A moperadic approach to cyclotomic associators
aa r X i v : . [ m a t h . QA ] A p r A MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS
DAMIEN CALAQUE AND MARTIN GONZALEZ
Abstract.
This is a companion paper to
Ellipsitomic associators [4]. We provide a(m)operadic description of Enriquez’s torsor of cyclotomic associators, as well as of itsassociated cyclotomic Grothendieck–Teichm¨uller groups.
Contents
Introduction 11. Moperads 32. Reminders on associators and G(R)T 53. Parenthesized braids with a frozen strand 104. The moperad of twisted parenthesized braids, and cyclotomic GT 135. The moperad of N -chord diagrams, and cyclotomic associators 24List of notation 35References 36 Introduction
Since the introduction of associators and Grothendieck–Teichm¨uller groups by Drinfeld [5],several variations of these have been considered; for instance ● following Grothendieck’s esquisse [10], Lochak–Nakamura–Schneps defined a new ver-sion of the Grothendieck–Teichm¨uller group [14], which acts on more general surfacemapping class groups than Drinfled’s original one; ● cyclotomic [6] and elliptic [7] variants of associators and Grothendieck–Teichm¨ullergroups were dicovered by Enriquez; ● ellipsitomic associators, which share both the features of cyclotomic and elliptic asso-ciators, have recently been introduced by the authors [4].It is known, after the original insight of Drinfeld [5] and Bar-Natan [2], and thanks tothe recent detailed proof of Fresse [9], that the torsor of associators can be understood asthe torsor of isomorphisms between two operads in groupoids. A similar result holds forEnriquez’s torsor of elliptic associators as well, as was recently proven in [4], where one hasto consider operadic modules instead of just operads. This need comes from the fact that,while compactified configuration spaces of points in the plane form an operad, compactifiedconfiguration spaces of points in a torus form an operadic module on the latter. Still in [4], ellipsitomic associators are defined as operadic module isomorphisms, and the description `ala Drinfeld is derived from it afterwards.In this companion paper to [4], we prove that Enriquez’s cyclotomic associators torsor(resp. Grothendieck–Teichm¨uller groups) can also be indentified with isomorphisms (resp. au-tomorphisms) of operadic gadgets. The appropriate notion here is the one of a moperad;it was introduced by Willwacher [18], and it typically encodes the structure of compactifiedconfiguration spaces of points in the punctured plane (or, equivalently, the annulus).After two reminders on moperads (Section 1) on the one hand, and associators (Section 2)on the other hand, we introduce in Section 3 the moperad PaB of parenthesized braids witha frozen strand (obtained as the fundamental groupoid of the configuration moperad of pointsin the punctured plane) and provide a generators and relations presentation for PaB : Theorem (Theorem 3.4) . The moperad in groupoid
PaB is generated by an arity arrow E and an arity arrow Ψ , with relations (cU) , (MP) , (RP) , and (O) . Unsurprisingly, these relations are completely analogous to the axioms for braided modulecategories from [3]; indeed, one can verify that a braided module category is nothing but arepresentation of
PaB in categories. In Section 4, we decorate the unfrozen strands of ourparenthesized braids with elements from a finite quotient Γ = Z / N Z of the fundamental groupof the punctured plane, giving rise to a moperad in groupoids PaB Γ . We show that PaB Γ admits a presentation by generators and relations similar to the one of PaB (Theorem 4.6),and thus identify the group of Γ-equivariant automorphisms of PaB Γ that are the identityon objects with Enriquez’s cyclotomic Grothendieck–Teichm¨uller group (Proposition 4.10).Finally, in Section 5, we put a moperad structure on the (parenthesized) horizontal N -chorddiagrams of [3], and prove the following Theorem (Theorem 5.5) . The set of Γ -equivariant moperad isomorphisms that are the iden-tity on objects between PaB Γ and parenthesized N -chord diagrams is in bijection with En-riquez’s cyclotomic associators. We moreover show that this identification respects the (bi)torsor structures (Theorem 5.13).
Acknowledgements.
We are deeply grateful to Adrien Brochier for numerous conversationsand suggestions. Discussions with Benjamin Enriquez have also been very helpful. Thefirst author has received funding from the Institut Universitaire de France, and from theEuropean Research Council (ERC) under the European Union’s Horizon 2020 research andinnovation programme (Grant Agreement No. 768679). This paper is part of the secondauthor’s doctoral thesis [12] at Sorbonne Universit´e, and part of this work has been done whilethe second author was visiting the Institut Montpelli´erain Alexander Grothendieck, thanksto the financial support of the Institut Universitaire de France. The second author warmlythanks the Max-Planck Institute for Mathematics in Bonn and Universit d’Aix-Marseille, fortheir hospitality and excellent working conditions.
Convention.
All along the paper, k is a field of characteristic zero. MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 3 Moperads
In this section we fix a symmetric monoidal category (C , ⊗ , ) having small colimits andsuch that ⊗ commutes with these. We borrow the notation and conventions for S -modulesand operads from [4].1.1. Moperads over an operad.
Let O be an operad. A moperad over an operad O is an S -module P carrying ● a unital monoid structure for the monoidal product ⊗ , ● and a left O -module structure for the monoidal product ○ , that are compatible in thefollowing sense: – One first observes that there is a natural map (O ○ P) ⊗ Q → O ○ (P ⊗ Q) . – Then the compatibility means that the following diagram commutes: (O ○ P) ⊗ P / / (cid:15) (cid:15) P ⊗ P ! ! ❉❉❉❉❉❉❉❉❉ O ○ (P ⊗ P) / / O ○ P / / P The map (O ○ P) ⊗ P → P one obtains decomposes into maps P( k ) ⊗ P( m ) ⊗ O( m ) ⊗ ⋯ ⊗ O( m k ) → P( m + ⋯ + m k ) satisfying certain associativity, unit and S -equivariance relations. We let the reader spell outthese conditions explicitely.We leave it as an exercise to check that, within the symmetric monoidal category of dif-ferential graded vector spaces, this definition coincides with Willwacher’s one from [18] (fromwhich we borrowed the name “moperad”). Note that the monoid structure for the monoidalproduct ⊗ encodes precisely the partial composition with respect to the second colour. Wewill denote this partial composition by ○ .1.2. Example of a moperad over an operad: coloured Stasheff polytopes.
To anyfinite set I we associate the configuration spaceConf ( R > , I ) = { x = ( x i ) i ∈ I ∈ ( R > ) I ∣ x i ≠ x j if i ≠ j } and its reduced version C ( R > , I ) ∶= Conf ( R > , I )/ R > . The Axelrod–Singer–Fulton–MacPherson compactification C ( R > , I ) of C ( R > , I ) is a disjointunion of ∣ I ∣ -th Stasheff polytopes with two kinds of colours, indexed by S I . The boundary ∂ C ( R > , I ) ∶= C ( R > , I ) − C ( R > , I ) is the union, over all partitions I = J ∐ J ∐ ⋯ ∐ J k , of ∂ J , ⋯ ,J k C ( R > , I ) ∶= C ( R > , k ) × C ( R > , J ) × k ∏ i = C ( R , J i ) . DAMIEN CALAQUE AND MARTIN GONZALEZ
The inclusion of boundary components provides C ( R > , −) with the structure of a C ( R , −) -moperad in topological spaces.One can see that C ( R > , I ) is a manifold with corners, and that considering only zero-dimensional strata of our configuration spaces we get a sub-moperad Pa ⊂ C ( R > , −) thatcan be shortly described as follows: ● Pa ( I ) is the set of pairs ( σ, p ) with σ is a linear order on I and p a maximalparenthesization of ⎛⎜⎜⎝ ●⋯●± ∣ I ∣ times ⎞⎟⎟⎠ such that there is no action of S n on 0, but this elementcan be inside a parenthesis. This means that we allow points to be near the origin. ● The C ( R , −) -moperad structure is given by substitution as above.Forgetting the C ( R , −) -moperad structure on C ( R > , −) and considering a C ( R , −) -modulestructure on it amounts to forbidding points to be close to the origin (i.e. the 0-strand cannotbe inside a parenthesis in this case).1.3. Pointing.
Recall the operad
U nit defined by
U nit ( n ) ∶= ⎧⎪⎪⎨⎪⎪⎩ if n = , ∅ elseBy convention, all our operads O will be pointed in the sense that they will come equipped witha specific operad morphism U nit → O . Morphisms of operads are required to be compatiblewith this pointing. Actually, all operads appearing in this paper are such that O( n ) ≃ if n = , M U nit over
U nit , which is such that
M U nit ( n ) = for all n ≥
0. By convention, all our moperads will be pointed, in the sense that they will comeequipped with a specific
U nit -moperad morphism
M U nit → Q . Morphisms of moperads arerequired to be compatible with the pointing. Remark 1.1.
In the category of sets,
M U nit is the sub-
U nit -moperad of Pa that consistsonly of the left-most maximal parenthesization.The main reason for these rather strange conventions is that we need the following features,that we have in the case of compactified configuration spaces: ● For operads and moperads, we want to have “deleting operations” O( n ) → O( n − ) that decrease arity. ● For a moperad, we want to be able to “see the operad inside” it, i.e. we want to havea distinguished morphism O → P of S -modules. Example 1.2.
For instance, being a Pa -moperad, Pa comes together with a morphism of S -modules Pa → Pa . We let the reader check that it sends a parenthesized permutation p to 0 ( p ) . MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 5
Group actions.
Let G be a group and O be an operad. We say that an O -module P carries a G -action if ● for every n ≥ G n acts S n -equivariantly on P( n ) , from the left. ● for every m ≥ n ≥
0, and 1 ≤ i ≤ n , the partial composition ○ i ∶ P( n ) ⊗ O( m ) Ð→ P( n + m − ) is equivariant along the group morphism G n Ð→ G n + m − ( g , . . . , g n ) z→ ( g , . . . , g i − , g i , . . . , g i ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ m times , g i + , . . . , g n ) If P is a moperad, we additionally require that the partial composition ○ ∶ P( n ) ⊗ P( m ) Ð→ P( n + m ) is G n + m -equivariant.A morphism P → Q of O -moperads with G -action is said G -equivariant if, for every n ≥ P( n ) → Q( n ) is G n -equivariant.2. Reminders on associators and G(R)T
In this Section we recollect some results from [5, 2, 9], following essentially the presentationof [4, Section2].2.1.
Compactified configuration space of the plane.
To any finite set I we associate the(reduced) configuration spaceC ( C , I ) ∶= { z = ( z i ) i ∈ I ∈ C I ∣ z i ≠ z j if i ≠ j }/ C ⋊ R > of points in the plane. We then consider its Axelrod–Singer–Fulton–MacPherson compactifi-cation ¯C ( C , I ) , whose boundary ∂ C ( C , I ) = C ( C , I ) − C ( C , I ) is made of irreducible components ∂ J , ⋯ ,J k C ( C , I ) indexed by partitions I = J ∐ ⋯ ∐ J k of I : ∂ J , ⋯ ,J k C ( C , I ) ≅ C ( C , k ) × k ∏ i = C ( C , J i ) . The inclusion of boundary components provides C ( C , −) with the structure of an operad intopological spaces. DAMIEN CALAQUE AND MARTIN GONZALEZ
The operad of parenthesized braids.
