YADDADEN KHALEF Abstract. In order to describe the double shuffle and regularization relations be- tween multiple polylogarithm values at N th roots of unity, Racinet attached to each finite cyclic group G of order N and each group embedding t : G → C × , a Q-scheme DMR t which associates to each commutative Q-algebra k , a set DMR t (k) that can be decomposed as a disjoint union of sets DMR f (k) with λ ∈ k. He also exhibited a Q- group scheme DMR 6  and showed, for any commutative Q-algebra k and any λ ∈ k × , that DMR i (k) is a torsor for the action of DMR i (k). Then, Enriquez and Furusho showed for N = 1 that a subscheme DMR ¢ of DMR ¢ is a torsor of isomorphisms re- lating "de Rham" and "Betti" objects. In previous work, we reformulated Racinet's construction in terms of crossed products and identified Racinet's coproduct with a coproduct Â M,DR  defined on a module ,MR over an algebra WPR , which is equipped with its own coproduct ÂW.DR. In this paper, we define the main ingredients for a generalization of Enriquez and Furusho's result to any N ≥ 1: we exhibit a module M B , over an algebra W B and we prove the existence of two compatible coproducts V,B and Δ M,B on WR and MR respectively such that DMR is contained in the ∀ N torsor of isomorphisms relating ˆ N ,B (resp. ˆ Δ ,M ,B ) to ˆ ΔW,DR (resp. ˆ ΔM,DR).

I NTRODUCTION
A multiple L-value (MLV in short) is a complex number defined by the following series
where r,k1,...,kr ∈ N* and z1,...,zr in μ N the group of N th roots of unity in C, where N is an integer ≥ 1. This series converges if and only if ( kr, zr)̸= (1, 1). These values satisfy a set of algebraic relations; our main interest here are the double shuffle and regularisation ones. Understanding the double shuffle and regularisation relations has been greatly im- proved thanks to Racinet's work [Rac]. He generalises the group μ N to a finite cyclic group G and attaches to each pair ( G, ι ) of a finite cyclic group G and a group injection ι : G → C × , a Q-scheme DMR ¢ which associates to each commutative Q-algebra k , a set DMR t (k) that can be decomposed as a disjoint union of sets DMR t (k) for λ ∈ k (see [Rac, Definition 3.2.1]). The double shuffle and regularisation relations on MLVs are then encoded in the statement that a suitable generating series of these values belongs to the set DMR t 2 π (C) where t can : G = μ N → C* is the canonical embedding. Racinet also proved that for any pair ( G, t ), the set DMR f (k) equipped with the "twisted Mag- nus" product (see (1.13)) is a group that is independent of the choice of ι .  It is therefore denoted DMR5 (k). The main result of Racinet in [Rac, Theorem I] is that, for each pair (λ,ι) where λ ∈ k× and ι : G → C×, the set DMR ζ (k) is equipped with a torsor structure for the action of the group (DMRF (k), ®). For any ι : G → C×, this yields a torsor structure on the set DMR's (k) :=  |  |  DMR's (k) for the action of a semidirect product λ∈k× group k × × DMR (k) (see Proposition 1.3.10). This gives rise to a torsor structure on |DMR ζ (k) (where ι runs over all group embeddings from G to C × ) for DMR x (k) := the action of the semidirect product group (Aut(G) × k × ) × DMR f (k) (see Proposition 1.4.14). On the other hand, we introduced in [Yad] a crossed product formalism and showed that Racinet's objects can be expressed within it. This constitutes the "de Rham" side of the double shuffle theory.  In this framework, the crossed product algebra is identified to a topological k-algebra VDR  defined by a presentation with generators and relations (see Proposition 1.1.1). Next, Racinet's objects are given in the form of a subalgebra WPR of VPR and a quotient module MPR of the left regular VPR-module. The algebra WDR is equipped with a Hopf algebra coproduct ÂW,DR and the module MPR is equipped with a compatible coalgebra coproduct ∂ M , DR Following the stabilizer interpretation of DMRF (k) given by Enriquez and Furusho in [EF0], we defined two stabilizers Stab( Â M,DR )(k) and Stab( Â W,DR )(k) for the action of grouplike elements equipped with the twisted Magnus product and therefore obtained the following chain of inclusions
2 the double shuffle torsor in terms of betti and de rham coproducts which is a generalisation of the G = {1} result of [EF2, Theorem 3.1]. This enables us to obtain the following semidirect product group chain of inclusions Result I (Corollary 2.2.5).
For G = {1} Enriquez and Furusho introduced a "Betti" formalism of the double shuffle theory in [EF1]. It is based on the filtered algebra V B , which denotes the group algebra over k of the free group of rank 2 denoted F 2 with generators X 0 and X 1 and equipped with the filtration induced by the augmentation ideal. The completion VB is a topological k-algebra. Next, we have a Hopf algebra ( WB,  ΔW,B ) which consists of a subalgebra WB of VB linearly generated by 1 ∈ VB and the left ideal generated by X 1 −1. It is presented as an algebra with generators X 1 , X 1 , Y n + = −(X 0 − 1) n−1 X 0 (X 1 − 1) and Y − = −( X − 1 − 1) n − 1 X − 1 ( X − 1 − 1) for n ∈ N*, with relation X 1 X − 1 = X − 1 X 1 = 1. In addition, we have a Hopf algebra coproduct ÂW,B : WB → (WB) ®² given by ˆ∀w,в(х †¹) = х †¹ ® х †¹ and for n ∈ N*, ˆ∀w,в(х ††) = ү † ® 1 + 1 ® ү †† ∑ ү † ® ү †. Finally, we have a coalgebra ( MB , ÂM , B ) which consists of a quotient module MB = VB /VB (Xo – 1) isomorphic to WB , as a k-module (see [EF1, (2.1.1)]) together with a coalgebra coproduct Â M,B compatible with the coproduct Â W,B In §3, we construct analogues of the Hopf algebra ( WB, ÂW,B ) and of the module- coalgebra ( MB , ÂM,B ), for a finite cyclic group G of order N . It is based on the filtered algebra V R , which denotes the group algebra k F 2 equipped with the filtration induced by the ideal ker(kF2 → kμN); where kF2 → kμN is the algebra morphism induced by the group morphism F 2 → μ N given by X 0 → e → and X 1 → 1. Its completion is the inverse limit of the projective system induced by the filtration and is denoted V R . It is a topological algebra isomorphic to V R (see Proposition-Definition 3.1.8). Next, we define a filtered algebra W R which is linearly generated by 1 ∈ V R and the left ideal generated by X 1 – 1 and whose filtration is induced by that of V B Its completion WB is isomorphic to WBR (see Proposition-Definition 3.1.15). We also define a filtered module M B  which consists of the quotient module k F 2 /k F 2 (X 0 − 1) and whose filtration is induced by that of V R . Its completion M R is isomorphic to MBR (see Proposition-Definition 3.1.25). We then have compatible Hopf algebra and coalgebra structures on WB, and MB, respectively thanks to the following result: Result II (Theorem 3.2.4 and Corollary 3.2.6). There exists a topological k-algebra morphism ÂW.B : WR → (WB)®² and a topological k-module morphism ÂM.B : MB, → (MB,) 82 that endows WB, and MB, respectively with compatible Hopf alge- bra and coalgebra structures. Finally, in §4, we deduce the following result:
YADDADEN KHALEF
Result III (Theorem 4.2.1). DMR x is contained in the torsor of isomorphisms relating
Â¼,¹ (resp. Â¼,¹) to Â¼,ðR  (resp. Â¼,ðR ). Acknowledgements. The author is grateful to Benjamin Enriquez for the helpful discussions, ideas and careful reading. Notation.  Throughout this paper, G is a (multiplicative) finite cyclic group of order N and k is a commutative Q-algebra. For a k-algebra A , an element x ∈ A and a left A-module M we consider: ● ℓ x : M → M to be the k-module endomorphism defined by m → xm and if x is invertible, then ℓ x is an automorphism. • r x : A → A to be the k-module endomorphism defined by a → ax and if x is invertible, then rx is an automorphism. ● Ad x : A → A to be the k-algebra automorphism defined by a → xax − 1 with x ∈ A × .
0. Some categories of algebra-modules
First, let us recall various categories introduced in [EF4] that will be used throughout this paper: ● k-mod is the category of k-modules. ● k-alg is the category of k-algebras. ● k-alg-mod is the category of pairs ( A, M ) where A is a k-algebra and M is a left A-module. ● k-coalg is the category of coassociative cocommutative coalgebras over k. ● k-Hopf is the category of Hopf algebras over k. ● k-HAMC is the category of Hopf-Algebra-Module-Coalgebras where objects are pairs (( A, Δ A ), ( M, Δ M )) where ( A, Δ A ) is a Hopf algebra and ( M, Δ M ) is a coalgebra such that ▶ The pair ( A, M ) is an algebra-module. ▶ For (a,m) ∈ A × M, we have △ M (am) = △ A (a) △ M (m). ● k-mod top is the category of topological k-modules with objects pairs ( M , ( F i M ) i ∈ N ), where M is a k-module and ( F t M ) i ∈ N is a decreasing filtration of M such that the map M → lim M/F i M is a k-module isomorphism, i.e.  M is complete and separated for the topology defined by the filtration ( F i M ) i ∈ N . It is equipped with a tensor product ®, with respect to which it is a symmetric tensor category. ● k-algtop is the category of topological k-algebras. i. e. algebras in the category k-modtop in the sense of [McL]. ● k-alg-modtop is the category of topological k-algebra-modules. i. e. k-algebra- modules in the category k-mod top in the sens of [McL]. ● k-coalgtop is the category of topological k-coalgebras. i. e. coalgebras in the category k-modtop in the sens of [McL]. ● k-Hopf top is the category of topological k-Hopf algebras. i. e. Hopf algebras in the category k-modtop in the sens of [McL]. ● k-HAMC top is the category of topological Hopf-Algebra-Module-Coalgebras. i. e. Hopf-Algebra-Module-Coalgebras in the category k-mod top in the sens of [McL]. 4 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS We define Finally, let C be a symmetric tensor category and O an object of C. Cop c ( O ) to be the set of morphisms D : O → O ®². One checks that the group Autc(0) acts on Copc(0) by (0.1) α · D := α 82 ° D ◦ α −1 , with α ∈ Autc(0) and D ∈ Copc(0). 1. THE DOUBLE SHUFFLE TORSORS In this section, we recall the various double shuffle torsors arising from Racinet's work in [Rac]. In §1.1, we recall the basic framework of Racinet's formalism. Namely, two Hopf algebras (k({X}), ˆ λ) and (k({Y}), ˆ λ α } δ ), a coalgebra (k({X}}/k({X}}xo, ˆ π od) and a group (G(k((X))),®). Additionally, we also recall the basic material of the crossed product formalism introduced in [Yad] which consists of an algebra VDR and its relation with a Hopf algebra ( WBR , ÂW , DR ) isomorphic to the Hopf algebra (k ((Y )) , Âal å ) and a coalgebra ( MBR, A M , DR ) isomorphic to the coalgebra (k ((X))/k((X))x0, A mod ). In §1.2, we introduce the double shuffle set DMR § (k) for λ ∈ k and ι : G → C × an injective group morphism, which is a torsor over the double shuffle group DMRF(k), a subgroup of G(k((X))), ®) ([Rac]). In §1.3, we define a set DMR ξ (k) = |  | DMR (k) and show that it is a torsor for a group given by a semidirect product arising from an action of k × . Finally, in §1.4, we define a set DMR × (k) = | |DMR × (k) where ι runs over all injections G → C× and show that it is a torsor for a group given by semidirect product arising from an action of Aut(G) × k×.
1.1. Preliminaries.
1.1.1. Basic objects of Racinet's formalism. Let k((X)) be the free noncommutative associative series algebra with unit over the alphabet X := {x 0 } ∪ {x 0 | g ∈ G}. It is complete graded with deg( x 0 ) = deg( x 9 ) = 1 for g ∈ G . This algebra provides an object in k-Hopf top when equipped with the coproduct ˆ α : k((X)) → k((X)) ®² , which is the unique topological k-algebra morphism given by ˆ ( x q ) = x q ⊗ 1 + 1 ⊗ x q , for any g ∈ G ∪ {0} ([Rac, §2.2.3]). Let then G(k((X)}) be the set of grouplike elements of k({X}} for the coproduct ∆, i.e. the set (1.1) G(k\\X))) = {Ψ ∈ k\\X)\× | Δ(Ψ) = Ψ옵Ψ}, where k((X))× denotes the set of invertible elements of k((X)). Since (k((X)), Â) is a Hopf algebra, G(k((X))) is a group for the product of k((X)). The group G acts on k((X)) by topological k-algebra automorphisms by g → tg, where for any g ∈ G , the topological k-algebra automorphism t q is given by t q ( x 0 ) = x0, tg(xh) = xgh for h ∈ G ([Rac, §3.1.1]). For any g ∈ G , we have (1.2) ˆ α t g = t g² ◦ ˆ α , this can be verified by checking this identity on generators since both sides are given as a composition of k-algebra morphisms. As a consequence, for any g ∈ G , the k-algebra automorphism t g : k((X)) → k((X)) restricts to a group automorphism of G(k((X))).
