aa r X i v : . [ m a t h . R A ] F e b Algebraic structures associated to operads
Loïc Foissy
Fédération de Recherche Mathématique du Nord Pas de Calais FR 2956Laboratoire de Mathématiques Pures et Appliquées Joseph LiouvilleUniversité du Littoral Côte d’Opale-Centre Universitaire de la Mi-Voix50, rue Ferdinand Buisson, CS 80699, 62228 Calais Cedex, FranceEmail: [email protected] bstract
We study different algebraic structures associated to an operad and their relations: to anyoperad P is attached a bialgebra, the monoid of characters of this bialgebra, the underlyingpre-Lie algebra and its enveloping algebra; all of them can be explicitely described with the helpof the operadic composition. non-commutative versions are also given.We denote by b ∞ the operad of b ∞ algebras, describing all Hopf algebra structures on asymmetric coalgebra. If there exists an operad morphism from b ∞ to P , a pair ( A, B ) of coin-teracting bialgebras is also constructed, that it to say: B is a bialgebra, and A is a gradedHopf algebra in the category of B -comodules. Most examples of such pairs (on oriented graphs,posets . . . ) known in the literature are shown to be obtained from an operad; colored versions ofthese examples and other ones, based on Feynman graphs, are introduced and compared. AMS classification. ontents Σ operads and operads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2 Algebras over an operad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Operadic species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 B ∞ algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.1 Definition and main property . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2 -associative algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.3 Quotients of B ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.4 Brace modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1.5 Dual construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 b ∞ algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.1 Definition and main property . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.2 Associative-commutative algebras . . . . . . . . . . . . . . . . . . . . . . . 272.2.3 Quotients of b ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.4 From B ∞ algebras to b ∞ algebras . . . . . . . . . . . . . . . . . . . . . . 292.2.5 Dual construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.2.6 Associated groups and monoids . . . . . . . . . . . . . . . . . . . . . . . . 312.2.7 pre-Lie modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Com , As and PreLie . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.1 The operads
Com and As . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.2 The operad PreLie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Feynman Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.1 Oriented Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.2 Lemmas on extraction-contraction . . . . . . . . . . . . . . . . . . . . . . 614.2.3 The operad of Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . . 644.2.4 Suboperads and quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3 Oriented graphs, posets, finite topologies . . . . . . . . . . . . . . . . . . . . . . . 684.3.1 Oriented graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682.3.2 Quasi-orders and orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.4 b ∞ structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.4.1 Operad morphisms and associated products . . . . . . . . . . . . . . . . . 734.4.2 Associated Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 ntroduction Operads –a terminology due to May– appear in the 70’s [32, 40, 35, 3, 8], to study loop spaces inalgebraic topology; see [27] for a historical review. They are now widely used in various contexts;our point here is to study certain constructions associated to an operad from a combinatorialHopf-algebraic point of view. To a given operad P , several objects are attached, such as monoidsand groups, pre-Lie and brace algebras, bialgebras and Hopf algebras, or pairs of interacting orcointeracting Hopf algebras.Let us precise the structures we obtain here. If P is an operad, then:1. the space P = M n ≥ P ( n ) is a brace algebra [19, 39] and a pre-Lie algebra [6, 7, 37]; inparticular, its pre-Lie product is given by: ∀ p ∈ P ( n ) , q ∈ P , p • q = n X i =1 p ◦ i q, where ◦ i is the i -th partial composition of the operad P . Moreover, the quotient of coin-variant coinv P is a pre-Lie (but generally not a brace) quotient of P .2. As observed in [5], this pre-Lie structure induces two monoid compositions ♦ and ♦ ′ onthe space P = Y n ≥ P ( n ) : ∀ p ∈ P ( n ) , q ∈ P , p ♦ ′ q = q + p ◦ ( q, . . . , q ) ,p ♦ q = q + X ≤ i <...
PreLie , seen as a quotient of b ∞ , through the canonical projection θ PreLie . • For the posets or quasi-posets of [11], use the operad on quasi-posets of [12]. • For the oriented graphs of [33], use an operad on graphs here introduced. • The example for Feynman graphs is here treated.As all these constructions are functorial, one also obtains morphisms of pairs of (co)interactingbialgebras, relating quasi-posets, graphs, Feynman graphs, and sub-families such as posets,graphs without cycles, simple graphs, etc.This paper is organized as follows. In the first, short, chapter, we give reminders on oper-ads. The examples of associative algebras As and commutative, associative algebras Com areintroduced. The second chapter is devoted to the study of the two operads B ∞ et b ∞ . Forthe reader’s comfort, we give a complete proof of the bijection between, for any vector space V , the set of B ∞ structures on V , and the set of products ∗ on T ( V ) making ( T ( V ) , ∗ , ∆ dec ) a Hopf algebra (theorem 2). Two quotients of B ∞ are described, namely Brace (in which casethe product ∗ comes from a dendrifrom Hopf algebra structure), and As (obtaining quasi-shuffleHopf algebras). The quite complicated operad B ∞ is shown to be isomorphic to a suboperad ofthe simpler , proving again Loday and Ronco’s rigidity result [29, 30]. A dual constructionis also given, needing the technical condition of -boundedness (definition 16). Cocommutativeversions of these results are given, replacing T ( V ) by S ( V ) , B ∞ by b ∞ , and by AsCom ;the operad
Brace is replaced by
PreLie , obtaining a diagram of operads: b ∞ Φ / / (cid:15) (cid:15) (cid:15) (cid:15) B ∞ (cid:15) (cid:15) (cid:15) (cid:15) PreLie / / % % % % ❑❑❑❑❑❑❑❑❑❑ Brace y y y y ssssssssss As (cid:15) (cid:15) (cid:15) (cid:15) Com
6t is also proved that, if ( V, ⌊ , ⌋ ) is a -bounded b ∞ , then its completion V is given a non-bilinearproduct ♦ defined by x ♦ y = ⌊ e x , e y ⌋ , making it a monoid, isomorphic to the monoid of charactersof the dual of S ( V ) (theorem 31).The next chapter applies these construction to operads. The brace algebra structure (hence, B ∞ structure) on P is defined in proposition 37; the implied pre-Lie product (hence, b ∞ struc-ture) in corollary 38; the associated dendriform products on T ( P ) and associative product on S ( P ) are given in proposition 40, giving two bialgebras and two Hopf algebras, namely D P , D P , B P and B P . If θ P : b ∞ −→ P is an operadic morphism, then any P -algebra A is also b ∞ ,which implies that S ( A ) becomes a Hopf algebra with a product ⋆ induced by the P -algebrastructure; in particular, if P = PreLie and θ PreLie is the canonical surjection, we obtain inthis way the Oudom-Guin construction. Applied to the free P -algebra in one variable F P , weprove that the graded, connected Hopf algebra A P = S ( F P ) , with this product ⋆ , is in inter-action with D P . There are two ways to obtain a duality; firstly, dualizing the composition of P gives bialgebras D ∗ P (non-commutative) and D ∗ P (commutative), such that A ∗ P and D ∗ P arein cointeraction (corollary 51). Secondly, using the -boundedness of P , one obtains bialgebras D ′ P (non-commutative) and D ′ P (commutative), different but isomorphic to the preceding ones(proposition 52, corollary 53). Consequently, we obtain a monoid M D P of characters of D ∗ P , whichcan be described with the product ♦ of chapter 2, acting on the completion of F P ; it is shownthat this is the monoid of formal endomorphisms of F P (proposition 58), that is to say a Faà diBruno-like monoid.Examples are treated in the last chapter. We start with Com and As , getting back Faà diBruno bialgebras of commutative or non-commutative formal diffeomorphisms coacting on K [ X ] .For the operad PreLie , we get back the two cointeracting bialgebras of rooted trees of [4], thefirst one being the Connes-Kreimer Hopf algebra, the second one being given by an extraction-contraction process. We then treat the case of Feynman graphs (definition 60), and give themtwo operadic compositions, ◦ and ∇ (theorem 67). It contains a suboperad of Feynman graphswithout oriented cycles and a suboperad of simple Feynman graphs; forgetting the externalstructure, we obtain operads on various families of graphs; considering Hasse graphs, we obtainoperads on quasi-posets (or equivalently, finite topologies) and posets, which were describedin [12]. Consequently, we obtain pairs of interacting bialgebras on these objects; for certainfamilies of graphs, this was done in [33]; for quasi-posets, in [11, 16]. The functoriality alsogives morphisms between these objects. The last chapter is a summary of the different objectsattached to an operad used in the paper. 7 otations • For all n ∈ N , we put [ n ] = { , . . . , n } . In particular, [0] = ∅ . • K is a commutative field of characteristic zero. All objects (vector spaces, algebras, coal-gebras, operad . . . ) in this text are taken over K . • Let V be a vector space. – We denote by T ( V ) the tensor algebra of V , that is to say: T ( V ) = ∞ M n =0 V ⊗ n . It is given an algebra structure with the concatenation product m conc , and a coalgebrastructure with the deconcatenation coproduct ∆ dec : ∀ x , . . . , x n ∈ V, ∆ dec ( x . . . x n ) = n X i =0 x . . . x i ⊗ x i +1 . . . x n . The shuffle product (cid:1) makes ( T ( V ) , ∆ dec ) a commutative Hopf algebra. For example,if x , x , x , x ∈ V : x (cid:1) x = x x + x x ,x (cid:1) x x = x x x + x x x + x x x ,x x (cid:1) x x = x x x x + x x x x + x x x x + x x x x + x x x x + x x x x . The augmentation ideal of T ( V ) is denoted by T + ( V ) . – We denote by S ( V ) the symmetric algebra of V , with its usual product. It is a Hopfalgebra, with the coproduct ∆ defined by: ∀ x , . . . , x k ∈ V, ∆( x . . . x k ) = X I ⊆ [ k ] Y i ∈ I x i ⊗ Y i/ ∈ I x i . The augmentation ideal of S ( V ) is denoted by S + ( V ) . • Let ( P ( n )) n ≥ be a family of vector spaces. We denote: P = M n ≥ P ( n ) , P = Y n ≥ P ( n ) . The family ( P + ( n )) n ≥ is defined by P + ( n ) = ( P ( n ) if n ≥ , (0) otherwise . Recall that a S -module is a family ( P ( n )) n ≥ such that for all n ≥ , P ( n ) is a right S n -module, where S n is the n -th symmetric group. We denote: coinv P ( n ) = P ( n ) V ect ( p − p σ | p ∈ P ( n ) , σ ∈ S n ) ,inv P ( n ) = { p ∈ P ( n ) | ∀ σ ∈ S n , p σ = p } . both ( coinv P ( n )) n ≥ and ( inv P ( n )) n ≥ are S -modules, with a trivial action of the sym-metric groups. 9 hapter 1 Reminders on operads
We here briefly recall the notions we shall use later on operads. We refer to [31, 34, 36, 41] formore details. Σ operads and operads A non- Σ operad P is a family ( P ( n )) n ≥ of vector spaces with, for all n, k , . . . , k n ≥ , acomposition ◦ : ◦ : (cid:26) P ( n ) ⊗ P ( k ) ⊗ . . . ⊗ P ( k n ) −→ P ( k + . . . + k n ) p ⊗ p ⊗ . . . ⊗ p n −→ p ◦ ( p , . . . , p n ) , such that for all n, k , . . . , k n , l , , . . . , l n,k n ≥ , for all p ∈ P ( n ) , p i ∈ P ( k i ) , p i,j ∈ P ( l i,j ) : ( p ◦ ( p , . . . , p n )) ◦ ( p , , . . . , p n,k n ) = p ◦ ( p ◦ ( p , , . . . , p ,k ) , . . . , p n ◦ ( p n, , . . . , p n,k n )) . There exists an element I ∈ P (1) , called the unit of P , such that for all n ≥ , for all p ∈ P ( n ) : I ◦ p = p, p ◦ ( I, . . . , I ) = p. An operad is a non- Σ operad P and a S -module; there is a compatibility between the actionof the symmetric groups and the composition we won’t detail here. We just mention that for all n, k , . . . , k n ≥ , for all p ∈ P ( n ) , p i ∈ P ( k i ) , for all σ ∈ S n , σ i ∈ S k i , there exists a particular σ ′ ∈ S k + ... + k n such that: p σ ◦ ( p σ , . . . , p σ n n ) = ( p ◦ ( p σ − (1) , . . . , p σ − ( n ) )) σ ′ . Notations.
Let P be a non- Σ operad, n ≥ , i , . . . , i k ∈ [ n ] , all distinct, p , . . . , p k ∈ P .We put: p ◦ i ,...,i k ( p , . . . , p k ) = p ◦ ( p ′ , . . . , p ′ n ) , where p ′ j = ( p l if j = i l ,I otherwise . For example, if p ∈ P (3) , p , p , p ∈ P : p ◦ p = p ◦ ( p , I, I ) , p ◦ p = p ◦ ( I, I, p ) ,p ◦ , ( p , p ) = p ◦ ( p , p , I ) , p ◦ , ( p , p ) = p ◦ ( p , p , I ) ,p ◦ , ( p , p ) = p ◦ ( p , I, p ) , p ◦ , ( p , p ) = p ◦ ( p , I, p ) ,p ◦ , , ( p , p , p ) = p ◦ ( p , p , p ) , p ◦ , , ( p , p , p ) = p ◦ ( p , p , p ) . Remark.
We shall here only consider operads P such that P (0) = (0) .10 .2 Algebras over an operad Let V be a vector space. Let us recall the construction of the operad L V of multilinear endo-morphisms of V : • For all n ≥ , L V ( n ) is the space of linear maps from V ⊗ n to V . • The operadic composition is defined by: ∀ f ∈ L V ( n ) , f i ∈ L V ( k i ) , f ◦ ( f , . . . , f n ) = f ◦ ( f ⊗ . . . ⊗ f n ) ∈ L V ( k + . . . + k n ) . The unit is Id V . • The action of S n on L V ( n ) is defined by: f σ ( x . . . x n ) = f ( x σ − (1) . . . x σ − ( n ) ) . Let P be an operad. An algebra over P is a pair ( V, ρ ) , where V is vector space and ρ : P −→ L V is an operad morphism. In other words, for all n ≥ , there exists a map: (cid:26) P ( n ) ⊗ V ⊗ n −→ Vp ⊗ v . . . v n −→ p. ( v . . . v n ) = ρ ( p )( v . . . v n ) , such that: • For all v ∈ V , I.v = v . • For all p ∈ P ( n ) , p i ∈ P ( k i ) , u i ∈ V ⊗ k i , p ◦ ( p , . . . , p n ) .u . . . u n = p. (( p .u ) . . . ( p n .u n )) . • For all p ∈ P n , σ ∈ S n , x , . . . , x n ∈ V , p σ . ( x . . . x n ) = p. ( x σ − (1) . . . x σ − ( n ) ) .Let P be an operad and V be a vector space. The free P -algebra generated by V is: F P ( V ) = M n ≥ P ( n ) ⊗ S n V ⊗ n , where for all n ≥ : P ( n ) ⊗ S n V ⊗ n = P ( n ) ⊗ V ⊗ n V ect ( p σ ⊗ x . . . x n − p ⊗ x σ − (1) . . . x σ − ( n ) ) | p ∈ P ( n ) , x , . . . , x n ∈ V ) . The P -algebra structure is given by: p. ( p ⊗ u ⊗ . . . ⊗ p n ⊗ u n ) = ( p ◦ ( p , . . . , p n )) ⊗ ( u . . . u n ) . Examples.
1. The operad
Com of commutative, associative algebras is generated by m ∈ Com (2) andthe relations: m (12) = m, m ◦ m = m ◦ m. • Consequently, the
Com -algebras are the commutative, associative (non necessarilyunitary) algebras; for any vector space V , F Com ( V ) = S + ( V ) . • For all n ≥ , Com ( n ) = V ect ( e n ) , and for all n, k , . . . , k n ≥ : e n ◦ ( e k , . . . , e k n ) = e k + ... + k n . • For all n ≥ , σ ∈ S n , e σn = e n . 11 For any associative, commutative algebra ( V, · ) , for any x , . . . , x n ∈ V : e n . ( x , . . . , x n ) = x · . . . · x n .
2. The operad As of associative algebras is generated by m ∈ As (2) and the relation: m ◦ m = m ◦ m. • Consequently, the As -algebras are the associative (non necessarily unitary) algebras;for any vector space V , F As ( V ) = T + ( V ) . • For all n ≥ , As ( n ) = V ect ( S n ) . Here are examples of compositions: (12) ◦ (12) = (123) , (12) ◦ (21) = (213) , (12) ◦ (12) = (123) , (12) ◦ (21) = (132) , (21) ◦ (12) = (312) , (21) ◦ (21) = (321) , (21) ◦ (12) = (231) , (21) ◦ (21) = (321) . • For any σ, τ ∈ S n , σ τ = σ ◦ τ . • For any associative algebra ( V, · ) , for any σ ∈ S n , for any x , . . . , x n ∈ V , σ. ( x , . . . , x n ) = x σ − (1) · . . . · x σ − ( n ) . We shall often work with operadic species [36]. A linear species P is a functor from the categoryof finite sets, with bijections as arrows, to the category of vector spaces. An operadic species isa species P with compositions, defined for all non-empty finite sets A and B , and a ∈ A : ◦ a : (cid:26) P ( A ) ⊗ P ( B ) −→ P ( A ⊔ B \ { a } ) p ⊗ q −→ p ◦ a q, such that: • For all finite sets
A, B, C , for all a = b ∈ A , p ∈ P ( A ) , q ∈ P ( B ) , r ∈ P ( C ) : ( p ◦ a q ) ◦ b r = ( p ◦ b r ) ◦ a q. • For all finite sets
A, B, C , for all a ∈ A , b ∈ B , p ∈ P ( A ) , q ∈ P ( B ) , r ∈ P ( C ) : ( p ◦ a q ) ◦ b r = p ◦ a ( q ◦ b r ) . • For all singleton { a } , there exists I a ∈ P ( { a } ) such that: – For all finite set A , for all a ∈ A , for all p ∈ P ( A ) , p ◦ a I a = p . – For all singleton { a } , for all finite set A , for all p ∈ P ( A ) , I a ◦ a p = p . Examples.
1. We take
Com ( A ) = V ect ( { A } ) for non-empty all finite set A . For all finite sets A , B , and a ∈ A : { A } ◦ a { B } = { A ⊔ B \ { a }} . For all singleton { a } , I a = { a } .2. For all non-empty finite set A , As ( A ) is the set of all words w with letters in A , such thatany element a ∈ A appears exactly one time in w . If a . . . a n ∈ As ( A ) and a ∈ A , thereexists a unique i ∈ [ n ] such that a i = a ; then: a . . . a n ◦ a w = a . . . a i − wa i +1 . . . a n . For any singleton { a } , I a = a . 12f P is an operadic species, one deduces an operad, also denoted by P , in the following way: • For all n ≥ , P ( n ) = P ([ n ]) . • For all p ∈ P ( m ) , q ∈ P ( n ) , i ∈ [ n ] : p ◦ i q = P ( σ ( i ) m,n )( p ◦ i q ) , with σ ( i ) m,n : ([ m ] \ { i } ) ⊔ [ n ] −→ [ m + n − defined by: σ ( i ) m,n (1) = 1 , σ ( i ) m,n ( ˙1) = i, σ ( i ) m,n ( i + 1) = n + i, ... ... ... σ ( i ) m,n ( i −
1) = i − , σ ( i ) m,n ( ˙ n ) = n + i − , σ ( i ) m,n ( m ) = m + n − , where the elements of [ m ] are denoted by , . . . , m , the elements of [ n ] by ˙1 , . . . , ˙ n . • The unit is I ∈ P ( { } ) . • For all p ∈ P ( n ) and σ ∈ S n , p σ = P ( σ − )( p ) .For example, the operad associated to the operadic species Com is the operad
Com ; the operadassociated to the operadic species As is the operad As .13 hapter 2 Infinitesimal structures on primitiveelements
Introduction
Let V be a vector space, m a product on T ( V ) making H = ( T ( V ) , m, ∆ dec ) a bialgebra. Isit possible to obtain m from a structure on the space V of primitive elements of H ? The an-swer is positive, this is the B ∞ structure (or multibrace, or LR, or Hirsch algebra structure[20, 19, 26, 30, 29]) on V . More precisely, there is a bijection between the sets of such products m and the set of B ∞ structures on V (theorem 2). As proved in [30], the operad B ∞ can beseen as a suboperad of the operad of -associative algebras (definition 5 and theorem 11; we heregive a complete proof of these well-known results. We will also be interested in several particularcases of such products m : right-sided in the sense of [30], or two-sided, leading to quotients ofthe operad of B ∞ algebras, such as the Brace operad (definition 12, [19, 39]) or the associativeoperad. In particular, if the B ∞ structure on V is brace, then m can be split into a dendriformstructure [39], which we here explicitly describe in theorem 13. We give a condition on a B ∞ algebra to obtain a dual bialgebra, namely the -boundedness condition (definition 16 and 17).There are cocommutative equivalents of these results, working on a symmetric coalgebrasinstead of tensor coalgebras; then B ∞ structures are replaced by b ∞ structures, and bracealgebras by pre-Lie algebras in the second part of this chapter. B ∞ algebras Definition 1
Let V be a vector space equipped with a map: h− , −i : (cid:26) T ( V ) ⊗ T ( V ) −→ Vx . . . x k ⊗ y . . . y l −→ h x . . . x k , y . . . y l i . For all k, l ∈ N , we put h− , −i k,l = h− , −i | V ⊗ k ⊗ V ⊗ l . We shall say that A is a B ∞ algebra if thefollowing axioms are satisfied: • For any k ≥ , h− , −i ,k = h− , −i k, = ( Id V if k = 1 , otherwise . for any tensors u, v, w ∈ T ( V ) : lg ( u )+ lg ( v ) X i =1 X u = u ...u i ,v = v ...v i hh u , v i . . . h u i , v i i , w i = lg ( v )+ lg ( w ) X i =1 X v = v ...v i ,w = w ...w i h u, h v , w i . . . h v i , w i ii . (2.1) The operad of B ∞ algebras is denoted by B ∞ . Remark. If i > lg ( u ) + lg ( v ) , u = u . . . u i and v = v . . . v i , there exists an index i suchthat u i = v i = 1 , so h u i , v i i = 0 . Consequently, (2.1) can be rewritten as: X i ≥ X u = u ...u i ,v = v ...v i hh u , v i . . . h u i , v i i , w i = X i ≥ X v = v ...v i ,w = w ...w i h u, h v , w i . . . h v i , w i ii . (2.2)A combinatorial description of free B ∞ -algebras can be found in [17]. Theorem 2
Let V be a vector space. Let Bialg ( V ) be the set of products ∗ on T ( V ) , making ( T ( V ) , ∗ , ∆ dec ) a bialgebra. Let B ∞ ( V ) be the set of B ∞ structures on V . These two sets are inbijections, via the maps: Φ V : B ∞ ( V ) −→ Bialg ( V ) h− , −i −→ ∗ defined by u ∗ v = X i ≥ X u = u ...u i ,v = v ...v i h u , v i . . . h u i , v i i Ψ V : (cid:26) Bialg ( V ) −→ B ∞ ( V ) ∗ −→ h− , −i defined by h u, v i = π ( u ∗ v ) , where π is the canonical projection on V . Remark. If ∗ = Φ V ( h− , −i ) , (2.1) can be rewritten as: h u, v ∗ w i = h u ∗ v, w i . (2.3)The proof of this theorem will need the following two lemmas: Lemma 3
Let C be a connected coalgebra and let φ, ψ : C −→ ( T ( V ) , ∆ dec ) be two coalgebramorphisms. Then φ = ψ if, and only if, π ◦ φ = π ◦ ψ . Proof.