The inclusions of topological operads Pa ⊂ C ( R , −) ⊂ C ( C , −) allows us to define PaB ∶= π ( C ( C , −) , Pa ) , which is an operad in groupoids. Example 2.1 (of arrows in small arity) . Recall from [4, Examples 2.1] that, in arity two,there is an arrow from ( ) to ( ) , we have an arrow R , going from ( ) to ( ) , that canbe depicted in the following ways:12 21 21There is another arrow ˜ R , ∶= ( R , ) − , having the same source and target, that can bedepicted as an undercrossing braid.In arity three, there is an arrow Φ , , , going from ( ) ( ) , that can be depicted inthe following ways: (
11 2 ) ( ) PaB . This was made more explicit by Bar-Natan [2] in a different language, and writtenin term of operads by Fresse [9, Theorem 6.2.4]. Following the convention and notation from[4, Section 2], it reads as follows
Theorem 2.2.
As an operad in groupoids having Pa as operad of objects, PaB is freelygenerated by R ∶= R , and Φ ∶= Φ , , together with the following relations: Φ ∅ , , = Φ , ∅ , = Φ , , ∅ = Id , ( in Hom PaB ( ) ( , )) , (U) R , Φ , , R , = Φ , , R , Φ , , ( in Hom PaB ( ) (( ) , ( ))) , (H1) ˜ R , Φ , , ˜ R , = Φ , , ˜ R , Φ , , ( in Hom PaB ( ) (( ) , ( ))) , (H2) Φ , , Φ , , = Φ , , Φ , , Φ , , ( in Hom PaB ( ) ((( ) ) , ( ( )))) . (P)In order to fix our braid group conventions we recall the following. The automorphismgroup Aut PaB ( n ) ( p ) of any parenthesized permutation p of lenght n is exactly the pure braidgroup PB n on n strands, which is generated by elementary pure braids x ij , 1 ≤ i < j ≤ n , MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 7 which satisfy a certain list of relations (see [4] for more details). In this article we will depictthe generator x ij in the following two equivalent ways:11 ii ...... jj nn ←→ ∢ i j n The group PB n is the kernel of the map B n → S n sending, for all 1 ≤ i ≤ n −
1, the generators σ i of B n to the permutation ( i, i + ) . The elements σ i are depicted in the same way as theR’s: ii + i + i i + i The operad of (parenthesized) chord diagrams.
Recall [16, 9] that the collectionof Kohno–Drinfeld Lie k -algebras t n ( k ) is provided with the structure of an operad in thecategory grLie k of positively graded finite dimensional Lie algebras over k , with symmetricmonoidal structure is given by the direct sum ⊕ . This is equivalent to Bar-Natan’s cablingoperation [2] on chord diagrams.Taking the degree completion of the universal enveloping algebra functor, we get an operad CD ( k ) ∶= ˆ U( t ( k )) in complete filtered cocommutative Hopf algebras which we view as cate-gories (with only one object) enriched in complete filtered cocommutative coalgebras, that wecall the operad of chord diagrams .By definition, the operad Ob ( CD ( k )) of object of CD ( k ) is the terminal operad in sets.We can thus define the PaCD ( k ) , that we call the operad of parenthesized chord diagrams asthe fake pull-back of CD ( k ) along the terminal morphism Pa → ∗ = Ob ( CD ( k )) . We refer to[4] for the definition of fake pull-back; it is enough to know that PaCD ( k ) has Pa as operadof objects, and that in arity n the complete filtered cocommutative coalgebra of morphismsbetween any pair of objects is always ˆ U( t n ( k )) .2.4. Drinfeld associators.
Recall that ● Grpd k denote the (symmetric monoidal) category of k -prounipotent groupoids. ● for C being Grpd , Grpd k , or Cat ( CoAlg k ) (see [4]), the notationAut + Op C ( resp . Iso + Op C ) refers to those automorphisms (resp. isomorphisms) which are the identity on objectswithin the category Op C of operads in C . DAMIEN CALAQUE AND MARTIN GONZALEZ A Drinfeld k -associator is an isomorphism between the operads ̂ PaB ( k ) and G PaCD ( k ) in Grpd k , which is the identity on objects. We denote by Assoc ( k ) ∶= Iso + Op Grpd k (̂ PaB ( k ) , G PaCD ( k )) the set of k -associators. Drinfeld already implicitely showed in [5] that there is a one-to-one correspondence between the set of Drinfeld k -associators and the set Ass ( k ) of couples ( µ, ϕ ) ∈ k × × exp ( ˆ f ( k )) , such that ● ϕ , , = ( ϕ , , ) − in exp ( ˆ t ( k )) , ● ϕ , , e µt / ϕ , , e µt / ϕ , , e µt / = e µ ( t + t + t )/ in exp ( ˆ t ( k )) , ● ϕ , , ϕ , , ϕ , , = ϕ , , ϕ , , in exp ( ˆ t ( k )) ,where ϕ , , = ϕ ( t , t ) is viewed as an element of exp ( ˆ t ( k )) via the inclusion ˆ f ( k ) ⊂ ˆ t ( k ) sending x to t and y to t . The proof of this result relies on the universal property of PaB from Theorem 2.2. In particular, a morphism F ∶ ̂ PaB ( k ) Ð→ G PaCD ( k ) is uniquelydetermined by a scalar parameter µ ∈ k and ϕ ∈ exp ( ˆ f ( k )) such that we have the followingassignment in the morphism sets of the parenthesized chord diagram operad PaCD : ● F ( R , ) = e µt / X , , ● F ( Φ , , ) = ϕ ( t , t ) a , , ,where R and Φ are the ones from Examples 2.1.An example of such an associator is the KZ associator Φ KZ . It is defined as the the renor-malized holonomy from 0 to 1 of G ′ ( z ) = ( t z + t z − ) G ( z ) , i.e., Φ KZ ∶= G + G − − ∈ exp ( ˆ t ( C )) ,where G + , G − are the solutions such that G + ( z ) ∼ z t when z → + and G − ( z ) ∼ ( − z ) t when z → − . We have that ( π i , Φ KZ ) is an element of Ass ( C ) .2.5. Grothendieck–Teichm¨uller group.
The
Grothendieck–Teichm¨uller group is definedas the group GT ∶ = Aut + Op Grpd ( PaB ) of automorphisms of the operad in groupoids PaB which are the identity of objects and its k -pro-unipotent version is ̂ GT ( k ) ∶ = Aut + Op Grpd k ( ̂ PaB ( k )) . In this article we will focus on the k -pro-unipotent version of this group in the cyclotomicsituation. The group ̂ GT ( k ) is isomorphic to Drinfeld’s Grothendieck–Teichm¨uller group ̂ GT ( k ) consisting of pairs ( λ, f ) ∈ k × × ̂ F ( k ) which satisfy the following equations: ● f ( x, y ) = f ( y, x ) − in ̂ F ( k ) , ● x ν f ( x , x ) x ν f ( x , x ) x ν f ( x , x ) = ̂ F ( k ) , ● f ( x x , x ) f ( x , x x ) = f ( x , x ) f ( x x , x x ) f ( x , x ) in ̂ PB ( k ) ,where x , x , x are 3 variables subject only to x x x = ν = λ − , and x ij is the elementarypure braid from Subsection 2.2. The multiplication law is given by ( λ , f )( λ , f ) = ( λ λ , f ( x λ , f ( x, y ) y λ f ( x, y ) − ) f ( x, y )) . MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 9
One obtains the couple ( λ, f ) from an automorphism F ∈ ̂ GT ( k ) , for λ = ν + ● F ( R , ) = ( R , R , ) ν R , , ● F ( Φ , , ) = f ( x , x ) Φ , , .2.6. Graded Grothendieck–Teichm¨uller group.
The graded Grothendieck–Teichm¨ullergroup is the group
GRT ( k ) ∶ = Aut + Op Grpd k ( G PaCD ( k )) of automorphisms of G PaCD ( k ) that are the identity on objects.Again, the operadic definition of GRT ( k ) happens to coincide with the one originaly givenby Drinfeld. Denote by GRT the set of elements in g ∈ exp ( ˆ f ( k )) ⊂ exp ( ˆ t ( k )) such that ● g , , = g − and g , , g , , g , , =
1, in exp ( ˆ t ( k )) , ● t + Ad ( g , , )( t ) + Ad ( g , , )( t ) = t + t + t , in ˆ t ( k ) , ● g , , g , , g , , = g , , g , , , in exp ( ˆ t ( k )) ,One has the following multiplication law on GRT : ( g ∗ g )( t , t ) ∶ = g ( t , Ad ( g ( t , t ))( t )) g ( t , t ) . Drinfeld showed in [5] that the above GRT is stable under ∗ , that it defines a group structureon it, and that rescaling transformations g ( x, y ) ↦ λ ⋅ g ( x, y ) = g ( λx, λy ) define an action of k × of GRT by automorphisms and we denote GRT ( k ) the corresponding semi-direct product.Then, as was shown in [9], the group GRT ( k ) is isomorphic to GRT ( k ) . In particular, weobtain the couple ( λ, g ) from an automorphism G ∈ GRT ( k ) by the assignment ● G ( X , ) = X , , ● G ( H , ) = e λt H , , ● G ( a , , ) = g ( t , t ) a , , .2.7. Bitorsor structure.
Recall first that there is a free and transitive left action of ̂ GT ( k ) on Ass ( k ) , defined, for ( λ, f ) ∈ ̂ GT ( k ) and ( µ, ϕ ) ∈ Ass ( k ) , by (( λ, f ) ∗ ( µ, ϕ ))( t , t ) ∶ = ( λµ, f ( e µt , Ad ( ϕ ( t , t ))( e µt )) ϕ ( t , t )) , where Ad ( f )( g ) ∶ = f gf − , for any symbols f, g .Recall also that there is a free and transitive right action of GRT ( k ) on Ass ( k ) defined asfollows: for ( λ, g ) ∈ GRT ( k ) and ( µ, ϕ ) ∈ Ass ( k ) , given by (( µ, ϕ ) ∗ ( λ, g ))( t , t ) ∶ = ( λµ, ϕ ( λt , Ad ( g )( λt )) g ( t , t )) . These two action commute with each other, and turn (̂ GT ( k ) , Ass ( k ) , GRT ( k )) into abitorsor. By its very definition, the triple ( ̂ GT ( k ) , Assoc ( k ) , GRT ( k )) is also a bitorsor,and it is proven in [9] that the above identifications from subsections 2.4, 2.5, and 2.6, can bepromoted to a bitorsor isomorphism(1) ( ̂ GT ( k ) , Assoc ( k ) , GRT ( k )) Ð→ (̂ GT ( k ) , Ass ( k ) , GRT ( k )) . Parenthesized braids with a frozen strand
Compactified configuration space of the annulus.