YADDADEN KHALEF
Let q be the k-module automorphism of k({X)} given on the topological k-module basis ( x n 1 x g1 x n 2 x g2 · · · x n r x gr x n r+1 )r,n1,...,nr+1 ∈ N of k((X)) ([Rac, §2.2.7]) by q(x π¹ x g 1 x n² x g 2 · · · x n r x g r x n r + 1 ) = x n 1 x g 1 x n 2 x g 2 g − 1 · · x n r x g − 1 x n r For (n, g) ∈ N>0×G, set yn.g := x 0− 1 x g . Let Y := { yn.g |(n, g) ∈ N>0×G} . We define
k((Y)) to be the topologically free k-algebra over Y, where for every (n, g) ∈ N>0× the element yn.g is of degree n . One shows that k((Y)) is equal to the k-subalgebra k ⊕ ⊕ k ((X))x g of k ((X)) ([Rac, §2.2.5] and [EF0, §2.2]). gEG Let Âªªª : k((Y)) → (k((Y))®² be the unique topological k-algebra morphism such that for any (n, g) ∈ N>0 × G
The element Âª¹§ ∈ Copk-algtop (k ((Y))) is called the harmonic coproduct ([Rac, §2.3.1]) and the pair (k((Y)), Âª¹g) is an object of k-Hopf top Let us consider the topological k-module quotient k((X))/k((X))x0. The pair (k\\Y\\,k\\X\\/k\\X\\x 0 ) is an object of the category k-alg-mod top . The restriction to k((Y)) of the projection morphism πγ : k((X)) → k((X))/k((X)) x 0 is an isomor- phism, therefore k({X)}/k({X)}x0 is free of rank 1 over k({Y)}. It follows that there is a topological k-module morphism Âmod ∈ Copk-modtop (k\\X\\/k\\X\\x0) uniquely defined by the condition that the diagram
commutes.  The pair ( ˆ λ ≠ l g , ˆถmod) is an element of Cop k -alg-mod top (k((Y)), k((X))/k((X))x0). The pair (k<{X)\/k({X){x0, Âmod) is an object of k-coalgtop and the pair ((k<{Y)}, Â$18), (k<{X}}/k<{X}}xo, Amod)) is an object of k-HAMCtop. 1.1.2. Basic objects of the crossed product formalism. Let VDR be the complete graded algebra generated by { e 0 , e 1 } ∪ G where e 0 and e 1 are of degree 1 and elements g ∈ G are of degree 0 satisfying the relations: (i) g × h = gh; (ii) 1 = 1 G ; (iii) g × eo = eo × g; for any g, h ∈ G ; where "×" is the algebra multiplication 1 ([Yad, §2.1.1]). Recall that the map g → tg defines an action of G on k{{X}} by k-algebra au- tomorphisms.  One then considers the crossed product algebra k((X)) × G for this 6 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS action, which is the k-module k((X)) ➤ kG equipped with the product given for any a,b ∈ k((X)) and any g,h ∈ G by ([Bou07, Chapter 3, Page 180, Exercise 11]) (1.5) (a 😍 g) ⋆ (b 😍 h) = a t g(b) 😍 gh. Proposition 1.1.1 ([Yad, Proposition 2.1.3]). There is a unique k-algebra isomor- phism β : k((X)) × G → VDR  such that x 0 S 1 → e 0 and for g ∈ G, x q S 1 → − ge1g−1 and 1 ⊗ g → g. Let WER be the complete graded k-subalgebra of VER given by
It is freely generated by the family Z = { zn,g := −en−1ge1 | (n,g) ∈ N>0 × G}, with deg( zn.g ) = n ([Yad, Proposition 2.1.5.(b)]). As a consequence, there is a unique k-algebra isomorphism ☞ : k ( Y ) → WB ® given for ( n, g ) ∈ N >0 × G by Yn,g ↔ zn,g One then has a unique topological k-algebra morphism ÂW,DR : WER → (WER) 2 such that for any (n,g) ∈ N>0 × G
The coproduct ÂW.DR is an element of Copk-alg,op (WBR). The pair (WBR, ÂW.DR) is an object in the category k-Hopf top and the k-algebra isomorphism ☞ : k((Y)) is an isomorphism between the Hopf algebras (k</Y)), ˆ a l s ) and ( WPR, ˆถW.DR, Let MDR be the complete graded k-module given by
The map — Let 1 DR be the class of 1 ∈ V DR in M DR · 1nr : VDR a surjective k-module morphism with kernel VPReo + VCR (g − 1). In addition, the pair ( VDR, MDR ) is an object in the category k-alg-modtop.  Moreover, one de- duces from [Yad, Proposition 2.1.6] that there is a unique k-module isomorphism k : k((X))/k((X))xo → MDR determined by the commutativity of the diagram

YADDADEN KHALEF
On the other hand, the following diagram
- ldr
commutes ([Yad, Corollary 2.1.8]). As a consequence, the map − · 1dr : WBR → MBR is a k-module isomorphism since all other arrows of Diagram (1.9) are isomorphisms. In addition, we obtain the following result Lemma 1.1.2. The pair (WBR, MBR) is an object of the category k-alg-modtop. More- over, MDR is free WDR-module of rank 1. Proof. The first statement follows from the fact that (WBR, MBR) is the pull-back of the k-algebra-module ( VDR , MDR ) by the k-algebra morphism WDR → VDR The second statement comes from the fact that k((X))/k((X))xo is a free k((Y)) module of rank 1 thanks to the commutativity of Diagram (1.9). This enables us to construct a topological k-module morphism Â M , DR ∈ Copk-mod top ( MDR uniquely defined such that the following diagram
commutes, thanks to Lemma 1.1.2 and the free rank 1 property of the WDR-module The pair ( MDR , AM,DR ) is an object in the category k-coalgtop and the k- MDR → MPR is an isomorphism of coalgebras module isomorphism κ : k((X))/k((X))xo (k\\X\\/k\\X\\xo, Âmod) and ( MDR , Âm,DR , Lemma 1.1.3.  The pair ( △W,DR , △M,DR ) is an element of Copk-alg-modtop WER, MDR Proof. Let w ∈ WDR and m ∈ MDR. Thanks to Lemma 1.1.2 there is a unique w' ∈ WDR such that m = w' · 1 DR . We have ˆM‚DR (w · m) = ˆM‚DR (ww′ · 1dr) = ˆ∀W,DR(w)ˆ∀W,DR(w' · 1dr) = ˆ∀W,DR(w)ˆ∀W,DR(m), where the second and fourth equalities come from the commutativity of Diagram (1.10). □ As a consequence, the pair ( (WDR, ZW,DR), ( MDR, ZM,DR) ) is an object of k-HAMC top In addition, the pain ( བ, κ ) : ((k((Υ)), ˆ ˆ ω δ), (k((X))/k((X)) x 0, ˆ α mod)) → (( ( ( WRR, ˆ ω , DR), ( MRR, ˆ ω , DR) is a morphism of objects of k-HAMCtop
8 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 1.2. The torsor DMR (k). Let us denote k((X)) → k{words in xo,(xg)gEG},V → ((V|W)),, the map such that v = ∑ w (v|w)w (the empty word is equal to 1). Let Γ : k((X)) → k[[x]]*, Ψ → Γψ the function² given by ([Rac, (3.2.1.2)]]
Definition 1.2.1 ([Rac, Definition 3.2.1]). Let λ ∈ k and ι : G → C× be a group embedding. We define DMR ζ (k) to be the set of Ψ ∈ G (k ((X)}) such that (i) (Ψ|xo) = (Ψ|x1) = 0; (ii)  Δ ⋆ mod (Ψ ⋆ ) = Ψ ⋆ ∅ Ψ ⋆ ;
(v) For k ∈ {1,...,|G|/2}, | ˜ | x g k  — x g − k where g ι := ι-1 (e l ण ) and Ψ ⋆ := πγ ◦ q (Γ q 1 (x 1 )Ψ ) ∈ k ((X))/k((X))x 0 . Remark 1.2.2. (i) Thanks to [Rac, §3.2.3], DMR չ (k) is a non-empty set. (ii) If |G| E {1, 2}, the embedding ι is unique. Proposition-Definition 1.2.3 ([Rac, Remark 3.2.2]). For λ = 0, the condition³ (iv) of Definition 1.2.1 does not depend of the choice of ι. The set DMR ρ (k) is then denoted DMR (k) instead. Proposition 1.2.4. Condition (ii) of Definition 1.2.1 is equivalent to ˆ M,DR(Ψ*) = Ψ* ◇ Ψ* (1.12) where Ψ* := (Γ~ 1 (−e 1 )β(Ψ ⊗ 1)) · 1dr ∈ MDR Proof. Thanks to Diagram (1.8), it follows that κ(Ψ ⋆ ) = Ψ * . Equality (1.12) then fol- lows from the fact that κ : (k((X))/k((X))xo, Δmod) → ( MDR, ΔM,DR ) is a coalgebra isomorphism. Recall the set G(k((X))) of grouplike elements of (k((X)), Δ) given in (1.1). In addition to its usual group structure, it is also a group for the "twisted Magnus" product denoted ☺ and given for any Ψ , Φ ∈ G (k ( ( X ) ) by Ψ ⊕ Φ := Ψaut φ (Φ), (1.13) where auty is the topological k-algebra automorphism of k((X)) given by ([EF0], §4.1.3 based on [Rac], §3.1.2) (1.14) Xo → Xo and for g ∈ G, x g → Ad t g (v − 1) (x g ). Proposition 1.2.5 ([Rac, Theorem I]). Let λ ∈ k and ι : G → C× be a group embed- ding. (i) The pair (DMRG (k), ⊕) is a subgroup of (G(k((X))), ⊕). (ii) The group (DMRf (k), ®) acts freely and transitively on DMRf (k) by left multi- plication 1.3. The torsor DMR ç (k). 1.3.1. Action of the group k × on k ((X)). The group k × acts on k ((X)) by k-algebra automorphisms by (1.15) k ×  —→ Autk-alg(k((X)));     λ → λ ● — : xg → λxg, for g ∈ G ∪ {0}. One checks that, for any λ ∈ k × , the automorphism λ ● − is a Hopf algebra automor- phism of (k ( { X } ), △ ). In addition, for any λ ∈ k × and any g ∈ G , we have (1.16) (λ ● −) o tg = tg o (λ ● −), this can be verified by checking this identity on generators since both sides are given as a composition of k-algebra morphisms. Proposition 1.3.1. For any λ ∈ k × , the map λ ● − : k((X)) → k((X)) restricts to a group automorphism of (G(k((X)), ☺). In order the prove this, we will need the following Lemma: Lemma 1.3.2. For any (λ,Ψ) ∈ k× × G(k((X))), we have (1.17) (λ ● −) ◦ aut ψ = aut λ ● (λ ● −). Proof. Let (λ,Ψ) ∈ k× × G(k((X))). Since all the morphisms are algebra automor- phisms of k((X)), it is enough to check this identity on generators. We have (λ ● −) o aut v (x 0) = λ ● x 0 = λx 0 = λaut λ ● v (x 0) = aut λ ● v (λx 0) = aut λ ● x 0) and for g ∈ G,
where the third equality comes from Identity (1.16).
Proof of Proposition 1.3.1. Let λ ∈ k × . Since λ ● − is a Hopf algebra automorphism of (k((X)), Δ), it restricts to a map G(k((X))) → G(k((X))). Let Ψ, Φ ∈ G(k((X))). We have
where the third equality comes from Lemma 1.3.2. This proves that λ ● – is a group endomorphism of (G(k((X)), ⊕). Finally, one has that (λ ● −)−1 = λ−1 ● − and the above computations shows that (λ ● − ) − 1 is an endomorphism of (G(k((X))), ®) thus口proving the statement. Proposition 1.3.1 enables us to define the following: 10 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 11 Definition 1.3.3. We denote k × G (k ((X))) the semi-direct product of k × and G(k((X))) with respect to the action given in Proposition 1.3.1. It consists of the set k × G(k((X))) endowed with a group law which will also be denoted ® and we have for (λ, Ψ), (ν, Φ) ∈ k× × G(k((X))), (1.18) ( λ, Ψ) ⊕ ( ν, Φ) := ( λν, Ψ ⊕ ( λ ● Φ )). 1.3.2. Action of the group k × on crossed product algebras and module. The group k × acts on VPR by k-algebra automorphisms by k* — (1.19) → Autk-alg(VGR); λ → λ ●ν − : e i → λe i ; g → g , for i ∈ {0,1} and g ∈ G . Lemma 1.3.4. Let λ ∈ k× (i) The following diagram
commutes. (ii) For Ψ ∈ k{{X}}, we have (1.21) Γ λ ●ψ (−e 1 ) = λ ●ψ Γψ (−e 1 ). Proof. (i) Since all arrows are k-algebra morphisms, one easily checks the commutativity of generators. (ii) It follows from the fact that, for n ∈ N >0, we have ( λ●Ψ| x n − 1x1) = λ n (Ψ| x n − 1x1 ). Lemma 1.3.5. Let λ ∈ k × . (i) The k-algebra automorphism λ● ν − of VDR restricts to a Hopf algebra automor- m λ ●w –  of (WDR, ΔW,DR). k-algebra automorphism λ ●ν – of VDR induces a coalgebra automorphism (ii) The of (MDR, AMDR). Proof. (i) For (n, g) ∈ N>0 × G we have λ ●v zn,g = λ ●v (−en−1ge1) = −λnen−1ge1 = λnzn,g Since the algebra WDR is freely generated by ( zn.g)(n.g)€N>0×G , it follows that WPR) ⊂ WPR. Similarly, (λ●ν −)− 1 (WPR) ⊂ WPR. Hence, λ● This implies that λ ●γ — restricts to a k-algebra automorphism λ ●γ — of
YADDADEN KHALEF
WDR and that the following diagram
λ•w WDR
commutes. Let us show that the following diagram
(ii)  One checks that λ●γ — preserves the submodule ∇ DRe 0 + VGR (g − 1). It follows gEG that there is a unique k-module automorphism λ ● M – of MDR such that the following diagram
commutes. Since λ●ν — restricts to the automorphism λ●ν WG ', we obtain the following commutative diagram
12
the double shuffle torsor in terms of betti and de rham coproducts We then have the following cube
Proposition 1.3.6. Let λ ∈ k× (i) The pair (λ ●ν −,λ ●м −) is an automorphism of ( ∇ BR , MBR ) in the category k-alg-modtop・ (ii) The pair (λ●W−,λ●,M−) is an automorphism of ((WDR, in the category k-HAMCtop Proof. (i) Let (v,m) ∈ VPR × MPR. Since − · 1dR : VPR → MPR is surjective, there exist
where the second and fourth equalities come from the commutativity of Diagram (1.24). (ii) It follows from (i) and from Lemma 1.3.5.