Let ( C n ) n ≥ be the coradical filtration of C , and deg the associated degree; that isto say: • C is the coradical of C , that is to say the sum of simple subcoalgebras of C . As C isconnected, it has a unique group-like element denoted by C , and C = V ect (1 C ) . • If n ≥ , C n is uniquely defined by: C n = { x ∈ C | ∆( x ) ∈ C ⊗ C n − + C n − ⊗ C ) . As the unique group-like element of T ( V ) is , φ (1 C ) = ψ (1 C ) = 1 . For all x ∈ Ker ( ǫ C ) , weput ˜∆( x ) = ∆( x ) − x ⊗ C − C ⊗ x . Then ˜∆ is a coassociative coproduct on Ker ( ǫ C ) and forall x ∈ Ker ( ǫ C ) ∩ C n : ˜∆( x ) ∈ C n − ⊗ C n − . We assume that π ◦ φ = π ◦ ψ . Let us prove that for all x ∈ C , φ ( x ) = ψ ( x ) , by induction on n = deg ( x ) . If n = 0 , then x = λ C for a certain λ ∈ K , so φ ( x ) = ψ ( x ) = λ . Let us assumethe result at all ranks < n . As φ (1 C ) = ψ (1 C ) = 1 , using the induction hypothesis: ˜∆ dec ◦ φ ( x ) = ( φ ⊗ φ ) ◦ ˜∆( x ) = ( ψ ⊗ ψ ) ◦ ˜∆( x ) = ˜∆ dec ◦ ψ ( x ) , so φ ( x ) − ψ ( x ) ∈ Ker ( ˜∆ dec ) = V . Hence, φ ( x ) − ψ ( x ) = π ( φ ( x ) − ψ ( x )) = 0 . (cid:3) emma 4 Let h− , −i : T ( V ) ⊗ T ( V ) −→ V be any linear map such that h , i = 0 . Thefollowing map is a coalgebra morphism: ∗ : T ( V ) ⊗ T ( V ) −→ T ( V ) u ⊗ v −→ X i ≥ X u = u ...u i ,v = v ...v i h u , v i . . . h u i , v i i . Moreover, for all u, v ∈ T ( V ) , π ( u ∗ v ) = h u, v i . Proof.
Note that, as h , i = 0 , for all u, v ∈ T ( V ) : u ∗ v = lg ( u )+ lg ( v ) X i =1 X u = u ...u i ,v = v ...v i h u , v i . . . h u i , v i i , so u ∗ v ∈ T ( V ) . Let u, v be two tensors of T ( V ) . ∆ dec ( u ∗ v ) = X i ≥ X u = u ...u i ,v = v ...v i i X p =0 h u , v i . . . h u p , v p i ⊗ h u p +1 , v p +1 i . . . h u i , v i i = X u = u (1) u (2) ,v = v (1) v (2) X i,j ≥ X u (1) = u (1)1 ...u (1) i ,v (1) = v (1)1 ...v (1) i ,u (2) = u (2)1 ...u (2) j ,v (2) = v (2) i ...v (2) j h u (1)1 , v (1)1 i . . . h u (1) i , v (1) i i ⊗ h u (2)1 , v (2)1 i . . . h u (2) i , v (2) j i = X u = u (1) u (2) ,v = v (1) v (2) u (1) ∗ v (1) ⊗ u (2) ∗ v (2) = ∆ dec ( u ) ∗ ∆ dec ( v ) . So ∗ is a coalgebra morphism. (cid:3) Proof. (Theorem 2). Let us first prove that Φ V is well-defined. By lemma 4, the product ∗ = Φ V ( h− , −i ) is indeed a coalgebra morphism. Moreover, by convention ∗ , and for all v = x . . . x k ∈ T ( V ) , with k ≥ : ∗ v = h , x i . . . h , x k i + 0 = x . . . x k = v,v ∗ h x , i . . . h x k , i + 0 = x . . . x k = v. So is a unit for the product ∗ . For all tensors u, v ∈ T ( V ) , if u or v is not a word of length ,then u ∗ v is a sum of words of length ≥ : this implies that ǫ ( u ∗ v ) = ǫ ( u ) ǫ ( v ) .We consider the two following morphisms: F : (cid:26) T ( V ) ⊗ T ( V ) ⊗ T ( V ) −→ T ( V ) u ⊗ v ⊗ w −→ ( u ∗ v ) ∗ w,G : (cid:26) T ( V ) ⊗ T ( V ) ⊗ T ( V ) −→ T ( V ) u ⊗ v ⊗ w −→ u ∗ ( v ∗ w ) . As ∗ is a coalgebra morphism, F and G are coalgebra morphisms. Moreover, for all tensors u, v, w ∈ T ( V ) : π ◦ F ( u ⊗ v ⊗ w ) = h u ∗ v, w i , π ◦ G ( u ⊗ v ⊗ w ) = h u, v ∗ , w i .
16y (2.1), π ◦ F = π ◦ G . By lemma 3, F = G , so ∗ is associative. Finally, ∗ ∈ Bialg ( V ) .Let us prove that Ψ V is well-defined. Let ∗ ∈ Bialg ( V ) and h− , −i = π ◦ ∗ . The unit of ∗ isa group-like of T ( V ) , so is equal to . Hence, if x , . . . , x k ∈ V : h , x . . . x k i = π (1 ∗ x . . . x k ) = π ( x . . . x k ) = δ ,k x . . . x k , so h− , −i ,k = Id V if k = 1 and otherwise. The condition on h− , −i k, is proved similarly.Let u, v, w ∈ T ( V ) . Then: h u ∗ v, w i = π (( u ∗ v ) ∗ w ) = π ( u ∗ ( v ∗ w )) = h u, v ∗ w i , so h− , −i ∈ B ∞ ( V ) .Let h− , −i ∈ B ∞ ( V ) and let ∗ = Φ V ( h− , −i ) . By lemma 3, for all u, v ∈ T ( V ) , π ( u ∗ v ) = h u, v i , so Ψ V ◦ Φ V = Id B ∞ ( V ) . Let ∗ ∈ Bialg ( V ) , h− , −i = Ψ V ( ∗ ) and ⋆ = Φ V ( h− , −i ) . Then ∗ and ⋆ are both coalgebra morphisms from T ( V ) ⊗ T ( V ) to T ( V ) , and for all u, v ∈ T ( V ) : π ( u ∗ v ) = h u, v i = π ( u ⋆ v ) . By lemma 4, ∗ = ⋆ , so Φ V ◦ Ψ V = Id Bialg ( V ) . (cid:3) Example.
Let x , x , y , y ∈ V . x ∗ y = x y + y x + h x , y i ,x x ∗ y = x x y + x y x + y x x + h x , y i x + x h x , y i + h x x , y i ,x ∗ y y = x y y + y x y + y y x + h x , y i y + y h x , y i + h x , y y i . Remarks.
1. For any vector space, there exists a trivial B ∞ structure on it, defined by: h− , −i k,l = ( Id if ( k, l ) = (0 , or (1 , , otherwise . The associated product Φ V (0) is the shuffle product (cid:1) .2. The coalgebra ( T ( V ) , ∆ dec ) is connected, so for any product ∗ ∈ Bialg ( V ) , ( T ( V ) , ∗ , ∆ dec ) is a Hopf algebra. -associative algebras Definition 5 A -associative algebra [30] is a family ( V, ∗ , m ) , where V is a vector space,and ∗ , m are both associative products on V . The operad of -associative algebras is denoted by . It is generated by ∗ and m , both in (2) , with the relations: ∗ ◦ ∗ = ∗ ◦ ∗ , m ◦ m = m ◦ m. Lemma 6
1. Let A be a -associative algebra. The products m and ∗ on A are extendedto U A = K ⊕ A , making it a -associative algebra: for all a, a ′ ∈ A , a ∗ a, ∗ a ′ = a ′ ,a a, a ′ = a ′ . . Let A and B be two -associative algebras. Then A ⊗ B = ( A ⊗ K ) ⊕ ( K ⊗ B ) ⊕ ( A ⊗ B ) isa -associative algebra: if a, a ′ ∈ A , b, b ′ ∈ B , ∗ a ′ ⊗ ⊗ b ′ a ′ ⊗ b ′ a ⊗ a ∗ a ′ ⊗ a ⊗ b ′ a ∗ a ′ ⊗ b ′ ⊗ b a ′ ⊗ b ⊗ b ∗ b ′ a ′ ⊗ b ∗ b ′ a ⊗ b a ∗ a ′ ⊗ b a ⊗ b ∗ b ′ a ∗ a ′ ⊗ b ∗ b ′ m a ′ ⊗ ⊗ b ′ a ′ ⊗ b ′ a ⊗ aa ′ ⊗ a ⊗ b ′ aa ′ ⊗ b ′ ⊗ b ⊗ bb ′ a ⊗ b a ⊗ bb ′ Moreover, if A , B and C are three -associative algebras, then ( A ⊗ B ) ⊗ C = A ⊗ ( B ⊗ C ) . Proof.
Direct verifications. (cid:3)
Definition 7 A -associative bialgebra is a family ( A, ∗ , m, ∆) , where:1. ( A, ∗ , m ) is a -associative algebra.2. ∆ : U A −→ U A ⊗ U A = ( K ⊗ K ) ⊕ ( A ⊗ A ) ≡ U ( A ⊗ A ) is a coassociative, counitarycoproduct, whose counit is the canonical projection on K , and sending to ⊗ .3. ∆ is a morphism of -associative algebras. Remark.
Let A be a -associative bialgebra.1. As the counit of U A is the canonical projection on K , for all x ∈ A : ∆( a ) = a (1) ⊗ a (2) = a ⊗ ⊗ a + a ′ ⊗ a ′′ , with a ′ ⊗ a ′′ ∈ A ⊗ A.
2. For all a, b ∈ A : ∆( a ∗ b ) = ∆( a ) ∗ ∆( b ) , ∆( ab ) = ( a ⊗ ⊗ a + a ′ ⊗ a ′′ )( b ⊗ ⊗ b + b ′ ⊗ b ′′ )= ab ⊗ a ⊗ b + ab ′ ⊗ b ′′ + 1 ⊗ ab + a ′ ⊗ a ′′ b = a (1) ⊗ a (2) b + ab (1) ⊗ b (2) − a ⊗ b. In other words, ( U A, ∗ , ∆) is a bialgebra and ( U A, m, ∆) is an infinitesimal bialgebra[2, 29, 30]. Definition 8
1. If A is a -associative algebra, we denote by P rim ( A ) the space of ele-ments a ∈ A such that ∆( a ) = a ⊗ ⊗ a .2. For all n ≥ , we denote by Prim2As ( n ) the space of elements p ∈ ( n ) such that forany -associative bialgebra A , for any a , . . . , a n ∈ P rim ( A ) , p. ( a , . . . , a n ) ∈ P rim ( A ) . Note that
Prim2As is a suboperad of . Proposition 9
Let X , . . . , X n be indeterminates, and V n be the vector space generated by X , . . . , X n . Recall that F ( V n ) is the free -associative algebra generated by V n . Let ∆ be theunique -associative algebra morphism such that: ∆ : U F ( V n ) −→ U F ( V n ) ⊗ U F ( V n ) X i , i ∈ [ n ] −→ X i ⊗ ⊗ X i , −→ ⊗ . Then ( F ( V n ) , ∗ , m, ∆) is a -associative bialgebra. Moreover, for all n ≥ : Prim2As ( n ) = { p ∈ ( n ) | p. ( X , . . . , X n ) ∈ P rim ( F ( V n )) } . roof.
1. We consider (∆ ⊗ Id ) ◦ ∆ and ( Id ⊗ ∆) ◦ ∆ . They are both morphisms of -associativealgebras from F ( V n ) to F ( V n ) ⊗ , sending X i to X i ⊗ ⊗ ⊗ X i ⊗ ⊗ ⊗ X i for all i ,so they are equal: ∆ is coassociative. Consequently, F ( V n ) is indeed a -associative bialgebra.2. ⊆ : immediate, as F ( V n ) is a -associative bialgebra and X , . . . , X n are primitiveelements of this bialgebra. ⊇ : let p ∈ ( n ) , such that p. ( X , . . . , X n ) ∈ P rim ( F ( V n )) . Let A be a -associativebialgebra and let a , . . . , a n be primitive elements of A . By universal property, there exists a -associative algebra morphism φ : F ( V n ) −→ A , sending X i to a i for all i . As a i is primitivefor all i , ∆ ◦ φ ( X i ) = ( φ ⊗ φ ) ◦ ∆( X i ) for all i ; as moreover both ( φ ⊗ φ ) ◦ ∆ and ∆ ◦ φ are -associative algebra morphisms from F ( V n ) to A ⊗ A , they are equal, so φ is a -associativebialgebra morphism. As p. ( X , . . . , X n ) is primitive, its image by φ also is: φ ( p. ( X , . . . , X n )) = p. ( φ ( X ) , . . . , φ ( X n )) = p. ( a , . . . , a n ) ∈ P rim ( A ) . So p ∈ Prim2As ( n ) . (cid:3) Remark.
The following map is injective: (cid:26) ( n ) −→ F ( V n ) p −→ p. ( X , . . . , X n ) . By restriction, the following map is injective: θ As : (cid:26) Prim2As ( n ) −→ P rim ( F ( V n )) p −→ p. ( X , . . . , X n ) . Lemma 10
Let A = ( A, ∗ , m, ∆) be a connected (as a coalgebra) -associative bialgebra.1. The following map is a coalgebra morphism: ξ : (cid:26) T ( P rim ( A )) −→ U Aa . . . a k −→ a . . . a k . From now on, we assume that A = ( T ( V ) , ∆ dec ) as a coalgebra, and that m is the concate-nation product m conc .2. For all k, l ∈ N , there exists a unique p k,l ∈ Prim2As ( k + l ) , independent of A , such thatfor all a , . . . , a k , b , . . . , b l ∈ P rim ( A ) : π ( a . . . a k ∗ b . . . b l ) = p k,l . ( a , . . . , a k , b , . . . , b l ) . Proof.
1. By the compatibility between ∆ and m , an easy induction proves that for all a , . . . , a k ∈ P rim ( A ) : ∆( a . . . a k ) = k X i =0 a . . . a i ⊗ a i +1 . . . a k . So ξ is a coalgebra morphism. As ξ | P rim ( T ( P rim ( A ))) = ξ | P rim ( A )) = Id P rim ( A ) is injective, ξ isinjective. Let us prove that ξ is surjective. Let ( A n ) be the coradical filtration of U A . We usethe notations of the proof of lemma 4. For all a ∈ A n , let us prove that a ∈ Im ( ξ ) by inductionon n . If n = 0 , we can assume that a is the unique group-like of A . Then a = ξ (1) . Let usassume the result at all rank < n . We can suppose that ǫ ( a ) = 0 , that is to say a ∈ A . Then: ˜∆ ( n − dec ( a ) = a ⊗ . . . ⊗ a n ∈ P rim ( A ) ⊗ n , ˜∆ ( n − dec ( a ) = ˜∆ ( n − ( a . . . a n ) . As a consequence, a − a . . . a n ∈ A n − . By the inductionhypothesis, a − a . . . a n ∈ Im ( ξ ) , so a ∈ Im ( ξ ) .2. Let us prove the existence of p k,l by induction on n = k + l . Firstly, if k = 0 , π (1 ∗ b . . . b l ) = π ( b . . . b l ) = ( b if l = 1 , otherwise.So we take p ,l = δ l, I . Similarly, we take p k, = δ k, I . We now assume that k, l ≥ . There isnothing more to prove if n ≤ . Let us assume that n ≥ . By lemma 4, if a , . . . , a k , b , . . . , b l ∈ V : a . . . a k ∗ b . . . b l = k + l X i =2 X a ...a k = u ...u i ,b ...b l = v ...v i p lg ( u ) ,lg ( v ) . ( u , v ) . . . p lg ( u i ) ,lg ( v i ) . ( u i , v i )+ π ( a . . . a k ∗ b . . . b l ) . So there exists p k,l ∈ ( k + l ) , such that π ( a . . . a k ∗ b . . . b l ) = p k,l . ( a , . . . , a k , b , . . . , b l ) . Bydefinition, p k,l ∈ Prim2As . By the injectivity of θ As , p k,l is unique. (cid:3) Theorem 11
The operad B ∞ is isomorphic to the operad Prim2As , through the morphism: (cid:26) B ∞ −→ Prim2As h− , −i k,l −→ p k,l . Proof.
By theorem 2, for any connected -associative bialgebra A , p k,l defines a B ∞ algebrastructure on P rim ( A ) , which gives the existence of this morphism.Let V be a space, ( W, h− , −i ) be a B ∞ algebra and f : V −→ W be a linear map.By restriction, F ( V ) contains F Prim2As ( V ) ; by definition of Prim2As , F Prim2As ( V ) ⊆ P rim ( F ( V )) . If ∗ = Φ W ( h− , −i ) , ( T ( W ) , ∗ , m, ∆) is a -associative algebra. There exists aunique -associative algebra morphism F : F ( V ) −→ T ( W ) , such that F ( x ) = f ( x ) for all x ∈ V . By construction, for all x , . . . , x k , y , . . . , y l ∈ V : F ( p k,l . ( x . . . x k , y . . . y l )) = F ◦ π ( x . . . x k ∗ y . . . y l )= π ( f ( x ) . . . f ( x k ) ∗ f ( y ) . . . f ( y l ))= h f ( x ) . . . f ( x k ) , f ( y ) . . . f ( y l ) i k,l . So the restriction of F to F Prim2As ( V ) is a morphism of B ∞ algebras taking its values in W .Finally, F Prim2As ( V ) satisfies a universal property in the category of B ∞ algebras: consequently,the operad morphism from B ∞ to Prim2As is an isomorphism. (cid:3) B ∞ Definition 12 [19, 39] The operad
Brace is the quotient of B ∞ by the operadic ideal gen-erated by the elements h− , −i k,l , k ≥ , l ≥ . Consequently, a brace algebra is a vector space V equipped with a map h− , −i : V ⊗ T ( V ) −→ V , such that: • For all x ∈ V , h x, i = x . • For all x, y , . . . , y k ∈ V , for all tensor w ∈ T ( V ) : hh x, y . . . y k i , w i = X w = w ...w k +1 h x, w h y , w i w . . . w k − h y k , w k i w k +1 i . (2.4)20ecall that a dendriform algebra is a family ( V, ≺ , ≻ ) , where V is a vector space, and ≺ , ≻ are bilinear products on V such that for any x, y, z ∈ V : ( x ≺ y ) ≺ z = x ≺ ( y ≺ z + y ≻ z ) , ( x ≻ y ) ≺ z = x ≻ ( y ≺ z ) ,x ≻ ( y ≻ z ) = ( x ≺ y + x ≻ y ) ≻ z. This implies that ∗ = ≺ + ≻ is associative. Theorem 13
Let V be a brace algebra. Then T + ( V ) is a dendriform Hopf algebra [28, 39,14], with the deconcatenation coproduct and the products given in the following way: for all x , . . . , x k ∈ V , for all v = y . . . y l ∈ T ( V ) , with ( k, l ) = (0 , , x . . . x k ∗ v = X v = v ...v k +1 v h x , v i v . . . v k − h x k , v k i v k +1 ,x . . . x k ≺ v = X v = v ...v k h x , v i v . . . v k − h x k , v k − i v k ,x . . . x k ≻ v = X v = v ...v k +1 ,v =1 v h x , v i v . . . v k − h x k , v k i v k +1 Proof.