For each finite set I , let usconsider the (reduced) configuration space of C × :C ( C × , I ) ∶ = { z = ( z i ) i ∈ I ∈ ( C × ) I ∣ z i ≠ z j , ∀ i ≠ j } / R > . We clearly have an isomorphism between C ( C × , n ) and C ( C , n + ) . We then consider theAxelrod–Singer–Fulton–MacPherson compactification C ( C × , n ) of C ( C × , n ) . The boundary ∂ C ( C × , n ) = C ( C × , n ) − C ( C × , n ) is made of the following irreducible components: for any partition [[ , n ]] = J ∐ ⋯ ∐ J k suchthat 0 ∈ J m , for some 0 ≤ m ≤ k , there is a component ∂ J , ⋯ ,J k C ( C × , n ) ≅ C ( C × , k ) × C ( C × , J m ) × k ∏ i = i ≠ m C ( C , J i ) . The PaB-moperad of parenthesized braids with a frozen strand.
We have inclu-sions of topological moperads Pa ⊂ C ( R > , − ) ⊂ C ( C × , − ) over Pa ⊂ C ( R , − ) ⊂ C ( C , − ) . We then define
PaB ∶ = π ( C ( C × , − ) , Pa ) , which is a moperad over the operad in groupoids PaB . Example 3.1 (Description of
PaB ( ) ) . First, observe that C ( C × , ) ≃ C ( C , ) ≃ S . More-over, Pa = {( )} . Hence PaB ( ) ≃ Z : it has only one object ( ) and is freely generatedby an automorphism E , of ( ) , which can be depicted as an elementary pure braid:0 10 1 0 1Two incarnations of E , Example 3.2 (Notable arrow in
PaB ( ) ) . Let us first recall that Pa ( ) = S × {( ●● ) ● , ● ( ●● )} and that C ( R > , ) ≅ S × [ , ] . Hence we have an arrow Ψ , , (the identity path in [ , ] )from ( ) ( ) in PaB ( ) , which can be depicted as follows: (
00 1 ) ( ) , , Remark 3.3.
Recall from § PaB -moperad,
PaB comes together with amorphism of S -modules PaB → PaB . In pictorial terms, this morphism sends a parentesizedbraid with n strands to a parenthesized braid with n + R , (a morphism in PaB ( ) ) and ofΦ , , (a morphism in PaB ( ) ) can be respectively depicted as follows:00 ( ( ) ) (( ( ) ( ) )) We will still denote these images by R , and Φ , , .3.3. The universal property of PaB . Our main goal in this § is to prove the followinggenerator and relation presentation of PaB . Theorem 3.4.
As a
PaB -moperad having Pa as Pa -moperad of objects, PaB is freelygenerated by E ∶ = E , ∈ PaB ( ) and Ψ ∶ = Ψ , , ∈ PaB ( ) together with the followingrelations: Ψ , ∅ , = Ψ , , ∅ = Id ( in Hom
PaB ( ) ( , )) , (cU)Ψ , , Ψ , , = Ψ , , Ψ , , Φ , , ( in Hom
PaB ( ) ((( ) ) , ( ( )))) , (MP)Ψ , , E , ( Ψ , , ) − = E , E , ( in Hom
PaB ( ) (( ) , ( ) ))) , (RP) E , = Ψ , , R , ( Ψ , , ) − E , Ψ , , R , ( Ψ , , ) − ( in Hom
PaB ( ) (( ) , ( ) )) . (O) Proof.
Let Q be the PaB -moperad with the above presentation. From Examples 3.1 and3.2 we deduce that, as a
PaB -moperad in groupoid,
PaB contains two morphisms E , (in PaB ( ) ) and Ψ , , (in PaB ( ) ). One easily shows, using the following pictures, that theysatisfy mixed pentagon and octogon relations, (MP) and (O), and relation (RP): (( ) ) ( ( )) = (( ) ) ( ( )) (MP) ( ( ) ) = ( ( ) ) ( ( ) ) = ( ( ) ) Q , there is a morphism of PaB -moperads Q → PaB , which is the identity on objects. In order to show that this is an isomorphism, itsuffices to show that it is an isomorphism at the level of automorphism groups of an objectarity-wise because all groupoids involved are connected. Let n ≥
0, and let p be the object ( ⋯ ( ) ⋯⋯ ) n of Q ( n ) and PaB ( n ) . We want to show that the induced group morphismAut Q ( n ) ( p ) Ð→ Aut
PaB ( n ) ( p ) = π ( ¯C ( C × , n ) , p ) is an isomorphism.On the one hand, we can replace the base-point p with p reg = ( , , . . . , n ) ∈ C ( C × , n ) , asthey are in the same path-connected component. Moreover, since the Axelrod–Singer–Fulton–MacPherson compactification does not change the homotopy type of our configuration spaces,we get an isomorphism π ( ¯C ( C × , n ) , p ) ≃ π ( C ( C × , n ) , p reg ) . On the other hand, in [6, § ● Given a braided module category M over a braided monoidal category C , an object X of C , and an object M of M , there is a group morphismB n → Aut M ( M ⊗ X ⊗ n ) , MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 13 where, by convention, M ⊗ X ⊗ n comes equipped with the left-most parenthesization (( M ⊗ X ) ⊗ ... ) ⊗ X , and B n = B n + × S n + S n is generated by elements σ i , for 1 ≤ i ≤ n − τ . Seen in B n + with generators σ , . . . σ n − , we have τ = σ = x . ● There is a universal braided module category
PaB ,Enr generated by a single object0, over the universal braided monoidal category PaB
Enr generated by a single object ● . Hence objects of PaB ,Enr are parenthesizations of 0 ● ⋯● , and thus p determinesan object (which we abusively still denote p ). ● the morphism B n → Aut
PaB ,Enr ( p ) is an isomorphism.One can moreover see that, by construction, Aut Q ( n ) ( p ) is exactly the kernel subgroupker ( Aut
PaB ,Enr ( n ) ( p ) → S n ) ≃ PB n + . Hence we have a commuting diagramPB n ≃ / / (cid:15) (cid:15) Aut Q ( n ) ( p ) / / (cid:15) (cid:15) π ( C ( C × , n ) , p ) (cid:15) (cid:15) π ( C ( C × , n ) , p reg ) ≃ o o (cid:15) (cid:15) B n ≃ / / (cid:15) (cid:15) Aut
PaB ,Enr ( p ) / / (cid:15) (cid:15) π ( C ( C × , n )/ S n , [ p ]) (cid:15) (cid:15) π ( C ( C × , n )/ S n , [ p reg ]) ≃ o o (cid:15) (cid:15) S n S n S n S n where all vertical sequences are short exact sequences. Thus, in order to get that the mapAut Q ( n ) ( p ) → π ( C ( C × , n ) , p ) is an isomorphism, we are left to prove that the composite mapB n Ð→ π ( C ( C × , n ) , p reg ) is indeed an isomorphism. But this map is, by its very construction,the isomorphism (from [15, 17]) exhibiting a presentation by generators and relations of thebraid group of a handlebody. (cid:3) The moperad of twisted parenthesized braids, and cyclotomic GT
Compactified twisted configuration space of the annulus.
Consider, for N ≥ = Z / N Z . To every finite set I let us associate the so-called Γ- twistedconfiguration space Conf ( C × , I, Γ ) = { z = ( z i ) i ∈ I ∈ ( C × ) I ∣ z i ≠ ζz j , ∀ i ≠ j, ∀ ζ ∈ µ N } ( µ N is the set of complex N th roots of unity) and its reduced versionC ( C × , I, Γ ) ∶ = Conf ( C × , I, Γ )/ R > . Remark 4.1.
Observe that Conf ( C × , I, Γ ) , resp. C ( C × , I, Γ ) , is an Γ I -covering space ofConf ( C × , I ) , resp. C ( C × , I ) , the covering map being given by ( z i ) i ∈ I ↦ ( z Ni ) i ∈ I .There are also inclusionsConf ( C × , I, Γ ) ↪ Conf ( C × , I × µ N ) and C ( C × , I, Γ ) ↪ C ( C × , I × µ N ) given by ( z i ) i ∈ I ↦ ( ζz i ) ( i,ζ )∈ I × µ N . This allows us to define the compactification C ( C × , I, Γ ) ofC ( C × , I, Γ ) , as the closure of C ( C × , I, Γ ) inside C ( C × , I × µ N ) . The irreducible components of its boundary ∂ C ( C × , I, Γ ) = C ( C × , I, Γ ) − C ( C × , I, Γ ) can be described as follows. For anarbitrary partition J ∐ ⋯ ∐ J k of { } ⊔ I there is a component ∂ J , ⋯ ,J k C ( C × , I, Γ ) ≅ C ( C × , k, Γ ) × C ( C × , J m , Γ ) × k ∏ i = i ≠ m C ( C , J i ) , where m ∈ { , . . . , k } is the index such that 0 ∈ J m . The inclusion of boundary componentssuch that m = ( C × , − , Γ ) with the structure of a moperad over the operad C ( C , − ) in topological spaces.We let the reader check that the covering map C ( C × , I, Γ ) → C ( C × , I ) from Remark 4.1extends to a continuous map φ n ∶ C ( C × , I, Γ ) → C ( C × , I ) between their compactifications, andthus leads to a morphism of moperads.Finally, one observes that the natural action of Γ I on each C ( C × , I × µ N ) , given by ( α ⋅ z ) ( j,ζ ) ∶ = z ( j,e − παjN ζ ) induces an action of Γ on the moperad C ( C × , − , Γ ) , in the sense of § The Pa-moperad of labelled parenthesized permutations.
Borrowing the notationfrom the previous subsection, we define Pa Γ0 ( n ) ∶ = φ − n ( Pa ( n )) . Explicitly, Pa Γ0 ( n ) is the setof parenthesized permutations of { , , . . . , n } that fix 0 and that are equipped with a label { , . . . , n } → Γ. Notation.
As a matter of notation, we will write the label as an index attached to each1 , . . . , n . For instance, ( β ) α belongs to Pa Γ0 ( ) for every α, β ∈ Γ.Observe that the S -module (in sets) Pa Γ0 carries the structure of a Pa -moperad. Indeed, itis a fiber product Pa Γ0 = Pa × C ( C × , −) C ( C × , − , Γ ) in the category of Pa -moperads (in topological spaces). Here are two self-explanatory examplesof partial compositions: ( α ) β ○ ( ) = ( (( α α ) α )) β and ( α ) β ○ ( γ ) δ = ((( γ ) δ ) α ) β . Remark 4.2.
As we have seen in Subsection 1.3, our conventions are such that the Pa -moperad structure on Pa Γ0 gives in particular a morphism of Pa -modules Pa → Pa Γ0 . Onecan see that it is the map that sends a parenthezised permutation p to 0 ( p ) together withthe trivial label function i ↦ Pa Γ0 is acted on by Γ in the following way: for n ≥
0, Γ n only acts on the labellings, via the group law of Γ. For instance, if f ∶ { , . . . , n } → Γ and α ∈ Γ n , then ( α ⋅ f )( i ) = f ( i ) + α i .4.3. The PaB-moperad of parenthesized cyclotomic braids.