1.3.3.  The torsor DMR ç (k).
Proposition 1.3.7. For any λ ∈ k × , the map λ ● − : k((X)) → k((X)) restricts to a group automorphism of (DMR) (k), ☺). 13 Proof. It follows from Proposition 1.3.1 that (λ●−)|G(k((x)}) is a group automorphism of (G(k((X))),®). It remains to prove that this permutation of G(k((X))) induces a permutation of its subset DMRG (k). Let λ ∈ k × and Ψ ∈ DMRG (k). Since λ● x 0 = λx 0 and λ ● x g = λx g for g ∈ G , Conditions (i), (iii), (iv) and (v) of Definition 1.2.1 are immediately satisfied by λ ● Ψ . In order to prove that Condition (ii) is satisfied, let us use Proposition 1.2.4. We have (λ ● Ψ)* = (Γ¯¬¹ν (−e1)β(λ ● Ψ ⊗ 1)) · 1dr = ((λ ●ν Γ¯¬¹(−e1)) (λ ●ν β(Ψ ⊗ 1))) · 1dr = (λ ●ν (Γ ç 1 (−e1)β(Ψ ⊗ 1))) · 1dr = λ ● ν (Γ ç 1 (−e1)β(Ψ ⊗ 1) · 1dr) = λ • M Ψ*, where the second equality comes from the commutativity of Diagram (1.20) and Identity (1.21), the third one from the fact that λ ● ν — is an algebra morphism and the fourth one from Lemma 1.3.5. (ii). Therefore, we obtain that
where the first and last equalities come from (1.26), the second one from Lemma 1.3.5. (ii) and the third one from Proposition 1.2.4 and the fact that Ψ ∈ DMR (k). This proves that λ ● – restricts to a self-map of DMR f (k). Following the same steps, one shows that (λ ● − )−1 = λ−1 ● − restricts to a self-map of DMRF (k) thus proving the ■ statement. Proposition 1.3.7 enables us to state the following definition: Definition 1.3.8. We denote k × DMRF(k) the semi-direct product of k × and DMR (k) with respect to the action given in Proposition 1.3.7. It is a subgroup of k × × G(k((X))). Definition 1.3.9. Let ι : G → C × be a group embedding. We define (1.27) DMR ç (k) := {(λ, Ψ) ∈ k × ∇ G(k((X)))|Ψ ∈ DMR ç (k)}. Proposition 1.3.10. Let ι : G → C× be a group embedding. The group k× × DMRG (k) acts freely and transitively on DMR ξ (k) by left multiplication ⊕ . In order to prove this, we will need the following Lemma: Lemma 1.3.11. Let ι : G → C× be α group embedding and λ,ν ∈ k×. If Φ ∈ DMRç,(k) then λν−¹ ● Φ ∈ DMR (k). Proof.  Let Φ ∈ DMR ζ (k).  Since λν− 1 ● x 0 = λν− 1 x 0 and λν− 1 ● x 9 = λν− 1 x 9 for 9 ∈ G, we have ● (λν−¹ ● Φ | x 0 ) = λν−¹( Φ | x 0 ) = 0 and (λν−¹ ● Φ | x 1 ) = λν−¹( Φ | x 1 ) = 0. ● (λν−1 ●Φ|x0x1) = λ²ν−12(Φ|x0x1) = −λ²ν−12 ν² = −λ² ● ( λν − 1 ● Φ | x g,  − x g,− 1 ) = λν − 1 (Φ | x g,  − x g,− 1 ) = λν − 1 (C − 2 ν =  (C − 2 ν =  (C − 2 λ . 14 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 15 This proves respectively conditions (i), (iii) and (iv) of Definition 1.2.1. Condition (v) follows from condition (iv). Finally, one shows that Condition (ii) is satisfied by using the same arguments as in the proof of Proposition 1.3.7. ■ Proof of Proposition 1.3.10. Since the action of the group k × × G(k((X))) on the set k × G(k((X))) by left multiplication ⊕ is free, so is its restriction to the action of k* × DMRC (k) on DMR ζ (k). Let us show that this action is transitive. Let (λ, Ψ) and (ν,Φ) ∈ DMR ζ (k). Set μ = λν− 1 . It follows from Lemma 1.3.11 that μ●Φ ∈ DMR ζ (k). Thanks to Proposition 1.2.5.(ii), the action of the group (DMRF(k),®) on DMR's (k) is transitive, therefore, there exists Λ ∈ DMR ξ (k) such that Λ ⊕ (μ ● Φ) = Ψ. Thus (μ, Λ) ∈ k× × DMR (k) is such that (μ, Λ) ⊕ (ν, Φ) = (λ, Ψ), which proves the transitivity.
1.4. The torsor DMR x (k). 1.4.1. Action of the group Aut(G) on k((X)). The group Aut(G) acts on k((X)) by k-algebra automorphisms, the element φ ∈ Aut( G ) acting by the automorphism η ό given by (1.28) X0 H X0, xg → xφ(g) for g ∈ G. One checks that, for any φ ∈ Aut( G ), the automorphism ηるis a Hopf algebra automor- phism of (k {{X}}, Δ). In addition, for any φ ∈ Aut( G ) and any g ∈ G , we have (1.29) ηφ ◦ tg = tg ◦ ηφ which can be verified by checking this identity on generators since both sides are given as a composition of k-algebra morphisms. Let us show that Proposition 1.4.1. Let φ ∈ Aut( G ). The map η δ restricts to a group automorphism of (G(k({X)}), ☺). In order to prove this, we will need the following Lemma: Lemma 1.4.2.  Let φ ∈ Aut(G), g ∈ G and Ψ ∈ G(k((X))).  We have (1.30) ηφ ◦ autφ = autηφ(¥) ◦ ηφ Proof.  Since all morphisms are algebra automorphisms of k((X)), it is enough to check this identity on generators. We have ηφ ◦ aut v (xo) = ηφ (xo) = xo = aut ηφ (xo) = aut ηφ (x) ◦ ηφ (xo) and for g ∈ G, ηφ ◦ aut v (x g) = ηφ (Ad t g (v − 1) (x g)) = Ad t φ (g) (πφ (v )− 1) (πφ (x g)) = Adt φ(g)(n φ(v)−1)(xφ(g)) = aut ηφ(v)(xφ(g)) = aut ηφ(v) ◦ ηφ(xg), where the second equality comes from identity (1.29). ■
YADDADEN KHALEF
Proof of Proposition 1.4.1. Let φ ∈ Aut( G ). Since ηφ is a Hopf algebra automorphism of (k((X)), Δ), it restricts to a map G(k((X))) → G(k((X))). Let Ψ, Φ ∈ G(k((X))). We have ηφ(Ψ ⊕ Φ) = ηφ(Ψautφ(Φ)) = ηφ(Ψ)ηφ(autφ(Φ)) = ηφ(Ψ)autηφ(Ψ)(ηφ(Φ)) = ηφ(Ψ) ⊕ ηφ(Φ), where the third equality comes from Lemma 1.4.2. This proves that η δ restricts to a group endomorphism of (G(k((X))), ®).  Finally, one has that η −1 = η δ −1 and the above computations shows that η_ 1 is an endomorphism of ( G (k ( X ))), ⊕ ), thus proving the口statement. Lemma 1.4.3.  For (φ,λ) ∈ Aut(G) × k×, we have ηφ ◦ (λ ● − ) = (λ ● − ) ◦ ηφ Proof. Let (φ,λ) ∈ Aut(G) × k × . Since all the morphisms are algebra automorphisms of k((X)), it is enough to check this identity on generators. We have ηφ ◦ (λ ● x 0) = ηφ (λx 0) = λx 0 = λ ● x 0 = λ ● ηφ (x 0) = (λ ● − ) ◦ ηφ (x 0) and for g ∈ G, ηφ ◦ (λ ● xg) = ηφ(λxg) = λxφ(g) = λ ● xφ(g) = λ ● ηφ(xg) = (λ ● −) ○ ηφ(xg). Propositions 1.3.1 and 1.4.1 and Lemma 1.4.3 enable us to define the following: Definition 1.4.4. We denote (Aut(G) × k × G(k((X))) the semi-direct product of Aut(G) × k × and G(k((X))) with respect to the action given in Propositions 1.3.1 and 1.4.1. It consists of the set Aut(G) × k × × G(k((X))) endowed with a group law which will also be denoted ⊕ and we have for (φ, λ, Ψ) , (φ ′ , ν, Φ) ∈ Aut(G) × k × × G(k((X))), (φ, λ, Ψ) ⊕ (φ′, ν, Φ) := (φ ◦ φ′, λν, Ψ ⊕ ηδ(λ ● Φ)). (1.31) 1.4.2. Action of the group Aut( G ) on Emb( G ). Let us denote (1.32) Emb(G) := { ι : G → C × | ι is a group embedding}. Lemma 1.4.5. The group Aut(G) acts freely and transitively on Emb(G) by (1.33) (φ,ι) →→ ι οφ−1, for (φ,ι) ∈ Aut(G) × Emb(G). Proof. One knows that for any ι ∈ Emb( G ), ι( G ) = μ N . That gives rise to a group isomorphism̃: G → μ N (C). Therefore, for any ι , ι ′ ∈ Emb(G) there is a unique group automorphism φ = ι′− o ῖ of G such that ι o φ− 1 = ι′. Corollary 1.4.6. The group (Aut(G) × k × G(k((X))) acts freely and transitively on Emb(G) × k× × G(k((X))) by (1.34) (φ, λ, Ψ) · (ι,ν, Φ) = (ι οφ− 1 ,λν, Ψ ⊕ ηφ(λ ● Φ)), for (φ,λ,Ψ) ∈ (Aut(G) × k×) × G(k((X)))  and (ι,ν,Φ) ∈ Emb(G) × k× × G(k((X))). □ THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 17 Proof. Let ( ι,ν,Φ),(ι′,ν′,Φ′) ∈ Emb(G ) × k × G (k ((X))). Thanks to Lemma 1.4.5, there is a unique φ ∈ Aut(G) such that ι′ Set ¹ν and Ψ = Φ′ ⊕ η φ (λ ● Φ) ® (−1) In conclusion, there is a unique (φ,λ,Ψ) ∈ (Aut(G) × k×) × G(k((X))) such that (ι,ν,Φ) = (ι',ν',Φ') which proves the statement.
1.4.3. Action of the group Aut(G) on crossed product algebras and module. The group Aut( G ) acts on VDR by k-algebra automorphisms the element φ ∈ Aut( G ) acting by the automorphism ηξ given by (1.35) and g ↔ φ(g) for g ∈ G. Lemma 1.4.7. Let φ ∈ Aut(G). The following diag
commutes. Proof. Since all arrows are k-algebra morphisms, one easily checks the commutativity of generators. Lemma 1.4.8. Let φ ∈ Aut(G). (i) The k-algebra automorphism ηξ of VBR restricts to a Hopf algebra automorphism algebra automorphism ηξ of VDR induces a coalgebra automorphism η (ii) T Proof. (i) For (n, g) ∈ N>0 × G we have Since the algebra WDR is freely general ted by the fa it follows that WDR Similarly. Wo This implies that η k restricts to a k-algebra automorphism of WPR which we denote η W and that we have the following commutative diagram

YADDADEN KHALEF
Let us show that the following diagram
commutes. Indeed, for (n,g) ∈ N >0 × G we have
(ii) Let φ ∈ Aut(G). One checks that ηξ preserves the submodule v BReo + It follows that there is a unique k-module automorphism η M of MPR such that the following diagram
commutes. Combined with (i), it gives the following commutative diagram
18 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS We then have the following cube Proposition 1.4.9. Let φ ∈ Aut(G). (i) The pair (ηξ,ηξ ν ) is an automorphism of (VBR, MBR) in the category k-alg-modtop (ii) The pair (n W , n & ) is an automorphism of ((WBR, ZW,DR), (MBR, ZM,DR)) in the category k-HAMCtop Proof. (i) Let (v,m) ∈ VPR × MPR. Since − · 1dr : VPR → MPR is surjective, there exist
where the second and fourth equalities come from Lemma 1.4.8. (ii). (ii) It follows from (i) and from Lemma 1.4.8.
1.4.4. The torsor DMR x (k). Lemma 1.4.5 sets up the following result: Proposition 1.4.10. Let λ ∈ k. For ι,ι′ ∈ Emb( G ), the element φ ∈ Aut( G ) such that ι′ = ι ○ φ is such that η δ is a bijection between DMR ζ (k) and DMR ζ (k). Proof. Since η φ is a Hopf algebra automorphism of (k ((X)), Δ), it restricts to group automorphism of G(k((X))). It remains to show that ηφ : DMR ζ (k) → DMR ζ ∞ − ζ (k) 19 is a bijection.  Let Ψ ∈ DMR k (k).  Since φ ( x 0 ) = x 0 and φ ( x 1 ) = x 1 , Conditions (i) and (iii) of Definition 1.2.1 are immediately satisfied by η δ (Ψ). Moreover, since 9ιοφ−1 = φ(gι), we have Then Identity (iv) of Definition 1.2.1 follows. One checks Identity (v) in a similar way. Let us prove that Condition (ii) is satisfied by η δ (Ψ). We have
where the second equality comes from the fact that η δ ( x 1 ) = x 1 , the third one from the commutativity of Diagram (1.36), the fourth one from the fact that η λ (e 1 ) = e 1 and the fifth one from the commutativity of Diagram (1.39). Therefore, thanks to Lemma 1.4.8.(ii) and the fact that Ψ ∈ DMR (k), we obtain that
which implies, by Proposition 1.2.4, that condition (ii) of Definition 1.2.1 is verified by ( η δ ( Ψ ))*. This proves that η δ restricts to a map DMR ζ (k) → DMR ζ ( φ − 1 (k). Finally, following the same steps, one shows that η − 1 = η δ − 1 restricts to a map DMR ί ｡ φ − 1 (k) → DMR x (k) thus proving the statement. Corollary 1.4.11. For any φ ∈ Aut( G ), the map η ь restricts to a group automorphism of (DMR ) (k), ⊕ ). Proof. From Proposition 1.4.10 it follows that for λ = 0 and any φ ∈ Aut( G ) the map ηφ restricts to a bijection from DMRF (k) to itself. In addition, since (DMRF (k), ®) is a subgroup of (G(k((X))),®), Proposition 1.4.1 states that this map is a group morphism. ■ Proposition 1.3.7, Corollary 1.4.11 and Lemma 1.4.3 enable us to state the following definition: Definition 1.4.12. We denote (Aut(G) × k × DMR (k) the semi-direct product of Aut( G )×k× and DMR G (k) with respect to the group action of Aut( G )×k× induced by Corollary 1.4.11 and Proposition 1.3.7. It is a subgroup of (Aut(G) × k × ) × G(k((X))). Definition 1.4.13. We define DMR x (k) := {( t, λ, ¥ ) ∈ Emb( G ) × k × G (k((X)}) | (λ, ¥ ) ∈ DMR x (k)}. (1.41) THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS Proposition 1.4.14. The group (Aut( G ) × k × DMR (k) acts freely and transitively on DMR x (k) by (1.42) (φ, λ, Ψ) · (ι,ν, Φ) = (ι ο φ− 1 ,λν, Ψ ⊕ ηφ(λ ● Φ)), for (φ,λ,Ψ) ∈ Aut(G) × k× × DMRξ (k) and (ι,ν,Φ) ∈ DMR x (k). Proof. Let ( ι,ν,Φ),( ι′,ν,Φ′ ) ∈ DMR x (k). Thanks to Lemma 1.4.5, there is a unique φ ∈ Aut(G) such that ι′ = ι 0 φ− 1 . Set λ = ν− 1 ν′ . Since Φ ∈ DMR ξ (k), thanks to Lemma 1.3.11 and Proposition 1.4.10, it follows that ηφ (λ ● Φ) ∈ DMR ζ (k). Thanks to Proposition 1.2.5.(ii), the set DMRi, (k) is a torsor for the action of the group (DMR f (k), ⊕). Therefore, there is a unique Ψ ∈ DMR f (k) such that Ψ ⊕ η δ (λ●Φ) = Φ ′ . In conclusion, there is a unique (φ,λ,Ψ) ∈ (Aut(G) × k×) × DMRG(k) such that
which proves the statement.