As brace algebras are also B ∞ algebras, theorem 2 can be applied. The product ∗ = Φ V ( h− , −i ) is given by the announced formula: we immediately obtain that ( T ( V ) , ∗ , ∆) isa Hopf algebra.Let u, v, w ∈ T ( V ) , u being a non-empty tensor. The formulas defining ∗ and ≺ give, withSweedler’s notations: u ∗ v = v (1) ( u ≺ v (2) ) , ( uv ) ∗ w = ( u ∗ w (1) )( v ≺ w (2) ) , ( uv ) ≺ w = ( u ≺ w (1) )( v ≺ w (2) ) . By subtraction, we also obtain that ( uv ) ≻ w = ( u ≻ w (1) )( v ≺ w (2) ) . We consider: A = { u ∈ T ( V ) | ∀ v, w ∈ T + ( V ) , ( u ≺ v ) ≺ w = u ≺ ( v ∗ w ) } . Firstly, ∈ A : indeed, for all v, w ∈ T + ( V ) , (1 ≺ v ) ≺ w = 0 = 1 ≺ ( v ∗ w ) . Let us assume that u ∈ A and let x ∈ V . For all v, w ∈ T + ( V ) : ( ux ≺ v ) ≺ w = (( u ≺ v (1) )( x ≺ v (2) )) ≺ w = (( u ≺ v (1) ) ≺ w (1) )(( x ≺ v (2) ) ≺ w (2) )= ( u ≺ ( v (1) ∗ w (1) ))(( h x, v (2) i v (3) ) ≺ w (2) )= ( u ≺ ( v (1) ∗ w (1) ))( h x, v (2) i ≺ w (2) )( v (3) ≺ w (3) )= ( u ≺ ( v (1) ∗ w (1) )) hh x, v (2) i , w (2) i w (3) ( v (3) ≺ w (4) )= ( u ≺ ( v (1) ∗ w (1) )) h x, v (2) ∗ w (2) i ( v (3) ∗ w (3) )= ( u ≺ ( v ∗ w ) (1) ) h x, ( v ∗ w ) (2) i ( v ∗ w ) (3) = ( u ≺ ( v ∗ w ) (1) )( x ≺ ( v ∗ w ) (2) )= ( ux ) ≺ ( v ∗ w ) . So ux ∈ A . As a consequence, A = T ( V ) : for all u, v, w ∈ T + ( V ) , ( u ≺ v ) ≺ w = u ≺ ( v ∗ w ) .21et u = x . . . x k ∈ T ( V ) , with k ≥ , v ∈ T ( V ) and y ∈ V . Then: ∆ dec ( yv ) = yv (1) ⊗ v (2) + 1 ⊗ yv, so: u ≻ yv = u ∗ yv − u ≺ v = yv (1) h x , v (2) v (3) . . . v (2 k − h x k , v (2 k ) i v (2 k +1) + h x , ( yv ) (1) i ( yv ) (2) . . . ( yv ) (2 k − h x k , ( yv ) (2 k − i ( yv ) (2 k ) − h x , ( yv ) (1) i ( yv ) (2) . . . ( yv ) (2 k − h x k , ( yv ) (2 k − i ( yv ) (2 k ) = y (cid:16) v (1) h x , v (2) i v (3) . . . v (2 k − h x k , v (2 k ) i v (2 k +1) (cid:17) = y ( u ∗ v ) . Let B = { w ∈ T ( V ) | ∀ u, v ∈ T + ( V ) , ( u ∗ v ) ≻ w = u ≻ ( v ≻ w ) } . Firstly, ∈ B : indeed, if u, v ∈ T + ( V ) , ( u ∗ v ) ≻ u ≻ ( v ≻ . Let w ∈ B and z ∈ V . For all u, v ∈ T + ( V ) : ( u ∗ v ) ≻ zw = z (( u ∗ v ) ∗ w ) = z ( u ∗ ( v ∗ w )) = u ≻ ( z ( v ∗ w )) = u ≻ ( v ≻ ( zw )) . So zw ∈ B : consequently, B = T ( V ) . For all u, v, w ∈ T + ( V ) , ( u ∗ v ) ≻ w = u ≻ ( v ≻ w ) . So ( T + ( V ) , ≺ , ≻ ) is a dendriform algebra.Let us prove the axioms of a dendriform Hopf algebra, that is to say for all u, v ∈ T + ( V ) : ˜∆ dec ( u ≺ v ) = u ⊗ v + u ′ ⊗ u ′′ ∗ v + u ≺ v ′ ⊗ v ′′ + u ′ ≺ v ⊗ u ′′ + u ′ ≺ v ′ ⊗ u ′′ ∗ v ′′ , ˜∆ dec ( u ≻ v ) = v ⊗ y + v ′ ⊗ u ∗ v + u ≻ v ′ ⊗ v ′′ + u ′ ≻ v ⊗ u ′′ + u ′ ≻ v ′ ⊗ u ′′ ∗ v ′′ . As we already know that ( T ( V ) , ∗ , ∆ dec ) is a bialgebra, it is enough to prove one of these tworelations, say the second one. Let y ∈ V and u ∈ T + ( V ) . ˜∆ dec ( u ≻ y ) = ˜∆ dec ( yu ) = y ⊗ u + yu ′ ⊗ u ′′ = y ⊗ u + u ′ ≻ y ⊗ u ′′ , which proves the relation for v ∈ V , as then ˜∆ dec ( v ) = 0 . If v is a tensor of length ≥ , we writeit as v = yw , with y ∈ V and w ∈ T + ( V ) . Then: ˜∆ dec ( u ≻ v ) = ˜∆ dec ( y ( u ∗ w ))= y ⊗ u ∗ w + yu ⊗ w + yw ⊗ u + y ( u ′ ∗ w ) ⊗ u ′′ + yu ′ ⊗ u ′′ ∗ w + y ( u ∗ w ′ ) ⊗ w ′′ + yw ′ ⊗ u ∗ w ′′ + y ( u ′ ∗ w ′ ) ⊗ u ′′ ∗ w ′′ = yw ⊗ u + ( y ⊗ u ∗ w + yw ′ ⊗ u ∗ w ′′ ) + ( yu ⊗ w + y ( u ∗ w ′ ) ⊗ w ′′ )+ y ( u ′ ∗ w ) ⊗ u ′′ + ( yu ′ ⊗ u ′′ ∗ w + y ( u ′ ∗ w ′ ) ⊗ u ′′ ∗ w ′′ )= yw ⊗ u + ( y ⊗ u ∗ w + yw ′ ⊗ u ∗ w ′′ ) + ( u ≻ y ⊗ w + u ≻ yw ′ ⊗ w ′′ )+ u ′ ≻ yw ⊗ u ′′ + ( u ′ ≻ y ⊗ u ′′ ∗ w + u ′ ≻ yw ′ ⊗ u ′′ ∗ w ′′ )= v ⊗ u + v ′ ⊗ u ∗ v ′′ + u ≻ v ′ ⊗ v ′′ + u ′ ≻ v ⊗ u ′′ + u ′ ≻ v ′ ⊗ u ′′ ∗ v ′′ . So the second compatibility is verified. (cid:3)
Remarks.
1. The products ≺ and ≻ can also be inductively defined: let x , . . . , x k , y , . . . , y l ∈ V , with22 , l ≥ . x . . . x k ≺ x . . . x k , ≺ y . . . y l = 0 ,x . . . x k ≺ y . . . y l = l X p =0 h x , y . . . y p i ( x . . . x k ∗ y p +1 . . . y l ) ,x . . . x k ≻ , ≻ y . . . y l = y . . . y l ,x . . . x k ≻ y . . . y l = y ( x . . . x k ∗ y . . . y l ) .
2. If ( V, h− , −i ) is brace, putting ∗ = Φ V ( h− , −i ) , (2.4) can be written in the following way: ∀ x ∈ V, u, v ∈ T ( V ) , hh x, u i , v i = h x, u ∗ v i . (2.5) Proposition 14
The quotient of B ∞ by the operadic ideal generated by the elements h− , −i k,l , k ≥ or l ≥ , is isomorphic to the operad As of associative algebras. Proof.
This quotient is generated by h− , −i , . The unique relation defining B ∞ algebraswhich does not become trivial in this quotient is: ∀ x, y, z ∈ V, hh x, y i , z i = h x, h y, z ii . So this quotient is indeed As . (cid:3) Consequently, if ( V, · ) is an associative algebra, it is also a B ∞ algebra; T ( V ) becomes abialgebra with the product ∗ = Φ V ( · ) . For all x , . . . , k k ∈ V , v a word in T ( V ) : x . . . x k ∗ v = X v = v ...v k +1 ,lg ( v ) ,...,lg ( v k ) ≤ v ( x · v ) v . . . v k − ( x k · v k ) v k +1 , with the convention x · x for all x ∈ V . This is the quasi-shuffle product associated to · [24, 18]. It is a dendriform algebra, with the following products: x . . . x k ≺ v = X v = v ...v k ,lg ( v ) ,...,lg ( v k − ) ≤ ( x · v ) v . . . v k − ( x k · v k − ) v k ,x . . . x k ≻ v = X v = v ...v k +1 ,v =1 , lg ( v ) ,...,lg ( v k ) ≤ v ( x · v ) v . . . v k − ( x k · v k ) v k +1 , Definition 15
Let V be a brace algebra. A brace module over V is a vector space M with amap: ← : (cid:26) M ⊗ T ( V ) −→ Mm ⊗ x . . . x k −→ m ← x . . . x k , such that: • For all m ∈ M , m ← m . • For all m ∈ M , y , . . . , y k ∈ V , for all tensor w ∈ T ( V ) : ( m ← y . . . y k ) ← w = m ← X w = w ...w k +1 w h y , w i w . . . w k − h y k , w k i w k +1 . (2.6)23 emark. Let ∗ = Φ V ( h− , −i ) . Then (2.6) can be rewritten as: ∀ m ∈ M, u, v ∈ T ( V ) , ( m ← u ) ← v = m ← ( u ∗ v ) . (2.7)So a brace module over V is a (right) module over the algebra ( T ( V ) , ∗ ) . Example. If V is a brace algebra, ( V, h− , −i ) is a brace module over itself. Notation . Let V be a graded space, such that the homogeneous components of V are finite-dimensional. We denote by V ⊛ the dual of V . The graded dual of V is: V ∗ = M n ≥ V ∗ n ⊆ V ⊛ . If V = (0) , then S ( V ) and T ( V ) are also graded spaces, and S ( V ) ∗ is isomorphic to S ( V ∗ ) ; T ( V ) ∗ is isomorphic to T ( V ∗ ) . Definition 16
Let V be a graded B ∞ algebra. We shall say that V is -bounded if: • For all n ≥ , V n is finite-dimensional. • For all m, n ≥ , there exists B ( m, n ) ≥ such that for all p , . . . , p k , q , . . . , q l ≥ with p + . . . + p k = m and q + . . . + q l = n : ♯ { i | p i = 0 } + ♯ { j | q j = 0 } > B ( m, n ) = ⇒ h V p . . . V p k , V q . . . V q l i = (0) . Examples.
1. If V is connected, that is to say if V = (0) , then V is -bounded, with B ( m, n ) = 0 for all m, n .2. If V is associative, then h− , −i k,l = 0 if k ≥ or l ≥ . Consequently, V is -bounded,with B ( m, n ) = 2 for all m, n .Let us assume that V is -bounded. We identify T ( V ∗ ) with a subspace of T ( V ) ∗ by thepairing ≪ − , − ≫ ′ such that for all x , . . . , x l ∈ V , f , . . . , f k ∈ V ∗ : ≪ f . . . f k , x . . . x l ≫ ′ = ( if k = l,f ( x ) . . . f k ( x k ) if k = l. Note that for all
F, G ∈ T ( V ∗ ) , X, Y ∈ T ( V ) : ≪ ∆ dec ( F ) , X ⊗ Y ≫ ′ = ≪ F, XY ≫ , ≪ F ⊗ G, ∆ dec ( X ) ≫ ′ = ≪ F G, X ≫ . Proposition 17
Let V be a -bounded B ∞ algebra. We define a coproduct ∆ ′∗ on T ( V ∗ ) asthe unique algebra morphism such that for all f ∈ V ∗ , for all X, Y ∈ T ( V ) , ≪ ∆ ′∗ ( f ) , X ⊗ Y ≫ ′ = ≪ f, h X, Y i ≫ ′ . Then ( T ( V ∗ ) , m conc , ∆ ′∗ ) is a graded bialgebra. Moreover, for all F, G ∈ T ( V ∗ ) , X, Y ∈ T ( V ) : ≪ ∆ ′∗ ( F ) , X ⊗ Y ≫ ′ = ≪ F, X ∗ Y ≫ ′ , ≪ F ⊗ G, ∆ dec ( Y ) ≫ ′ = ≪ F G, X ≫ ′ . In other words, ≪ − , − ≫ ′ is a Hopf pairing. roof. The B ∞ bracket can be dualized into a map δ from V ∗ to ( T ( V ) ⊗ T ( V )) ∗ . Un-fortunately, if V = (0) , the homogeneous components of T ( V ) are not finite-dimensional, so T ( V ∗ ) ⊗ T ( V ∗ ) is (identified to) a strict subspace of ( T ( V ) ⊗ T ( V )) ∗ . But, by the -boundedcondition, for all N ≥ : δ ( V ∗ N ) ⊆ X m + n = N X p + ... + p k = m,q + ... + q l = n,♯ { i | p i =0 } + ♯ { j | q j =0 }≤ B ( m,n ) V ∗ p . . . V ∗ p k ⊗ V ∗ q . . . V ∗ q l . As this is a finite sum, δ ( V ∗ N ) ⊆ T ( V ∗ ) ⊗ T ( V ∗ ) . We can define an algebra morphism ∆ ′∗ from T ( V ∗ ) to T ( V ∗ ) ⊗ T ( V ∗ ) by ∆ ′∗ ( f ) = δ ( f ) for all f ∈ V ∗ .Let us consider: A = { F ∈ T ( V ∗ ) | ∀ X, Y ∈ T ( V ) , ≪ ∆ ′∗ ( F ) , X ⊗ Y ≫ ′ = ≪ F, X ∗ Y ≫ ′ . } . Firstly, ∈ A : for all X, Y ∈ T ( V ) , ≪ ∆ ′∗ (1) , X ⊗ Y ≫ ′ = ≪ ⊗ , X ⊗ Y ≫ ′ = ε ( X ) ε ( Y ) = ε ( X ∗ Y ) = ≪ , X ∗ Y ≫ ′ . Let
F, G ∈ A . For all X, Y ∈ T ( V ) : ≪ ∆ ′∗ ( F G ) , X ⊗ Y ≫ ′ = ≪ F (1) ∗ G (1) ∗ ⊗ F (2) ∗ G (2) ∗ , X ⊗ Y ≫ ′ = ≪ F (1) ∗ ⊗ G (1) ∗ ⊗ F (2) ∗ ⊗ G (2) ∗ , X (1) ⊗ X (2) ⊗ Y (1) ⊗ Y (2) ≫ ′ = ≪ F ⊗ G, X (1) ∗ Y (1) ⊗ X (2) ∗ Y (2) ≫ ′ = ≪ F ⊗ G, ( X ∗ Y ) (1) ⊗ ( X ∗ Y ) (2) ≫ ′ = ≪ F G, X ∗ Y ≫ ′ . So A is a subalgebra of ( T ( V ∗ ) , m conc ) . In order to prove that ≪ − , − ≫ ′ is a Hopf pairing, itis enough to prove that V ∗ ⊆ A . Let f ∈ V ∗ . For all X, Y ∈ T ( V ) : ≪ ∆ ′∗ ( f ) , X ⊗ Y ≫ ′ = ≪ f, h X, Y i ≫ ′ = ≪ f, π ( X ∗ Y ) ≫ ′ = ≪ f, X ∗ Y ≫ ′ +0= ≪ f, X ∗ Y ≫ ′ . So ≪ − , − ≫ ′ is a Hopf pairing.Let F ∈ T ( V ∗ ) , X, Y, Z ∈ T ( V ) . ≪ (∆ ′∗ ⊗ Id ) ◦ ∆ ′∗ ( F ) , X ⊗ Y ⊗ Z ≫ ′ = ≪ F, ( X ∗ Y ) ∗ Z ≫ ′ = ≪ F, X ∗ ( Y ∗ Z ) ≫ ′ = ≪ ( Id ⊗ ∆ ′∗ ) ◦ ∆ ′∗ ( F ) , X ⊗ Y ⊗ Z ≫ ′ . As the pairing is non degenerate, ∆ ′∗ is coassociative: ( T ( V ∗ ) , m conc , ∆ ′∗ ) is a bialgebra. (cid:3) Remark. If V is connected, then ( T ( V ∗ ) , m conc , ∆ ′∗ ) is a graded, connected bialgebra, so isa Hopf algebra, isomorphic to the graded dual of ( T ( V ) , ∗ , ∆ dec ) . Corollary 18
Let V be a -bounded B ∞ algebra. • T ( V ∗ ) is a subbialgebra of T ( V ∗ ) . Let I be the ideal of T ( V ∗ ) generated by V ∗ . Then I is a biideal of T ( V ∗ ) , and T ( V ∗ ) /I is a graded, connected Hopf algebra, isomorphic to the graded dual of T ( V + ) . Proof.
We already noticed that: δ ( V ∗ ) ⊆ X k + l ≤ B (0 , V ⊗ k ⊗ V ⊗ l , so ∆ ′∗ ( V ∗ ) ⊆ T ( V ∗ ) ⊗ T ( V ∗ ) : T ( V ∗ ) is a subbialgebra.As V + is a B ∞ subalgebra of V , T ( V + ) is a Hopf subalgebra of T ( V ) : its orthogonal J is abiideal of T ( V ∗ ) . Moreover, if x , . . . , x k ∈ V + , for all f ∈ V ∗ : ≪ f, x . . . x k ≫ ′ = ( if k = 1 ,f ( x ) if k = 1= 0 . So I ⊆ J . Moreover, the following map is an algebra isomorphism: (cid:26) T ( V ∗ + ) −→ T ( V ∗ ) /I f . . . f k −→ f . . . f k . For all f , . . . , f k ∈ V ∗ + , x , . . . , x l ∈ V + : ≪ f . . . f k , y . . . y l ≫ ′ = ≪ f . . . f k , y . . . y l ≫ ′ = ( if k = l,f ( x ) . . . f k ( x k ) if k = l. So the pairing induced by ≪ − , − ≫ ′ between T ( V ∗ ) /I and T ( V + ) is non degenerate: therefore, I = J . Finally, T ( V ∗ ) /I is the graded dual of ( T ( V + ) , ∗ , ∆) , so is a Hopf algebra. (cid:3) b ∞ algebras Definition 19
Let V be a vector space equipped with a map: ⌊− , −⌋ : (cid:26) S ( V ) ⊗ S ( V ) −→ Vx . . . x k ⊗ y . . . y l −→ ⌊ x . . . x k , y . . . y l ⌋ . For all k, l , let us put ⌊− , −⌋ k,l = ⌊− , −⌋ | S k ( V ) ⊗ S l ( V ) . We shall say that V is a b ∞ algebra if thefollowing properties are satisfied: • ⌊− , −⌋ , = ⌊− , −⌋ , = ( Id V if k = 1 , otherwise . • We shall need the following notations: let u = x . . . x k ∈ S k ( V ) and v = y . . . y l ∈ S l ( V ) .Let I ⊆ { , . . . , k + l } . We put I = { i , . . . , i p , j , . . . , j q } , with i < . . . < i p ≤ k < j < . . . < j q . We then put: ⌊ u, v ⌋ I = ⌊ x i . . . x i p , y j . . . y j q ⌋ . For all u = x . . . x k ∈ S k ( V ) , v = y . . . y l ∈ S l ( V ) , w = z . . . z m ∈ S m ( V ) , X { I ,...,I p } partition of [ l + m ] ⌊ u, ⌊ v, w ⌋ I . . . ⌊ v, w ⌋ I p ⌋ = X { I ,...,I p } partition of [ k + l ] ⌊⌊ u, v ⌋ I . . . ⌊ u, v ⌋ I p , w ⌋ . (2.8)26 he operad of b ∞ algebras is denoted by b ∞ . Theorem 20
Let V be a vector space. Let bialg ( V ) be the set of products ⋆ on T ( V ) , making ( S ( V ) , ⋆, ∆) a bialgebra. Let b ∞ ( V ) be the set of b ∞ structures on V . These two sets are inbijections, via the maps: φ V : b ∞ ( V ) −→ bialg ( V ) ⌊− , −⌋ −→ ⋆ defined by u ⋆ v = X { I ,...,I p } partition of [ k + l ] ⌊ u, v ⌋ I . . . ⌊ u, v ⌋ I p ψ V : (cid:26) bialg ( V ) −→ b ∞ ( V ) ⋆ −→ ⌊− , −⌋ defined by ⌊ u, v ⌋ = π ( u ⋆ v ) , where π is the canonical projection on V . The proof of the following theorem is similar to the proof of theorem 2. In particular, itsproof uses the following lemma:
Lemma 21
Let C be a connected coalgebra and let φ, ψ : C −→ S ( V ) be two coalgebramorphisms. Then φ = ψ if, and only if, π ◦ φ = π ◦ ψ . Example.
Let x , x , y , y ∈ V . x ⋆ y = x y + ⌊ x , y ⌋ ,x ⋆ y y = x y y + ⌊ x , y ⌋ y + ⌊ x , y ⌋ y + ⌊ x , y y ⌋ x x ⋆ y y = x x y y + ⌊ x , y ⌋ x y + ⌊ x , y ⌋ x y + ⌊ x , y y ⌋ x + ⌊ x , y ⌋ x y + ⌊ x , y ⌋ x y + ⌊ x , y y ⌋ x + ⌊ x x , y ⌋ y + ⌊ x x , y ⌋ y + ⌊ x x , y y ⌋ + ⌊ x , y ⌋⌊ x , y ⌋ + ⌊ x , y ⌋⌊ x , y ⌋ . Remarks.
1. The coalgebra ( S ( V ) , ∆) is connected so, for any ⋆ ∈ bialg ( V ) , ( S ( V ) , ⋆, ∆) is a Hopfalgebra.2. Any vector space V admits a trivial b ∞ structure, defined by: ⌊− , −⌋ k,l = ( Id V if ( k, l ) = (1 , or (0 , , otherwise . The associated product is the usual one of S ( V ) . Definition 22
1. An associative-commutative algebra is a -associative algebra ( A, ⋆, m ) ,such that m is commutative. The operad of associative-commutative algebras is denoted by AsCom . It is generated by m and ⋆ , both in AsCom (2) , with the relations: m (12) = m, m ◦ m = m ◦ m, ⋆ ◦ ⋆ = ⋆ ◦ ⋆.
2. An associative-commutative algebra is a family ( A, ⋆, m, ∆) , such that ( A, ⋆, m ) is anassociative-commutative algebra, and both ( U A, ⋆, ∆) and ( U A, m, ∆) are bialgebras. . For all n ≥ , we denote by PrimAsCom ( n ) the subspace of elements p ∈ AsCom ( n ) such that for any associative-commutative bialgebra A : ∀ a , . . . , a n ∈ P rim ( A ) , p. ( a , . . . , a n ) ∈ P rim ( A ) . This is a suboperad of
AsCom . As for B ∞ algebras: Theorem 23
The operads b ∞ and PrimAsCom are isomorphic. b ∞ Recall that a (right) pre-Lie algebra –also called a right symmetric algebra or a Vinberg algebra–is a vector space V together with a bilinear product • such that: ∀ x, y, z ∈ V, x • ( y • z ) − ( x • y ) • z = x • ( z • y ) − ( x • z ) • y. Proposition 24
The quotient of b ∞ by the operadic ideal generated by ⌊− , −⌋ k,l , k ≥ , isisomorphic to the operad PreLie , generated by • ∈
PreLie (2) and the relation: • ◦ • − • ◦ • = ( • ◦ • − • ◦ • ) (23) . Proof.