We define
PaB Γ ∶ = π ( C ( C × , − , Γ ) , Pa Γ0 ) . It is a
PaB -moperad (in groupoids), that carries an action of the group Γ. The maps φ n ∶ C ( C × , n, Γ ) → C ( C × , n ) induce a PaB -moperad morphism
PaB Γ → PaB . MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 15
Example 4.3 (Description of
PaB Γ ( ) ) . First observe that Pa Γ0 ( ) → Pa ( ) is the terminalmap µ N ≃ { α ∣ α ∈ Γ } → { } = ∗ . Then observe that the map C ( C × , , Γ ) → C ( C × , ) isnothing but the path-connected Γ-cover S → S . Hence we in particular have morphisms E , α , α ∈ Γ from 01 α to 01 α + ¯1 in PaB Γ ( ) , being the unique lift of E , that starts at01 α ∈ Pa Γ0 ( ) . Pictorially:0 1 ¯0 z e − π / N z z N z z N Two incarnations of E , ¯0 In the above picture, on the right we have pictured a path in the twisted configuration space,together with its image under the covering map, which is a loop. Diagrammatically (see theleft of the above picture), we depict it as a pure braid (a loop in the base configuration space)together with appropriate base points (which uniquely determines the lift in the coveringtwisted configuration space).
Example 4.4 (Notable arrow in
PaB Γ ( ) ) . Let Ψ , ¯0 , ¯0 be the unique lift of Ψ , , (a mor-phism in PaB ( ) ) starting at ( ¯0 ) ¯0 . It can be depicted as follows: (
00 1 ¯0 ) ( ¯0 ¯0 ¯0 ) Remark 4.5.
As in Remark 3.3, one can see from § S -modules PaB → PaB Γ . In pictorial terms, it sends a parenthesized braid with n strands to a labelledparenthesized braid with n + R ¯0 , ¯0 of R , and Φ ¯0 , ¯0 , ¯0 of Φ , , canbe respectively depicted as follows:00 ( ¯0 ( ¯0 ¯0 ) ¯0 ) (( ¯0 ( ¯0 ¯0 ) ( ¯0 ¯0 ) ¯0 )) Notation. (i) First of all, for any arrow X = X , ¯0 ,...,n ¯0 in PaB Γ ( n ) starting at a paren-thesized permutation x equipped with the constant labelling equal to ¯0, and for any α = ( α , . . . , α n ) ∈ Γ n , we write X , α ,...,n αn ∶ = α ⋅ X , which starts now at the same parenthesizedpermutation x equipped with the labelling α . (ii) Second of all, for p ≥
0, if X ends at the same parenthesized permutation x , but equippedwith a possibly non-trivial labelling α , then we write X ( p ) ∶ = → ∏ k = ,...,p − ( kα ) ⋅ X = X , ¯0 ,...,n ¯0 X , α ,...,n αn ⋯ X , ( p − ) α ,...,n ( p − ) αn , which starts at ( x, ¯0 ) and ends at ( x, pα ) .(iii) Finally, if γ ∈ Γ and 1 ≤ i ≤ n , then we write γ i ∶ = ( ¯0 , . . . , ¯0 , γ i , ¯0 , . . . , ¯0 ) . In particular, ( E , ¯0 ) ( p ) ∶ = → ∏ k = ,...,p − E , ¯ k = E , ¯0 E , ¯1 ⋯ E , p − , which is an element in Hom PaB Γ ( ) (( , ¯0 ) , ( , ¯ p )) .4.4. The universal property of PaB Γ . We are now ready to provide an explicit presenta-tion for the
PaB -moperad
PaB Γ : Theorem 4.6.
As a
PaB -moperad in groupoids with a Γ -action having Pa Γ0 as Pa Γ0 -moperadof objects, PaB Γ is freely generated by E , ¯0 and Ψ , ¯0 , ¯0 together with the following relations: Ψ , ∅ , ¯0 = Ψ , ¯0 , ∅ = Id , ¯0 ( in Hom
PaB Γ ( ) ( ¯0 , ¯0 )) , (tU)Ψ ¯0 , ¯0 , ¯0 Ψ , ¯0 , ¯0 ¯0 = Ψ , ¯0 , ¯0 Ψ , ¯0 ¯0 , ¯0 Φ ¯0 , ¯0 , ¯0 ( in Hom
PaB Γ ( ) ((( ¯0 ) ¯0 ) ¯0 , ( ¯0 ( ¯0 ¯0 )))) , (MP)Ψ , ¯0 , ¯0 E , ¯0 ¯0 ( Ψ , ¯1 , ¯1 ) − = E , ¯0 E ¯1 , ¯0 ( in Hom
PaB Γ ( ) (( ¯0 ) ¯0 , ( ¯1 ) ¯1 )) , (tRP) E ¯0 , ¯0 = Ψ , ¯0 , ¯0 R ¯0 , ¯0 ( Ψ , ¯0 , ¯0 ) − E , ¯0 Ψ , ¯1 , ¯0 R ¯1 , ¯0 ( Ψ , ¯0 , ¯1 ) − ) , (tO) ( in Hom
PaB Γ ( ) (( ¯0 ) ¯0 , ( ¯0 ) ¯1 ) . Proof.
Let Q Γ be the PaB -moperad with the above presentation, and recall that Q is the PaB -moperad with the presentation of Theorem 3.4. Our first goal is to show that there isa morphism Q Γ → PaB Γ of PaB -moperads, commuting with the Γ-action. We have alreadyseen in the Examples above that there are morphisms E , ¯0 and Ψ , ¯0 , ¯0 , in PaB Γ ( ) and PaB Γ ( ) , respectively. We have to prove that they satisfy the mixed pentagon and twistedoctogon relation, (MP) and (tO) and (tRP).These relations are the unique lifts of the similar relations (MP), (RP) and (O) in PaB from Theorem 3.4, starting at (( ¯0 ) ¯0 ) ¯0 and ( ¯0 ) ¯0 , respectively. They can be depictedas follows: MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 17 (( ¯0 ) ¯0 ) ¯0 ( ¯0 ( ¯0 ¯0 )) = (( ¯0 ) ¯0 ) ¯0 ( ¯0 ( ¯0 ¯0 )) (MP) ( ( ¯0 ) ¯1 ) ¯0 ¯1 = ( ( ¯0 ) ¯1 ) ¯0 ¯1 (tRP)and ( ( ¯0 ) ¯0 ) ¯0 ¯1 = ( ( ¯0 ) ¯0 ) ¯0 ¯1 (tO)By universal property of Q Γ there is a Γ-equivariant morphism of PaB -moperads Q Γ Ð→ PaB Γ , which is the identity on objects. As before, in order to show that this is an isomorphism,it suffices to show that it is an isomorphism at the level of automorphism groups of an objectarity-wise (because all groupoids involved are connected). Let n ≥
0, and let ˜ p be the object ( ⋯ ( ¯0 ) ¯0 ⋯⋯ ) n ¯0 of Q Γ ( n ) and PaB Γ ( n ) , which lifts the object p = ( ⋯ ( ) ⋯⋯ ) n of Q ( n ) ≃ PaB ( n ) . We want to show that the induced group morphismAut Q Γ ( n ) ( ˜ p ) Ð→ Aut
PaB Γ ( n ) ( ˜ p ) = π ( ¯C ( C × , n, Γ ) , ˜ p ) is an isomorphism. We claim that it fits into a commuting diagramAut Q Γ ( n ) ( ˜ p ) / / (cid:15) (cid:15) π ( C ( C × , n, Γ ) , ˜ p ) (cid:15) (cid:15) π ( C ( C × , n, Γ ) , ˜ p reg ) ≃ o o (cid:15) (cid:15) Aut Q ( n ) ( p ) ≃ / / (cid:15) (cid:15) π ( C ( C × , n ) , p ) (cid:15) (cid:15) π ( C ( C × , n )) , p reg ) ≃ o o (cid:15) (cid:15) Γ n Γ n Γ n where only the left-most vertical arrows remain to be described.The morphism Aut Q ( n ) ( p ) → Γ n . Let ∗ be the terminal operad in groupoids. We have a ∗ -moperad structure on the following S -module in groupoids: Γ = { Γ n } n ≥ , where we view agroup as a groupoid with only one object, and where the action of the symmetric group is bypermutation. The moperad structure is described as follows: ● ○ ∶ Γ n × Γ m → Γ n + m is the concatenation of sequences. ● for every i ≠ ○ i ∶ Γ n → Γ n + m − is the partial diagonal ( α , . . . , α n ) z→ ( α , . . . , α i − , α i , . . . , α i ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ m times , α i + , . . . , α n ) . We let the reader check that sending E to ¯1 ∈ Γ and Ψ to ( ¯0 , ¯0 ) ∈ Γ defines a moperadmorphism PaB → Γ along the terminal operad morphism
PaB → ∗ . This in particularinduces a group morphism Aut Q ( n ) ( p ) Ð→ Γ n , for every n ≥
0. Heuristically, this morphism counts, for every i , and modulo N , the numberof times that E ,i appears in an element of Aut Q ( n ) ( p ) . It is obviously surjective, and we letthe reader check that the following triangle commutes:Aut Q ( n ) ( p ) ≃ / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ π ( C ( C × , n ) , p ) (cid:15) (cid:15) Γ n The morphism Aut Q Γ ( n ) ( ˜ p ) → Aut Q ( n ) ( p ) . We have a Γ-equivariant morphism of PaB -moperads Q Γ → Q , where Γ acts trivially on Q , which forgets the label on objects,and sends the generators E , ¯0 and Ψ , ¯0 , ¯0 to E and Ψ, respectively. It obviously fits into acommuting square Q Γ / / (cid:15) (cid:15) PaB Γ (cid:15) (cid:15) Q / / PaB of PaB -moperads. This induces in particular a group morphismAut Q Γ ( n ) ( ˜ p ) Ð→ Aut Q ( n ) ( p ) , MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 19 for every n ≥
0, that fits into a commuting squareAut Q Γ ( n ) ( ˜ p ) / / (cid:15) (cid:15) π ( C ( C × , n, Γ ) , ˜ p ) (cid:15) (cid:15) Aut Q ( n ) ( p ) ≃ / / π ( C ( C × , n ) , p ) We now turn to the proof of the fact that the left-most vertical sequence is a short exactsequence, which shows thatAut Q Γ ( n ) ( ˜ p ) Ð→ Aut
PaB Γ ( n ) ( ˜ p ) = π ( ¯C ( C × , n, Γ ) , ˜ p ) is an isomorphism. We already know that the morphism Aut Q ( n ) ( p ) → Γ n is surjective.The morphism Aut Q Γ ( n ) ( ˜ p ) → Aut Q ( n ) ( p ) is injective. Indeed, an automorphism of ˜ p in Q Γ ( n ) can be represented by a finite sequence ˜ S of R ’s, Φ’s, E ’s, Ψ’s, and their imagesunder the action of Γ n . The image of such an automorphism under Q Γ → Q is representedby the corresponding finite sequence S of R ’s, Φ’s, E ’s and Ψ’s. Every modification of S usingthe relations (MP), (RP) and (O) can be lifted (uniquely) to a modification of ˜ S using (MP),(tRP) and (tO) or their images under the action of Γ n . Hence, if an automorphism has trivialimage, then it must be trivial.The sequence is exact. We already know from the commuting diagram that the image ofAut Q Γ ( n ) ( ˜ p ) in Aut Q ( n ) ( p ) lies in the kernel of Aut Q ( n ) ( p ) → Γ n . We finally can showthat the image is exactly the kernel. Indeed: ● Using (O), every element g in Aut Q ( n ) ( p ) can be written as a product of Φ’s, R ’s,Ψ’s and E ’s, where the only E ’s appearing are of the form E ,i . ● Such an element admits a unique lift to a morphism ˜ g in Q Γ ( n ) , with source being ˜ p (one just replace Φ’s, R ’s, Ψ’s and E ’s in the expression for g by the same symbols,maybe acted on by Γ n in order to get the correct starting objects). ● An element g as above lies inker ( Aut Q ( n ) ( p ) Ð→ Γ n ) if and only if for every i , the number of occurence of E ,i (counted in an algebraic way)is a multiple of N . This tells us in particular that the target of the lifted morphismshall be the same as its source, so that ˜ g lies in the kernel.This ends the proof of the Proposition. (cid:3) Cyclotomic Grothendieck-Teichm¨uller groups.