Corollary 1.4.15. The pair ((Aut(G) × k × DMR (k),DMR × (k) ) is a subtorsor of ((Aut(G) × k×) × G(k((X))), Emb(G) × k× × G(k((X))) | Proof. It follows from Propositions 1.4.6 and 1.4.14.
2. THE DOUBLE SHUFFLE GROUP AS A STABILIZER OF A “DE RHAM” COPRODUCT In this section, we recall the action of the group (G(k((X))),®) on the algebra- module ( WER , MDR ) given in [Yad]. This action enables us in §2.1, to construct an ac- tion of the group (Aut(G)×k×)×G(k((X))) on the algebra-module (WPR, MDR). This and  Δ leads us in §2.2 to define the stabilizer groups of the coproducts ΔW. DR These stabilizers are related to stabilizers arising from the action of ( G ( k ( ( X ))), ®), which contain DMR (k) thanks to [EF0]. Thanks to the main result of [Yad], we conclude in Corollary 2.2.5 a chain of inclusions involving the former stabilizers and (Aut(G) × k × ) × DMR (k). 2.1. Group actions on the algebra-module (WPR, MPR) 2.1.1. Actions of the group (G(k((X))), ®). For ¥ ∈ G(k((X))), there is a unique topo- logical k-algebra automorphism Γ aut ν .(1) of V D R such that ([Yad, Definition 2.3.1]

YADDADEN KHALEF
This automorphism restricts to a topological k-algebra automorphism ([Yad, Proposition-Definition 2.3.2]). It is such that the following diagram WDR
commutes. Next, one defines the topological k-module automorphism by
of MDR This automorphism induces a topological k-module automorphism such that the following diagram ([Yad, Definition 2.3.4])
The group (G(k({X)}), ☺) acts on V Proposition 2.3.3])
Thanks to this and the commutativity of Diagram (2.2), the group (G(k((X))), ®) acts on WDR by ([Yad, Proposition 2.3.3])
commutes. Lemma 2.1.1 ([Yad, Lemma 2.3.5]). For any Ψ ∈ G(k((X))), the following pairs are automorphisms in the category k-alg-modtop: On the other hand, the action (2.5) induces an action of X }}), B ) on or by (2.7) (G(k({X)}), ☺) → Autk-mod Thanks to the commutativity of Diagram (2.4), the action action (2.7) induces an action of the group (G(k((X)), ⊕) on MPR by ([Yad, Proposition 2.3.6]) (2.8) (G(k((X))), ®) — > Autk-mo Proposition 2.1.2. - 1DR ldr the double shuffle torsor in terms of betti and de rham coproducts 23
( Γ aut ν ,(1) , Γ aut Μ ,(10)
Proof.  This follows from Lemma 2.1.1 and the fact that (2.5) – (2.8) define actions. ● 2.1.2. Actions of the group k × G(k((X))). Definition 2.1.3. For (λ, Ψ) ∈ k× × G(k((X))), we define the topological k-algebra-
with (λ ●ν Proposition-Definition 2.1.4. For (λ,Ψ) ∈ k××G(k((X))), we define the topological M (10) \ Γ aut ( λ, ν ) ,  Γ aut ( λ, ν ) k-algebra-module automorphism of ( WDR given by Гаиt V .(1) , Гаиt M .(10) o ((λ • It is such that the following diagrams
– 1dr commute. Proof.  From Propositions 1.3.6.(ii) and 2.1.2.(iii) we have that the pairs (λ●W−, λ●W−) ( Γ aut W ,(1) , Γ aut M ,(10) ) are morphisms in k-alg-modtop ; the composition is then a and morphism in k-alg-mod top. ´Next, the commutativity of the diagrams follows from the commutativity of Diagrams (1.22) and (2.2) and Diagrams (1.24) and (2.4). ● YADDADEN KHALEF Lemma 2.1.5.  For (λ, Ψ) ∈ k × G(k((X))), we have
Γ aut ν :(1) ο (λ ●ν −) = (λ ●ν −) ο Γ aut ν :(1) Proof. Since both sides are given as a composition of k-algebra morphisms of VPR , it is enough to verify this identity on generators. We have
where the fourth equality comes from the commutativity of Diagram (1.20) and Identity (1.21) and the fifth one from the fact that λ ●ν — is an algebra morphism. Next,
where the fourth equality comes from Identity (1.21) and the fifth one from the fact that λ ●ν — is an algebra morphism. Finally, for g ∈ G,
where the fourth equality comes from the commutativity of Diagram (1.20) and Identity (1.21) and the fifth one from the fact that λ ●γ — is an algebra morphism. Corollary 2.1.6. For (λ,Ψ) ∈ k× × G(k((X))), we have −) = (λ ●w −) o Γaut M,(10) o (λ ●M −) = (λ ●M −) o raut m,(10) (ii) 「aut」 . Proof. (i) This follows from Lemma 2.1.5 thanks to Lemma 1.3.5.(i) and to the commuta- tivity of Diagram (2.2). (ii) This follows from Lemma 2.1.5 thanks to Lemma 1.3.5.(ii) and to the commuta- tivity of Diagram (2.4).
■
24 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 25
where the third equality comes from the fact that Ψ → group actions and the fourth one from Corollary 2.1.6.(i). Next, we have
where the third equality comes from the fact that Ψ → group actions and the fourth one from Corollary 2.1.6.(ii). ■ 2.1.3. Actions of the group (Aut(G) × k × G(k((X))). Definition 2.1.8.  For (φ, λ, Ψ) ∈ Aut( G ) × k × G (k (( X ))), we define the topological ( Γ aut ) raut, (10) VDR  VDR k-algebra-module automorphism given by of (るλ.Y) (φ,λ,Ψ)
given in (1.35). with ηξ ∈ Autk-algtor Proposition-Definition 2.1.9. For ( φ , λ, Ψ ) ∈ Aut( G ) × k × × G (k((X))), we define M,(10) ( Γ aut , (1) the topological k-algebra-module automorphism given by
It is such that the following diagrams
commute. Proof. From Proposition 1.4.9.(ii) and Proposition-Definition 2.1.4, we have that the pairs ( η φ , η φ ν ) and ( Γ aut ν , ν ) ,  Γ aut ν , (10) are morphisms in k-alg-mod top ; the com- position is then a morphism in k-alg-modtop.  Next, the commutativity of the diagrams follows from the commutativity of Diagrams (1.37) and (2.9) and Diagrams (1.39) and (2.10). ■
Proof. Since both sides are given as composition of k-algebra morphisms of VBR , it is enough to verify this identity on generators. We have
where the second equality comes from the identity Γη o(¥)(−e1) = Γ ¥(−e1) and from the commutativity of Diagram (1.36) and the third one from the fact that η δ is an algebra morphism and from the equality ηξ(Γ ν(− e 1 )) = Γ ν(− e 1 ). Next,
where the second equality comes from the identity Γ η δ ( ν ) (− e 1 ) = Γ ν (− e 1 ) and the third one from the fact that ηξ is an algebra morphism and from the equality ηξ (Γ ν (− e 1 )) = Γ v (−e 1 ). Finally, for g ∈ G,
26 and THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 27 where the second equality comes from the identity Γηφ(Ψ)(−e1) = Γψ(−e1) and from the commutativity of Diagram (1.36), the third one from the fact that ηδ is an algebra morphism and from the equality ηδ(Γν(−e1)) = Γψ(−e1) and the fourth and sixth ■ ones from the fact that (ηξ)- 1 ( g ) = φ − 1 ( g ). Corollary 2.1.11. For (φ,λ,Ψ) ∈ Aut(G) × k× × G(k((X))) we have W o ΓautW,(1) o (nY)-1 Proof. This follows from Lemma 2.1.10 thanks to Proposition-Definition 2.1.4 and Lemma 1.4.8. ■ Corollary 2.1.12. The group (Aut(G) × k × G(k((X))) acts on (WBR,MPR) by Гаиt (ф.λ. (Aut(G)×k×)×G(k((X)) → Autk-alg-mod(W) 「aut (6入力)
where the second equality comes from Lemma 1.4.3, the third one from the fact that ηW : Aut(G) → Autk-alg(WGR) is a group morphism, the fourth one from Corollary 2.1.7 and the fifth one from Corollary 2.1.11.(i). Next, we have
where the second equality comes from Lemma 1.4.3, the third one from the fact that η M : Aut(G) → Aut top ( MBR ) is a group morphism, the fourth one from Corollary 2.1.7 and the fifth one from Corollary 2.1.11.(ii). ■
YADDADEN KHALEF
2.2. The double shuffle group as a stabilizer of a "de Rham" coproduct. Proposition 2.2.1. (i) The group (Aut(G) × k × G (k ((X))) acts on Copk-algtop (WPR) by (φ,λ,Ψ) · DW := ( Γ autW :(1)
( 「 aut W (1) (ii) The group (Aut(G) × k × G(k((X))) acts on Copk-modtop(MPR) by
Proof. (i) This is the formula for the pull-back of the action (0.1) with C = k-alg top and O = WDR by the group morphism (φ, λ, Ψ) → (ii) This is the formula for the pull-back of the action (0.1) with C = k-mod top and O = MPR by the group morphism (φ, λ, Ψ) → □
Definition 2.2.2.
(i) We denote Stab(Aut(C)×k×)×G(k((X)))( ÔW,DR )(k) the stabilizer subgroup of the coproduct Â W , DR  ∈  Copk-alg top ( W G R ) for the action of Proposition 2.2.1.(i). Namely, Stab(Aut(G)×k×)×G(k((X)))( ÔW,DR )(k) := (φ,λ,Ψ) ∈ (Aut(G) × k × G(k((X)})| ( r aut w ,(1) } ∞ (ii) We denote Stab(Aut(G)×k×)×G(k((X))( Z G ,DR)(k) the stabilizer subgroup of the coproduct Â G , DR E Copk-modtop ( MDR ) for the action of Proposition 2.2.1.(ii). Namely, Stab(Aut(G)×k×)×G(k((X))( Ô M,DR )(k) := ( Γ aut M , (10) §2 (φ, λ, Ψ) ∈ (Aut(G) × k × G(k((X))) | 。 Âª,dr = Âª,dr o raut M,(10) Since (G(k((X))), ®) is a subgroup of (Aut(G) × k × G(k((X))), the actions of Proposition 2.2.1 induce actions of (G(k((X)), ®) on the spaces Copk-algtop (WPR) and Copk-modtop ( MDR ). This enables us to define the stabilizer subgroups (see [Yad, (2.29) and (2.31)]) Ψ ∈ G(k((X)}) | ( Γ aut W,(1) ) ∞ Stabς(k((x)))(∂W,DR)(k) := { and Stabς(κ((x)))( ˆ∀｀∀｀∀｀∀｀(k((x)))| ( Γ aut ¥ ,(10) θ ² 。 ˆ 28 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS Moreover, we have ([Yad, Theorem 2.4.1]) Stabg(k((x)))( ˆ ω,DR)(k) ⊂ Stabg(k((x)))( ˆ ω,DR)(k). (2.14) Proposition 2.2.3. We have (i) Stab(Aut(G)×k×)×G(k((X)))( △ G ,DR )(k) = (Aut(G)×k×)×StabG(k((X)))( △ G ,DR )(k). (ii) Stab(Aut(G)×k×)×G(k((X)))( A G )(k) = (Aut(G)×k×)×StabG(k((X)))( A G )(k). It is a consequence of the following general lemma Lemma 2.2.4.  Let us consider the semidirect product group H × R.  If K  is a subgroup of H × R containing H , then K = H × (K ∩ R). Proof of Proposition 2.2.3. Set X = W or M .  We use Lemma 2.2.4 where H = Aut( C ) × k × , R = G(k((X))) and K = Stab(Aut( C )×k×)×G(k((X)))( ˆ ζ ,DR)(k). We have that K ∩ R = Stab(Aut(G)×k×)×G(k((X)))( Ô ¥,DR)(k) ∩ G(k((X))) = StabG(k((X)))( Ô ¥,DR)(k). Additionally, Stab(Aut(G)×k×)×G(k((x)))( Z X,DR)(k) contains Aut(G) × k×.  Therefore,
the condition of Lemma 2.2.4 is met and the result then follows. Finally, one has from [EF0, Theorem 1.2] that DMRG(k) = {Ψ ∈ Stabg(k((x)))( ζ M,DR)(k)|(Ψ | x 0) = (Ψ | x 1) = 0}. (2.15) This establishes an inclusion DMRG (k) ⊂ Stabg(k((x))( Â M,DR )(k) of subgroups of (G(k((X))), ®). We then have the following result: Corollary 2.2.5. We have (Aut(G) × k × ) × DMRG (k) ⊂   Stab(Aut(G)×k×)×G(k((X)))( ˆ C ′ C ′ C ′ (k) Stab(Aut(G)×k×)×G(k((X)))( ÔW,DR )(k) Proof. Thanks to Proposition 2.2.3.(ii), we have (2.16) Stab(Aut(G)×k×)×G(k((X)))( Z M,DR)(k) = (Aut(G) × k×) × StabG(k((X)))( Z M,DR)(k). On the other hand, using equality (2.15), we obtain (2.17) (Aut(G) × k × ) × DMRG (k) ⊂ (Aut(G) × k × ) × Stabg(k((x)))( ˆ C ′ C ′ DR )(k). From equality (2.16) and inclusion (2.17), we obtain the inclusion (Aut(G) × k × ) × DMRG (k) ⊂ Stab(Aut(G)×k × G(k((X)))( ˆ G,DR)(k), which is the wanted first inclusion. For the second inclusion, thanks to inclusion (2.14), we have that (Aut(G) × k × ) × Stab G(k((x)))( Ô G ,DR )(k) ⊂ (Aut(G) × k × ) × Stab G(k((x)))( Â W ,DR )(k). Thanks to Proposition 2.2.3, this inclusion implies that Stab(Aut(G)×k×)×G(k((X)))( Â M,DR)(k) ⊂ Stab(Aut(G)×k×)×G(k((X)))( Â W,DR)(k). □
YADDADEN KHALEF
3. Construction of “Betti” coproducts
In this section, we construct a "Betti" version of the double shuffle formalism. The relevant algebras and modules are introduced in §3.1 : (i) an algebra V R de- fined as the inverse limit of an algebra V N endowed with a suitable filtration; (ii) an algebra-module ( WB , MB ) composed of a subalgebra WB of VB and a k-module MB which has a V B -module structure inducing a free rank one W B -module structure on it. In proposition 3.1.27, we construct algebra-module isomorphisms (isoW,r,isoM,r) from ( WB, MB, ) to ( WBR, MBR ) indexed by ι ∈ Emb( G ). This gives rise to a fam- ily of algebra-module isomorphisms ( Γ comρ ( ι,λ,ψ),  comρ(ι,λ,Ψ) indexed by elements ( ι, λ, Ψ) ∈ Emb( G ) × k × G(k((X)}). In §3.2, we show that the transport by this (t,X, F)とLins(8) γ . II (1) of the element ( ι, λ, Ψ) ∈ DMR x (k) (see Theorem 3.2.4). This is derived from the chain of inclusions of Corollary 2.2.5 and from the torsor structure of DMR x (k) over (Aut(G) × k × ) × DMR C (k) (see Proposition 1.4.14). The resulting pair of coproducts is denoted ( Δ W , Δ N , B ) and equips W N and M N with Hopf algebra and coalgebra structures respectively (see Corollary 3.2.6).