We denote by b ∞′ this quotient of b ∞ . If V is a b ∞′ algebra and x, y, z ∈ V , in S ( V ) , x ⋆ y = xy + ⌊ x, y ⌋ . So: ⌊⌊ x, y ⌋ , z ⌋ − ⌊ x, ⌊ y, z ⌋⌋ = ⌊ x ⋆ y, z ⌋ + ⌊ xy, z ⌋ − ⌊ x, y ⋆ z ⌋ − ⌊ x, yz ⌋ = ⌊ x ⋆ y, z ⌋ − ⌊ x, y ⋆ z ⌋ − ⌊ x, yz ⌋ = ⌊ x, yz ⌋ . As yz = zy in S ( V ) , ⌊− , −⌋ , is pre-Lie. We obtain an operad morphism: Φ : (cid:26)
PreLie −→ b ′∞ • −→ ⌊− , −⌋ , . Moreover, in a b ∞′ algebra V , if x, y , . . . , y k +1 ∈ V : y . . . y k ⋆ y k +1 = y . . . y k +1 + k X i =1 y . . . y i − ⌊ y i , y k +1 ⌋ y i +1 . . . y k , so: ⌊ x, y . . . y k +1 ⌋ = ⌊ x, y . . . y k ⋆ y k +1 ⌋ − k X i =1 ⌊ x, y . . . y i − ⌊ y i , y k +1 ⌋ y i +1 . . . y k ⌋ = ⌊⌊ x, y . . . y k ⌋ , y k +1 ⌋ − k X i =1 ⌊ x, y . . . y i − ⌊ y i , y k +1 ⌋ y i +1 . . . y k ⌋ . An easy induction proves that b ′∞ is generated by ⌊− , −⌋ , , so Φ is surjective.Let ( V, • ) be a pre-Lie algebra. The Oudom-Guin construction [37] allows to construct aproduct ⋆ on S ( V ) , making it a Hopf algebra, isomorphic to the enveloping algebra of V . So V is b ∞ . Moreover, for all x , . . . , x k , y , . . . , y l ∈ V , with the notations of [37]: π ( x . . . x k ⋆ y . . . y l ) = ( if k = 1 ,x • y . . . y l if k = 1 . V is a b ′∞ -algebra: we obtain an operad morphism: φ ′ : (cid:26) b ∞′ −→ PreLie ⌊− , −⌋ , −→ • . Then Φ ′ ◦ Φ = Id PreLie , so Φ is injective. (cid:3) Remark.
As noticed in the preceding proof, if ( V, • ) is a pre-Lie algebra, ⋆ = φ V ( • ) is theOudom-Guin construction. In particular, its b ∞ brackets can be inductively computed: • For all x ∈ V , ⌊ x, ⌋ = x • x . • For all x, y ∈ V , ⌊ x, y ⌋ = x • y . • For all x, x , . . . , x k ∈ V , ⌊ x, x . . . x k ⌋ = ⌊⌊ x, x . . . x k − ⌋ , x k ⌋ − k − X i =1 ⌊ x, x . . . ⌊ x i , x k ⌋ . . . x k − ⌋ . ⌊ x, x . . . x k ⌋ is denoted by x • x . . . x k in [37]. Corollary 25
The quotient of b ∞ by the operadic ideal generated by ⌊− , −⌋ k,l , k ≥ or l ≥ , is isomorphic to the operad As . Proof.
The antecedent by Φ of the element ⌊− , −⌋ , is • ◦ • − • ◦ • . So this quotient of b ∞ is isomorphic to the quotient of PreLie by the element • ◦ • − • ◦ • . This quotient is generatedby the class m of • and the relation m ◦ m = m ◦ m , so it is As . (cid:3) B ∞ algebras to b ∞ algebras Notations.
We denote by coS ( V ) the subalgebra of ( T ( V ) , (cid:1) ) generated by V ; it is a Hopfsubalgebra of ( T ( V ) , (cid:1) , ∆ dec ) . As the characteristic of the base field K is zero, this Hopf algebrais isomorphic to S ( V ) via the morphism: ι V : (cid:26) S ( V ) −→ coS ( V ) x . . . x k −→ x (cid:1) . . . (cid:1) x k . Lemma 26 coS ( V ) is the greatest cocommutative subcoalgebra of T ( V ) .2. Let ∗ ∈ Bialg ( V ) . The subalgebra of ( T ( V ) , ∗ ) generated by V is coS ( V ) . Proof. coS ( V ) is indeed a cocommutative coalgebra of T ( V ) . Let C be a cocommutativecoalgebra of T ( V ) . The subalgebra C ′ generated for the shuffle product by C is then a cocom-mutative subbialgebra of T ( V ) . As T ( V ) is connected as a coalgebra, C ′ also is: by the Cartier-Quillen-Milnor-Moore theorem, C ′ is generated by P rim ( C ′ ) . As P rim ( C ′ ) ⊆ P rim ( T ( V )) = V , C ⊆ C ′ ⊆ coS ( V ) .2. As ∗ is a coalgebra morphism and coS ( V ) ⊗ is a cocommutative subcoalgebra of T ( V ) ⊗ , coS ( V ) ∗ coS ( V ) is a cocommutative subcoalgebra of T ( V ) . By the first point, it is included in coS ( V ) . So coS ( V ) is a subalgebra of ( T ( V ) , ∗ ) , containing V . Denoting by A the subalgebraof T ( V ) , ∗ ) generated by V , A ⊆ coS ( V ) . Moreover, for all x , . . . , x k ∈ V , by theorem 2: x ∗ . . . ∗ x k = x (cid:1) . . . (cid:1) x k + a sum of words of length < k. By a triangularity argument, coS ( V ) ⊆ A . (cid:3) V be a B ∞ algebra, and let ∗ = φ V ( h− , −i ) . By the preceding lemma, coS ( V ) is asubbialgebra of ( T ( V ) , ∗ , ∆ dec ) . Via ι V , S ( V ) is also a bialgebra, so V is a b ∞ algebra. Thisstructure is given by: ⌊ x . . . x k , y . . . y l ⌋ = h x (cid:1) . . . (cid:1) x k , y (cid:1) . . . (cid:1) y l i . At the operadic level, we obtain:
Proposition 27
There is an operad morphism Φ from b ∞ to B ∞ , such that: ∀ k, l ≥ , Φ( ⌊− , −⌋ k,l ) = X σ ∈ S k ,τ ∈ S l h− , −i σ ⊗ τk,l . We finally obtain a commutative diagram of operads: b ∞ Φ / / (cid:15) (cid:15) (cid:15) (cid:15) B ∞ (cid:15) (cid:15) (cid:15) (cid:15) PreLie / / % % % % ❑❑❑❑❑❑❑❑❑❑ Brace y y y y ssssssssss As (cid:15) (cid:15) (cid:15) (cid:15) Com
Definition 28
Let V be a graded b ∞ algebra. We shall say that V is -bounded if: • For all n ≥ , V n is finite-dimensional. • For all m, n ≥ , there exists B ( m, n ) ≥ such that: k + l ≥ B ( m, n ) = ⇒ ⌊ S k ( V ) S ( V ) m , S l ( V ) S ( V ) n ⌋ = (0) . Examples.
1. If V is connected, that is to say if V = (0) , then V is -bounded, with B ( m, n ) = 0 for all m, n .2. If V is associative, then h− , −i k,l = 0 if k ≥ or l ≥ . Consequently, V is -bounded,with B ( m, n ) = 2 for all m, n .3. If V is a -bounded B ∞ algebra, it is also a -bounded b ∞ algebra.Let us assume that V is -bounded. We identify S ( V ∗ ) with a subspace of S ( V ) ∗ by thepairing ≪ − , − ≫ ′ induced by the pairing between V ∗ and V . More precisely, let us choose abasis ( x i ) i ∈ I of V , made of homogeneous elements of V . The dual basis of V ∗ is denoted by ( f i ) i ∈ I . We shall need the following notations: • We denote by Λ the set of sequences of positive integers ( α i ) i ∈ I whose support is finite. • For all α = ( α i ) i ∈ I , we put: x α = Y i ∈ I x α i i , f α = Y i ∈ I f α i i , α ! = Y i ∈ I α i ! ( x α ) α ∈ Λ is a basis of S ( V ) , ( f α ) α ∈ Λ is a basis of S ( V ∗ ) , and the pairing is given by: ≪ f α , x β ≫ = α ! δ α,β . Note that for all
F, G ∈ S ( V ∗ ) , X, Y ∈ S ( V ) : ≪ ∆( F ) , X ⊗ Y ≫ = ≪ F, XY ≫ , ≪ F ⊗ G, ∆( Y ) ≫ = ≪ F G, X ≫ . Proposition 29
Let V be a -bounded b ∞ algebra. We define a coproduct ∆ ∗ on S ( V ∗ ) asthe unique algebra morphism such that for all f ∈ V ∗ , for all X, Y ∈ S ( V ) , ≪ ∆ ∗ ( f ) , X ⊗ Y ≫ = ≪ f, ⌊ X, Y ⌋ ≫ . Then ( S ( V ∗ ) , ∆ ∗ ) is a graded bialgebra. Moreover, for all F, G ∈ S ( V ∗ ) , X, Y ∈ S ( V ) : ≪ ∆ ∗ ( F ) , X ⊗ Y ≫ = ≪ F, X ∗ Y ≫ , ≪ F ⊗ G, ∆( Y ) ≫ = ≪ F G, X ≫ . In other words, ≪ − , − ≫ is a Hopf pairing. Proof.
Similar to the proof of proposition 17. (cid:3)
Remarks.
1. If V is a -bounded B ∞ algebra, then ( S ( V ∗ ) , m, ∆ ∗ ) is the abelianization of the bialgebra ( T ( V ∗ ) , m conc , ∆ ∗ ) .2. If V is connected, then ( S ( V ∗ ) , m, ∆ ∗ ) is a graded, connected Hopf algebra, isomorphic tothe graded dual of ( S ( V ) , ∗ , ∆) . Corollary 30
Let V be a -bounded b ∞ algebra. • S ( V ∗ ) is a subbialgebra of S ( V ∗ ) . • Let J be the ideal of S ( V ∗ ) generated by the elements f ∈ V ∗ . Then J is a biideal of S ( V ∗ ) , and S ( V ∗ ) /J is a graded, connected Hopf algebra, isomorphic to the graded dualof S ( V + ) . Proof.
Similar to the proof of corollary 18. (cid:3)
Remark. If V is a -bounded B ∞ algebra, then ( S ( V ∗ ) , m, ∆ ∗ ) is the abelianization of ( T ( V ∗ ) , m conc , ∆ ∗ ) , whereas S ( V ∗ ) /J is the abelianization of T ( V ∗ ) /I . Theorem 31
Let ( V, ⌊− , −⌋ ) be a -bounded b ∞ algebra. Then V is given a monoid structurewith the product defined by: ∀ x, y ∈ V , x ♦ y = ⌊ e x , e y ⌋ . It is isomorphic to the monoid of characters of both ( S ( V ∗ ) , m, ∆ ∗ ) and ( T ( V ∗ ) , m conc , ∆ ∗ ) . Proof. As S ( V ∗ ) is the abelianization of T ( V ∗ ) , these two bialgebras have the same monoidof characters. Let us determine the monoid of characters of S ( V ∗ ) . For all α, β ∈ Λ , we put: ⌊ x α , y β ⌋ = X i ∈ I a ( i ) α,β x i . j ∈ I , for all α, β ∈ λ : ≪ ∆ ∗ ( f j ) , x α ⊗ x β ≫ = ≪ f j , x α ∗ x β ≫ = ≪ f j , π ( x α ∗ x β ) ≫ = ≪ f j , ⌊ x α , y β ⌋ ≫ = a ( j ) α,β . Hence: ∆ ∗ ( f j ) = X α,β ∈ Λ a ( j ) α,β α ! β ! f α ⊗ f β . The monoid of characters of S ( V ∗ ) is identified, as a set, with the (complete) dual ( V ∗ ) ⊛ of V ∗ ,that is to say: M n ≥ V ∗ n ⊛ = Y n ≥ V ∗∗ n = Y n ≥ V n = V .
The identification is through the map: ϕ : V −→ Char ( S ( V ∗ )) x −→ ϕ x : S ( V ∗ ) −→ K f α −→ Y i ∈ I ≪ f i , x ≫ α i . Let x = P λ i x i and y = P µ i x i ∈ V . For all j ∈ I : ϕ x ∗ ϕ y ( f j ) ϕ ⌊ e x ,e y ⌋ ( f j )= ( ϕ x ⊗ ϕ y ) ◦ ∆ ∗ ( f j ) = ≪ f j , ⌊ e x , e y ⌋ ≫ = X α,β ∈ Λ a ( j ) α,β α ! β ! ϕ x ( f α ) ϕ y ( f β ) = X α,β ∈ Λ Y i ∈ I λ α i i α i ! µ β i i β i ! ≪ f j , ⌊ x α , y β ⌋ ≫ = X α,β ∈ Λ a ( j ) α,β α ! β ! Y i ∈ I λ α i i Y i ∈ I µ β i i ; = X α,β ∈ Λ a ( j ) α,β α ! β ! Y i ∈ I λ α i i Y i ∈ I µ β i i . As ϕ x ∗ ϕ y and ϕ ⌊ e x ,e y ⌋ are characters which coincide on V ∗ , they are equal. Through the bijec-tion ϕ , we obtain the monoid structure on V . (cid:3) Remark. If V is connected, then S ( V ∗ ) is a Hopf algebra, and in this case ( V , ♦ ) is a group. Corollary 32
1. Let ( V, • ) be a -bounded pre-Lie algebra. Then V is a monoid, with theproduct defined by: ∀ x, y ∈ V , x ♦ y = y + x • e y .
2. Let ( V, · ) be a graded associative algebra. Then V is a monoid, with the product defined by: ∀ x, y ∈ V , x ♦ y = x + y + x · y. Proof.
1. Here, ⌊− , −⌋ k,l = 0 if k ≥ . So: x ♦ y = X k ≥ k ! ⌊ x k , e y ⌋ = ⌊ , e y ⌋ + ⌊ x, e y ⌋ = X l ≥ l ! ⌊ , y l ⌋ + x • e y = y + x • e y .
32. Here, ⌊− , −⌋ k,l = 0 , except for ( k, l ) = (0 , , (1 , and (1 , . Hence: x ♦ y = X k,l ≥ k ! l ! ⌊ x k , y l ⌋ k,l = ⌊ x, ⌋ + ⌊ , y ⌋ + ⌊ x, y ⌋ = x + y + x · y. Note that, V can be identified with the monoid of elements of the associative, unitary algebra K ⊕ V , whose constant terms are equal to . (cid:3) Using the graduation:
Corollary 33
Let V be a -bounded b ∞ algebra.1. Then V and V + are submonoids of ( V , ♦ ) . Moreover, ( V + , ♦ ) is a group, isomorphic tothe group of characters of both ( S ( V ∗ + ) , m, ∆ ∗ ) and ( T ( V ∗ + ) , m, ∆ ∗ ) .2. Let x = x + x + ∈ V , with x ∈ V and x + ∈ V + . Then x is a unit of V if, and only if, x is a unit in V . Proof.
1. Immediate.2. = ⇒ . The canonical projection on V is a monoid morphism from V to V , which impliesthe first point. ⇐ = . Let y be the inverse of x in V . We put y = x ♦ y . Then: y = ( ⌊ e x , e y ⌋ ) = ⌊ e x , e y ⌋ = x ♦ y = 0 . So y ∈ V + , so is a unit of V ; by composition, x is a unit of V . (cid:3) Definition 34
Let ( V, • ) be a pre-Lie algebra. A pre-Lie module over V is a vector space M , with a map: : (cid:26) M ⊗ V −→ Mm ⊗ x −→ m x, such that for all m ∈ M , x , x ∈ V : ( m a ) a − ( m a ) a = m ( a • a − a • a ) . Example. If V is a pre-Lie algebra, ( V, • ) is a pre-Lie module over itself. Remarks.
1. A pre-Lie module over V is a module over the Lie algebra associated to V , so is a moduleover the Hopf algebra ( S ( V ) , ∗ , ∆) , where ∗ = φ V ( • ) . The action is given in the followingway: • For all m ∈ M , m m . • For all m ∈ M , x , . . . , x k ∈ V : m x . . . x k = ( m x . . . x k − ) x k − k − X i =1 m ( x . . . ( x i • x k ) . . . x k − ) .
2. Let V be a brace algebra and M be a brace module over V . Then V is also a pre-Liealgebra, with • = h− , −i , . Moreover, for all m ∈ M , x , x ∈ V : m ← ( x ∗ x ) = m ← ( x x + x x + x • x ) = ( m ← x ) ← x , ( m ← x ) ← x − ( m ← x ) ← x = m ← ( x • x − x • x ) .
33o the restriction of ← to M ⊗ V makes M a pre-Lie module over V . By the isomorphismbetween S ( A ) and coS ( A ) , for all x , . . . , x k ∈ V , m ∈ M : m x . . . x k = m ← ( x (cid:1) . . . (cid:1) x k ) . Definition 35
Let V be a -bounded pre-Lie algebra and M be a graded pre-Lie module over V . We shall say that M is -bounded if for all k, l ≥ , there exists B ( k, l ) ≥ such that: p > B ( k, l ) = ⇒ M k S p ( V ) S ( V ) l = (0) . Note that if V is connected, then any graded pre-Lie module over V is -bounded, with B ( k, l ) = 0 for all k, l . Proposition 36
Let V be a -bounded pre-Lie algebra and M be a -bounded pre-Lie moduleover V . M is a module over the monoid ( V , ♦ ) , with the action defined by: ∀ m ∈ M , ∀ x ∈ V , m ⊳ x = x e y . By restriction, it is also a module over the group ( V + , ♦ ) . Proof.
By transposition of the action of ( S ( V ); ∗ ) on M , we obtain thanks to the -boundedcondition a coaction of ( S ( V ∗ ) , ∆ ∗ ) on M ∗ ; consequently, the dual of M ∗ , identified with M ,becomes a module over the monoid of characters of ( S ( V ∗ ) , m, ∆ ∗ ) , identified with ( V , ♦ ) . Theend of the proof is similar to the proof of theorem 31. (cid:3) hapter 3 Brace and pre-Lie structures on operads
Introduction
We now study the brace and pre-Lie structure on an operad P induced by the operadic com-position (proposition 37 and corollary 38). We have seen in the preceding chapter that thesestructures imply a product on T ( P ) making it a graded, non connected dendriform Hopf algebra,named D P (proposition 40). By the -boundedness condition, we construct a dual bialgebra D ∗ P .Considering a connected Hopf subalgebra of D P , we obtain a graded, connected Hopf algebraon T ( P + ) , named B P , and its graded dual B ∗ P , as a quotient of D ∗ P .Using the pre-Lie product induced by the brace structure, we obtain a graded, non connectedHopf algebra D P , which underlying coalgebra is S ( coinv P ) with its usual coproduct; a graded,connected Hopf algebra B P , which underlying coalgebra is S ( coinv P + ) with its usual coproduct;and bialgebras D ∗ P and B ∗ P , in duality with the preceding ones. All these objects admit coloredversions by any vector space V , see propositions 42, 43 and 44. As D ∗ P and B ∗ P are commu-tative, they can be seen as coordinates bialgebras of a monoid: these monoids are described inproposition 54, with the help of the pre-Lie product of P , as in theorem 31; they appeared in [5].A specially interesting case is obtained by operads P equipped with an operad morphism θ P : b ∞ −→ P . In this case, for any vector space V , the coalgebra S ( F P ( V )) , where F P ( V ) is the free P -algebra generated by V , becomes a graded, connected Hopf algebra denoted by A P ( V ) . We prove in theorem 46 that A P ( V ) and D P ( V ) are bialgebras in interaction (definition41), that is to say that A P ( V ) is a bialgebra in the categroy of D P ( V ) -modules; dually, A ∗ P ( V ) and D ∗ P ( V ) are cointeracting bialgebras, in the sense of [33], that is to say that A ∗ P ( V ) is a Hopfalgebra in the category of D ∗ P ( V ) -comodules. Proposition 37
Let P be a non- Σ operad. We define a brace structure on P = M n ≥ P ( n ) by: ∀ p ∈ P ( n ) , p , . . . , p k ∈ P , h p, p . . . p k i = X ≤ i <...
Let p, p , . . . , p k , q , . . . , q l ∈ P . Then, using the associativity of the operadic com-35osition: hh p, p . . . p k i , q . . . q l i = X h p ◦ ( I, . . . , p , . . . , p k , . . . , I ) , q . . . q l i = X ( p ◦ ( I, . . . , p , . . . , p k , . . . , I )) ◦ ( I, . . . , q , . . . , q l , . . . , I )= X p ◦ ( I, . . . , q , . . . , q i , p ◦ ( I, . . . , q i +1 , . . . , q i + i , . . . , I ) , . . . ,p k ◦ ( I, . . . , q i + ... + i k − +1 , . . . , q i + ... + i k , . . . , I ) , . . . ,q i + ... + i k +1 , . . . , q l , . . . , I )= X q ...q l = Q ...Q k +1 h p, Q h p , Q i Q . . . Q k − h p k , Q k i Q k +1 i . So h− , −i is a brace structure on P .Let p, p , . . . , p k in P , homogeneous of respective degrees n , n , . . . , n k . Then p ∈ P ( n + 1) and p i ∈ P ( n i + 1) for all i . Then h p, p . . . p k i is a linear span of element of P ( m ) , with: m = n + 1 − k + n + 1 + . . . + n k + 1 = n + n + . . . + n k + 1 . So h p, p . . . p k i is homogeneous of degree n + n + . . . + n k : the brace algebra P is graded.Let m, n ≥ . If p ∈ P ( m + 1) , and if k > m + 1 , then h p, p . . . p k i = 0 for all p , . . . , p k ∈ P .So P is -bounded, with B ( m, n ) = m + 1 . (cid:3) Remark.
Consequently, P (1) is a brace subalgebra of P . For all p, p , . . . , p k ∈ P (1) , h p, p . . . p k i = ( p ◦ p if k = 1 , if k ≥ . So the brace algebra P (1) is the associative algebra ( P (1) , ◦ ) .By the morphism from PreLie to Brace , we immediately obtain:
Corollary 38
Let P be a non- Σ operad. It is a graded pre-Lie algebra, with: ∀ p ∈ P ( n ) , q ∈ P , p • q = h p, q i = n X i =1 p ◦ i q. Its b ∞ brackets are given by: ∀ p ∈ P ( n ) , p , . . . , p k ∈ P , ⌊ p, p . . . p k ⌋ = X ≤ i ,...,i k ≤ n, all distincts p ◦ i ,...,i k ( p , . . . , p k ) . Proof.
Indeed, ⌊ p, p . . . p k ⌋ = h p, p (cid:1) . . . (cid:1) p k i = X σ ∈ S k h p, p σ (1) . . . p σ ( k ) i . (cid:3) Corollary 39
Let P be an operad. Then coinv P is a graded pre-Lie algebra, quotient of P . Proof.