We let Mop C be the category ofpairs ( O , M ) , with O an operad and M a O -moperad, in a symmetric monoidal category C .A morphism ( O , M ) → ( P , N ) is a pair ( F, G ) , with F ∶ O → P an operad morphism and G ∶ M → N a O -moperad morphism, where the O -moperad structure on N is defined fromits P -moperad structure by applying F .In addition to the superscript “ + ”, wich means, as in § Definition 4.7.
The ( k -pro-unipotent version of the) cyclotomic Grothendieck-Teichm¨ullergroup is defined as the group ̂ GT Γ ( k ) ∶ = Aut + Mop
Grpd k (̂ PaB ( k ) , ̂ PaB Γ ( k )) Γ of Γ-equivariant automorphisms of the pair (̂ PaB ( k ) , ̂ PaB Γ ( k )) which are the identity onobjects.Our main goal in this subsection is to relate this cyclotomic Grothendieck-Teichm¨ullergroup with one of those introduced by Enriquez in [6].Let us recall that PB ≃ PB is identified with the free group F generated by a singlegenerators x (being x in PB , and x in PB ). Let us also recall that the quotient ofPB ≃ PB by its center (which is freely generated by a single element) is a free groupF generated by two elements x, y . As usual, we will consider the inclusion of F in PB (resp. PB ) sending x to x (resp. x ), and y to x (resp. x ). Recall finally that PB Γ n isthe kernel of the morphism PB n → Γ n sending x j to ¯1 j , and the other generators to ( ¯0 , . . . , ¯0 ) .Whenever n =
1, this is nothing but the morphism F → Γ sending x to ¯1, having kernelfreely generated by X = x N . Finally notice that the morphism φ N ∶ F → Γ sending x to ¯1and y to ¯0 fits into the following commuting square:F / / (cid:15) (cid:15) PB (cid:15) (cid:15) Γ ( id, ¯0 ) / / Γ It induces a morphism between the kernels F N + ≃ ker φ N → PB Γ2 . The generators of ker φ N are X = x N and y ( a ) = x − a yx a , 1 ≤ a ≤ N − ̂ GT Γ ( k ) first depends on anautomorphism F of ̂ PaB ( k ) , which is determined by a pair ( λ, f ) , where λ ∈ k × and f ∈ ˆF ( k ) satisfying the relations from § ● F ( R , ) = x λ − R , , ● F ( Φ , , ) = f ( x , x ) Φ , , .Then we have an automorphism G of ̂ PaB Γ ( k ) , compatible with F , which is likewise deter-mined by the images of E , ¯0 ∈ Hom ̂ PaB Γ ( k )( ) ( ¯0 , ¯1 ) and Ψ , ¯0 , ¯0 ∈ Hom ̂ PaB Γ ( k )( ) (( ¯0 ) ¯0 , ( ¯0 ¯0 )) : ● G ( E , ¯0 ) = uE , ¯0 , with u = X µ = x Nµ for some µ ∈ k × , necessarily, ● G ( Ψ , ¯0 , ¯0 ) = v Ψ , ¯0 , ¯0 ,where v ∈ ̂ PB Γ2 ( k ) ⊂ ̂ PB ( k ) can be written as C µ g ( x , x ) , with C a central generator ofker φ N and g ∈ ̂ ker φ N ( k ) ⊂ ˆF ( k ) . Notation.
We will also write g ( X, y ( ) , . . . , y ( N − )) when we want to view g in ˆF N + ( k ) ≃ ̂ ker φ N ( k ) . MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 21
Relation (tU) tells us that X µ = v , ∅ , =
1, and thus that µ =
0. Indeed, the morphism ( − ) , ∅ , ∶ PB Γ2 → PB Γ1 ≃ F sends ker φ N to 1, and x N (as well as the central generator) to X = x N . We conclude that v = g ( x , x ) = g ( X, y ( ) , . . . , y ( N − )) . Proposition 4.8.
The elements ( λ, f, µ, g ) satisfy (2) g ( x x , x ) g ( x , x x ) = g ( x , x ) g ( x x , x x ) f ( x x ) , and, for α = ¯1 ∈ Γ , (3) x µ g ( x, y ) y λ + g ( z, y ) − z µ α ⋅ ( g ( z, y ) y λ − g ( x, y ) − ) = ( in ˆF ( φ N , k )) xyz = . Proof.
First of all, the fact that relation (2) is satisfied is straightforward. Second of all,suppose N = G : G ( E , ) = g ( x , x ) Ψ , , x λ − R , ( Ψ , , ) − g − ( x , x ) G ( E , ) (4) g ( x , x ) Ψ , , x λ − R , ( Ψ , , ) − g − ( x , x ) . Now, by using x = x , σ x σ − = x and σ x σ − = x , we get g ( x , x ) Ψ , , x λ − R , ( Ψ , , ) − g − ( x , x ) = σ − g ( x , x ) σ x λ − σ g − ( x , x ) σ − Ψ , , R , ( Ψ , , ) − = σ − g ( x , x ) x λ + g − ( x , x ) σ − Ψ , , R , ( Ψ , , ) − . Plugging this into equation (4) we obtain G ( E , ) = σ − g ( x , x ) x λ + g − ( x , x ) σ − Ψ , , R , ( Ψ , , ) − G ( E , ) (5) Ψ , , R , ( Ψ , , ) − σ − g ( x , x ) x λ + g − ( x , x ) σ − . Now, since Ψ , , R , ( Ψ , , ) − is nothing but σ , we obtain G ( E , ) = σ − g ( x , x ) x λ + g − ( x , x ) σ − ⋅ G ( E , ) g ( x , x ) x λ + g − ( x , x ) . Next, G ( E , ) = x so, by using relation σ − x = x σ − , we obtain G ( E , ) = σ − g ( x , x ) x λ + g − ( x , x ) x µ σ − g ( x , x ) x λ + g − ( x , x ) = σ − g ( x , x ) x λ + g − ( x , x ) x µ g ( x , x ) σ − x λ + σ − g − ( x , x ) = σ − g ( x , x ) x λ + g − ( x , x ) x µ g ( x , x ) x λ − g − ( x , x ) σ . The above equation is then equivalent to g ( x , x ) x λ + g − ( x , x ) x µ g ( x , x ) x λ − g − ( x , x ) = σ G ( E , ) σ − = σ ( zx − ) µ σ − = ( zx − ) µ . Finally, by writing x = zx − x − , we obtain, by absorbing the central element z and simplify-ing it from the equation, the following result:1 = x µ g ( x , x ) x λ + g − ( x , x ) x − µ x − µ g ( x , x ) x λ − g − ( x , x ) . By denoting x = x , y = x and ˜ z = y − x − we obtain x µ g ( x, y ) y λ + g ( ˜ z, y ) − ˜ z µ g ( ˜ z, y ) y λ − g ( x, y ) − = . Finally, when N ≥
1, one takes the chosen lifts of each term of the above equation to obtainequation (3). (cid:3)
Lemma 4.9.
We have λ = + µ N .Proof. It is proven in [6] that, if we have a quadruple ( λ, µ, f, g ) , with ( λ, f ) ∈ ̂ GT ( k ) , µ = ( a, µ ) ∈ Γ × k , and g ∈ ̂ ker φ N ( k ) , satisfying the above two equations (2) and (3), then λ = [ µ ] ∶ = ˜ a + µ N , where 0 ≤ ˜ a ≤ N − a ∈ Γ. In our case, we are in thesituation where µ = ( ¯1 , µ ) . (cid:3) As a consequence, we can identify the underlying set of our operadicly defined cyclotomicGrothendieck-Teichm¨uller group ̂ GT Γ ( k ) with the underlying set of the group GTM ¯1 ( N, k ) introduced in [6].Indeed, GTM ¯1 ( N, k ) is defined as the set of triples ( λ, f, g ) with ( λ, f ) ∈ ̂ GT ( k ) and g ∈ ̂ ker φ N ( k ) , and satisying equations (2) and (3) with µ = λ − N . It carries a group structure,which is defined by ( λ , f , g ) ∗ ( λ , f , g ) = ( λ, f, g ) with ● λ = λ λ , ● f ( x, y ) = f ( f ( x, y ) − x λ f ( x, y ) , y λ ) ⋅ f ( x, y ) , ● g ( x, y ) = g ( g ( x, y ) − x µ g ( x, y ) , y λ ) ⋅ g ( x, y ) .Therefore it follows from the above discussion that we get an assignement(6) ̂ GT Γ ( k ) Ð→ GTM ¯1 ( N, k ) ; ( F, G ) z→ ( λ, f, g ) . It is obviously injective.
Proposition 4.10.
The injective map (6) is a group morphism.
We will see later in Theorem 5.13 that (6) is actually an isomorphism.
Proof.