3.1. The topological algebra-module (WR, MR).
3.1.1. The filtered algebra V N . Let F 2 be the free group generated by two elements denoted X 0 and X 1 . We consider the group morphism F 2 → μ N given by X 0 → ζ N and X 1 → 1; where ζ N := e * Lemma 3.1.1. The group ker( F 2 → μ N ) is isomorphic to the free group of rank N + 1 denoted F N +1・ In order to prove this, we use the following result: Proposition 3.1.2 (Nielsen-Schreier Theorem, see [Ste, Theorem 3]). Let F be a free group on a non-empty set X and let H be a subgroup of F . Let σ : H\F → F be a section of the canonical projection F → H\F such that T := σ(H\F) is stable under left prefixation. Then H is freely generated by {tx(tx)-1 | (t,x) ∈ T × X  and tx(tx)-1 ≠ 1} , where for g ∈ F , g the image of g under the composition F → H\F → F . Proof of Lemma 3.1.1. We apply the Nielsen-Schreier Theorem for X = { X 0 , X 1 }, F = F 2 , H = ker( F 2 → μ N ) and σ : ker( F 2 → μ N )\ F 2 ~ μ N → F 2 where the first map is the isomorphism induced by the surjective morphism F 2 → μ N and the second map given by e i ∂ The theorem then states that ker( F 2 → μ N ) is freely generated by:
THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS Finally, ker( F 2 → μ N ) is freely generated by the N + 1 elements { X N , (X n X 1 X n ) n ∈ [0,N − 1] } . the generators of the free group F N +1 Moreover, if we denote , Xo, ( Xc n ) n ∈ [0,N −1] x of rank N + 1, one checks that correspondence Xo → Xo N , Xc N → Xo X1 Xo n for n ∈ [0, N − 1] defines a free group isomorphism from F N +1 to ker(F 2 → μ N ). We then obtain the following short exact sequence (3.1) {1} → F N +1 → F 2 → μ N → {1} Next, let σ : μ N → F 2 be the set-theoretic section of F 2 → μ N given by e α n π ↓→ X n for n ∈ [0, N − 1]. Thanks to the exact sequence (3.1) we obtain a bijection (3.2) Σ : μ N × F N +1 → F 2 ,    (ζ, x) → σ(ζ)x ; where FN+1 is seen as ker(F2 → μN) ⊂ F2 thanks to Lemma 3.1.1 The set μ N × F N +1 is equipped with a right F N +1-set structure by (ζ, x) * y := (ζ, xy), for (ζ, x) ∈ μ N × F N +1 and y ∈ F N +1. The group F 2 is also equipped with a right F N +1-set structure given by x * y := xy, for x ∈ F2 and y ∈ FN+1; where F N +1 is seen as ker( F 2 → μ N ) ⊂ F 2 thanks to Lemma 3.1.1. One checks that (3.2) upgrades to a right F N +1-set isomorphism. Let us consider the tensor functor k(−) : {right F N +1-sets} −→ {right kF N +1-modules} taking X to kX , the set of finitely supported maps X → k. Applying this functor to the isomorphism of right F N +1 -sets (3.2), one obtains the right kF N +1 -module isomorphism (3.3) kΣ : kμ N 😊 kF N +1 → kF 2 , where both the source and the target are equipped with the right kFN+1-module struc- ture given by the right F N +1-set structure on μ N × F N +1 and F 2 respectively. Let us denote I := ker(kF2 → kμ N ) where kF2 → kμ N is the k-algebra morphism induced from the group morphism F 2 → μ N . Then I is a two-sided ideal of k F 2 . In particular, I is a right kFN+1-module. Let ε : kFN+1 → k be the augmentation morphism of the group algebra kFN+1. It is equipped with a right regular kFN+1-module structure. Lemma 3.1.3. (i) The k-module isomorphism k Σ : k μ N ⑧ k F N +1 → k F 2 sets up a right k F N +1 - module isomorphism of I with kμ N ➤ ker(ε). (ii) The ideal I is linearly generated by σ(ζ)( x − 1) where ζ ∈ μ N and x ∈ F N +1. Proof. ■
YADDADEN KHALEF
(i) The following commutative diagram of F N +1-set morphisms
induces a commutative diagram of kFN+1-module morphisms
→ kFっOne checks that the associated group algebra morphism of the first projection P1 : μ N × F N +1 → μ N is identified with id ⊗ ε : kμ N ⊗ kF N +1 → kμ N thanks to the identification kμ N ⊗ k F N +1 ~ k(μ N × F N +1). Therefore, the ideal I is mapped by the isomorphism k Σ to the ideal ker(id ⊗ ε ) = k μ N ⊗ ker(ε). (ii) Since ε : kF N +1 → k is the augmentation morphism, its kernel is generated by elements x − 1 with x ∈ F N +1. Therefore, taking the image of the generators by kΣ, we obtain generators of the ideal I as announced. ■ Proposition-Definition 3.1.4. Let V B be the group algebra of F 2 over k endowed with the filtration FmVB = Im, for m ∈ N, where I m is the m th -power of the ideal I with the convention that I 0 = V B The filtration (FmVR)men is an algebra filtration. Proof. Immediate. Lemma 3.1.5. Let m ∈ N. The k-module isomorphism kΣ : kμ N ⊕ kF N +1 → kF 2 sets up a right kF N+1-module isomorphism of F n V R with k μ N B (kF N+1) π , where (kF N +1)0 is the augmentation ideal of the group algebra kF N +1. Proof. If m = 0, we have k μ N ⊗ k F N +1 ∼ − k(μ N × F N +1) ∼ − → V B = F o ν B = Next, if m = 1, we have F¹V B = I ~ k N S ker(ε) = k μ N S (kFN+1)0, where the identification is given by Lemma 3.1.3 (i). Now, let m ≥ 2. Since k μ N ⊗ kF N +1 is a right k F N +1 -module, we have that (3.4) kμ N ⊗ (kF N +1)m = (kμ N ⊗ (kF N +1)m) · (kF N +1)m−1 The composition k μ N ® k F N +1 ~ k( μ N × F N +1) k ≥ V B is a right k F N +1 -module isomorphism which, combined with the identification I ~ k μ N ⊗ (k F N +1)0 and equality (3.4), gives us kμ N & (kF N +1)φ ~ I · (kF N +1)φ where ( kFN+1) m-1 is seen as a subset of kFN+1 = k ker( F2 → μN) ⊂ kF2 . It remains to show that I · (kFN+1) m-1 = Im. First, since (kFN+1) o ⊂ I, we have THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 33 I · (kF N +1)m−1 ⊂ I m . Conversely, thanks to Lemma 3.1.3 (ii), I m is linearly generated by elements Π((ζι, x 1 ), . . . , (ζm, x m)) := σ(ζι)(x 1 − 1) . . . σ(ζm)(x m − 1) with (ζ 1 , x 1 ), . . . , (ζm, x m) ∈ μ N × F N +1.  Moreover, we have that Π((ζ1, x1), . . . , (ζm, xm)) = σ(ζ1) · · σ(ζm) (Adσ(ζm)-¹··σ(ζ2)-¹(x1) − 1) (Ad σ (ζm)− 1 …σ(ζ )− 1 ( x 2 ) − 1 ) … (Ad σ (ζm)− 1 ( xm− 1 ) − 1 ) ( xm − 1 ) . Next, since FN+1 is a normal subgroup of F2, we have that (Ad σ (ζm)− 1…σ(( 3 )− 1 ( x 2 ) − 1 ) · · · (Ad σ (ζm)− 1 ( xm− 1 ) − 1 ) ( xm − 1 ) ∈ ( kFN+1)m− 1 . In addition, thanks to Lemma 3.1.3 (ii), we have σ(ζ1) · · · σ(ζm) (Adσ(ζm)− ¹ · σ(ζ2)− 1 ( x1) − 1) ∈ kF 2 · (kF N+1)0· Since (kFN+1)o ⊂ I, it follows that kF2 · (kFN+1)o ⊂ kF2 · I and since I is a two-sided ideal of kF2, we have kF2 · I = I. Therefore, σ(ζι)···σ(ζm) (Adσ(ζm)−¹··σ(ζ2)−¹(x1) − 1) ∈ I, and then Π((ζ1, x1), . . . , (ζm, xm)) ∈ I · (kF N+1) m−1 , thus proving the wanted inclusion. ■ 3.1.2. The topological algebra VR .  The decreasing filtration ( FmVR )men given in Proposition-
Definition 3.1.4 induces an algebra morphism VR / Fm+1VR → VR / FmVR . One d Definition 3.1.6. We denote VB := lim VB /FmVB the inverse limit of the system (VB/FmVB,VB/Fm+1VB, → VB/FmVB). The algebra V B is equipped with the filtration FmV B := lim FmV B /Fmax(m,l)V B . When equipped with the topology defined by this filtration, 𝑠𝑗𝑗 is a complete separated topological algebra. Recall that kFN+1 is a group algebra equipped with a filtration given by the powers of its augmentation ideal. Let us denote kFN+1 the completion of this group algebra with respect to this filtration.
Lemma 3.1.7.
(i) The k-algebra morphism kΣ 0 (1 ⊗ −) : kF N +1 → V B  gives rise to a topological k-algebra morphism kFN+1 → VB. (ii) The k-module morphism kΣ : kμ N ⊗ k F N +1 → V B  gives rise to an isomorphism of topological right kFN+1-module kE : kμN ® kFN+1 → VR, (iii) The k-algebra morphism k F N +1 → V R is injective. Proof. (i)  This follows from the fact that the k-algebra morphism k Σ ◦ (1 ◕ − ) : k F N +1 → V B is compatible with filtrations, which follows from Lemma 3.1.5. (ii) This follows from the fact that k∑ : kµ N ® kF N +1 → V N is an isomorphism of filtered right module over kFN+1 (see (3.3)). (iii) By (i), the topological k-algebra morphism k F N +1 → V B is equal to the compo- sition k Σ ◦ (1 ⊗ −) : k F N +1 → V R . The map 1 ⊗ − : k F N +1 → k μ N ⊗ k F N +1 is trivially injective and k ∑ : k μ N ➤ k F N +1 → V B is injective by (ii). This implies that their composition is injective, implying the claim.
Proposition-Definition 3.1.8. Let ι ∈ Emb( G ). There is a unique topological algebra isomorphism iso V x : V B → V B  given by
where g t = t-1(e * ). Proof. Recall that the set Mork-alg(kF2,VBR) is identified with Morgrp(F2,(VBR) * ) As a consequence, there is an algebra morphism V N → V DR given by
since the images of X 0 and X 1 are invertible. Composing the k-algebra morphism VB → VBR with the k-module isomorphism kE : kµN ® kFN+1 → VB and the inverse of the k-algebra isomorphism β : k((X)) × G → VDR respectively from the left and from the right, we obtain a k-module morphism
One checks that morphism (3.5) is a right module morphism over the k-algebra mor- phism kF N +1 → k ( X ) } given by
In addition, ( ζ λ ν ⊗ 1) ι∈ [0, ν − 1] and (exp ( ζ x 0) ⊗ δ ί ) ι∈ [0, ν − 1] are bases of k μ N ⊗ k F N + 1 and k〈〈X〉〉× G respectively and the morphism (3.5) induces the following bijection between the bases
Furthermore, there is a topological k-algebra isomorphism kFN+1 → k((X)) such that the following diagram 34 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 35 commutes, where kFN+1 ↔ kFN+1 is the canonical k-algebra morphism. Indeed, such an isomorphism is obtained by composing the topological k-algebra iso- morphism kF N +1 → k ((X)) obtained from [Qui, Example A2.12] and the topological k-algebra automorphism of k F N+1 given by
On the other hand, one checks that k μ N ⊗ k F N +1 is a free right k F N +1-module with basis ( ζ ′ ν ⊗ 1) ι∈ [0, N − 1] and recall that k ( X ) ⟩ × G is a free right k ( 〈 X ) 〉 -module with basis (exp (̸ν x 0 ) ⊗ 9 ′ ) ιε[0, ν − 1] · Therefore, there is a unique module isomorphism k μ N ⊗ kF N+1 → k ((X)) × G over the k-algebra isomorphism kF N+1 → k ((X)) which extends bijection (3.6) between bases. Therefore, the restriction to the bases of the following diagram
commutes, where k μ N ® kF N +1 → k μ N ® kF N +1 is the tensor product of the identity of k μ N with k F N +1 → k F N +1 . This implies that the diagram commutes. Next, by composing the k-module isomorphism kμ N ⑧ kF N +1 → k ((X)) × G from the left and from the right with the isomorphisms k Σ− 1 : ∇ B → k μ N ⊗ k F N +1 and β : k((X)) × G → V B respectively, we obtain a k-module isomorphism V B → V B Let us prove that this k-module isomorphism is a k-algebra isomorphism. It is, there- fore, enough to show that it is a k-algebra morphism. Let us consider the following prism The left, right and middle squares commute by definition of k Σ , V R → V R and k μ N ⊗ kF N +1 → k ((X)) × G respectively and the upper triangle is Diagram (3.7), so is commutative.  Additionally, the arrows going from the upper triangle to the lower triangle are isomorphisms. Therefore, the lower triangle is commutative. The restriction of the topological k-module isomorphism V N → V DR to V N is an algebra morphism, which by the density of V B in V B implies that V B → V B B is a topological
YADDADEN KHALEF
k-algebra morphism and therefore a topological k-algebra isomorphism. Finally, the commutativity of the triangle also implies that the k-algebra isomorphism VB → VBR is as announced. Proposition 3.1.9.  Let ι ∈ Emb(G) and φ ∈ Aut(G).  We have isoV,ιοφ− = ηξ ο isoV, with ηξ ∈ Autk-algeon (VDR) given in (1.35). Proof. Since both sides are given as a composition of topological k-algebra morphisms, let us the equality by checking on the family of topological generators:
3.1.3. The filtered algebra WB Proposition-Definition 3.1.10. Let us denote W ;= k ⊕ V ;(X 1 − 1). It is a subalgebra of V B endowed with the filtration F™WB := WB ∩ F™VB (3.9) for m ∈ N. The filtration ( F mWR ) m ∈ N is an algebra filtration. Proof. Immediate. □
Proof.