Let us prove that I = V ect ( p − p σ | p ∈ P ( n ) , σ ∈ S n ) is a pre-Lie ideal. Let p ∈ P ( n ) , σ ∈ S n and q ∈ P ( m ) . There exist permutations σ ′ i such that: ( p σ − p ) ∗ q = n X i =1 p σ ◦ i q − p ◦ i q = n X i =1 ( p ◦ σ ( i ) q ) σ ′ i − n X i =1 p ◦ i q = n X i =1 ( p ◦ i q ) σ ′ σ − i ) − p ◦ i q ∈ I. I is a right pre-Lie ideal. There exists permutations σ ′′ i such that: q ∗ ( p σ − p ) = m X i =1 ( q ◦ i p σ − q ◦ i p ) = X i =1 (cid:16) ( q ◦ i p ) σ ′′ i − q ◦ i p (cid:17) ∈ I. So I is also a left pre-Lie ideal. (cid:3) By theorems 13 and 20, and by proposition 24:
Proposition 40
Let P be an operad.1. The brace structure on P induces a product ∗ = ≺ + ≻ , making D P = ( T ( P ) , ∗ , ∆ dec ) adendriform bialgebra. The graded, connected Hopf subalgebra T ( P + ) is denoted by B P .2. The pre-Lie product on P induces products ∗ making ( S ( P ) , ∗ , ∆) , ( S ( P + ) , ∗ , ∆) , D P =( S ( coinv P ) , ∗ , ∆) and B P = ( S ( coinv P + ) , ∗ , ∆) bialgebras.3. There is a commutative diagram of bialgebras: D P " " " " ❉❉❉❉❉❉❉❉ B P .(cid:14) = = ③③③③③③③③ " " " " ❉❉❉❉❉❉❉❉ D P B P .(cid:14) < < ③③③③③③③③ Examples.
Let p ∈ P ( n ) , p ∈ P ( n ) , q , q ∈ P . In D P : p ≺ q = p q + X ≤ i ≤ n p ◦ i q ,p ≻ q = q p ,p ≺ q q = p q q + X ≤ i ≤ n ( p ◦ i q ) q + X ≤ i Let A and B be two bialgebras. . We shall say that A and B are in interaction if A is a B -module-bialgebra, or equivalentlyif A is a bialgebra in the category of B -modules, that is to say: • B is acting on A , via a map : A ⊗ B −→ A . • A is a bialgebra in the category of B -modules, that is to say: – For all b ∈ B , A b = ǫ ( b )1 A . – For all a ∈ A , b ∈ B , ε ( a b ) = ε ( a ) ε ( b ) . – For all a , a ∈ A , b ∈ B , or, ( a a ) b = m (( a ⊗ a ) ∆( b )) or, withSweedler’s notation, ( a a ) b = ( a b (1) )( a b (2) ) . – For all a ∈ A , b ∈ B , ∆( a b ) = ∆( a ) ∆( b ) or, with Sweedler’s notation, ∆( a b ) = a (1) b (1) ⊗ a (2) b (2) .2. We shall say that A and B are in cointeraction if if A is a B -comodule-bialgebra, orequivalently if A is a bialgebra in the category of B -comodules, that is to say: • B is coacting on A , via a map ρ : (cid:26) A −→ A ⊗ Ba −→ ρ ( a ) = a ⊗ a . • A is a bialgebra in the category of B -comodules, that is to say: – ρ (1 A ) = 1 A ⊗ B . – m , ◦ ( ρ ⊗ ρ ) ◦ ∆ = (∆ ⊗ Id ) ◦ ρ , where: m , : (cid:26) A ⊗ B ⊗ A ⊗ B −→ A ⊗ A ⊗ Ba ⊗ b ⊗ a ⊗ b −→ a ⊗ a ⊗ b b . Equivalenlty, for all a ∈ A : ( a (1) ) ⊗ ( a (2) ) ⊗ ( a (1) ) ( a (2) ) = ( a ) (1) ⊗ ( a ) (2) ⊗ a . – For all a, b ∈ A , ρ ( ab ) = ρ ( a ) ρ ( b ) . – For all a ∈ A , ( ε A ⊗ Id ) ◦ ρ ( a ) = ε A ( a )1 B . Remark. If A and B are in interaction, the action map is a coalgebra morphism; if A and B are in cointeraction, the coaction map ρ is an algebra morphism.For examples and applications of cointeracting bialgebras, see [4, 11, 16, 15]. Proposition 42 1. Let V be a vector space. We define the operad C V by: • For all n ≥ , C V ( n ) = End K ( V, V ⊗ n ) . • For all f ∈ C V ( m ) , g ∈ C V ( n ) and ≤ i ≤ m : f ◦ i g = ( Id ⊗ ( i − ⊗ g ⊗ Id ⊗ ( n − i ) ) ◦ f ∈ C V ( m + n − . The unit is Id V . • For all f ∈ C V ( n ) , σ ∈ S n , and x ∈ V , if f ( x ) = x . . . x n : f σ ( x ) = x σ (1) . . . x σ ( n ) . 2. The tensor algebra T ( V ) is a brace module over the brace algebra ( C V , h− , −i ) with, for all x , . . . , x n ∈ V , f , . . . f k ∈ C V : x . . . x n ← f . . . f k = X ≤ i <...
Let P be an operad and V be a vector space.1. The following space is a graded brace module over the brace algebra associated to the operad P ⊗ C V : M = ∞ M n =1 P ( n ) ⊗ V ⊗ n . 2. By restriction, M is a pre-Lie module on P ⊗ C V . This structure induces a graded pre-Lie coinv ( P ⊗ C V ) -module structure over the vector space F P ( V ) , such that, for all p ∈ P ( k ) ,for all x , . . . , x k ∈ V , for all q ∈ P ( n ) , f ∈ End K ( V, V ⊗ n ) : p. ( x . . . x k ) q ⊗ f = k X i =1 p ◦ i q. ( x . . . x i − f ( x i ) x i +1 . . . x n ) . Proof. 1. As P is a N -graded brace module over P and V is a N -graded brace module overthe brace algebra C V , M is a graded brace module over P ⊗ C V . Moreover, for all p ∈ P ( k ) , forall x , . . . , x k ∈ V , for all q ∈ P ( n ) , f ∈ End K ( V, V ⊗ n ) : p ⊗ x . . . x k q ⊗ f = k X i =1 p ◦ i q ⊗ ( x . . . x i − f ( x i ) x i +1 . . . x n ) . 2. Let p ∈ P ( k ) , x , . . . , x k ∈ V , σ ∈ S k , q ∈ P ( n ) , f ∈ End K ( V, V ⊗ n ) . There existpermutations σ i , σ ′ j such that: ( p ⊗ x . . . x k ) σ q ⊗ f = p σ ⊗ x σ (1) . . . x σ ( k ) p ⊗ f = k X i =1 p σ ◦ i q ⊗ x σ (1) . . . f ( x σ ( i ) ) . . . x σ ( n ) = k X i =1 ( p ◦ σ ( i ) q ⊗ x . . . f ( x σ ( i ) ) . . . x n ) σ i = k X j =1 ( p ◦ j q ⊗ x . . . f ( x j ) . . . x n ) σ ′ j , so: (( p ⊗ x . . . x k ) σ − p ⊗ x . . . x k ) q ⊗ f = k X j =1 ( p ◦ j q ⊗ x . . . f ( x j ) . . . x n ) σ ′ j − p ◦ j q ⊗ x . . . f ( x j ) . . . x n . P ⊗ C V induces a pre-Lie action on the quotient of M by the ideal I : I = V ect (( p ⊗ x . . . x k ) σ − p ⊗ x . . . x k | k ≥ , p ∈ P ( k ) , x , . . . , x k ∈ V, σ ∈ S k )= V ect ( p σ ⊗ x σ (1) . . . x σ ( k ) − p ⊗ x . . . x k | k ≥ , p ∈ P ( k ) , x , . . . , x k ∈ V, σ ∈ S k )= V ect ( p σ ⊗ x . . . x k − p ⊗ x σ − (1) . . . x σ − ( k ) | k ≥ , p ∈ P ( k ) , x , . . . , x k ∈ V, σ ∈ S k ) . Note that the quotient M/I = F P ( V ) .Let p ∈ P ( k ) , x , . . . , x k ∈ V , q ∈ P ( n ) , f ∈ End K ( V, V ⊗ n ) , σ ∈ S n . p.x . . . x k ( q ⊗ f ) σ = k X i =1 p ◦ iq σ . ( x . . . f σ ( x i ) . . . x k )= k X i =1 p ◦ iq. ( x . . . ( f σ ) σ − ( x i ) . . . x k )= k X i =1 p ◦ iq. ( x . . . f ( x i ) . . . x k )= p.x . . . x k q ⊗ f. so the action of P ⊗ C V on F P ( V ) induces an action of coinv ( P ⊗ C V ) on F P ( V ) . (cid:3) Definition 44 We put: D P ( V ) = D P ⊗ C V = ( S ( coinv ( P ⊗ C V )) , ∗ , ∆) ,B P ( V ) = B P ⊗ C V = ( S ( coinv ( P ⊗ C V ) + ) , ∗ , ∆) ,A P ( V ) = S ( F P ( V )) . Note that if V is one-dimensional, then D P ( V ) , respectively B P ( V ) , is isomorphic to D P ,respectively to B P . Lemma 45 1. The pre-Lie action of coinv ( P ⊗ C V ) on F P ( V ) is extended to A P ( V ) : ∀ v , . . . , v k ∈ F P ( V ) , ∀ q ∈ coinv ( P ⊗ C V ) , v . . . v k q = k X i =1 v . . . ( v i q ) . . . v k . This induces an action of D P ( V ) on A P ( V ) , such that: • For all a ∈ A P ( V ) , b ∈ D P ( V ) , ∆( a ← b ) = a (1) ← b (1) ⊗ a (2) ← b (2) . • For all b ∈ D P ( V ) , b = ε ( b ) . • For all a , a ∈ A P ( V ) , b ∈ D P ( V ) , a a b = ( a b (1) )( a b (2) ) .In other words, ( A P ( V ) , m, ∆) is a Hopf algebra in the category of D P ( V ) -modules.2. For all p ∈ P ( n ) , v , . . . , v n ∈ F P ( V ) , Q ∈ D P ( V ) : p. ( v , . . . , v n ) Q = p. ( v Q (1) , . . . , v n Q ( n ) ) . In other words, F P ( V ) is a P -algebra in the category of D P ( V ) -modules. Proof. 1. We consider: X = { a ∈ A P ( V ) | ∀ b ∈ D P ( V ) , ∆( a b ) = a (1) b (1) ⊗ a (2) b (2) } . ∈ X : ∀ b ∈ D P ( V ) , ∆(1 b ) = ε ( b )1 ⊗ ε ( b (1) ) ε ( b (2) )1 ⊗ b (1) ⊗ b (2) . Let a , a ∈ X . For all b ∈ D P ( V ) , by the cocommutativity of D P ( V ) : ∆(( a a ) b ) = ∆(( a b (1) )( a b (2) ))= ( a (1)1 b (1) )( a (1)2 b (3) ) ⊗ ( a (2)1 b (2) )( a (2)2 b (4) )= ( a (1)1 b (1) )( a (1)2 b (2) ) ⊗ ( a (2)1 b (3) )( a (2)2 b (4) )= (( a (1)1 a (1)2 ) b (1) ) ⊗ (( a (2)1 a (2)2 ) b (2) )= (( a a ) (1) b (1) ) ⊗ (( a a ) (2) b (2) ) . So X is a subalgebra of A P ( V ) for its usual product. Let a ∈ F P ( V ) and b ∈ D P ( V ) . Then a b ∈ F P ( V ) , so: ∆( a b ) = a b ⊗ ⊗ a b = a b (1) ⊗ ε ( b (2) )1 + ε ( b (1) )1 ⊗ a b (2) = a b (1) ⊗ b (2) + 1 b (1) ⊗ a b (2) , so a ∈ X . As X is a subalgebra containing F P ( V ) , it is equal to A P ( V ) : A P ( V ) is a coalgebrain the category of D P ( V ) -modules.2. As this pre-Lie action comes from a brace action, if p ∈ P ( n ) , x , . . . , x n ∈ V , q ⊗ f , . . . , q k ⊗ f k ∈ P ⊗ C V : p.x . . . x n q ⊗ f . . . q k ⊗ f k = X ≤ i ,...,i k ≤ n, all distinct p ◦ i ,...,i k ( q , . . . , q k ) . ( x . . . f ( x i ) . . . f k ( x i k ) . . . x n ) . Let us consider: C = (cid:26) Q ∈ D P ( V ) | ∀ p ∈ P ( n ) , v , . . . , v n ∈ F P ( V ) ,p. ( v , . . . , v n ) Q = p. ( v Q (1) , . . . , v n Q ( n ) ) (cid:27) . Obviously, ∈ C . Let us take Q , Q ∈ C . For all p ∈ P ( n ) , v , . . . , v n ∈ F P ( V ) : p. ( v , . . . , v n ) Q ∗ Q = ( p. ( v , . . . , v n ) Q ) Q = p. (( v Q (1)1 ) Q (1)2 , . . . , ( v n Q ( n )1 ) Q ( n )2 )= p. ( v ( Q (1)1 ∗ Q (1)2 ) , . . . , v n ( Q ( n )1 ∗ Q ( n )2 ))= p. ( v ( Q ∗ Q ) (1) , . . . , v n ( Q ∗ Q ) ( n ) ) . So Q ∗ Q ∈ C : C is a subalgebra of D P ( V ) .Let us take p ∈ P ( n ) and v i = p i . ( x i, , . . . , x i,l i ) ∈ F P ( V ) for all ≤ i ≤ n . If q ⊗ f ∈ P ⊗ C V ,by the associativity of the operadic composition: p. ( v , . . . , v n ) q ⊗ f = p ◦ ( p , . . . , p n ) . ( x , , . . . , x n,l n ) q ⊗ f = n X i =1 l i X j =1 p ◦ ( p , . . . , p i ◦ j q, . . . , p n ) . ( x , , . . . , f ( x i,j ) , . . . , x n,l n )= n X i =1 p. ( v , . . . , v i q ⊗ f , . . . , v n ) . So coinv ( P ⊗ C V ) ⊆ C . As coinv ( P ⊗ C V ) generates D P ( V ) , C = D P ( V ) . (cid:3) heorem 46 Let θ P : b ∞ −→ P be an operad morphism. Any P -algebra is also b ∞ , andwe denote: ⋆ = φ F P ( V ) ( θ P ( ⌊− , −⌋ )) . Then ( A P ( V ) , ⋆, ∆) and D P ( V ) are two bialgebras in interaction. Proof. We already proved that ( A P ( V ) , m, ∆) is a Hopf algebra in the category of D P ( V ) -modules. Let us consider the two following maps: Φ : (cid:26) A P ( V ) ⊗ A P ( V ) ⊗ D P ( V ) −→ A P ( V ) a ⊗ a ⊗ b −→ ( a ∗ a ) b, Φ : (cid:26) A P ( V ) ⊗ A P ( V ) ⊗ D P ( V ) −→ A P ( V ) a ⊗ a ⊗ b −→ ( a b (1) ) ∗ ( a b (2) ) . For all a , a ∈ A P ( V ) , b ∈ D P ( V ) : ∆ ◦ Φ ( a ⊗ a ⊗ b ) = ( a (1)1 ∗ a (1)2 ) b (1) ⊗ ( a (2)1 ∗ a (2)2 ) b (2) = (Φ ⊗ Φ ) ◦ ∆( a ⊗ a ⊗ b ) , ∆ ◦ Φ ( a ⊗ a ⊗ b ) = ( a (1)1 b (1) ) ∗ ( a (1)2 b (3) ) ⊗ ( a (2)1 b (2) ) ∗ ( a (2)2 b (4) )= ( a (1)1 b (1) ) ∗ ( a (1)2 b (2) ) ⊗ ( a (2)1 b (3) ) ∗ ( a (2)2 b (4) )= (Φ ⊗ Φ ) ◦ ∆( a ⊗ a ⊗ b ) . So both Φ and Φ are coalgebra morphisms. In order to prove that their equality, by lemma21, it is enough to prove that π ◦ Φ = π ◦ Φ , where π is the canonical projection on F P ( V ) in A P ( V ) . As for all k , S k ( F P ( V )) D P ( V ) ⊆ S k ( F P ( V )) : π ◦ Φ ( a ⊗ a ⊗ b ) = π (( a ∗ a ) b ) = π ( a ∗ a ) b = ⌊ a , a ⌋ b. The b ∞ structure is induced by θ P : denoting q k,l = θ P ( ⌊− , −⌋ k,l ) and q = X k,l ≥ q k,l , for all a , a ∈ A P ( V ) , ⌊ a , a ⌋ = q. ( a , a ) . By lemma 45, for all b ∈ D P ( V ) : ⌊ a , a ⌋ b = q. ( a , a ) b = q. ( a b (1) , a b (2) )= ⌊ a b (1) , a b (2) ⌋ = π ◦ Φ ( a ⊗ a ⊗ b ) . As a conclusion, Φ = Φ , and A P ( V ) is a bialgebra in the category of D P ( V ) -modules. (cid:3) We assume now that for all n ≥ , P is finite-dimensional. The composition can be seen as amap: ◦ : M n ≥ P ( n ) ⊗ P ⊗ n −→ P . Moreover, for any n ≥ : ◦ − ( P ( n )) = n M p =1 M k + ... + k p = n P ( p ) ⊗ P ( k ) . . . P ( k p ) . 42y duality, we obtain a map δ : P ∗ −→ ( P ⊗ T ( P )) ∗ , such that for any n ≥ : δ ( P ( n )) ⊆ n M p =1 M k + ... + k p = n P ∗ ( p ) ⊗ P ∗ ( k ) . . . P ∗ ( k p ) . So δ ( P ∗ ) ⊆ P ∗ ⊗ T ( P ∗ ) . Proposition 47 1. We define a coproduct ∆ ∗ on T ( P ∗ ) as the unique algebra morphism(for the concatenation product m conc ) such that for all f ∈ P ∗ : ∆ ∗ ( f ) = δ ( f ) . Then D ∗ P = ( T ( P ∗ ) , m conc , ∆ ∗ ) is a bialgebra. It is graded, the elements of P ∗ ( n ) beinghomogeneous of degree n − for all n ≥ .2. There exists a nondegenerate pairing ≪ − , − ≫ : T ( P ∗ ) ⊗ T ( P ) −→ K , such that for all F, G ∈ T ( P ∗ ) , for all X, Y ∈ T ( P ) : ≪ , X ≫ = ε ( X ) , ≪ F ⊗ G, ∆( X ) ≫ = ≪ F G, X ≫ , ≪ F, ≫ = ε ( F ) , ≪ ∆ ∗ ( F ) , X ⊗ Y ≫ = ≪ F, X ∗ Y ≫ . In other words, ≪ − , − ≫ is a Hopf pairing between D ∗ P and D P . Proof. The following map is an algebra isomorphism: (cid:26) T ( P ∗ + ) −→ T ( P ∗ ) /I f . . . f k −→ f . . . f k . We define a first pairing between T ( P ∗ ) and T ( P ) by: ∀ x , . . . , x k ∈ P , ∀ f , . . . , f l ∈ P ∗ , ≪ f . . . f l , x . . . x k ≫ ′ = ( if k = l,f ( x ) . . . f k ( x k ) if k = l. We shall need the completion: T ( P ) = ∞ Y k =0 P ⊗ k . We can extend the concatenation product, deconcatenation coproduct and the pairing as maps: m conc : T ( P ) ⊗ T ( P ) −→ T ( P ) , ∆ dec : T ( P ) −→ T ( P ) ⊗ T ( P ) , ≪ − , − ≫ ′ : T ( P ∗ ) ⊗ T ( P ) −→ K . We put: J = ∞ X n =0 I n = 11 − I ∈ T ( P ) . Then: ∆( J ) = ∞ X n =0 X k + l = n I k ⊗ I l = X k,l ≥ I k ⊗ I l = J ⊗ J. This implies that the following map is a coalgebra morphism: φ : (cid:26) T ( P + ) −→ T ( P ) x . . . x k −→ J x J . . . J x k J. 43e now define the pairing ≪ − , − ≫ : ∀ F ∈ T ( P ∗ ) , X ∈ T ( P ) , ≪ F, X ≫ = ≪ F, φ ( X ) ≫ ′ . If l > k , ≪ f . . . f k , x . . . x k ≫ = 0 . If k = l : ≪ f . . . f k , x . . . x k ≫ = ≪ f . . . f k , J x J . . . J x k J ≫ ′ = ≪ f . . . f k , x . . . x k ≫ ′ +0= f ( x ) . . . f k ( x k ) . By a triangularity argument, this pairing is non degenerate.Let X = x . . . x k ∈ T ( P ) . ≪ , X ≫ = ( if k ≥ ≪ , J ≫ ′ if k = 0= ( if k ≥ if k = 0= ε ( X ) . Let F, G ∈ T ( P ∗ ) , and X ∈ T ( P ) . ≪ F ⊗ G, ∆( X ) ≫ = ≪ F ⊗ G, ( φ ⊗ φ ) ◦ ∆( X ) ≫ ′ = ≪ F ⊗ G, ∆ ◦ φ ( X ) ≫ ′ = ≪ F G, φ ( X ) ≫ ′ = ≪ F G, X ≫ . Let us consider now: A = { F ∈ T ( P ∗ ) | ∀ X, Y ∈ T ( P ) , ≪ ∆ ∗ ( F ) , X ⊗ Y ≫ = ≪ F, X ∗ Y ≫} . For all X, Y ∈ T ( P ) : ≪ , X ∗ Y ≫ = ε ( X ∗ Y ) = ε ( X ) ε ( Y ) = ≪ ⊗ , X ⊗ Y ≫ , so ∈ A . Let F, G ∈ A . For all X, Y ∈ T ( P ) : ≪ ∆ ∗ ( F G ) , X ⊗ Y ≫ = ≪ ∆ ∗ ( F )∆ ∗ ( G ) , X ⊗ Y ≫ = ≪ ∆ ∗ ( F ) ⊗ ∆ ∗ ( G ) , X (1) ⊗ Y (1) ⊗ X (2) ⊗ Y (2) ≫ = ≪ F ⊗ G, X (1) ∗ Y (1) ⊗ X (2) ∗ Y (2) ≫ = ≪ F ⊗ G, ( X ∗ Y ) (1) ⊗ ( X ∗ Y ) (2) ≫ = ≪ F G, X ∗ Y ≫ . So A is a subalgebra of T ( P ∗ ) . In order to prove that A = T ( P ∗ ) , it is now enough to provethat P ∗ ⊆ A . Let f ∈ P ∗ , X = x . . . x k , y = y . . . y l ∈ T ( P ) . • Let us assume that k ≥ . As ∆ ∗ ( P ∗ ) ⊆ P ∗ ⊗ T ( P ∗ ) , ≪ ∆ ∗ ( f ) , X ⊗ Y ≫ = 0 . Moreover: M n ≥ P ⊗ n ∗ T ( P ) ⊆ M n ≥ P ⊗ n , so ≪ f, X ∗ Y ≫ = 0 . 