We have to prove that the assignment ( F, G ) → ( λ, f, g ) constructed above is a groupmorphism. As we already know, the composition of automorphisms F and F in Aut + Op Grpd k (̂ PaB ( k )) corresponds to the composition law in ̂ GT ( k ) , that is, the associated couples ( λ , f ) and ( λ , f ) in k × × ˆF ( k ) satisfy ( F ○ F )( R , ) = ( R , ) λ λ and ( F ○ F )( Φ , , ) = f ( x λ , f ( x, y ) y λ f ( x, y ) − ) f ( x, y ) Φ , , (here F is generated by x ∶ = x and y ∶ = x ). We also already showed that any twoautomorphisms G and G in the group ̂ GT Γ ( k ) , depending on F and F respectively, areassociated to couples ( µ , g ( X ∣ y ( ) , . . . , y ( N − ))) and ( µ , g ( X ∣ y ( ) , . . . , y ( N − ))) where g and g are elements of in ˆF N + ( k ) . MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 23
In the group A ∶ = Aut ̂ PaB Γ ( k )( ) (( ¯0 ) ¯0 ) , we have X = x N = ( E , ¯0 ) ( N ) and G ( X ) = X λ for λ ∈ k × . Next, we want to compute G ( y ( )) , where y ( ) = x decomposes in A as follows: ( ¯0 ) ¯0 Ψ , , / / ( ¯0 ¯0 ) R , R , / / ( ¯0 ¯0 ) ( Ψ , , ) − / / ( ¯0 ) ¯0 . Then, as G ( Ψ , ¯0 , ¯0 ) = g ( X ∣ y ( ) , . . . , y ( N − )) Ψ , ¯0 , ¯0 and G ( R ¯0 ¯0 R ¯0 ¯0 ) = ( x ) λ , we obtain G ( y ( )) = g ( X ∣ y ( ) , . . . , y ( N − )) y ( ) λ g − ( X ∣ y ( ) , . . . , y ( N − )) . More generally, y ( a ) = x − a x x a decomposes as ( ¯0 ) ¯0 ( E , ) ( a ) / / ( ¯ a ) ¯0 Ψ , a, / / ( ¯ a ¯0 ) R a, R , a / / ( ¯ a ¯0 ) ( Ψ , a, ) − / / ( ¯ a ) ¯0 ( E , ) (− a ) / / ( ¯0 ) ¯0 . Therefore, by Γ-equivariance we get x a g ( X ∣ y ( ) , . . . , y ( N − ))) = g ( X ∣ y ( a ) , . . . , y ( a + N − ))) x a , and thus G ( y ( a )) = G (( E , ¯0 ) ( a ) Ψ , ¯ a , ¯0 R ¯ a , ¯0 R ¯0 , ¯ a ( Ψ , ¯ a , ¯0 ) − ( E , ¯0 ) (− a ) ) = G (( E , ¯0 ) ( a ) ) G ( Ψ , ¯ a , ¯0 ) G ( R ¯ a , ¯0 R ¯0 , ¯ a ) G ( Ψ , ¯ a , ¯0 ) − ) G (( E , ¯0 ) (− a ) ) = Ad (( X Nµ E , ¯0 ) ( a ) g ( X ∣ y ( ) , . . . , y ( N − )))( x λ ) = Ad ( X aNµ ( E , ¯0 ) ( a ) g ( X ∣ y ( ) , . . . , y ( N − )))( x λ ) = Ad ( X ak g ( X ∣ y ( a ) , . . . , y ( a + N − )))( y ( a ) λ ) . Finally we obtain ( G ○ G )( Ψ , ¯0 , ¯0 ) = G ( g ( X ∣ y ( ) , . . . , y ( N − )) Ψ , ¯0 , ¯0 ) = g ( G ( X )∣ G ( y ( )) , . . . , G ( y ( N − ))) g ( X ∣ y ( ) , . . . , y ( N − ) Ψ , ¯0 , ¯0 = g ( X λ ∣ Ad ( g ( X ∣ y ( ) , . . . , y ( N − )))( y ( ) λ ) , Ad ( X λ g ( X ∣ y ( ) , . . . , y ( N )))( y ( ) λ ) , . . . , Ad ( X ( N − ) k g ( X ∣ y ( N − ) , . . . , y ( N − )))( y ( N − ) λ )) g ( X ∣ y ( ) , . . . , y ( N − )) Ψ , ¯0 , ¯0 . which is nothing but the composition law in the group GTM ¯1 ( N, k ) . This concludes the proof,as the composite of moperad morphisms G ○ G is compatible with the composition of operadmorphisms F ○ F . (cid:3) The moperad of N -chord diagrams, and cyclotomic associators Infinitesimal cyclotomic braids.
Let Γ = Z / N Z , I a finite set, and let t Γ I ( k ) be theLie k -algebra with generators t i , ( i ∈ I ), and t αij , ( i ≠ j ∈ I , α ∈ Z / N Z ), and relations: t αij = t − αji , (tS) [ t i , t αjk ] = [ t αij , t βkl ] = , (tL) [ t αij , t α + βik + t βjk ] = , (t4T) [ t i , t j + ∑ α ∈ Z / N Z t αij ] = , (t4 T ) [ t i + t j + ∑ β ∈ Z / N Z t βij , t αij ] = , (t6 T )where i, j, k, l ∈ I are pairwise distinct and α, β ∈ Z / N Z . We will call it the k -Lie algebra of infinitesimal cyclotomic braids . This definition is obviously functorial with respect to bijec-tions, exhibiting t Γ ( k ) ∶ = { t Γ I ( k )} I as an S -module. It moreover also has the structure of a t ( k ) -moperad, where partial compositions are defined as follows : for i ∈ I , ○ i ∶ t Γ I ( k ) ⊕ t J ( k ) Ð→ t Γ J ⊔ I −{ i } ( k )( , t pq ) z→ t pq ( t αjk , ) z→ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ t αjk if i ∉ { j, k } ∑ r ∈ J t αrk if j = i ∑ r ∈ J t αjr if k = i ( t j , ) z→ ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩ t j if j ≠ i ∑ p ∈ J t p + ∑ γ ∈ Γ ∑ q,r ∈ Jq ≠ r t γqr if j = i and ○ ∶ t Γ I ( k ) ⊕ t Γ J ( k ) Ð→ t Γ J ⊔ I ( k )( , t p ) z→ t p ( , t αpq ) z→ t αpq ( t αjk , ) z→ t αjk ( t i , ) z→ t i + ∑ γ ∈ Γ ∑ j ∈ J t γji We call t Γ ( k ) the moperad of infinitesimal cyclotomic braids. It is acted on by Γ: for γ ∈ Γand 1 ≤ i ≤ n , γ i ∈ Γ n acts as γ i ⋅ t p = t p ( p ∈ { , . . . , n }) ,γ i ⋅ t αqr = t αqr ( α ∈ Γ and q, r ≠ i ) ,γ i ⋅ t αir = t α + γir ( α ∈ Γ and r ≠ i ) , ,γ i ⋅ t αqi = t α − γqi ( α ∈ Γ and q ≠ i ) . We just re-package Enriquez’s insertion-coproduct morphisms [6, § MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 25
Horizontal N -chord diagrams. We now consider the CD ( k ) -moperad CD Γ0 ( k ) ∶ = ˆ U ( t Γ ( k )) in Cat ( CoAlg k ) . Morphisms in CD Γ0 ( k )( n ) can be represented as linear combina-tions of diagrams of chords on n + N -diagrams according to the terminologyof [3, Definition 6.4] (where the relation to Vassiliev invariants has been explored): more pre-cisely, those horizontal N -diagrams for which the sum of labels on each strand is ¯0. Using therepresentation from [3], i.e. the one with labels on (non frozen) strands rather than on chords,the diagram corresponding to t i is0 i i = i i α − α while the one corresponding to t αij = t − αji is i ji jα − α = i ji j α − α and relations can be depicted as follows: j ki li lj kα − α β − β = j ki li lj kα − α β − β (tL) i j k ki jα − α = i j k ki jα − α i j ki j kαβ − α − β + i j ki j kα − α − β − β = i j ki j kα + β − β − α + i j ki j kβ − βα − α (t4T) 0 i j i j + ∑ α i j i jα − α = i j i j + ∑ α i j i jα − α (t4 T ) 0 i j i jα − α + i j i jα − α + ∑ β i j i jαβ − α − β (t6 T ) = i j i jα − α + i j i jα − α + ∑ β i j i jβα − β − α Let us now introduce another CD ( k ) -moperad CD Γ ( k ) , which will be made of all horizon-tal N -diagrams. In arity n , objects of CD Γ ( k ) are just labellings { , . . . , n } → Γ, Ob ( CD Γ ( k ))( n ) = Γ n , and the ∗ -moperad structure is given as follows: ● ○ ∶ Γ n × Γ m → Γ n + m is the concatenation of sequences. ● for every i ≠ ○ i ∶ Γ n → Γ n + m − is the partial diagonal ( α , . . . , α n ) z→ ( α , . . . , α i − , α i , . . . , α i ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ m times , α i + , . . . , α n ) . Given two labellings α = ( α , . . . , α n ) and β = ( β , . . . , β n ) , the k -vector space of morphismsfrom α to β in CD Γ ( k ) is the vector space of horizontal N -chord diagrams such that, on the MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 27 i -th strand, the sum of labels i β i − α i . The CD ( k ) -moperad structure on morphisms is theexact same as the one for CD Γ0 ( k ) .We call CD Γ ( k ) the CD ( k ) -moperad of N -chord diagrams . It carries an obvious action ofΓ, by translation on the labelling of objects. Example 5.1 (Notable arrows of CD Γ ( k )( ) ) . We have the following arrows in CD Γ ( k )( ) : K , α ∶ = t ⋅ α α = α α L , α ∶ = α α + ¯1 ¯100Note that, by definition, we have t αij = L ,i ¯0 t ij ( L ,i ¯0 ) − . There is an obvious Γ-equivariant morphism of moperads ω ∶ Pa Γ0 → Ob ( CD Γ ( k )) , over theterminal operad morphism Pa → ∗ = Ob ( CD ( k )) , that forgets the underlying parenthsizedpermutation and just remembers the labelling. Hence we can consider the fake pull-back PaCD ( k ) -moperad PaCD Γ ( k ) ∶ = ω ⋆ CD Γ ( k ) of parenthesized N -chord diagrams , which is still acted on by Γ. Example 5.2 (Notable arrow in
PaCD Γ ( k )( ) ) . We also have the following arrow in
PaCD Γ ( k )( ) : b , ¯0 , ¯0 ∶ = ⋅ ¯0 ) ( ¯0 ¯0 ¯0 )( Remark 5.3.
As explained in subsection 1.3, there is a map of S -modules PaCD Ð→ PaCD Γ and we denote (borrowing our previous conventions) X ¯0 , ¯0 , H ¯0 , ¯0 and a ¯0 , ¯0 , ¯0 the imagesin PaCD Γ of the corresponding arrows in PaCD . The elements K , ¯0 , L , ¯0 and b , ¯0 , ¯0 aregenerators of the PaCD ( k ) -moperad with Γ-action PaCD Γ ( k ) . They satisfy the following relations: ( L , ¯0 ) ( N ) ∶ = L , ¯0 ⋅ L , ¯1 ⋯ ⋅ L , N − = Id ¯0 , (7) L , ¯0 K , ¯1 = K , ¯0 L , ¯1 , (8) b ¯0 , ¯0 , ¯0 b , ¯0 , ¯0 ¯0 = b , ¯0 , ¯0 b , ¯0 , ¯0 a ¯0 , ¯0 , ¯0 , (9) b , ¯0 , ¯0 L , ¯0 ¯0 ( b , ¯1 , ¯1 ) − = L , ¯0 L ¯1 , ¯0 , (10) L ¯0 , ¯0 = b , ¯0 , ¯0 X ¯0 , ¯0 ( b , ¯0 , ¯0 ) − L , ¯0 b , ¯1 , ¯0 X ¯1 , ¯0 ( b , ¯0 , ¯1 ) − , (11) K , ¯0 ¯0 = ( b , ¯0 , ¯0 ) − K , ¯0 b , ¯0 , ¯0 + X ¯0 , ¯0 ( b , ¯0 , ¯0 ) − K , ¯0 b , ¯0 , ¯0 X ¯0 , ¯0 (12) + N − ∑ k = ( L , ¯0 ) ( k ) H ¯ k , ¯0 ( L , ¯0 ) (− k ) ,K ¯0 , ¯0 = b , ¯0 , ¯0 ( X ¯0 , ¯0 ( b , ¯0 , ¯0 ) − K , ¯0 b , ¯0 , ¯0 X ¯0 , ¯0 )( b , ¯0 , ¯0 ) − (13) + N − ∑ k = (( L , ¯0 ) ( k ) b , ¯ k , ¯0 ) H ¯ k , ¯0 (( L , ¯0 ) ( k ) b , ¯ k , ¯0 ) − . Cyclotomic associators.