where the second equality follows from the inclusion FmVB ⊂ ker(VB → k) since FmVB = ker(VB → kμ N ) m ⊂ ker(VB → kμ N ) ⊂ ker(VB → k), (3.10) where the last inclusion of (3.10) is a consequence of the fact that V ξ → k is the composition V ξ → k μ N → k (the maps with target k being the augmentation morphisms). The third equality follows from
36 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 37 which, in turn, follows from the fact that ker(V N → k) ∩ (k ⊕ V N (X 1 − 1)) is the k and takes V R (X 1 − 1) to 0. Its kernel is therefore V R (X 1 − 1). (ii) Recall from Lemma 3.1.5 that, for m ∈ N*, the k-module isomorphism kΣ : kμ N ➤ kF N +1 → V N induces an isomorphism (3.11) F™V N ~ k N S (kFN+1) ~ where ( kFN+1 )0 is the augmentation ideal of the group algebra kFN+1. The isomorphism k∑ also induces an isomorphism V N (X 1 − 1) ~ k μ N $ k F N +1(X c ∞ − 1). Thanks to Lemma 3.11 (i), this induces the isomorphism F™WB ~ kµn ® ((kFN+1)m ∩ kFN+1(Xco, − 1) Next, thanks to [Wei, Proposition 6.2.6], we have a kF N +1 -module isomorphism (kFN+1) B (N+1) → (kFN+1) 0 . This isomorphism induces the following isomor- phisms kF N+1 ⊕ {0} ⊕ N ～ kF N+1 (Xçç, − 1) and (kF N+1) m− 1) ⊕ ( N+1) ～ (kF N+1) m ; where for the latter one we use the fact that (kFN+1) σ n = (kFN+1) σ − 1 (kFN+1) σ and the fact that (kFN+1)m-1 is an ideal of kFN+1. On the other hand, using the inclusion (kFN+1)""-1 ⊂ kFN+1 and the isomor- phism kFN+1 ⊕ {0} ⊕ N ~ kFN+1 (Xççç − 1), one obtains (kF N +1)m−1)⊕(N+1) ∩ (kF N +1 ⊕ {0}⊕N ) = (kF N +1)m−1 ⊕ {0}⊕N Finally, one checks that the isomorphism (kFN+1) ⊕ ( N+1) → (kFN+1) 0 induces an isomorphism (kF N+1)m−1 ⊕ {0}⊕ N ～ (kF N+1)m−1 ( X ζφ − 1) and using (3.11) for m replaced by m — 1, together with the fact that k Σ in- tertwines right multiplication by X 1 – 1 on V R with the tensor product of the identity on k μ N with right multiplication by X c 0 , − 1 on k F N +1 implies thus proving the wanted result. ■ 3.1.4. The topological algebra WB. The decreasing filtration ( F m W N )mEN given in (3.9) induces an algebra morphism W B / Fm+1W B → W B / FmW B Definition 3.1.12. We denote W = lim W = / F m W = the inverse limit of the projective system (WB/FmWB,WB/Fm+1WB → WB/FmWB). The algebra WB equipped with filtration F™WB := lim F™WB / Fmax(m,l)WB and endowed with the topology defined by this filtration is a complete separated topo- logical algebra. Lemma 3.1.13. The k-algebra inclusion W B ⊂ V B gives rise to an injective morphism of topological k-algebras WB → VB Proof. This follows from the compatibility of the inclusion WR ⊂ VR with filtrations and the fact that the filtration on W is induced by that of V B口Proposition 3.1.14. The topological algebra WB is isomorphic to the topological sub- algebra k ⊕ V B (X 1 − 1) of V B Proof. This will be done following this program: Step 1: Construction of the topological k-module WB + Let us define a k-submodule W R + := V R (X1 - 1) ⊂ W R . It is equipped with the filtration F™WR,+ := WR,+ ∩ F™WR, for m ∈ N induced by the inclusion W N + ⊂ W N . Denote as follows the associated inverse limit WN + := lim WN +/F™WN One checks that the k-module inclusion W k + ⊂ W k is compatible with the filtrations, which induces a morphism of topological k-modules WB,+ → WB. As the filtration of WB + is induced by that of WB , this morphism is injective. Thanks to Lemma 3.1.13, we then have a chain of injections (3.12) WB + → WB → VB On the other hand, for any m ∈ N*, we have F™WR+=WR+∩F™WR = VR(X1 − 1)∩Fm−1VR(X1 − 1) = Fm−1VR(X1 − 1), where the second equality comes from Lemma 3.1.11 (ii). Therefore, for any m ∈ N,
Moreover, let us notice that W B = k ⊕ W B + . Using (3.13) we obtain Fown = k ⊕ Fown +; F™W® = F™W®,+, for m ∈ N*. These equalities induce the following topological k-algebra isomorphism = lim W B / F m W B ≈ k ⊕ lim W B + / F m W B + = k ⊕ W B + WB (3.14) where, on the right, the algebra structure is defined by the conditions that 1 E k is a unit and that the inclusion W B + ⊂ k ⊕ W B + is a non-unital algebra morphism. THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 39 Step 2: The existence of a topological k-module morphism φ : VR → WR + such that the triangle
commutes. First, let us consider the k-module morphism ϕ : V k → W k + given by v → v(X 1 − 1). For any m ∈ N *, one has
where the first equality follows from the definition of φ , the inclusion follows from decreasing character of ( F m V n )m ∈ N and the last equality follows from (3.13).  One also has
This implies that the morphism φ : V B → W B + is compatible with filtrations. This induces a k-module morphism ϕ m : V N / F m V N → W N + / F m W N In the following prism the upper triangle commutes by definition of ϕ : V k → W k ,+ and all the squares commute thanks to the compatibility of the maps ϕ : V B → W B +, − · (X 1 − 1) : → V B and W B + ⊂ V B with filtrations. Therefore, thanks to the surjectivity of VB the projection V N → V B / FmV B , the lower triangle commutes. As a consequence, the morphism φ : V N → W N + induces a morphism of topological k-modules O : V N W R + such that Diagram (3.15) commutes. Finally, the commutativity of the latter diagram implies

YADDADEN KHALEF
Step 3: The existence of a topological k-module morphism φ : W R + → V R such that the triangle
commutes. First, one notices that ϕ : V B → W B + is a surjective k -module morphism. Therefore, It is injective thanks to the integral domain status of the algebra V B the map ϕ : V F → W F + is a k-module isomorphism whose inverse will be denoted Thanks to (3.13), the k-module isomorphism φ : W R + → V R φ : W N + → V N → Fm−1VB, for any m ∈ N*. This induces a restricts to an isomorphism FmWB k-module isomorphism φm : W N +/F™W N + → V B / Fm-1V B , for any m ∈ N* and, via a prism similar to the one of Step 2, one checks that the following triangle
commutes where the morphism − (X 1 − 1) : V k / F m − 1V k → V k / F m V k is well-defined thanks to the inclusion Fm-1VR(X1 – 1) ⊂ FmVR being a consequence of (3.13). On the other hand, we have, for any m ∈ N*, the following triangle
where πm : V B / F m V B → V B / F m − 1 V B is the morphism which associates to the class of an element modulo FmVB , its class modulo Fm-1VB ; this is well-defined and sur- jective thanks to the inclusion FmVB ⊂ Fm-1VB . One then checks that this triangle commutes.  By linking the two triangles and doing the inverse limit we obtain the following diagram → lim V N / F m − 1 V N is obtained by degree shifting and is where  π  :=  lim π m  :  V N therefore a topological k-module isomorphism. Let us set φ := π−1 o φ : W5 It is a topological k-module morphism such that Diagram (3.17) commutes. Finally, the double shuffle torsor in terms of betti and de rham coproducts 41 the commutativity of the latter diagram implies
(3.18) Finally, combining inclusions (3.16) and (3.18) we obtain
In addition, thanks to 3.14, the topological k-algebras k⊕WB ↓ and WB are isomorphic. One then obtains the isomorphism of topological k-algebras
Proposition-Definition 3.1.15. Let ι ∈ Emb( G ). There exists a topological algebra isomorphism isoW,ι : WB, → WB, such that the following diagram
commutes. Proof. We have iso ν , t (X 1 − 1) = exp(e 1 ) − 1 = ue 1 , where u = f(e 1 ) with f(x) being the invertible formal series exp(x) −1 . Moreover, since iso ν
This implies that iso : V B (X 1 −1) → V D R e 1 is a surjective k -module morphism VB(X1-1) which is trivially injective, therefore, is a k-module isomorphism.  Taking the direct sum with k, we obtain a k-module isomorphism
which is a k-algebra isomorphism. Finally, thanks to Lemma 3.1.14, this isomorphism is the wanted k-algebra isomorphism isoW.t : WB → WBR ■ Corollary 3.1.16. Let ι ∈ Emb(G) and φ ∈ Aut(G). We have = η V o iso V , with η V ∈ Autk-alg top (WBR) given in Lemma 1.4.8.(i). Proof. The statement follows from Proposition 3.1.9 thanks to the commutativity of ■ diagrams (3.19) and (1.37). 42 3.1.5. The filtered module M N . Proposition-Definition 3.1.17. The quotient k-module MB := VR /VR(Xo – 1) (3.20) is a V R -module.  Moreover, if we denote 1 B the class of 1 ∈ V R in M R , then the canonical projection − · 1 B : V N → M N is a surjective V B -module morphism and its restriction to W B is a W B -module iso- morphism. Proof. This follows from the direct sum decomposition VB = k ⊕ VB (X1 − 1) ⊕ VB (X0 − 1) = WB ⊕ VB (X0 − 1) given by [Wei, Proposition 6.2.6]. □ Remark. The statement implies that (− · 1в) | w B : W B → M B is a W B -module isomorphism, therefore MB is a free WB -module of rank 1. Proposition-Definition 3.1.18. The k-module M n is endowed with the decreasing k-module filtration given by F™™™ := F™W♬ · 1B   for m ∈ N. (3.21) Moreover, the pair (MB, (Fm MB)mEN) is a filtered module over the filtered algebra /m=N Proof. Immediate. Lemma 3.1.19. (i) For any m ∈ N, the k-module isomorphism − · 1 B : W R → M N induces a k-modules isomorphism FmWB → Fm MB (ii) For any m ∈ N, we have Fm MB, = FmVB, 1b. Proof. (i) By definition of Fm M , the isomorphism − · 1 B : W R → M R restricts to a surjective k-module morphism FmWR → FmMR . In addition, since − · 1 B : WB → MB, is injective, so is the restriction FmWB → FmMB, (ii) First, if m = 0, the equality follows from Proposition-Definition 3.1.17. From now on, let m ∈ N*. Since F m W B ⊂ F m V B , we have that F™ M N ⊂ F™ V N · 1b. Conversely, let us prove the inclusion FmV N · 1 B ⊂ Fm M N . This inclusion is equivalent to FmVN ⊂ FmWN + VN(X 0 − 1). Since FmVB = Tm = ker(VB → kµN)m and by Lemma 3.1.11 (i), this inclusion is also equivalent to (3.22) T™ ⊂ (I™ ∩ VN (X1 − 1)) + VN (X0 − 1). ■ THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 43 We have I = ker(V N → k μ N ) ⊂ ker(V N → k) = V N (X 0 − 1) + V N (X 1 − 1), with ker(V B → k) being the augmentation ideal of the group algebra V B = kF 2 and the last equality being a consequence of [Wei, Proposition 6.2.6]. This implies Tm = Tm−1T ⊂ Tm−1 (V N (X 1 − 1) + V N (X 0 − 1)) ⊂ Im−1VN(X1 − 1) + VN(X0 − 1). Moreover, V B (X 1 − 1) ⊂ ker(V B → k μ N ) since X 1 − 1 → 0 through the map V N → kμ N . This implies Tm-1VB(X1 – 1) ⊂ Tm-1T = Tm. On the other hand, we have Tm−1VB(X1 − 1) ⊂ VB(X1 − 1). From (3.24) and (3.25) we obtain Tm-1VR(X1 – 1) ⊂ (Im ∩VR(X1 – 1)) . Finally, from (3.23) and (3.26), we obtain Im ⊂ (Im ∩ VN (X1 − 1)) + VN (X0 − 1), which is the wanted inclusion. (3.23) (3.24) (3.25) (3.26) 3.1.6. The topological module MB . The decreasing filtration ( Fm MB ) men given in (3.21) induces a k-module morphism MB / Fm+1 MB → MB / Fm MB . Definition 3.1.20. We denote M N  := lim M N / F m M N . the limit of the projective system ( M fi / Fm M fi , M fi / Fm +1 M fi → M fi / Fm M fi ). The k-module MB, is a VB,-module equipped with the filtration F™ M := lim F™ M n / Fmax(m,l) M n , for m ∈ N . When equipped with the topology defined by this filtration, MB  is a complete separated topological k-module. Lemma 3.1.21. (i) The surjective k-module morphism − · 1 B : V R → M R induces a topological (ii) The k-module morphism − · 1в : VR → MB.