44 Let us assume that k = 0 . Then: ≪ ∆ ∗ ( f ) , ⊗ Y ≫ = ≪ ∆ ∗ ( f ) , J ⊗ φ ( Y ) ≫ ′ = ≪ δ ( f ) , I ⊗ φ ( Y ) ≫ ′ +0= ≪ f, I ◦ φ ( Y ) ≫ ′ = ≪ f, φ ( Y ) ≫ ′ = ≪ f, ∗ Y ≫ . • Let us finally assume that k = 1 . Then: ≪ ∆ ∗ ( f ) , ⊗ Y ≫ = ≪ δ ( f ) , x ⊗ J y J . . . J y l J ≫ ′ = ≪ f, x ◦ ( J y J . . . J y l J ) ≫ ′ = ≪ f, h x , y . . . y l i ≫ ′ . Moreover: ≪ f, x ∗ y . . . y l ≫ = ≪ f, h x , y . . . y l i ≫ + terms ≪ f, z . . . z l ≫ ′ , l ≥ ≪ f, h x , y . . . y l i ≫ +0= ≪ f, h x , y . . . y l i ≫ ′ . Finally, A is equal to T ( P ∗ ) : ≪ − , − ≫ is a Hopf pairing.Let F ∈ T ( P ∗ ) , X, Y, Z ∈ T ( P ) . ≪ (∆ ∗ ⊗ Id ) ◦ ∆ ∗ ( F ) , X ⊗ Y ⊗ Z ≫ = ≪ F, ( X ∗ Y ) ∗ Z ≫ = ≪ F, X ∗ ( Y ∗ Z ) ≫ = ≪ ( Id ⊗ ∆ ∗ ) ◦ ∆ ∗ ( F ) , X ⊗ Y ⊗ Z ≫ . As the pairing is nondegenerate, ∆ ∗ is coassociative. (cid:3) Remarks. P ∗ (0) is a subcoalgebra of D ∗ P ; it is the dual of the algebra ( P (0) , ◦ ) . Its counit is denotedby ε ; for all f ∈ P ∗ (0) , ε ( f ) = f ( I ) .2. In general, D ∗ P is not a Hopf algebra. Let us take the example where P (1) = V ect ( I ) . Wedefine X ∈ P ∗ (1) by X ( I ) = 1 . Then ∆ ∗ ( X ) = X ⊗ X . As X has no inverse, D ∗ P is not aHopf algebra. Corollary 48 The abelianized algebra S ( P ∗ ) of D ∗ P inherits a coproduct ∆ ∗ , making it abialgebra. Moreover, D P = S ( inv P ∗ ) is a subbialgebra of S ( P ∗ ) . Proof. We denote by I ab the ideal of T ( P ∗ ) generated by all the commutators. Then S ( P ∗ ) = T ( P ∗ ) /I ab . Let f ∈ inv P ∗ . We denote by f (1) ⊗ f (2)1 . . . f (2) n the component of ∆ ∗ ( f ) belonging to P ∗ ( n ) ⊗ P ∗ ( k ) . . . P ∗ ( k n ) . Let σ ∈ S n , σ i ∈ S k i . There exists τ ∈ S k + ... + k n such45hat for all p ∈ P ( n ) , p i ∈ P ( k i ) : ≪ ( f (1) ) σ ⊗ ( f (2)1 ) σ . . . ( f (2) n ) σ n , p ⊗ p . . . p n ≫ = ≪ ( f (1) ) σ ⊗ ( f (2)1 ) σ . . . ( f (2) n ) σ n , p ⊗ p . . . p n ≫ ′ = ≪ f (1) ⊗ f (2)1 . . . f (2) n , p σ − ⊗ p σ − . . . p σ − n n ≫ ′ = ≪ f (1) ⊗ f (2)1 . . . f (2) n , p σ − ⊗ p σ − . . . p σ − n n ≫ ′ = ≪ f, p σ − ◦ ( p σ − , . . . , p σ − n n ) ≫ ′ = ≪ f, ( p ◦ ( p σ (1) , . . . , p σ ( n ) )) τ ≫ ′ = ≪ f τ − , p ◦ ( p σ (1) , . . . , p σ ( n ) ) ≫ ′ = ≪ f, p ◦ ( p σ (1) , . . . , p σ ( n ) ) ≫ ′ = ≪ f (1) ⊗ f (2) σ − (1) . . . f (2) σ − ( n ) , p ⊗ p . . . p n ≫ . This implies that ∆ ∗ ( f ) ∈ T ( inv P ∗ ) ⊗ T ( inv P ∗ ) + T ( P ∗ ) ⊗ I ab . So, in the quotient S ( P ∗ ) , ∆ ∗ ( f ) ∈ S ( inv P ∗ ) ⊗ S ( inv P ∗ ) , so S ( inv P ∗ ) is a subbialgebra of S ( P ∗ ) . (cid:3) Corollary 49 1. Let I be the ideal of D ∗ P generated by the elements f − ε ( f )1 , f ∈ P ∗ (1) . This is a biideal; the quotient B ∗ P = D ∗ P /I is a graded, connected Hopf algebra,and its graded dual is ( T ( P + ) , ∗ , ∆) .2. Let J be the ideal of S ( P ∗ ) generated by the elements f − ε ( f )1 , f ∈ P ∗ (1) . This is abiideal; the quotient S ( P ∗ ) /J is a graded, connected Hopf algebra, and its graded dual is ( S ( P + ) , ∗ , ∆) .3. B ∗ P = D ∗ P /J ∩ D ∗ P is a graded, connected Hopf subalgebra of S ( P ∗ ) /J , and its graded dualis B P . Proof. 1. Firstly, observe that for all f ∈ I : ∆( f − ε ( f )1) = f (1) ⊗ f (2) − ε ( f )1 ⊗ f (1) ⊗ ( f (2) − ε ( f (2) )1) + ε ( f (2) ) f (1) ⊗ − ε ( f )1 ⊗ f (1) ⊗ ( f (2) − ε ( f (2) )1) + f ⊗ − ε ( f )1 ⊗ f (1) ⊗ ( f (2) − ε ( f (2) )1) + ( f − ε ( f )1) ⊗ ∈ I ⊗ D ∗ P + D ∗ P ⊗ I . So I is a biideal of D ∗ P . Moreover, D ∗ P /I is isomorphic to the algebra T ( P ∗ + ) via the morphism: (cid:26) T ( P ∗ + ) −→ D ∗ P /I f . . . f k −→ f . . . f k . Let f ∈ P . If x , . . . , x k ∈ P + : ≪ f − ε ( f )1 , x . . . x k ≫ = ≪ f − ε ( f )1 , J x J . . . J x k J ≫ ′ = if k ≥ , ≪ f, x ≫ ′ if k = 1 , ≪ f, I ≫ ′ − ε ( f ) ≪ , ≫ ′ if k = 0;= if k ≥ , if k = 1 ,ε ( f ) − ε ( f ) = 0 if k = 0;= 0 . 46o the pairing ≪ − , − ≫ induces a Hopf pairing between D ∗ P /I and B P , which is a Hopfsubalgebra of ( T ( P ) , ∗ ∆) . Moreover, for all f , . . . , f k ∈ P ∗ + , x , . . . , x k ∈ P ∗ : ≪ f . . . f k , x . . . x l ≫ = ≪ f . . . f k , x . . . x l ≫ = ≪ f . . . f k , J x J . . . J x l J ≫ = if k > l (as ≪ f i , I ≫ ′ = 0 for all i ) , if k < l,f ( x ) . . . f k ( x k ) if k = l. Hence, we have a nondegenerate Hopf pairing between D ∗ P /I and B P . It is clearly homoge-neous, so D ∗ P /I is isomorphic to the graded dual of B P .2. Taking the abelianization of D ∗ P /I , we obtain the graded, connected Hopf algebra S ( P ∗ ) /J . Its graded dual is isomorphic to the largest cocommutative Hopf subalgebra of T ( P + ) ,that is to say coS ( P + ) , or up to an isomorphism S ( P + ) .3. This is implied by the second point, noticing that the dual of coinv P ( n ) is inv P ∗ ( n ) forall n . (cid:3) Definition 50 For any finite-dimensional vector space V , we put: D ∗ P ( V ) = D ∗ P ⊗ C V , D ∗ P ( V ) = D ∗ P ⊗ C V , B ∗ P ( V ) = B ∗ P ⊗ C V , B ∗ P ( V ) = B ∗ P ⊗ C V . Remark. If V is one-dimensional: D ∗ P ( V ) ≈ D ∗ P , D P ( V ) ≈ D P , B ∗ P ( V ) ≈ B ∗ P , B ∗ P ( V ) ≈ B ∗ P . Let θ : b ∞ −→ P be an operad morphism. Then A P ( V ) is a graded Hopf algebra inthe category of B P ( V ) -modules. Let us consider its graded dual A ∗ P ( V ) . Using the pairing ≪ − , − ≫ , one can define a coaction of D P ( V ) over the graded dual A ∗ P ( V ) , which we canquotient to obtain a coaction of B P ( V ) . We obtain: Corollary 51 Let V be a finite-dimensional space and θ : b ∞ −→ P be an operad morphism.Then A ∗ P ( V ) and D ∗ P ( V ) are in cointeraction, via the transposition ρ of the action of D P ( V ) over A P ( V ) . As P is a -bounded brace algebra, there exists a second coproduct ∆ ′∗ on T ( P ∗ ) , induced bythe brace product, as defined in proposition 17. It is a different from ∆ ∗ (see section 4.1.1 foran example), but there is a bialgebra isomorphism: Proposition 52 The following map is a bialgebra isomorphism: Ψ P : D ∗ P = ( T ( P ∗ ) , m conc , ∆ ∗ ) −→ D ′ P = ( T ( P ∗ ) , m conc , ∆ ′∗ ) f ∈ P ∗ −→ f − ε ( f )1 ,f ∈ P ∗ + −→ f. Proof. The coproducts ∆ ′∗ and ∆ ∗ are defined by: ∀ F ∈ P ∗ , ∀ X, Y ∈ P , ≪ ∆ ′∗ ( F ) , X ⊗ Y ≫ ′ = ≪ F, X ∗ Y ≫ ′ , ≪ ∆ ∗ ( F ) , X ⊗ Y ≫ ′ = ≪ F, X ∗ Y ≫ . F ∈ T ( P ∗ ) , X ∈ T ( P ) : ≪ Ψ P ( F ) , X ≫ ′ = ≪ F, X ≫ . Let: A = { F ∈ T ( P ∗ ) | ∀ X ∈ T ( P ) , ≪ Ψ P ( F ) , X ≫ ′ = ≪ F, X ≫} . Clearly, ∈ A . Let F, G ∈ A . For all X ∈ T ( P ) : ≪ Ψ P ( F G ) , X ≫ ′ = ≪ Ψ P ( F )Ψ P ( G ) , X ≫ ′ = ≪ Ψ P ( F ) ⊗ Ψ P ( G ) , ∆( X ) ≫ ′ = ≪ F ⊗ G, ∆( X ) ≫ = ≪ F G, X ≫ . So A is a subalgebra of T ( P ∗ ) . Let us take f ∈ P ∗ . For all x , . . . , x k ∈ P : ≪ Ψ P ( f ) , x . . . x k ≫ ′ = ≪ f, x . . . x k ≫ ′ − ε ( f ) ε ( x . . . x k )= f ( x ) if k = 1 , if k ≥ ,ε ( f ) − ε ( f ) if k = 0= ( f ( x ) if k = 1 , otherwise = ≪ f, x . . . x k ≫ . Consequently, P ∗ ⊆ A , so A = T ( P ∗ ) .Let F ∈ T ( P ∗ ) . For all X, Y ∈ T ( P ) : ≪ (Ψ P ⊗ Ψ P ) ◦ ∆ ∗ ( F ) , X ⊗ Y ≫ ′ = ≪ ∆ ∗ ( F ) , X ⊗ Y ≫ = ≪ F, X ∗ G ≫ = ≪ Ψ P ( F ) , X ∗ G ≫ ′ = ≪ ∆ ′∗ ◦ Ψ P ( F ) , X ⊗ Y ≫ ′ . As the pairing ≪ − , − ≫ ′ is non degenerate, (Ψ P ⊗ Ψ P ) ◦ ∆ ∗ = ∆ ′∗ ◦ Ψ P . (cid:3) Considering the abelianization: Corollary 53 The following map is a bialgebra isomorphism: ψ P : D ∗ P = ( S ( P ∗ ) , m, ∆ ∗ ) −→ D ′ P = ( S ( P ∗ ) , m, ∆ ′∗ ) f ∈ P ∗ −→ f − ε ( f )1 ,f ∈ P ∗ + −→ f. Proposition 54 1. (a) The monoid of characters of both D ′ P and ( D ′ P ) ab is identifiedwith ( P , ♦ ′ ) , where for all x = P x n ∈ P , y ∈ P : x ♦ ′ y = y + X n ≥ X ≤ i <...
1. We use the description of the monoid of characters of ( D ′ P ) ab = ( S ( P ∗ ) , ∗ , ∆ ∗ ) from corollary 32; its product ♦ ′ is given by: x ♦ ′ y = y + x • e y = y + X n ≥ X ≤ i ,...,i k ≤ n, all distincts ,j ,...,j k ≥ ∞ Y l =0 ♯ { p | j p = l } ! x ◦ i ,...,i k ( y j , . . . , y j k )= y + X n ≥ X ≤ i <...
2. Using the isomorphism ψ P , we obtain an explict isomorphism of monoids: (cid:26) ( M D P , ♦ ′ ) −→ ( M D P , ♦ ) x −→ x + I, (cid:26) ( M D P , ♦ ) −→ ( M D P , ♦ ′ ) x −→ x − I. Definition 55 Let V be a vector space and P be an operad. We put: M D P ( V ) = M D P ⊗ C V , G B P ( V ) = G B P ⊗ C V . Let θ P : b ∞ −→ P be an operad morphism. We obtain: • A group structure on G A P ( V ) = F P ( V ) , given by: ∀ x, y ∈ G A P ( V ) , x (cid:7) y = ⌊ e x , e y ⌋ . In particular, if the morphism θ P is trivial, that is to say: θ P ( ⌊− , −⌋ k,l ) = ( I if ( k, l ) = (1 , or (0 , , otherwise , then x (cid:7) y = x + y . • The monoid M D P = ( coinv ( P ⊗ C V ) , ♦ ) and the group G B P = ( coinv ( P ⊗ C V ) + , ♦ ) .50 By theorem 46 and proposition 36, there exists right actions ⊳ of ( M D P ( V ) , ⋄ ) and ⊳ ′ of ( M D P ( V ) , ⋄ ′ ) on G A P ( V ) by group endomorphisms; by restriction, there exists a right action ⊳ of G B P ( V ) on G A P ( V ) by group automorphisms.Let us first describe these two actions. Proposition 56 The vector spaces M D P ( V ) and F P ( V ) ⊗ V ∗ are canonically isomorphic. Forall x = P p n . ( x , . . . , x n ) ∈ G A P ( V ) , for all y = q ⊗ f ∈ M D P ( V ) , with q ∈ F P ( V ) and f ∈ V ∗ : x ⊳ ′ y = X n ≥ X ≤ i <...
We naturally identify End K ( V, V ⊗ n ) with V ⊗ n ⊗ V ∗ . For all n ≥ : coinv P ⊗ C V ( n ) = P ( n ) ⊗ S n ( V ⊗ n ⊗ V ∗ ) = ( P ( n ) ⊗ S n V ⊗ n ) ⊗ V ∗ = F P ( V )( n ) ⊗ V ∗ , so: M D P ( V ) = Y n ≥ F P ( V )( n ) ⊗ V ∗ = Y n ≥ F P ( V )( n ) ⊗ V ∗ = F P ( V ) ⊗ V ∗ . The first formula comes from proposition 36, as x ⊳ ′ y = x e y . The second formula is obtainedby the application of the isomorphism between D P ( V ) and D ′ P ( V ) , inducing the isomorphismbetween ( M D P ( V ) , ♦ ′ ) and ( M D P ( V ) , ♦ ) . (cid:3) The graduation of F P ( V ) induces a distance d on F P ( V ) : denoting val the valutation asso-ciated to this graduation, ∀ x, y ∈ F P ( V ) , d ( x, y ) = 2 − val ( x − y ) . Proposition 57 For all y ∈ M D P ( V ) , we consider: φ y : (cid:26) F P ( V ) −→ F P ( V ) x −→ φ y ( x ) = x ⊳ y. Then φ y is a continuous endomorphism of P -algebras. Moreover: ∀ y, z ∈ M D P ( V ) , φ y ◦ φ z = φ z ♦ y . Proof. For all x ∈ G A P ( V ) , y ∈ M D P ( V ) , val ( x ⊳ y ) ≥ val ( x ) , so φ y is continuous.Un to the isomorphism between ( M D P ( V ) , ⋄ ) and ( M D P ( V ) , ⋄ ′ ) , we work with the action ⊳ ′ .For all y ∈ M D P ( V ) , x ∈ F P ( V ) , we put φ y ( x ) = x ⊳ y . By lemma 45, for all p ∈ P ( n ) , for all v , . . . , v n ∈ F P ( V ) , as y is primitive, e y is a group-like element and: φ ′ y ( p. ( v , . . . , v n )) = p. ( v , . . . , b n ) e y = p. ( v e y , . . . , v n e y )= p. ( φ ′ y ( v ) , . . . , φ ′ y ( v n )) . By continuity of φ ′ y , this is still true if v , . . . , v n ∈ F P ( V ) , so φ ′ y is indeed a continuous morphismof P -algebras. Up to an automorphism, this is also the case for φ y .51or all x ∈ G A P ( V ) , y, z ∈ M D P ( V ) : φ y ◦ φ z ( x ) = ( x ⊳ z ) ⊳ y = x ⊳ ( z ♦ y ) = φ z ♦ y ( x ) , so φ y ◦ φ z = φ z ♦ y . (cid:3) Notations. 1. We denote by M P ( V ) the monoid of continuous P -algebra endomorphisms of F P ( V ) andby G P ( V ) the group of continuous P -algebra automorphisms of F P ( V ) .2. We put: F P ( V ) = Y n ≥ P ( n ) .V ⊗ n . If φ ∈ M P ( V ) , then φ ( F P ( V ) ) ⊆ F P ( V ) , so φ induces a map: φ ′ : (cid:26) V = F P ( V ) /F P ( V ) −→ F P ( V ) /F P ( V ) v −→ φ ( v ) . We obtain in this way a monoid morphism ̟ : M P ( V ) −→ End K ( V ) , and by restrictiona group morphism ̟ : G P ( V ) −→ GL ( V ) . The kernel of this morphism is denoted by G (1) P ( V ) : this is the group of continuous automorphisms of F P ( V ) tangent to the identity.3. If ϕ ∈ GL ( V ) , let ι ( ϕ ) be the unique continuous endomorphism of P -algebras of F P ( V ) such that ι ( ϕ )( x ) = ϕ ( x ) for all x ∈ V . Then ι : GL ( V ) −→ G P ( V ) is a group morphism,such that ̟ ◦ ι = Id GL ( V ) . So: G P ( V ) = G (1) P ( V ) ⋊ GL ( V ) . Proposition 58 1. The morphism φ is an anti-isomorphism from M D P ( V ) to M P ( V ) .2. Its restriction to G D P ( V ) is an anti-isomorphism from G B P ( V ) to G (1) P ( V ) . Proof. By proposition 57, φ is antimorphism from M B P ( V ) to M P ( V ) . Let y = q ⊗ f ∈ M D P ( V ) , with q ∈ F P ( V ) and f ∈ V ∗ . For all x ∈ V , φ y ( x ) = f ( x ) q, so φ is injective. Let φ ∈ M P ( V ) . We fix a basis ( x i ) i ∈ I of V . For all i ∈ I , we put q i = φ ( x i ) ,and y = P q j ⊗ x ∗ j . For all i ∈ I : φ y ( x i ) = X j x ∗ j ( x i ) q j = q i = φ ( x i ) . As both φ y and φ are continuous morphisms of P -algebras, they are equal, so φ is surjective.Moreover, for all y = q ⊗ f ∈ M D P ( V ) : φ y ∈ G (1) P ( V ) ⇐⇒ ∀ x ∈ V, φ y ( x ) − x ∈ F P ( V ) ⇐⇒ ∀ x ∈ V, f ( x ) q − x ∈ F P ( V ) ⇐⇒ y = Id V ⇐⇒ y ∈ G B P ( V ) . So φ induces an anti-isomorphism from G B P ( V ) to G (1) P ( V ) . (cid:3) .4 Set bases In numerous cases, there exists, for all n ≥ , a basis B n of P ( n ) such that for for all x ∈ B n , forall σ ∈ S n , x σ ∈ B n . We shall say in this case that the basis B = F B n of P is a set basis, andwe shall identify the vector spaces P and P ∗ through the map: (cid:26) P −→ P ∗ x ∈ B −→ x ∗ ∈ B ∗ . We do not expect these bases to be stable under the operadic composition (we shall not alwayswork here with set operads). For example, ( e n ) n ≥ and ( σ ) σ ∈⊔ S n are set bases of respectively Com and As .Let us consider a set basis B of P . • For all n ≥ , we denote by O n the set of orbits of the action of S n on B n . This is the setof isoclasses of elements of B n . • For all x ∈ B n , its orbit will be denoted by b x ∈ O n . • For all ω ∈ O n , we denote by s ω the quotient cardinal of N ! / | ω | (number of symmetries of ω ).Let us fixe a system ( x ω ) ω ∈O n of representants of the orbits. For all ω ∈ O n , we denote by ω the class of x ω in coinv P ( n ) : this does not depend of the choice of x ω , and ( ω ) ω ∈O n is a basis of coinv ( P ) . We denote by B ∗ n the dual basis of P ∗ ( n ) . For all ω ∈ O n , we put: f ω = X σ ∈ S n ( x σω ) ∗ = s ω X x ∈ ω x ∗ . Then ( f ω ) ω ∈⊔O n is a basis of inv P ∗ . For all ω , ω ′ ∈ O n : f ω ( ω ′ ) = δ ω,ω ′ s ω . We now fix a finite-dimensional vector space V . Let us choose a basis ( X , . . . , X N ) of V and let us denote by ( ǫ , . . . , ǫ N ) the dual basis of V ∗ . A basis of P ⊗ V ⊗ n is given by ( x ⊗ X i . . . X i n ) b ∈B n , ≤ i ,...,i n ≤ N , which can be seen as the set of elements of B n decorated byelements of [ N ] . The action of the symmetric group is given by: ( x ⊗ X i . . . X i n ) σ = x σ ⊗ X i σ (1) . . . X i σ ( n , so this is also a set basis. The set of orbits of this action is interpreted as the set of isoclasses of el-ements of B n decorated by [ N ] . It is a basis of the homogeneous component of degree n of F P ( V ) .A basis of End K ( V, V ⊗ n ) is given by ( ǫ j X i . . . X i n ) ≤ i ,...,i n ,j ≤ N , where for all v ∈ V : ǫ j X i . . . X i n ( v ) = ǫ j ( v ) X i . . . X i n . A basis of P ⊗ C V ( n ) is given by ( x ⊗ ǫ j X i . . . X i n ) b ∈B n , ≤ j,i ,...,i n ≤ N . The action of the symmetricgroup is given by: ( x ⊗ ǫ j X i . . . X i n ) σ = x σ ⊗ ǫ j X i σ (1) . . . X i σ ( n , so it is a set basis. Moreover, the orbits of this action can be seen as pairs ( ω, j ) , where ω is anisoclasse of elements of B decorated by [ N ] and j ∈ [ N ] .53 hapter 4 Examples and applications Let us now give examples of these constructions. We start with classical operads Com , As and PreLie . We obtain that A ∗ Com is the coordinate Hopf algebra of the group G = ( K [[ X ]] + , +) and B ∗ Com is the coordinate bialgebra of the Faà di Bruno monoid of formal continuous maps M = ( K [[ X ]] + , ◦ ) ; the coaction of B ∗ Com and A ∗ Com corresponds to the action of M on G bycomposition. Similarly, for As , we obtain groups and monoids of non-commutative formal se-ries. Moreover, A ∗ PreLie is the Connes-Kreimer Hopf algebra [9, 25, 13, 38], and A PreLie is theGrossman-Larson Hopf algebra [21, 22, 23], both of them based on trees; D ∗ PreLie is anotherbialgebra of rooted trees, whose coproduct is given by extraction-contraction operations, definedin [4], as well as the coaction of D ∗ PreLie on A ∗ PreLie .We then introduce an operadic structure on Feynman graphs, inducing operadic structureson other combinatorial objects as simple graphs, simple graphs without cycle, posets. All theseoperads give pairs of (co)-interacting bialgebras, as well as non-commutative versions of them;we recover in particular in this process the bialgebras on graphs without cycle of [33], or thebialgebras of quasiposets used in [16]. Com , As and PreLie Com and As The brace and pre-Lie structures of Com are given by: h e i , e j . . . e j k i = (cid:18) i + 1 k (cid:19) e i + j + ... + j k , e i • e j = ( i + 1) e i +1 . Let us fix the vector space V = ( X , . . . , X N ) . We denote by ( ǫ i ) i ≥ the dual basis of ( X , . . . , X N ) .We consider the morphism θ Com : b ∞ −→ Com given in section 2.2.3. Then: A Com ( V ) = S ( F Com ( V )) = S ( K [ X , . . . , X N ] + ) , with the quasi-shuffle product ∗ induced by the product of K [ X , . . . , X N ] + . For all α ∈ N N , weput X α = X α . . . X α N N . Then, for example, if α, β, γ, δ ∈ N N − { } : X α ∗ X β = X α X β + X α + β ,X α ∗ X β X Γ = X α X β X γ + X α + β X γ + X α X β + γ ,X α X β ∗ X γ X δ = X α X β X γ X δ + X α + γ X β X γ + X α + δ X β X γ + X α X β + γ X δ + X α X β + δ X γ + X α + γ X β + δ + X α + δ X β + γ . Dually, A ∗ Com ( V ) is identified with S ( K [ X , . . . , X N ] + ) as an algebra. Its coproduct ∆ ∗ isgiven by: ∀ α ∈ N N , ∆( X α ) = X α = β + γ X β ⊗ X γ . ( { P ( X , . . . , X N ) | P ∈ K [[ X , . . . , X N ]] + } , · ) . Moreover, B ∗ Com ( V ) is the Faà di Bruno Hopf algebra on N variables, which group of charactersis the group of formal diffeomorphisms of K [[ X , . . . , X N ]] which are tangent to the identity, thatis to say: ( { ( X i + P ( X . . . , X N )) i ∈ [ N ] | P i ( X , . . . , X N ) ∈ K [[ X , . . . , X N ]] ≥ } , ◦ ) , where K [[ X , . . . , X N ]] ≥ is the subspace of formal series in K [[ X , . . . , X N ]] of valuation ≥ .Here are examples of coproducts ∆ ∗ and ∆ ′∗ on D ∗ Com : ∆ ′∗ ( e ) = e ⊗ ⊗ e + e ⊗ e , ∆ ∗ ( e ) = e ⊗ e , ∆ ′∗ ( e ) = e ⊗ ⊗ e + e ⊗ e e + e ⊗ e + 2 e ⊗ e , ∆ ∗ ( e ) = e ⊗ e e + e ⊗ e . Let us describe D ∗ As . Definition 59 Let σ ∈ S n .1. We shall write I ⊔ . . . ⊔ I k = σ [ n ] if: • For all p ∈ [ k ] , both I p and σ ( I p ) are intervals of [ n ] . • [ n ] = I ⊔ . . . ⊔ I k . • For all ≤ p < q ≤ k , for all i ∈ I p , j ∈ I q , i < j .2. Let us assume that I ⊔ . . . ⊔ I k = σ [ n ] . As σ ( I ) , . . . , σ ( I k ) are intervals, there existsa unique permutation τ ∈ S k such that σ ( I τ (1) ) ⊔ . . . ⊔ σ ( I τ ( k ) ) = σ − [ n ] . We denote σ/ ( I , . . . , I k ) = τ − . The bialgebra D ∗ As is freely generated by the set G n ≥ S n . For all permutation σ ∈ S n , ∆ ∗ ( σ ) = X I ⊔ ... ⊔ I k = σ [ n ] σ/ ( I , . . . , I k ) ⊗ Std ( σ | I ) . . . Std ( σ | I k ) , where Std is the usual standardization of permutations. For example: ∆ ∗ ((1)) = (1) ⊗ (1) , ∆ ∗ ((12)) = (12) ⊗ (1)(1) + (1) ⊗ (12) , ∆ ∗ ((21)) = (21) ⊗ (1)(1) + (1) ⊗ (21) , ∆ ∗ ((123)) = (123) ⊗ (1)(1)(1) + (1) ⊗ (123) + (12) ⊗ (12)(1) + (12) ⊗ (1)(12) , ∆ ∗ ((132)) = (132) ⊗ (1)(1)(1) + (1) ⊗ (132) + (12) ⊗ (1)(21) , ∆ ∗ ((213)) = (213) ⊗ (1)(1)(1) + (1) ⊗ (213) + (12) ⊗ (21)(1) , ∆ ∗ ((231)) = (231) ⊗ (1)(1)(1) + (1) ⊗ (231) + (21) ⊗ (12)(1) , ∆ ∗ ((312)) = (312) ⊗ (1)(1)(1) + (1) ⊗ (312) + (21) ⊗ (1)(12) , ∆ ∗ ((321)) = (321) ⊗ (1)(1)(1) + (1) ⊗ (321) + (21) ⊗ (21)(1) + (21) ⊗ (1)(21) . Let us consider the vector space V = V ect ( X , . . . , X N ) . Then D ∗ As ( V ) is generated by theelements ( X i . . . X i k ǫ j ) k ≥ ,i ,...,i k ,j ∈ [ N ] . For all word w in letters X , . . . , X N , for all i ∈ [ N ] : ∆ ∗ ( wǫ i ) = X k ≥ X w = u v u ...v k u k ,i ,...,i k ∈ [ n ] u X i u . . . u k − X i k u k ǫ i ⊗ ( v ǫ i ) . . . ( v k ǫ k ) . D ∗ Com ( V ) is D ∗ As ( V ) . Examples. In D ∗ Com ( V ) or in D ∗ As ( V ) , if i, j, k, l ∈ [ N ] : ∆ ∗ ( X i ǫ j ) = N X p =1 X p ǫ j ⊗ X i ǫ p , ∆ ∗ ( X i X j ǫ k ) = N X p =1 X p ǫ k ⊗ X i X j ǫ p + N X p,q =1 X p X q ǫ k ⊗ ( X i ǫ p )( X j ǫ q ) , ∆ ∗ ( X i X j X k ǫ l ) = N X p =1 X p ǫ l ⊗ X i X j X k ǫ p + N X p,q =1 X p X q ǫ l ⊗ ( X i X j ǫ p )( X k ǫ q )+ N X p,q =1 X p X q ǫ l ⊗ ( X i ǫ p )( X j X k ǫ q ) + N X p,q,r =1 X p X q X r ǫ l ⊗ ( X i ǫ p )( X j ǫ q )( X k ǫ r ) . In order to obtain the Hopf algebra B ∗ As ( V ) , we quotient by the relations X i ǫ j = δ i,j . Thecoproduct becomes: ∆ ∗ ( X i X j ǫ k ) = 1 ⊗ X i X j ǫ p + X i X j ǫ k ⊗ , ∆ ∗ ( X i X j X k ǫ l ) = 1 ⊗ X i X j X k ǫ l + N X p =1 X p X k ǫ l ⊗ X i X j ǫ p + N X q =1 X i X q ǫ l ⊗ X j X k ǫ q + X i X j X k ǫ l ⊗ . We consider the morphism θ As : b ∞ −→ As defined in section 2.2.3. The Hopf alge-bra A ∗ As ( V ) is generated by the elements ( X i . . . X i k ) k ≥ ,i ,...,i k ∈ [ N ] . For all word w in letters X , . . . , X N , for all i ∈ [ N ] : ∆ ⋆ ( w ) = w ⊗ ⊗ w + X w = uv, u,v = ∅ u ⊗ v. The coaction of D ∗ As ( V ) over A ∗ As ( V ) is given by: ρ ( w ) = X k ≥ X w = u v u ...v k u k ,i ,...,i k ∈ [ n ] u X i u . . . u k − X i k u k ⊗ ( v ǫ i ) · . . . · ( v k ǫ k ) . For example: ρ ( X i ) = N X p =1 X p ⊗ X i ǫ p ,ρ ( X i X j ) = N X p =1 X p ⊗ X i X j ǫ p + N X p,q =1 X p X q ⊗ ( X i ǫ p )( X j ǫ q ) ,ρ ( X i X j X k ) = N X p =1 X p ⊗ X i X j X k ǫ p + N X p,q =1 X p X q ⊗ ( X i X j ǫ p )( X k ǫ q )+ N X p,q =1 X p X q ⊗ ( X i ǫ p )( X j X k ǫ q ) + N X p,q,r =1 X p X q X r ⊗ ( X i ǫ p )( X j ǫ q )( X k ǫ r ) . B ∗ As ( V ) : ρ ( X i ) = X i ⊗ ,ρ ( X i X j ) = X i X j ⊗ N X p =1 X p ⊗ X i ǫ p ,ρ ( X i X j X k ) = X i X j X k ⊗ N X p =1 X p X k ⊗ X i X j ǫ p + N X q =1 X i X q ⊗ X j X k ǫ q + N X p =1 X p ⊗ X i X j X k ǫ p . PreLie We now consider the operad PreLie , as described in [6, 7]. This operad comes from an operadicspecies; for all finite set A , PreLie ( A ) is the vector space generated by rooted trees whose setof vertices is A . For example: PreLie ( { } ) = V ect ( q ) , PreLie ( { , } ) = V ect ( qq , qq ) , PreLie ( { , , } ) = V ect ( qqq , qqq , qqq , qqq , qqq , qqq , qqq ∨ , q qq ∨ , q qq ∨ ) . The composition is given by insertion at vertices in all possible ways. For example: qq ◦ qq = qqq ∨ + qqq , qq ◦ qq = qqq . The morphism θ PreLie : b ∞ −→ PreLie is described in section 2.2.3.Let us fix V = V ect ( X , . . . , X N ) . • A basis of A PreLie ( V ) is given by forests of rooted trees decorated by [ N ] ; in particular,if i ∈ [ N ] , X i is identified with q i . The product is given by graftings; for example, if i, j, k ∈ [ N ] : q i ∗ q j = q i q j + qq ij , q i ∗ qq jk = q i qq jk + qqq ijk , qq ij ∗ q k = qq ij q k + q qq ∨ i kj + qqq ijk , q i q j ∗ q k = q i q j q k + qq ik q j + q i qq jk , q i ∗ q j q k = q i q j q k + qq ij q k + qq ik q j + q qq ∨ i kj . In other terms, this is the Grossman-Larson Hopf algebra of decorated rooted trees [21, 22,23]. Its dual is (the coopposite of) the Connes-Kreimer Hopf algebra of decorated rootedtrees [9, 13, 38, 25], which coproduct is given by admissible cuts. If i, j, k ∈ [ N ] : ∆ ⋆ ( q i ) = q i ⊗ ⊗ q i , ∆ ⋆ ( qq ij ) = qq ij ⊗ ⊗ qq ij + q i ⊗ q j , ∆ ⋆ ( q qq ∨ i kj ) = q qq ∨ i kj ⊗ ⊗ q qq ∨ i kj + qq ij ⊗ q k + qq ik ⊗ q j + q i ⊗ q j q k , ∆ ⋆ ( qqq ijk ) = qqq ijk ⊗ ⊗ qqq ijk + qq ij ⊗ q k + q i ⊗ qq jk . • A basis of B PreLie ( V ) is given by forests of pairs ( t, j ) , where t is a rooted tree decoratedby [ N ] and j ∈ [ N ] . The underlying pre-Lie product • is given by insertion at a vertex, asthe operadic composition is. For example, if i, j, k ∈ [ N ] and N ≥ : ( qq , i ) • ( qq jk , 1) = ( q qq ∨ j k , i ) + ( qqq jk , i ) , ( qq , i ) • ( qq jk , 2) = ( qqq jk , i ) , ( qq , i ) • ( qq jk , 1) = ( q qq ∨ j k , i ) + ( qqq jk , i ) + ( qqq jk , i ) , ( qq , i ) • ( qq jk , 2) = 0 . The bialgebra D ∗ PreLie ( V ) has the same basis. Its coproduct is given by extraction-contraction of subtrees. For example, in the non decorated case (or equivalently if N = 1 ): ∆ ∗ ( q ) = q ⊗ q , ∆ ∗ ( qq ) = qq ⊗ q q + q ⊗ qq , ∆ ∗ ( q qq ∨ ) = q qq ∨ ⊗ q q q + qq ⊗ qq q + qq ⊗ q qq + q ⊗ q qq ∨ , ∆ ∗ ( qqq ) = qqq ⊗ q q q + qq ⊗ qq q + qq ⊗ q qq + q ⊗ qqq . This is the extraction-contraction coproduct of [4]. More generally, in D ∗ PreLie ( V ) , if a, b, c, d ∈ [ N ] : ∆ ∗ (( q a , d )) = N X p =1 ( q p , d ) ⊗ ( q a , p ) , ∆ ∗ (( qq ab , d )) = N X p,q =1 ( qq pq , d ) ⊗ ( q a , p )( q b , q ) + N X p =1 ( q p , d ) ⊗ ( qq ab , p ) , ∆ ∗ (( q qq ∨ acb , d )) = N X p,q,r =1 ( q qq ∨ prq , d ) ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 ( qq pq , d ) ⊗ ( qq ab , p )( q c , q )+ N X p,q =1 ( qq pq , d ) ⊗ ( qq ac , p )( q b , q ) + N X p =1 ( q p , d ) ⊗ ( q qq ∨ acb , p ) , ∆ ∗ (( qqq abc , d )) = N X p,q,r =1 ( qqq pqr , d ) ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 ( qq pq , d ) ⊗ ( qq ab , p )( q c , q )+ N X p,q =1 ( qq pq , d ) ⊗ ( q a , q )( qq bc , p ) + N X p =1 ( q p , d ) ⊗ ( qqq abc , p ) . After taking the quotient by q i ǫ j X − δ i,j , in B ∗ PreLie ( V ) : ∆ ∗ (( qq ab , d )) = ( qq ab , d ) ⊗ ⊗ ( qq ab , d ) , ∆ ∗ (( q qq ∨ acb , d )) = ( q qq ∨ abc , d ) ⊗ N X p =1 ( qq pc , d ) ⊗ ( qq ab , p ) + N X p =1 ( qq pb , d ) ⊗ ( qq ac , p ) + 1 ⊗ ( qqq ∨ acb , d ) , ∆ ∗ (( qqq abc , d )) = ( qqq abc , d ) ⊗ N X p =1 ( qq pc , d ) ⊗ ( qq ab , p ) + N X q =1 ( qq aq , d ) ⊗ ( qq bc , p ) + 1 ⊗ ( qqq abc , d ) . D ∗ PreLie ( V ) over A ∗ PreLie ( V ) is given in a similar way. For example: ρ ( q a ) = N X p =1 q p ⊗ ( q a , p ) ,ρ ( qq ab ) = N X p,q =1 qq pq ⊗ ( q a , p )( q b , q ) + N X p =1 q p ⊗ ( qq ab , p ) ,ρ ( qqq ∨ acb ) = N X p,q,r =1 q qq ∨ prq ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 qq pq ⊗ ( qq ab , p )( q c , q )+ N X p,q =1 qq pq ⊗ ( qq ac , p )( q b , q ) + N X p =1 q p ⊗ ( q qq ∨ acb , p ) ,ρ ( qqq abc ) = N X p,q,r =1 qqq pqr ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 qq pq ⊗ ( qq ab , p )( q c , q )+ N X p,q =1 qq pq ⊗ ( q a , q )( qq bc , p ) + N X p =1 q p ⊗ ( qqq abc , p ) . We shall use the following formalism for oriented Feynman graphs : Definition 60 1. A Feynman graph is a family Γ = ( V (Γ) , Int (Γ) , OutExt (Γ) , InExt (Γ) , S Γ , T Γ ) , where: • V (Γ) is a finite, non-empty set, called the set of vertices of Γ . • Int (Γ) is a finite set, called the set of internal edges of Γ . • OutExt (Γ) is a finite set, called the set of external outgoing edges of Γ . • InExt (Γ) is a finite set, called the set of internal ingoing edges of Γ . • S Γ : Int (Γ) ⊔ OutExt (Γ) −→ V (Γ) is the source map. • T Γ : Int (Γ) ⊔ InExt (Γ) −→ V (Γ) is the target map.2. Let Γ and Γ ′ be two Feynman graphs. We shall say that Γ and Γ ′ are equivalent if thefollowing conditions hold: • V (Γ) = V (Γ ′ ) . • There exist bijections: φ Int : Int (Γ) −→ Int (Γ ′ ) ,φ OutExt : OutExt (Γ) −→ OutExt (Γ ′ ) ,φ InExt : InExt (Γ) −→ IntExt (Γ ′ ) , such that: ∀ e ∈ Int (Γ) , S Γ ( e ) = S Γ ′ ◦ φ Int ( e ) and T Γ ( e ) = T Γ ′ ◦ φ Int ( e ) , ∀ e ∈ OutExt (Γ) , S Γ ( e ) = S Γ ′ ◦ φ OutExt ( e ) , ∀ e ∈ InExt (Γ) , T Γ ( e ) = T Γ ′ ◦ φ InExt ( e ) . oughly speaking, two Feynman graphs Γ and Γ ′ are equivalent if one obtains Γ from Γ ′ bychanging the names of its edges.3. Let A be a finite set. The set of all equivalence classes of Feynman graphs Γ such that V (Γ) is denoted by F ( A ) and the space generated by F ( A ) will be denoted by F ( A ) . We shall work only with equivalence classes of Feynman graphs, which we now simply callFeynman graphs. Remarks. Int (Γ) , OutExt (Γ) or InExt (Γ) may be empty.2. Restricting to Int (Γ) , Feynman graphs are also oriented graphs, possibly with multipleedges and loops.We shall represent Feynman graphs by a diagram, such as: ?>=<89:; O O O O y y rrrrrrrrrrrrrr ?>=<89:; / / ?>=<89:; o o O O O O (cid:15) (cid:15) O O O O Definition 61 Let Γ be a Feynman graph, and I ⊆ V (Γ) , non-empty.1. (Extraction). We define the Feynman graph Γ | I by: V (Γ | I ) = I,Int (Γ | I ) = { e ∈ Int (Γ) | S Γ ( e ) ∈ I, T Γ ( e ) ∈ I } ,OutExt (Γ | I ) = { e ∈ Int (Γ) | S Γ ( e ) ∈ I, T Γ ( e ) / ∈ I } ⊔ { e ∈ OutExt (Γ) | S Γ ( e ) ∈ I } ,InExt (Γ | I ) = { e ∈ Int (Γ) | S Γ ( e ) / ∈ I, T Γ ( e ) ∈ I } ⊔ { e ∈ InExt (Γ) | T Γ ( e ) ∈ I } . For all e ∈ Int (Γ | I ) ⊔ OutExt (Γ | I ) , for all f ∈ Int (Γ | I ) ⊔ IntExt (Γ | I ) : S Γ | I ( e ) = S Γ ( e ) , T Γ | I ( f ) = T Γ ( f ) . Roughly speaking, Γ | I is the Feynman graph obtained by taking all the vertices in I and thehalf edges attacted to them.2. (Contraction). Let b / ∈ V (Γ) . We define the Feynman graph Γ /I → b by: V (Γ /I → b ) = ( V (Γ) \ I ) ⊔ { b } , OutExt (Γ /I → b ) = OutExt (Γ) ,Int (Γ /I → b ) = Int (Γ) \ Int (Γ | I ) , InExt (Γ /I → b ) = InExt (Γ) . For all e ∈ Int (Γ /I → b ) ⊔ OutExt (Γ /I → b ) : S Γ /I → b ( e ) = ( S Γ ( e ) if S Γ ( e ) / ∈ I,b if otherwise. or all e ∈ Int (Γ /I → b ) ⊔ InExt (Γ /I → b ) : T Γ /I → b ( e ) = ( T Γ ( e ) if T Γ ( e ) / ∈ I,b if otherwise. Roughly speaking, Γ /I → b is obtained by contracting all the vertices of I and the internaledges between them to a single vertex b .3. We shall say that I is Γ -convex if for any oriented path x → y → . . . → y k → z in Γ : x, z ∈ I = ⇒ y , . . . , y k ∈ I. Lemma 62 Let Γ be a Feynman graph, I a , I b ⊆ V (Γ) , non-empty and disjoint.1. (Γ /I a → a ) | I b = Γ | I b .2. (Γ /I a → a ) /I b → b = (Γ /I b → b ) /I a → a .3. The following conditions are equivalent:(a) I a is Γ -convex and I b is (Γ /I a → a ) -convex.