_≥-1)^⊗2∧_B_Φ·(-)[r]
       
    (ℳ^[1/e_1]_≥-1)^⊗2∧

The commutativity of subdiagram (M1) (resp. (M2), (M3), (M4), (M5)) follows from Lemma <ref> 
(resp. Lemma <ref>, Lemma <ref>, Lemma <ref>, 
Lemma <ref>). 

Using the commutativities of these diagrams in the following order M1-M4-M3-M5-M2, we obtain that the composition of the external 
square with the morphism π^:𝒱̂^→ℳ̂^ is commutative. Since this morphism is surjective, 
this implies that the external diagram commutes. 

The commutativity of the announced diagram then follows from Lemma <ref> and 
from the fact that the injection (ℳ^)^⊗2∧↪(ℳ^[1 e_1])^⊗2∧ 
intertwines the left multiplication B_Φ·(-) on both sides. 





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