(ii) The k-module isomorphism − · 1в : WR → MB induces a topological k-module isomorphism − · 1 B : W B → M B Proof. (i) By definition of V B and M B , this follows from Lemma 3.1.19 (ii). □
YADDADEN KHALEF
(ii) By definition of M R , this follows from Lemma 3.1.19 (i). Corollary 3.1.22. (i) The pair (VR, MB) is an object in the category k-alg-modtop (ii) The pair (WB,MB) is an object in the category k-alg-modtop. Moreover, MB is a free WB-module of rank 1. Proof. It immediately follows from Lemma 3.1.21. ■ Proposition 3.1.23. The topological k-module morphism – · 1 B : V R → M R induces an isomorphism V B /V B (X 0 − 1) → M B , of topological k-modules. In order to prove this, we will need the following Lemma: Lemma 3.1.24. Let V be a k-module and u be an endomorphism of V . Let f : V N → V N be the endomorphism given by (vo, . . . , vn − 1) → (u(vn − 1) − vo, vo − v1, v1 − v2, . . . , vn − 2 − vn − 1). Then we have an isomorphism coker( f ) ~ coker( u – id). Proof of Lemma 3.1.24. Let us consider the k-module morphism sum : VN → V given by ( vo, . . . , v N −1) → vo + . . . + v N −1.  This morphism sends Im( f ) to Im( u − id). Therefore, there is a unique k -module morphism V N /Im( f ) → V /Im( u − id) such that the diagram
→ V
phism. First, the surjectivity of the morphism VN /Im( f ) → V/Im( u − id) is an isomor- phism V N /Im( f ) → V /Im( u −id) is surjective as well. Second, let ( w 0 , . . . , w N −1) ∈ V N such that there exists an element v ∈ V such that w 0 + · · · w N − 1 = u(v) − v . The element (vo, . . . , v n − 1 ) ∈ V N given by vN-1 = v, vN-2 = wN-1 + v, vN-3 = wN-2 + wN-1 + v, . . , v0 = w1 + . . wN-1 + v is such that ( w 0 , . . . , w N − 1 ) = ( u ( v N − 1 ) − v 0 , v 0 − v 1 , . . , v N − 2 − v N − 1 ) ∈ Im( f ). Thus proving the injectivity of V N /Im( f ) → V /Im( u − id). □ Proof of Proposition 3.1.23. The proof consists of the following steps: Step 1: Construction of the k-module isomorphism P 0 : (kFN+1)N → V N As in (3.3), one defines the k-module isomorphism kFN+1 S kµN → V N such that for (x, ζ) ∈ F N +1 × μ N (3.27) x $ ζ → x σ (ζ), THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 45 where F N +1 is seen as ker( F 2 → μ N ) ⊂ F 2 thanks to Lemma 3.1.1. Moreover, one checks there is a k-module isomorphism (kFN+1)N → kFN+1   kμN given by
Therefore, the composition P 0 : (kFN+1)N → kFN+1 ® k l N → V B is a k-module isomorphism and is given by
Step 2: Identification of MB One checks that the endomorphism f : (kFN+1)N → (kFN+1)N given by (3.29) (vo, . . . , vn − 1 ) → (vn − 1 X 0 − vo, vo − v1, v1 − v2, . . . , vn − 2 − vn − 1 ), is such that the following diagram
commutes. This induces a k-module isomorphism coker( f ) ~ coker( − · ( X 0 − 1)). On the other hand, by applying Lemma 3.1.24 with V = kFN+1 and u = − · ∑ 0 , we obtain an isomomorphism coker( f ) ~ coker( u — id). It then follows that
~ coker( f ) ~ coker( u − id) = k F N +1 /k F N +1 ( X 0 − 1). Step 3: Compatibility of the isomorphism kFN+1/kFN+1(Xo − 1) → MB, with filtra- tions. Let us show that for any m ∈ N , we have
If m = 0, this has been proved in Step 2. From now on, let us assume that m ∈ N*. The isomorphism kF N +1/kF N +1( X 0 − 1) → M N fits in the following commutative diagram
where kFN+1 → V N is the group algebra morphism induced by the group morphism F N +1 ~ ker( F 2 → μ N ) ⊂ F 2 obtained in Lemma 3.1.1. This group algebra morphism induces the injection
46 Then, thanks to the commutativity of Diagram (3.31), the k-module isomorphism kF N +1/kF N +1( X 0 − 1) → M N induces an injection (kF N +1) or /  ( (kF N +1) or  n kF N +1(X 0 − 1) )  → F m V k · 1 B = F m M k where the equality comes from Lemma 3.1.19.(ii). This implies that Im ((kFN+1)‰ / ((kFN+1)‰ ∩ kFN+1(X0 − 1) ) → MB ) ⊂ F™MB Conversely, let us show the opposite inclusion. Thanks to Lemma 3.1.11(ii), we have F™™™ = FmW N · 1b = Fm-1V N (X1 − 1) · 1b. Moreover, we have by definition that Fm-1VR = (F1VR)m-1. This implies, thanks to Lemma 3.1.3 (ii) that Fm-1VB(X1 – 1) · 1B is linearly generated by elements σ(ζ1)( x 1 − 1) · · σ(ζm−1)( x m−1 − 1)( X 1 − 1) · 1 B with (ζ 1 , x 1 ), . . . , (ζm−1 , x m−1 ) ∈ μ N × F N +1 . Additionally, we have that σ(ς1)(x1 − 1) ···σ(ςm−1)(xm−1 − 1)(X1 − 1) · 1 B = (Ad σ (ς 1 ) ( x 1 ) − 1) … (Ad σ (ς 1 )…σ (ς m − 1 ) ( x m − 1 ) − 1) (Ad σ (ς 1 )…σ (ς m − 1 ) ( X 1 ) − 1) σ(ζ1)···σ(ζm–1)· 1b = (Ad σ (ζ 1 )( x 1 ) − 1) ··· (Ad σ (ζ 1 )···σ (ζ m −1)( x m −1) − 1) (Ad σ (ζ 1 )···σ (ζ m −1)( X 1 ) − 1) · 1 B which belongs to the image of (kFN+1) σι / ((kFN+1) σι ∩ kFN+1(X0 − 1) ) by the iso- morphism k F N +1/k F N +1( ∑ 0 − 1) → M N as (Adσ(ζ 1 )(x 1 ) − 1) ··· (Adσ(ζ 1 )···σ(ζ m −1)(x m −1) − 1) (Adσ(ζ 1 )···σ(ζ m −1)(X 1 ) − 1) , seen as an element of kFN+1, belongs to (kFN+1)™. This implies that F™ M N ⊂ Im ((kFN+1)‰ / ((kFN+1)‰ ∩ kFN+1(X0 − 1) ) → M N ) . Therefore, one has equality Im ((kFN+1)‰ / ((kFN+1)‰ ∩ kFN+1(¥0 − 1) ) → MB, ) = F™MB, which establishes the wanted isomorphism. Step 4: Identification of MB Thanks to Step 3, one has for any m ∈ N M B / Fm M N ~ k F N +1 / (k F N +1 ( X 0 − 1) + (k F N +1 / m ) and, on the other hand, for any m ∈ N*, kF N +1 / (kF N +1 (X 0 − 1) + (kF N +1 ) σ ′ ) ～ coker (kF N +1 / (kF N +1 ) σ − 1 → kF N +1 / (kF N +1 ) σ ′ ) , where the morphism (3.32) kF N +1/(kF N +1) m −1 → kF N +1/(kF N +1) m is induced by the endomorphism − · ( X 0 − 1) of k F N +1 . Therefore, M B   ~ lim coker (k F N +1) (k F N +1) n −1  → k F N +1/(k F N +1) n −1) THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS Step 5:  Identification of V B /V B (X 0 – 1). As in Lemma 3.1.5, one proves that the k-module isomorphism kFN+1 ® kμN → V ® given in (3.27) allows us to identify (kFN+1) σ ' ® kμN with FmV B for any m ∈ N . Recall the isomorphism (kFN+1)N → kFN+1 SkμN given in (3.28). One checks that it is compatible with the filtration of (kFN+1)N given for any m ∈ N by Π i =1 (kFN+1) π Therefore the isomorphism P 0 : (kFN+1) N → kFN+1 S kμN → V R of Step 1 is com- patible with filtrations. Therefore, it extends to a topological k-module isomorphism
On the other hand, the endomorphism f : (kFN+1)N → (kFN+1)N given in (3.29) is compatible with filtrations and then extends to a topological endomorphism f : (k F N +1) N → (k F N +1) N and, thanks to Diagram (3.30), it is such that the following diagram commutes. This induces a k-module isomorphism coker( f ) ~ coker( − · ( X 0 − 1)). Similarly to Step 1, by applying Lemma 3.1.24 with V = kF N+1 and u = − · X 0 , we obtain an isomomorphism coker( f ) ~ coker( u - id). It then follows that ′∀ ( X 0 − 1) = coker(− · ( X 0 − 1))
On the other hand, we have kF N +1/kF N +1( X 0 − 1) ≥ coker ( lim kF N +1/(kF N +1) π −1 → lim kF N +1/(kF N +1) π ), where the morphism lim k F N +1/(k F N +1) σ − 1 → lim k F N +1/(k F N +1) σ  is induced by the morphism kF N +1/(kF N +1) m −1 → kF N +1/(kF N +1) m given in (3.32). Step 6: Cokernel of limits and limit of cokernels coincide. For any m ∈ N*, the morphism
kF N +1/(kF N +1)m−1 → kF N +1/(kF N +1)m induced by the endomorphism − · ( X 0 − 1) of k F N +1 is injective. Indeed, let x ∈ kF N+1 such that x ( X 0 − 1) ∈ (kF N+1) α . Let l to be the smallest integer such that x ∈ (kFN+1)0. Let us show that l ≥ m − 1. Otherwise, since x ∈ (kFN+1)0, we have that [ x ] ∈ gr i ( k F N +1). Moreover, we have that [ X 0 − 1] ∈ gr i ( k F N +1). Therefore, [x( X 0 − 1)] ∈ gr l +1 (k F N +1).  Since, by assumption, l + 1 ≤ m − 1, the condition x( X 0 − 1) ∈ (k F N +1) π  implies that [ x ( X 0 − 1)] = 0.  Finally, since gr l+1 (k F N +1 ) is an integral domain we would obtain that [ x ] = 0, contradicting the minimality of l . Therefore, l ≥ m − 1 and the morphism (3.33) is injective. In addition, the image of morphism (3.33) is the same as the image of the morphism
YADDADEN KHALEF
kF N +1/(kF N +1) σι → kF N +1/(kF N +1) σι induced by the endomorphism − · ( ∑ 0 − 1) of kFN+1. We then have the short exact sequence {0} → kF N +1/(kF N +1)0 −1 → kF N +1/(kF N +1)0 n → coker((kF N +1/(kF N +1) ∞ → kF N +1/(kF N +1) ∞ )) → {0} which, by applying the inverse limit functor, gives us {0} → lim k F N +1/(k F N +1) m −1 → lim k F N +1/(k F N +1) m → lim coker((k F N +1/(k F N +1) n  → k F N +1/(k F N +1) n ))  → lim 1 k F N +1/(k F N +1) n −1. where lim ← ¹ is the functor given in [BK72, §IX.2.1]. Since the transition maps of the inverse system (kFN+1/(kFN+1)m-1)mEN+ are surjec- tive, this implies that lim 1 kFN+1/(kFN+1)m−1 = 0 (see, for example, [BK72, Propos- tion IX.2.4]). As a consequence,
Thanks to Step 4 and Step 5, this proves the wanted result.
Proposition-Definition 3.1.25. Let ι ∈ Emb( G ). There exists an unique topological k-module isomorphism iso M t : M B → M B R such that the following diagram
commutes. Proof. Let us construct a topological module morphism iso M,t : M,t → M,t over the topological algebra morphism iso ν : ∀ B → ∀ DR . We consider the composition
This composition sends the k-submodule V B (X 0 − 1) to 0. Indeed, this comes from the fact that (3.35) is a module morphism over the algebra morphism iso ν, t and the following computation iso V it (X 0 − 1) = g t exp ( f ∈ 0 ) − 1 = g t (exp ( f ∈ 0 ) − 1) + (g t − 1) ∈ f ∈ f ∈ r ∈ 0 + f ∈ r (g t − 1) Therefore, thanks to Proposition 3.1.23, the composition (3.35) factorises into a k- module morphism iso M,t : MR, → MR which is a module morphism over the algebra morphism iso V x : V N → V DR Next, let us show that iso M x : M x → M R is an isomorphism. Recall from Proposi- tion 3.1.15 that iso V . t : V B → WDR is an algebra submorphism of iso V . t : V B → VDR . THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 49 As a result, iso M x : M B → M B R is a module morphism over the algebra isomorphism isow. : W B → W B R . In addition, M B and M B R are both free rank 1 modules over WB and WDR respectively and iso M,r sends 1 B to 1 DR and therefore a basis of the source to a basis of the target. Thus iso M,t : M, → MP,R is a module isomorphism over iso V ; t and then a k-module isomorphism. ■ Remark 3.1.26. Let us notice that we have the following equality of k-submodules of VDR: VBR(g, exp(eo) – 1) = VBReo + VBR(g, – 1). Indeed, since we have that g, exp(eo) − 1 = g, (exp(eo) − 1) + (g, − 1), this gives us the inclusion VDR(g, exp(eo) – 1) ⊂ VDReo + VDR(g, – 1). Conversely, the inclusion VPReo + VPR(g, − 1) ⊂ VPR(g, exp(eo) − 1) follows from exp( Ne 0 ) − 1 (1 + g t exp(e 0 ) + · · · + g t N − 1 exp((N − 1)e 0 ))(g t exp(e 0 ) − 1) and from g, − 1 = exp(−eo)(g, exp(eo) − 1) + (exp(−eo) − 1) =exp(−eo)(g, exp(eo) − 1)+ exp(—en) — 1 exp(Neo) − 1 (1 + g ι exp(eo) + · · · + g N − 1 exp((N − 1)eo))(g ι exp(eo) − 1) ( exp( − e 0 ) + exp( − e 0 ) − 1 ( 1 + g t exp( e 0 ) + · · · + g t ′ N − 1 exp(( N − 1 ) e 0 ) ) (g t exp( e 0 ) − 1 ) . Proposition 3.1.27. For any (λ, Ψ) ∈ k × G(k((X))), the following pairs are iso- morphisms in the category k-alg-modtop: (i)  (iso ν · ν , iso ν · ν ) : ( (ii)  (iso W , r , iso M , r ) : (W , M , M , ) → (W , R , M , R ). Proof (i) The fact that iso ν, ν (resp. iso M, ν ) is a k-algebra (resp k-module) isomorphism follows from Proposition-Definition 3.1.8 (resp. Proposition-Definition 3.1.25). Let (a,m) ∈ V B × M B . There exists v ∈ V B such that m = v · 1 B . We have iso M , t (am) =iso M , t (av · 1 B ) = iso V , t (av) · 1 DR = iso V , t (a) iso V , t (v) · 1 DR =isoV i (a) isoM i (v · 1 B ) = isoV i (a) isoM i (m), where the second and fourth equalities come from Proposition-Definition 3.1.25. (ii) The fact that iso V , t (resp. iso M , t ) is a k-algebra (resp k-module) isomorphism follows from Proposition-Definition 3.1.15 (resp. Proposition-Definition 3.1.25). One proves, for any (w,m) ∈ W B × M B , that iso M x (wm) = iso V x (w)iso M x (m) using the argument of (i) and Proposition-Definition 3.1.15.