We borrow the notation from § Definition 5.4.
A cyclotomic associator is a couple ( F, G ) where F is in Assoc ( k ) and G isa Γ-equivariant isomorphism of ̂ PaB ( k ) -moperads from ̂ PaB Γ ( k ) to G PaCD Γ ( k ) which isthe identity on objects (the ̂ PaB ( k ) -moperad structure on G PaCD Γ ( k ) is given by F ). Wedenote by Assoc Γ ( k ) ∶ = Iso + Mop ((̂
PaB ( k ) , ̂ PaB Γ ( k )) , ( G PaCD ( k ) , G PaCD Γ ( k ))) Γ the set of cyclotomic associators. Notation.
The Lie algebra t Γ2 ( k ) is the direct sum of its center, that is one dimensional andgenerated by c ∶ = t + t + ∑ α ∈ Γ t α , with the free Lie algebra f N + ( k ) = f ( k )( t , t , ..., t N − ) generated by t and the t α ’s ( α ∈ Γ). The quotient of t Γ2 ( k ) by its center will be denoted ¯ t Γ2 ( k ) ,and is thus isomorphic to f N + ( k ) . For every ψ ∈ ¯ t Γ2 ( k ) , we write ψ , , ∶ = ψ ( t , t , ..., t N − ) .We then have the following theorem: Theorem 5.5.
There is a one-to-one correspondence between elements of
Assoc Γ ( k ) andthose of the set Ass Γ ( k ) consisting of triples ( µ, ϕ, ψ ) ∈ k × × exp ( ˆ¯ t ( k )) × exp ( ˆ¯ t Γ2 ( k )) , such that ( µ, ϕ ) ∈ Ass ( k ) and ψ satisfies ψ , , ψ , , = ψ , , ψ , , ϕ , , , in exp ( ˆ¯ t Γ3 ( k )) (14) e µN t ψ , , e µ t ( ψ , , ) − e µN t α ⋅ ( ψ , , e µ t ( ψ , , ) − ) = in exp ( ˆ¯ t Γ2 ( k )) , (15) where α = ( ¯0 , ¯1 ) ∈ Γ .Proof. Let ˜ F be a k -associator ̂ PaB ( k ) Ð→ G PaCD ( k ) , and let ˜ G be a Γ-equivariant iso-morphism ̂ PaB Γ ( k ) Ð→ G PaCD Γ ( k ) MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 29 of ̂ PaB ( k ) -moperads, which is the identity on objects. It corresponds to a unique Γ-equivariantmorphism G ∶ PaB Γ Ð→ G PaCD Γ ( k ) . From the presentation of PaB Γ , we know that G isuniquely determined by the images of the morphisms E , ¯0 ∈ Hom
PaB Γ ( k )( ) ( , ) andΨ , ¯0 , ¯0 ∈ Hom
PaB Γ ( k )( ) (( ) , ( )) . Thus, there are elements u ∈ exp ( ˆ t Γ1 ( k )) and v ∈ exp ( ˆ t Γ2 ( k )) such that ● G ( E , ¯0 ) = uL , ¯0 , with u = e µ t for some µ ∈ k , necessarily; ● G ( Ψ , ¯0 , ¯0 ) = vb , ¯0 , ¯0 .These elements must satisfy the following relations, that are images of (tRP), (MP) and (tO),respectively: v , , u , ( ¯1 , ¯1 ) ⋅ ( v , , ) − = u , ( ¯1 , ¯0 ) ⋅ u , ( in exp ( ˆ t Γ2 ( k ))) , (16) v , , v , , = v , , v , , ϕ , , ( in exp ( ˆ t Γ3 ( k ))) , (17) u , = v , , e µ t ( v , , ) − u , α ⋅ ( v , , e µ t ( v , , ) − ) ( in exp ( ˆ t Γ2 ( k ))) . (18) Lemma 5.6.
Equation (16) is satisfied for arbitrary u and v .Proof. First of all, observe that the diagonal action of Γ on t Γ n ( k ) is trivial. Hence ( ¯1 , ¯1 ) ⋅ v , , = v , , . Second of all, recall the formulæ for the moperadic structure on t Γ ( k ) :(19) ( t ) , = t + t + ∑ γ ∈ Γ t γ = t + ( t ) , . Therefore, ( ¯1 , ¯0 ) ⋅ u , = u , , and u , = u , u , . Finally, the above element (19) is centralin t Γ2 ( k ) , thus so is u , = u , u , , and equation (16) is satisfied. (cid:3) Considering that we have a Lie algebra isomorphism t Γ2 ( k ) ≃ k c ⊕ f N + ( k ) , where c = t + t + ∑ a ∈ Γ t a , then v is of the form e µ c ψ ( t , t , ..., t N − ) for some µ ∈ k . Lemma 5.7.
We have µ = µN . In particular, u = e µN t .Proof of the Lemma. Denote by ε ∶ t Γ n Ð→ t n − the Lie algebra morphism sending t i to 0, and t αij to N t i − ,j − if 0 < i < j . We have ε ( u , , ) = e st ∈ exp ( ˆ t ) , for some s ∈ k . Now, the imageof the mixed pentagon relation (17) by ε yields:(20) u , , e s ( t + t ) = e st e s ( t + t ) ϕ , , . Moreover, as [ t , t + t ] =
0, we further have e st e s ( t + t ) = e s ( t + t + t ) . Thus equation(20) is equivalent to(21) ψ , , = e s ( t + t + t ) ϕ , , e − s ( t + t ) = ϕ , , e st , where we have used that t + t + t is central in t ( k ) . Now, we consider the image ofequation (18) in ¯ t = t /( t + t + t ) . Using that ψ , , = ϕ , , e st we have e − µ t = ϕ , , e st e µ t N ( ϕ , , e st ) − e µ t ϕ , , e st e µ t N ( ϕ , , e st ) − ⇔ = e µ t ϕ , , e µ N t ( ϕ , , ) − e µ t ϕ , , e µ N t ( ϕ , , ) − ⇔ = e µ t ϕ , , e µ N t ( ϕ , , ) − e µ t e µ t ϕ , , e µ N t ( ϕ , , ) − e µ t ⇔ = e Nµ − µ N t ϕ , , e Nµ − µ N t e Nµ − µ N t ψ , , e Nµ − µ N t where for the last equivalence we have used the hexagon relation for the couple ( µ, ϕ ) twice.This is now equivalent to 1 = ϕ , , e ( µ − µN ) t ϕ , , e ( µ − µN ) t Sending the generator t to 0, we get that e ( µ − µN ) t =
1, and thus µ = µN . (cid:3) End of the proof of the Theorem.
Finally, relation (tU) is equivalent to e µ ∂ ( c ) ∂ ψ ( t , t , . . . , t N − ) = , where we consider the restriction operator ∂ ∶ t Γ2 ( k ) → t Γ1 ( k ) . We have ∂ ( t ) = t and ∂ ( t ) = ∂ ( t α ) =
0. Therefore this equation reduces to e µ ∂ ( c ) = µ = v = ψ ( t , t , . . . , t N − ) . This automatically shows that relations (17) and(14) are equivalent. Now, in order to show that relation (18) is equivalent to relation (15),we notice that [ c, − ] = ψ ( t , t , ..., t N − ) has no component in weight 1 so onecan collect the factors e µ c in equation (18). The element u , is of the form e µN ( t +∑ a ∈ Γ t a ) .Then, by noticing that − t = t + ∑ a ∈ Γ t a in ¯ t Γ2 ( k ) , we obtain the equivalence between (18)and (15). (cid:3) Remark 5.8.
The set Ass Γ ( k ) is denoted Pseudo ¯1 ( N, k ) in [6]. One more generally has aset Pseudo γ ( N, k ) for every γ ∈ ( Z / N Z ) × : one just has to replace α = ( ¯0 , ¯1 ) with ( ¯0 , γ ) in thedefinition appearing in the statement of Theorem 5.5. A variation on the proof of Theorem5.5 shows that Pseudo γ ( N, k ) can be identified with Γ-equivariant isomorphisms between the ̂ PaB ( k ) -moperad ̂ PaB Γ ( k ) and the G PaCD ( k ) -moperad G PaCD Γ ( k ) which, on objects,is the global relabeling given by the automorphism of Γ sending ¯1 to γ . Example 5.9 (Cyclotomic KZ Associator) . Consider the differential equation(22) dd z H ( z ) = ⎛⎝ t z + ∑ α ∈ Z / N Z t α z − ζ α ⎞⎠ H ( z ) , where ζ is a primitive N th root of unity, and let H + , H − be the solutions such that H + ( z ) ∼ z t when z → + and H − ( z ) ∼ z t when z → − . Then, in our conventions, the renormalizedholonomy Ψ KZ = H + H − − ∈ exp ( ˆ¯ t Γ2 ( k ) from 0 to 1 of the above differential equation is thecyclotomic KZ associator constructed by Enriquez in [6]. More precisely, Enriquez showedthat the triple ( iπ, Φ KZ , Ψ KZ ) is in Pseudo − ¯1 ( N, C ) . MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 31
Graded cyclotomic Grothendieck-Teichm¨uller groups.Definition 5.10.
The graded cyclotomic Grothendieck–Teichm¨uller group is the group
GRT Γ ( k ) ∶ = Aut + Mop ( PaCD ( k ) , PaCD Γ ( k )) Γ of Γ-equivariant automorphisms of ( PaCD ( k ) , PaCD Γ ( k )) which are the identity on objects.In the rest of this subsection, we again compare our operadic definition with the ones givenby Enriquez in [6]. Definition 5.11.
Define GRT Γ ( ¯1 , ) ( k ) as the set of pairs ( g, h ) , with g ∈ GRT ( k ) and h ∈ exp ( ˆ¯ t Γ2 ( k )) , such that(23) h , , ( h , , ) − h ( t ∣ t , t , . . . , t − N ) h ( t ∣ t , . . . , t N ) − = ( ˆ¯ t Γ2 ( k )) , (24) t + N − ∑ a = Ad ( h ( t ∣ t a , . . . , t a + N − ))( t a ) + Ad ( h , , ( h , , ) − ) ( t ) = t Γ2 ( k ) , (25) h , , h , , = h , , h , , g , , in exp ( ˆ t Γ3 ( k )) . One can show that GRT Γ ( ¯1 , ) ( k ) is a group when equipped with the product ( g , h ) ∗ ( g , h ) = ( g, h ) , where ● g ( t , t ) = g ( t , Ad g ( t , t )( t )) g ( t , t ) , ● h , , = h ( t ∣ Ad (( h , , ))( t ) , . . . , Ad ( h ( t ∣ t N − , . . . , t N − ))( t N − )) h , , . The action of ( Z / N Z ) × × k × by automorphisms of t Γ3 (resp. t ) given by ( c, γ ) ⋅ t i = γt i , ( c, γ ) ⋅ t αij = γt cαij (resp. ( c, γ ) ⋅ t ij = γt ij ) induces its action by automorphisms of GRT Γ ( ¯1 , ) ( k ) . Wedenote by GRT Γ ( k ) the corresponding semidirect product, and GRT Γ¯1 ( k ) = GRT Γ ( ¯1 , ) ( k ) ⋊ k × . Proposition 5.12.