L
YADDADEN KHALEF
Corollary 3.1.28. Let ι ∈ Emb(G) and φ ∈ Aut(G). We have η ω  o iso M,p with η M ∈ Autk-algtop ( MDR ) given in Lemma 1.4.8.(ii). Proof.  It follows from Proposition 3.1.9 thanks to the commutativity of diagrams (3.34) and (1.39). ■ 3.2. The coproducts Â N B and Â N , 3.2.1. Comparison isomorphisms. Definition 3.2.1.  For ( t, λ, Ψ) ∈ Emb( G ) × k × G (k (( X ))), we define the topological k-algebra-module isomorphism
given by
Proposition-Definition 3.2.2. For ( t, λ, Ψ) ∈ Emb( G ) × k × G (k((X))), we define the topological k-algebra-module isomorphism
given by
It is such that the following diagrams
and
50 commute. THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 51 Proof. From Proposition-Definition 2.1.4 and Proposition 3.1.27.(ii), we have that the pairs ( ﾟ r autW.(1)  r autW.(10)  and (isoW.(, isoW.( ) are isomorphisms in k-alg-modtop; the composition is then an isomorphism in k-alg-modtop. Next, the commutativity of the diagrams follows from the commutativity of Diagrams (2.9) and (3.19) and Diagrams (2.10) and (3.34). ■ Recall the action of the group (Aut(G) × k × G (k((X))) on Emb(G) × k × G (k((X))) given in Corollary 1.4.6. One has the following result: Proposition 3.2.3.  For (φ,λ,Ψ) ∈ Aut(G)×k××G(k((X))) and (ι,ν,Φ) ∈ Emb(G)× k× × G(k((X))), we have o 「comp W ,(1), ι l aut W (1) i) 「comp」(1) φ,λ,Ψ) (ι,ν,Φ) o 「comp 「aut (ii) 「comp(φ,λ,¥) (ι,ν,Φ) Proof. (i) We have =  Гаиt ( λ,Ψ)®( ν,ηδ ( φ )) ◦ isoW,ιοφ=
where the second equality follows from Lemma 1.4.3, the third equality from Corollary 2.1.7 and Corollary 3.1.16 and the fourth one from Corollary 2.1.11.(i). (ii) We have
=  Γ aut (λ,Ψ)®(ν,ηφ(Φ))  o iso M,ιοφ=¹ M  o iso M i M,(10) o iso M,( where the second equality follows from Lemma 1.4.3, the third equality from Corollary 2.1.7 and Corollary 3.1.28 and the fourth one from Corollary 2.1.11.(ii). 3.2.2. The coproducts ÂW,B and ÂM,B Theorem 3.2.4. The composition
We denote it ( ˆ N , B , ˆ N , B ). is independent of the choice of (ι,ν,Φ) ∈ DMR x (k). Moreover, the pair ( ˆ N,B , ˆ N,B ) is an element of Copk-alg-modtop WB, MB Proof. Let ( ι,ν,Φ) and ( ι ′,ν ′,Φ′ ) ∈ DMR x (k). Thanks to Proposition 1.4.14, there exists a unique (φ, λ, Ψ) ∈ (Aut(G) × k × ) × DMRG (k) such that ( ι ′ , ι ′ , Φ′ ) = (φ, λ, Ψ) · (ι,ν,Φ). We have
where the second equality comes from Proposition 3.2.3.(ii) and the last equality from the inclusion (Aut(G) × k × ) × DMRF(k) ⊂ Stab(Aut(G)×k × )×G(k((X)))( Â M,DR)(k) of Corollary 2.2.5. Similary, we prove that 。 ▲W,DR 。 Γ'compW,(1) 三o ZW,DR o LcompW,(1) ( Γ comp W , ( 1 ) ( Γ compW , ( 1 ) by replacing M , (10) (resp. M , DR) by W , (1) (resp. W , DR) in the exponents and the use of Proposition 3.2.3.(ii) by that of Proposition 3.2.3.(i); and using the the inclusion (Aut(G) × k×) × DMRG(k) ⊂ Stab(Aut(G)×k×)×G(k((X)))( ( ´ W.DR)(k) of Corollary 2.2.5. ( ˆ∀｀∀｀, ˆ∀｀ Finally, since the pair is an element of Copk-alg-modtop WB, MB ∀ M,DR WDR, MDR thanks to Lemma is an element of Copk-alg-modtop 1.1.3 and the pair is a k-algebra-module isomorphism Гсотр W , (1) Гсотр,М,(10) thanks to Proposition-Definition 3.2.2. ■ Corollary 3.2.5. We have Δ M ,B (1b) = 1 B2 Proof. From Theorem 3.2.4, let us compute ˆ λ M,B (1 B ) by considering an element (ι, λ, Ψ) ∈ DMR x (k). First, we have Г comp ( t,λ,¥) (1b) =  Г comp ( t,λ,¥) (1) · 1dr = Г v 1 (−e 1)β(¥ 8 1) · 1dr = ¥*. (3.40) Therefore,
52 the double shuffle torsor in terms of betti and de rham coproducts 53 where the first and last equalities come from (3.40) and the second one from the fact that Ψ ∈ DMR (k). ■ Corollary 3.2.6. (i) The pair (WB, AW,B) is an object in the category k-Hopf top (ii) The pair ( MB, ÂM,B ) is an object in the category k-coalgtop (iii) The pair ( (WB, ΔW, B), (MB, ΔW, B)) is an object in the category k-HAMCtop Proof. (i) From Theorem 3.2.4, it follows that △ W , B is an algebra morphism. In addition, one checks that the coassociativity of AN,B follows from the coassociativity of Δ W DR (ii) From Theorem 3.2.4, it follows that Â M , B is a k-module morphism. In addition, one checks that the coassociativity of Δ N follows from the coassociativity of V M DR (iii) It follows from (i) and (ii) and the fact that the pair ( ˆ N ′ , ˆ N ′ B ′ ) is an element WB, MB of Copk-alg-modtop 4. EXPRESSION OF THE TORSOR DMR x (k) IN TERMS OF THE BETTI AND DE RHAM COPRODUCTS In this section, we show that DMR x (k) is a subtorsor of a stabilizer torsor of ( ∀ M,B , ∀ M,DR ) the pair of coproducts In §4.1, we define the setwise stabilizers Stabemb( G )×k××G(k((X))) ( Z M ,B , Z M ,DR ) (k) and Stabemb( G )×k××G(k((X))) ( Z W ,B , Z W ,DR ) (k) and show that they are equipped with a torsor structure for the actions of the stabilizer |͡∀ W,DR ) (k) groups Stab(Aut(G)×k×)×G(k((X))) ( ˆ M,DR ) (k) and Stab(Aut(G)×k×)×G(k((X))) respectively. In §4.2.1, we obtain a chain of inclusions of torsors involving these stabi- lizers and DMR x (k). 4.1. The stabilizer subtorsors. Definition 4.1.1. ( Δ W,B, Δ W,DR ) (k) the setwise stabilizer of (i) We denote StabEmb(G)×k××G(k((X))) the pair of coproducts ( ∂ W,B , ∂ W,DR ) E Copk-algton given by ( â StabEmb(G)×k××G(k((X))) ° ÂW,B = ÂW,DR 。 Γ compW,(1) (ι,ν,Φ) ∈ Emb(G) × k× × G(k((X)))| ( Γcomp(ι,ν,Φ) | (ii) We denote StabEmb(G)×k××G(k((X))) ( △ M,B , △ M,DR ) (k) the setwise stabilizer of the pair of coproducts ( ´∀ M,B , ˆ∀ M,DR ) ∈ Copk-modtop ( M N ) × Copk-modtop ( W BR ) □
YADDADEN KHALEF
given by StabEmb(G)×k××G(k((X))) ( ∀ M,B , ∀ M,DR ) (k) := ( ι,ν,Φ) ∈ Emb(G ) × k × G(k((X))) | ( Γ comp,Μ,(10) | ∇ 2 。 ∇ M,B = ∇ M,DR o Γ comp,Μ,(10) ( ∂∀ , ∀∀ , ∀∀ , ∀∀ ) (k) Remark 4.1.2. Theorem 3.2.4 implies that StabEmb(C)×k××G(k((X))) and StabEmb(G)×k××G(k((X))) ( ˜ M.B , ˜ M.DR ) (k) contain DMR x (k), which implies that these are nonempty sets. Proposition 4.1.3. i. The pair (Stab(Aut(G)×k×)×G(k((X))) ( ÂG,DR ) (k), StabEmb(G)×k××G(k((X))) ( ÂW,B,ÂW,DR ) (k) is a subtorsor of ((Aut(G) × k×) × G(k((X))),Emb(G) × k× × G(k((X)))). ii. The pair (Stab(Aut(G)×k×)×G(k((X))) ( ˆ △M,DR) (k), StabEmb(G)×k××G(k((X))) ( ˆ △M,B, ˆ △M,DR) (k) is a subtorsor of ((Aut(G) × k×) × G(k((X))),Emb(G) × k× × G(k((X)))). In order the prove this, we will need the following Lemma: Lemma 4.1.4 ([EF2, Lemma 2.6]). Let (H,T) be a torsor, and let V,V' be k-modules. Let ρ : H → Autk-mod(V) be a group morphism and let ρ ′ : T → Isok-mod(V ′ , V) be a map such that for any h ∈ H, x ∈ T, one has ρ′(h · x ) = ρ(h) ◦ ρ′(x). Let ν ∈ V and ν′ ∈ V′.  Then StabH(ν) := {h ∈ H |ρ(h)(ν) = ν} is a subgroup of H , and either Stab τ( v, v ′ ) := { x ∈ T | p ( v ′ ) = v } is empty, or (Stab H ( v ), Stab τ( v, v ′ )) is a subtorsor of (H,T). Proof of Proposition 4.1.3. It follows from Lemma 4.1.4 by setting : ● (H,T) = ((Aut(G) × k × G(k((X))),Emb(G) × k × G(k((X)))); ● V = Copk-mod(WBR) (resp. V = Copk-mod(MBR)); ● V' = Copk-mod(WB) (resp. V' = Copk-mod(MB)); ● v = ˆ ∀ ∀ , DR  (resp. v = ˆ ∀ ∀ , DR); ● v' = ˆ ν , B (resp. v' = ˆ ν , B ); (resp.
54 THE DOUBLE SHUFFLE TORSOR IN TERMS OF BETTI AND DE RHAM COPRODUCTS 55
follows from Proposition 3.2.3.
4.2. Inclusion of stabilizer torsors. Theorem 4.2.1. We have the following inclusions of torsors ((Aut(G) × k × ) × DMR (k), DMR × (k))
In order the prove this, we will need the following Lemmas: Lemma 4.2.2 ([EF2, Lemma 2.3]). Let (H,T) be a torsor and let (H',T') and (H',T'') be subtorsors of (H,T) such that T ′ ∩ T ′′̸= ∅ . Then (H ′ ∩ H ′′ ,T ′ ∩ T ′′ ) is a subtorsor of both (H ′ ,T ′ ) and (H ′ ,T ′′ ), therefore of (H ,T ′ ). Lemma 4.2.3 ([EF2, Lemma 2.7]). Let (H,T) be a torsor and let (H 0 ,T 0 ) and (H 1 ,T 1 ) be subtorsors of (H,T) such that T 0 ⊂ T 1 . Then (H 0 ,T 0 ) is a subtorsor of (H 1 ,T 1 ). If, moreover, H 0 = H 1 then the subtorsors (H 0 , T 0 ) and (H 1 , T 1 ) are equal. Proof of Theorem 4.2.1. The group-part inclusion is shown in Corollary 2.2.5. The first and last set-part inclusions are immediate. It remains to show that Stabemb( G )×k××G(k((x))) ( Â M ,B , Â M ,DR ) (k) ⊂ Stabemb( G )×k××G(k((x))) ( Â W ,B , Â W ,DR ) (k). In Lemmas 4.2.2 and 4.2.3, set (H,T) = ((Aut(G) × k × G(k((X)),Emb(G) × k × G(k((X)))) . First, let us apply Lemma 4.2.2 for ● (H',T') = (Stab(Aut(G)×k×)×G(k((X)) ( Z M,DR ) (k), StabEmb(G)×k××G(k((X)) ( Z M,B, Z M,DR ) (k)); ● (H",T") = (Stab(Aut(G)×k×)×G(k((X)) ( ZW,DR) (K),StabEmb(G)×k××G(k((X))) ( ZW,B,ZW,DR) (k)). From Remark 4.1.2, we have that T ′ ∩ T ′̸≠ ∅. Therefore, ( H ′ ∩ H ′ , T ′ ∩ T ′ ′ ) is a subtorsor of ( H ′′ , T ′′ ). Second, let us apply Lemma 4.2.3 for ● (H 0 , T 0 ) = (H ′ ∩ H ′ , T ′ ∩ T ′′ );
We have that To = T' ∩ T'' ⊂ T' = T 1 . In addition, H 0 = H ′ ∩ H ′ = H ′ = H 1 , where the second equality follows from the stabilizer group inclusion in Corollary 2.2.5. Finally, it follows that T ′ ∩ T ′′ = T 0 = T 1 = T ′′ . Thus T ′ ⊂ T ′′ , which is the wanted inclusion of setwise stabilizers. ■