There is an injective group morphism
GRT Γ ( k ) → GRT
Γ¯1 ( k ) .Proof. Let ( G, H ) be an element in GRT Γ ( k ) . We have ● G ( X , ) = X , , ● G ( H , ) = λH , , ● G ( a , , ) = g ( t , t ) a , , , ● H ( b , ¯0 , ¯0 ) = wb , ¯0 , ¯0 , ● H ( K , ¯0 ) = µK , ¯0 , ● H ( L , ¯0 ) = L , ¯0 ,where ( λ, g ) ∈ GRT ( k ) and ( µ, w ) ∈ k × × exp ( ˆ t Γ2 ( k )) . Borrowing the notation from the previoussubsection, let us write w = e νc h ( t , t , . . . , t N − ) , with h ∈ ˆF N + ( k ) ≃ exp ( ˆ¯ t Γ2 ( k )) . First ofall, observe that one can show, along the same lines as in the previous subsection , that µ = λ See the proof of Lemma 5.7, and the end of the proof of Theorem 5.5. and ν =
0. Second of all, (25) follows directly from (9), and (23) follows directly from (11).Finally, (13) implies that t + N − ∑ a = t a = Ad ( h , , ( h , , ) − ) ( t ) + N − ∑ a = Ad ( h ( t ∣ t a , . . . , t a + N − )) ( t a ) , and since t + ∑ N − a = t a = − t in ˆ¯ t Γ2 ( k ) , we get (24).As a consequence of the above discussion, the assignment(26) ( G, H ) z→ ( λ, g ( t , t ) , h ( t ∣ t , . . . , t N − )) defines a map GRT Γ ( k ) → GRT
Γ¯1 ( k ) , that is obviously injective. It remains to prove that thecomposition of automorphisms in GRT Γ ( k ) corresponds to the composition law of the groupGRT Γ¯1 ( k ) . We already know that the composition of automorphisms G and G in GRT ( k ) corresponds to the composition law in GRT ( k ) , that is, the associated couples ( λ , g ) and ( λ , g ) in k × × exp ( ˆ t ( k )) satisfy ● ( G ○ G )( H , ) = λ λ H , , ● ( G ○ G )( a , , ) = g ( λ t , g ( t , t )( λ t ) g ( t , t ) − ) g ( t , t ) a , , .We also already showed that any two H and H such that ( G , H ) and ( G , H ) lie in GRT Γ ( k ) are determined by elements h ( t ∣ t , . . . , t N − ) and h ( t ∣ t , . . . , t N − ) whichrepresent automorphisms of the parenthesized word ( ¯0 ) ¯0 in the groupoid G PaCD Γ ( k )( ) .Note that the group Aut G PaCD Γ ( k )( ) (( ¯0 ) ¯0 ) is canonically identified with exp ( ˆ t Γ2 ) . Withinthis identification, t = K , ¯0 for instance, but t = b , ¯0 , ¯0 H ¯0 , ¯0 ( b , ¯0 , ¯0 ) − . Therefore H i ( t ) = λ i t and H i ( t ) = Ad ( h i ( t ∣ t , . . . , t N − ))( λ i t ) . More generally, t a = ( L , ¯0 ) ( a ) b , ¯ a , ¯0 H ¯ a , ¯ a ( b , ¯ a , ¯0 ) − ( L , ¯0 ) (− a ) , and thus H i ( t a ) = Ad ( h i ( t , t a , . . . , t α + N − ))( λ i t a ) . Therefore, we compute ( H ○ H )( b , ¯0 , ¯0 ) = H ( h ( t , t , . . . , t N − ) b , ¯0 , ¯0 ) = h ( H ( t ) , H ( t ) , . . . , H ( t N − )) h ( t , t , . . . , t N − ) b , ¯0 , ¯0 = h ( λ t , Ad ( h ( t , t , . . . , t N − ))( λ t ) , . . .. . . , Ad ( h ( t , t N − , . . . , t N − ))( λ t N − )) h ( t ∣ t , . . . , t N − ) b , ¯0 , ¯0 , which is nothing but the composition law in the group GRT Γ¯1 ( k ) . This concludes the proof, asthe composite of moperad morphisms H ○ H is compatible with the composition of operadmorphisms G ○ G . (cid:3) In the next § we will show, among other things, that this injective morphism is actuallyan isomorphism. We could prove surjectivity by proving that the relations from Remark 5.3completely determine PaCD Γ ( k ) (which we believe is true), but we use instead the torsorstructure. MOPERADIC APPROACH TO CYCLOTOMIC ASSOCIATORS 33
Bitorsors.
Our main goal in this final § is to promote the one-to-one correspondencefrom Theorem 5.5 to a bitorsor isomorphism.Recall from [6] that the group GTM ¯1 ( N, k ) acts freely and transitively on Ass Γ ( k ) fromthe left, in the following manner: ( λ, f, g ) ∗ ( λ ′ , ϕ ′ , ψ ′ ) = ( λλ ′ , ϕ ′′ , ψ ′′ ) , where ϕ ′′ ( t , t ) ∶ = ϕ ′ ( t , t ) f ( e λ ′ t , Ad ( ϕ ′ ( t , t ))( e λ ′ t )) , and ψ ′′ ( t ∣ t , . . . , t N − ) ∶ = ψ ′ ( t ∣ t , . . . , t N − ) g ( e λ ′ t ∣ Ad ( ψ ′ ( t ∣ t , . . . , t N − ))( e λ ′ t ) , Ad ( e ( λ ′ / N ) t ψ ′ ( t ∣ t , . . . , t N ))( e λ ′ t ) , . . .. . . , Ad (( e (( N − ) λ ′ / N ) t ψ ′ ( t ∣ t N − , . . . , t N − ))( e λ ′ t N − )) Enriquez also constructed [6] a free and transitive right action of GRT
Γ¯1 ( k ) on Ass Γ ( k ) ,which commutes with the above left action of GTM ¯1 ( N, k ) , thus turning the triple ( GTM ¯1 ( N, k ) , Ass Γ ( k ) , GRT
Γ¯1 ( k )) into a bitorsor.For the sake of completeness, let us recall how this right action is defined. For every µ ∈ k × ,the group GRT Γ ( ¯1 , ) ( k ) acts onAss Γ µ ( k ) ∶ = {( ϕ, ψ ) ∈ exp ( ˆ t ( k )) × exp ( ˆ¯ t Γ2 ( k )) ; ( µ, ϕ, ψ ) ∈ Ass Γ ( k )} . from the right by ( ϕ, ψ ) ∗ ( g, h ) = ( ϕ ′ , ψ ′ ) , where ϕ ′ ( t , t ) = ϕ ( t , Ad ( g ( t , t ))( t )) g ( t , t ) and ψ ′ ( t ∣ t , . . . , t N − ) = ψ ( t ∣ Ad ( h ( t ∣ t , . . . , t N − ))( t ) , . . .. . . , Ad ( h ( t ∣ t N − , . . . , t N − ))( t N − )) h ( t ∣ t , . . . , t N − ) . This naturally extends to an action of GRT
Γ¯1 ( k ) on Ass Γ ( k ) , which is compatible with thescaling action of k × on k .We already know that, by definition, ( ̂ GT Γ ( k ) , Assoc Γ ( k ) , GRT Γ ( k )) has a natural bitorsor structure. Theorem 5.13.
There is a bitorsor isomorphism ( ̂ GT Γ ( k ) , Assoc Γ ( k ) , GRT Γ ( k )) Ð→ ( GTM ¯1 ( N, k ) , Ass Γ ( k ) , GRT
Γ¯1 ( k )) . Proof.
This is a summary of most of the above results. Indeed, we proved that ● There is an injective group morphism ̂ GT Γ ( k ) → GTM ¯1 ( N, k ) (Proposition 4.10); ● There is an injective group morphism
GRT Γ ( k ) → GRT
Γ¯1 ( k ) (Proposition 5.12); ● There is a bijection
Assoc Γ ( k ) → Ass Γ ( k ) (Theorem 5.5).Hence, in order to conclude it is sufficient to prove that the three above maps take therespective actions ̂ GT Γ ( k ) and GRT Γ ( k ) on Assoc Γ ( k ) , to the ones of GTM ¯1 ( N, k ) andGRT Γ¯1 ( k ) on Ass Γ ( k ) . The proof is similar to the proofs that ̂ GT Γ ( k ) → GTM ¯1 ( N, k ) and GRT Γ ( k ) → GRT
Γ¯1 ( k ) are group morphisms (see the proofs of Propositions 4.10 and 5.12),and is left to the reader. (cid:3) lossary 35 List of notation
Operads.PaB:
Operad of parenthesized braids. 5
PaCD ( k ) : Operad of parenthesized chord diagrams. 7
PaB : PaB -moperad of parenthesized braids with a frozen strand. 10 PaB Γ : PaB -moperad of parenthesized cyclotomic braids. 14 PaCD Γ ( k ) : PaCD ( k ) -moperad of parenthesized N -chord diagrams. 27 Groups.GT:
Operadic Grothendieck-Teichm¨uller group. 8 ̂ GT ( k ) : k -pro-unipotent Grothendieck-Teichm¨uller group. 8 GRT ( k ) : Operadic graded Grothendieck-Teichm¨uller group. 9 ̂ GT Γ ( k ) : Operadic k -pro-unipotent cyclotomic Grothendieck-Teichm¨uller group. 19GTM ¯1 ( N, k ) : k -pro-unipotent ( ¯1 , N ) -cyclotomic Grothendieck-Teichm¨uller group. 22 GRT Γ ( k ) : Operadic graded cyclotomic Grothendieck-Teichm¨uller group. 30GRT
Γ¯1 ( k ) : Graded ( ¯1 , N ) -cyclotomic Grothendieck-Teichm¨uller group. 31 Spaces. C ( C , I ) : Reduced configuration space of I -indexed points in C . 5¯C ( C , I ) : Fulton-MacPherson compactification of C ( C , I ) . 5C ( C × , I ) : Reduced configuration space of I -indexed points in C × . 9C ( C × , I, Γ ) : Reduced Γ-decorated configuration space of I -indexed points in C × . 13 Torsor sets.Assoc ( k ) : Operadic k -associators. 7Ass ( k ) : k -associators. 8 Assoc Γ ( k ) : Operadic cyclotomic k -associators. 28Ass Γ ( k ) : Cyclotomic ( ¯1 , k ) -associators. 28 Series. Φ KZ : KZ associator. 8Ψ KZ : Cyclotomic KZ associator. 30
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E-mail address : [email protected] Martin GONZALEZIMM, Universit´e d’Aix-Marseille, 39, rue F. Joliot Curie, 13453, Marseille, France.
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