(b) I b is Γ -convex and I a is (Γ /I b → b ) -convex. Proof. 1. Let us put Γ ′ = (Γ /I a → a ) | I b . Then: V (Γ ′ ) = I b ,Int (Γ ′ ) = { e ∈ Int (Γ) | S Γ ( e ) ∈ I b , T Γ ( e ) ∈ I b } ,OutExt (Γ ′ ) = { e ∈ Int (Γ) | S Γ ( e ) ∈ I b , T Γ ( e ) / ∈ I b } ⊔ { e ∈ OutExt (Γ) | S Γ ( e ) ∈ I b } ,InExt (Γ ′ ) = { e ∈ Int (Γ) | S Γ ( e ) / ∈ I b , T Γ ( e ) ∈ I b } ⊔ { e ∈ OutExt (Γ) | T Γ ( e ) ∈ I b } . For all e ∈ Int (Γ ′ ) ⊔ OutExt (Γ ′ ) , S Γ ′ ( e ) = S Γ ( e ) . For all e ∈ Int (Γ ′ ) ⊔ InExt (Γ ′ ) , T Γ ′ ( e ) = T Γ ( e ) .So Γ ′ = Γ | I b .2. Let us put Γ ′′ = (Γ /I a → a ) /I b → b . Then: V ( γ ′′ ) = V ( γ ) ⊔ { a, }¯ \ ( I a ⊔ I b ) ,Int (Γ ′′ ) = { e ∈ Int (Γ) | S Γ ( e ) / ∈ I a ⊔ I b or T Γ ( e ) / ∈ I a ⊔ I b } ,OutExt (Γ ′′ ) = OutExt (Γ) ,InExt (Γ ′′ ) = InExt (Γ) . For all e ∈ Int (Γ ′′ ) ⊔ OutExt (Γ ′′ ) : S Γ ′′ ( e ) = a if S Γ ( e ) ∈ I a ,b if S Γ ( e ) ∈ I b ,S Γ ( e ) otherwise . For all e ∈ Int (Γ ′′ ) ⊔ IntExt (Γ ′′ ) : T Γ ′′ ( e ) = a if T Γ ( e ) ∈ I a ,b if T Γ ( e ) ∈ I b ,T Γ ( e ) otherwise . 61y symmetry between a and b , Γ ′′ = (Γ /I b → b ) /I a → a .3. = ⇒ . Let us assume that x → y → . . . → y k → z in Γ , with x, z ∈ I b . For all y ∈ V (Γ) ,we put: y = ( a if y ∈ I a ,y otherwise . Then x → y → . . . → y k → z in Γ /I a → a . As I b is (Γ /I a → a ) -convex, all the y i belong to I b ,so are different from a : hence, y , . . . , y k ∈ I b .Let us assume x → y → . . . → y k → z in Γ /I b → b , with x, z ∈ I a . If at least one of the y p isequal to b , let us consider the smallest index i such that x i = b and the greatest index j such that x j = b . There exists y ′ i , y ′′ j ∈ I b , such that x → y → . . . → y i − → y ′ i and y ′′ j → y j +1 → . . . → z in Γ . As a consequence, in Γ /I a → a : y ′′ j → y j +1 → . . . → y k → a → y → . . . → y i − → y ′′ i . As I b is (Γ /I a → a ) -convex, a ∈ I b , which is absurd. So none of the y p is equal to b , whichimplies that x → y → . . . → y k → z in Γ . As I a is Γ -convex, y , . . . , y p ∈ I a . ⇐ = : by symmetry between a and b . (cid:3) Lemma 63 Let Γ be a Feynman graph, and I a ⊆ I b ⊆ V ( G ) be non-empty sets.1. (Γ /I a → a ) / ( I b ⊔ { a } \ I a ) → b = Γ /I b → b .2. (Γ /I a → a ) | I b ⊔{ a }\ I a = (Γ | I b ) /I a → a .3. (Γ | I b ) | I a = Γ | I a .4. The following conditions are equivalent:(a) I b is Γ -convex and I a is Γ | I b -convex.(b) I a is Γ -convex and I b ⊔ { a } \ I a is Γ /I a → a -convex. Proof. 1. Let Γ ′ = (Γ /I a → a ) / ( I b ⊔ { a } \ I a ) → b . Then: V (Γ ′ ) = V (Γ) ⊔ { a } \ I b ,Int (Γ ′ ) = { e ∈ Int (Γ) | S Γ ( e ) / ∈ I b or T Γ ( e ) / ∈ I b } ,OutExt (Γ ′ ) = OutExt (Γ) ,InExt (Γ ′ ) = InExt (Γ) . For all e ∈ Int (Γ ′ ) ⊔ OutExt (Γ ′ ) : S Γ ′ ( e ) = ( S Γ ( e ) if S Γ ( e ) / ∈ I b ,b otherwise . For all e ∈ Int (Γ ′ ) ⊔ InExt (Γ ′ ) : T Γ ′ ( e ) = ( T Γ ( e ) if T Γ ( e ) / ∈ I b ,b otherwise . So Γ ′ = Γ /I b → b . 62. Let Γ ′′ = (Γ /I a → a ) | I b ⊔{ a }\ I a . Then: V (Γ ′′ ) = I b ⊔ { a } \ I a ,Int (Γ ′′ ) = { e ∈ Int (Γ) | S Γ ( e ) ∈ I b and T Γ ( e ) ∈ I b }\ { e ∈ Int (Γ) | S Γ ( e ) ∈ I a and T Γ ( e ) ∈ I a } ,OutExt (Γ ′′ ) = { e ∈ OutExt (Γ) | S Γ ( e ) ∈ I b } ⊔ { e ∈ Int (Γ) | S Γ ( e ) ∈ I b and T Γ ( e ) / ∈ I b } ,InExt (Γ ′′ ) = { e ∈ InExt (Γ) | T Γ ( e ) ∈ I b } ⊔ { e ∈ Int (Γ) | S Γ ( e ) / ∈ I b and T Γ ( e ) ∈ I b } . For all e ∈ Int (Γ ′′ ) ⊔ OutExt (Γ ′′ ) : S Γ ′′ ( e ) = ( S Γ ( e ) if S Γ ( e ) ∈ I b \ I a ,a otherwise . For all e ∈ Int (Γ ′′ ) ⊔ InExt (Γ ′′ ) : T Γ ′′ ( e ) = ( T Γ ( e ) if T Γ ( e ) ∈ I b \ I a ,a otherwise . So Γ ′′ = (Γ | I b ) /I a → a .3. Immediate.4. = ⇒ . Let x → y → . . . → y k → z in Γ , with x, z ∈ I a . Then x, z ∈ I b . As I b is Γ -convex, y , . . . , y k ∈ I b . As I a is Γ | I b -convex, y , . . . , y k ∈ I a .Let x → y → . . . → y k → z in Γ /I a → a , with x, z ∈ I b ⊔ { a } \ I a . Let i < . . . < i l be theindices such that y i = a . There exists elements y ′ i p , y ′′ i p ∈ I a , such that, in Γ : x → . . . → y ′ i , y ′′ i → y i +1 → . . . → y ′ i p , y ′′ i p → . . . → z. As I a ⊆ I b and I b is Γ -convex, all the y i except the y i p are elements of I b . So y , . . . , y k ∈ I b ⊔ { a } \ I a . ⇐ = . Let x → y → . . . → y k → z in Γ , with x, z ∈ I b . For all y ∈ V (Γ) , we put: y = ( y if y / ∈ I a ,a otherwise . Then x → y → . . . → y k → z in Γ /I a → a . As I b ⊔ { a } \ I a is Γ /I a → a -convex, y , . . . , y k ∈ I b ⊔ { a } \ I a , so y , . . . , y k ∈ I b .Let x → y → . . . → y k → z in Γ | I b , with x, z ∈ I a . Then Let x → y → . . . → y k → z in Γ ;as I a is Γ -convex, y , . . . , y k ∈ I a . (cid:3) Definition 64 Let Γ be a Feynman graph.1. We shall say that Γ is simple if the two following conditions hold: • For all v, v ′ ∈ V (Γ) , there exists at most one internal edge e in Γ with S Γ ( e ) = v and T Γ ( e ) = v ′ . • For all e ∈ Int (Γ) , S Γ ( e ) = T Γ ( e ) . Lemma 65 Let Γ be a Feynman graph, I ⊆ V (Γ) , non-empty.1. If Γ | I and Γ /I → a has no oriented cycle, then Γ has no oriented cycle and I is Γ -connex.2. We assume that I is Γ -convex. If Γ | I or Γ /I → a has an oriented cycle, then Γ has anoriented cycle. roof. 1. Let x → y → . . . → y k → z in Γ , with x, z ∈ I . We assume that at least oneof the y i is not in I . Let i be the smallest index such that y i / ∈ I and j be the smallest indexgreater than i such that y j ∈ I , with the convention y k +1 = z . Then a → y i → . . . → y j − → a in Γ /I → a , and Γ /I → a has an oriented cycle: this is a contradiction, so y , . . . , y k ∈ I and I is Γ -convex.Let us consider an oriented cycle x → . . . → x k → x in Γ . If one of the x i belongs to I ,as I is Γ -convex, all the x i belongs to I , so Γ | I has an oriented cycle: this is a contradiction. Ilnone of the x i belong to I , they form an oriented cycle in Γ /I → a : this is a contradiction. Asa conclusion, Γ has no oriented cycle.2. If Γ | I has an oriented cycle, obviously Γ has an oriented cycle. Let us assume that Γ /I → a has an oriented cycle. If this cycle does not contain a , then obviously Γ has an ori-ented cycle. If not, there exists an oriented cycle a → y → . . . → y k → a in Γ /I → a , with y , . . . , y k / ∈ V (Γ) \ I . Note that k ≥ , by definition of Γ /I → a . Hence, there exists x, z ∈ I ,such that x → y → . . . → y k → z in Γ : I is not Γ -convex. (cid:3) Remark. If I is not Γ -convex, Γ may have no oriented cycle, whereas Γ /I → a may haveone. Take for example: Γ = ?>=<89:; / / ! ! ?>=<89:; / / ?>=<89:; , If I = { , } , then: Γ /I → a = ?>=<89:; a $ $ ?>=<89:; e e . Proposition 66 Let Γ ∈ F ( A ) , Γ ′ ∈ F ( B ) , and b ∈ A . We put: Γ ∇ b Γ ′ = X Γ ′′ ∈F ( A ⊔ B \{ b } ) , Γ ′′| B =Γ ′ , Γ ′′ /B → b =Γ Γ ′′ , Γ ◦ b Γ ′ = X Γ ′′ ∈F ( A ⊔ B \{ b } ) , Γ ′′| B =Γ ′ , Γ ′′ /B → b =Γ ,B Γ ′′ -convex Γ ′′ . For all Γ ∈ F ( A ) , Γ ′ ∈ F ( B ) , Γ ′′ ∈ F ( C ) , if b = c ∈ A : (Γ ∇ b Γ ′ ) ∇ c Γ ′′ = (Γ ∇ c Γ ′′ ) ∇ b Γ ′ , (Γ ◦ b Γ ′ ) ◦ c Γ ′′ = (Γ ◦ c Γ ′′ ) ◦ b Γ ′ . If b ∈ B and c ∈ C : (Γ ∇ b Γ ′ ) ∇ c Γ ′′ = Γ ∇ b (Γ ′ ∇ c Γ ′′ ) , (Γ ◦ b Γ ′ ) ◦ c Γ ′′ = Γ ◦ b (Γ ′ ◦ c Γ ′′ ) . Proof. If b = c ∈ A : (Γ ∇ b Γ ′ ) ∇ c Γ ′′ = X Υ ∈F ( A ⊔ B ⊔ C \{ b,c } ) , Υ | C =Γ ′′ , (Υ /C → C ) | B =Γ ′ , (Υ /C → c ) /B → b =Γ Υ = X Υ ∈F ( A ⊔ B ⊔ C \{ b,c } ) , Υ | C =Γ ′′ , Υ | B =Γ ′ , (Υ /B → b ) /C → c =Γ Υ = (Γ ∇ c Γ ′′ ) ∇ b Γ ′ ;(Γ ◦ b Γ ′ ) ◦ c Γ ′′ = X Υ ∈F ( A ⊔ B ⊔ C \{ b,c } ) , Υ | C =Γ ′′ , (Υ /C → C ) | B =Γ ′ , (Υ /C → c ) /B → b =Γ ,C Υ -convex, B Υ /C → c -convex Υ = X Υ ∈F ( A ⊔ B ⊔ C \{ b,c } ) , Υ | C =Γ ′′ , Υ | B =Γ ′ , (Υ /B → b ) /C → c =Γ ,B Υ -convex, C Υ /B → b -convex Υ = (Γ ◦ c Γ ′′ ) ◦ b Γ ′ . We used lemma 62, with I a = B and I b = C . 64et b ∈ A and c ∈ B . (Γ ∇ b Γ ′ ) ∇ c Γ ′′ = X Υ ∈F ( A ⊔ B ⊔ C \{ b,c } ) , Υ | C =Γ ′′ , (Υ /C → c ) | B =Γ ′′ , (Υ /C → c ) /B → b =Γ Υ = X Υ ∈F ( A ⊔ B ⊔ C \{ b,c } ) , (Υ | B ⊔ C \{ c } ) | C =Γ ′′ , (Υ | B ⊔ C \{ c } ) /C → c =Γ ′′ , Υ /B ⊔ C \{ c }→ b =Γ Υ = Γ ∇ b (Γ ′ ∇ c Γ ′′ );(Γ ◦ b Γ ′ ) ◦ c Γ ′′ = X Υ ∈F ( A ⊔ B ⊔ C \{ b,c } ) , Υ | C =Γ ′′ , (Υ /C → c ) | B =Γ ′′ , (Υ /C → c ) /B → b =Γ ,C Υ -convex, B Υ | C -convex Υ = X Υ ∈F ( A ⊔ B ⊔ C \{ b,c } ) , (Υ | B ⊔ C \{ c } ) | C =Γ ′′ , (Υ | B ⊔ C \{ c } ) /C → c =Γ ′′ , Υ /B ⊔ C \{ c }→ b =Γ ,B ⊔ C \ { c } Υ -convex ,C Υ | B ⊔ C \{ c } -convex Υ = Γ ◦ b (Γ ′ ◦ c Γ ′′ ) . We used lemma 63, with I a = C and I b = B ⊔ C \ { c } . (cid:3) Although the associativity of the composition is satisfied, F is not an operad: it contains nounit. In order to obtain it, we must take a completion. For all finite set A , we put: F ( A ) = Y Γ ∈F ( A ) K Γ . It contains F ( A ) . Its elements will be denoted under the form: X Γ ∈F ( A ) a Γ Γ . Theorem 67 The compositions ∇ and ◦ are naturally extended to F , making it an operad.Its unit is: I = X Γ ∈F ( { } ) Γ . Proof. Let X = P a Γ Γ ∈ F ( A ) and Y = P b Γ Γ in F ( B ) and b ∈ A . Then: X ∇ b Y = X Υ ∈F ( A ⊔ B \{ b } ) a Υ /B → b b Υ | B Υ , X ◦ b Y = X Υ ∈F ( A ⊔ B \{ b } ) ,B Υ -convex a Υ /B → b b Υ | B Υ . These formulas also make sense if X ∈ F ( A ) and Y ∈ F ( B ) . Proposition 66 implies the associa-tivity of ∇ and ◦ .Let Γ ∈ F ( A ) and b ∈ A . If Γ ′ ∈ F ( { b } ) : Γ ∇ b Γ ′ = ( Γ if InExt (Γ |{ b } ) = InExt (Γ ′ ) and OutExt (Γ |{ b } ) = OutExt (Γ ′ ) , otherwise . Summing over all possible Γ ′ , Γ ∇ b I = Γ . Hence, for all X ∈ F ( A ) , X ∇ b I = X .Let Γ ∈ F ( { } ) and Γ ′ ∈ F ( A ) . Then: Γ ∇ Γ ′ = ( Γ if InExt (Γ ′ ) = InExt (Γ) and OutExt (Γ) = OutExt (Γ ′ ) , otherwise . Summing over all possible Γ , I ∇ Γ ′ = Γ ′ . Hence, for all X ∈ F ( A ) , I ∇ X = X , so I is the unitof the operad ( F , ∇ ) . The proof is similar for ( F , ◦ ) . (cid:3) emark. The unit of F is: I = X i,j,k ≥ i / / ?>=<89:; j / / k (cid:10) (cid:10) , where the integers on the edges and half-edges indicate their multiplicity. Proposition 68 1. For all finite space A , we denote by N c F ( A ) the set of Feynmangraphs Γ ∈ F ( A ) with no oriented cycle. We also put: NcF ( A ) = M Γ ∈N c F ( A ) K Γ , NcF ( A ) = Y Γ ∈N c F ( A ) K Γ . Then NcF is a suboperad of ( F , ∇ ) and ( F , ◦ ) . Moreover, ( NcF , ∇ ) = ( NcF , ◦ ) .2. For all finite set A , we put: I ( A ) = Y Γ ∈F ( A ) \N c F ( A ) K Γ . Then I is an operadic ideal of ( F , ◦ ) . Moreover, ( F /I, ◦ ) is isomorphic to ( NcF , ◦ ) . Proof. This is a direct consequence of lemma 65. (cid:3) Definition 69 Let Γ ∈ F ( A ) . We define an equivalence relation on Int (Γ) : ∀ e, f ∈ Int (Γ) , e ∼ f ⇐⇒ ( S Γ ( e ) , T Γ ( e )) = ( S Γ ( f ) , T Γ ( f )) . We define a Feynman graph s (Γ) by: V ( s (Γ)) = V (Γ) , OutExt ( s (Γ)) = OutExt (Γ) ,Int ( s (Γ)) = { e ∈ Int (Γ) | S Γ ( e ) = T Γ ( e ) } / ∼ , InExt ( s (Γ)) = InExt (Γ) . For all e ∈ Int ( s (Γ)) : S s (Γ) ( e ) = S Γ ( e ) , T s (Γ) ( e ) = T Γ ( e ) . For all e ∈ OutExt ( s (Γ)) , for all f ∈ InExt ( s (Γ)) : S s (Γ) ( e ) = S Γ ( e ) , T s (Γ) ( f ) = T Γ ( f ) . Roughly speaking, s (Γ) is obtained by deleting the loops of Γ and reducing multiple edges to singleedges. Note that s (Γ) is a simple Feynman graph. Proposition 70 For all finite set A , we denote by SF ( A ) the set of simple Feynman graphs Γ such that V (Γ) = A . We also put: SF ( A ) = M Γ ∈SF ( A ) K Γ , SF ( A ) = Y Γ ∈SF ( A ) K Γ . We define two operadic composition on SF : if Γ ∈ SF ( A ) , Γ ′ ∈ SF ( B ) and b ∈ A : Γ ∇ b Γ ′ = X Γ ′′ ∈SF ( A ⊔ B \{ b } ) , Γ ′′| B =Γ ′ , s (Γ ′′ /B → b )=Γ Γ ′′ , Γ ◦ b Γ ′ = X Γ ′′ ∈SF ( A ⊔ B \{ b } ) , Γ ′′| B =Γ ′ , s (Γ ′′ /B → b )=Γ ,B Γ ′′ -convex Γ ′′ . he following map is an injective operad morphism from ( SF , ∇ ) to ( F , ∇ ) and from ( SF , ◦ ) to ( F , ◦ ) : ψ : SF ( A ) −→ F ( A )Γ −→ X Γ ′ ∈F ( A ) , s (Γ ′ )=Γ Γ ′ . Proof. The map ψ is clearly injective. Let Γ ∈ SF ( A ) , Γ ′ ∈ SF ( B ) , b ∈ A . Note that ψ (Γ) ∇ b ψ (Γ ′ ) is a sum of Feynman graphs with multiplicity , more precisely: ψ (Γ) ∇ b ψ (Γ ′ ) = X Υ ∈F ( A ⊔ B \{ b } ) ,s (Υ | B )=Γ , s (Υ /B → b )=Γ ′ Υ . Let us assume that Υ appears in ψ (Γ) ∇ b ψ (Γ) and that s (Υ) = s (Υ ′ ) . Then, obviously, s (Υ ′| B ) = s (Υ | B ) = Γ ′ , and g (Υ ′ /B → b ) = g (Υ /B → b ) = Γ , so Υ ′ also appears in ψ (Γ) ∇ b ψ (Γ ′ ) .Therefore: ψ (Γ) ∇ b ψ (Γ ′ ) = ψ X Υ ∈F ( A ⊔ B \{ b } ) ,s (Υ | B )=Γ ′ , Υ /B → b =Γ Υ . As ψ is injective, this defines an operadic composition on SF , making ψ an operad morphism.The proof is similar for ◦ , observing that if Γ , Γ ′ ∈ F ( A ⊔ B \ { b } ) are such that s (Γ) = s (Γ ′ ) ,then B is Γ -convex if, and only if, B is Γ ′ -convex. (cid:3) Remark. The unit of SF is: I = X i,j ≥ i / / ?>=<89:; j / / . Restricting to Feynman graphs with no oriented cycle: Proposition 71 1. For all finite set A , we denote by N c SF ( A ) the set of simple Feynmangraphs Γ with no oriented cycle such that V (Γ) = A . We also put: NcSF ( A ) = M Γ ∈N c SF ( A ) K Γ , NcSF ( A ) = Y Γ ∈N c SF ( A ) K Γ . Then NcSF is a suboperad of both ( SF , ∇ ) and ( SF , ◦ ) , and ψ ( NcSF ) ⊆ NcF . Moreover, ( NcSF , ∇ ) = ( NcSF , ◦ ) .2. For all finite set A , we put: J ( A ) = Y Γ ∈SF ( A ) \N c SF ( A ) K Γ . Then J is an operadic ideal of ( SF , ◦ ) . Moreover, ( SF /J, ◦ ) is isomorphic to ( NcSF , ◦ ) . We obtain two commutative diagrams of operads.For ∇ : For ◦ : FSF -(cid:13) ; ; ✇✇✇✇✇✇✇✇✇ NcF d d ■■■■■■■■■ NcSF ,(cid:12) : : ✉✉✉✉✉✉✉✉✉ c c ●●●●●●●● F / / / / NcFSF / / / / -(cid:13) ; ; ✇✇✇✇✇✇✇✇✇ NcSF NcF d d ■■■■■■■■■ Id O O NcSF ,(cid:12) : : ✉✉✉✉✉✉✉✉✉ c c ●●●●●●●● Id O O .3 Oriented graphs, posets, finite topologies We shall identify oriented graphs (possibly with loops and multiple edges) with Feynman graphswith no external edge. For all finite set A , we denote by G ( A ) the set of graphs G such that V ( G ) = A and we put: G ( A ) = M G ∈G ( A ) K G, G ( A ) = Y G ∈G ( A ) K G. Definition 72 Let Γ be a Feynman graph. The graph g (Γ) is defined by: • V ( g (Γ)) = V (Γ) . • Int ( g (Γ)) = Int (Γ) . • S g (Γ) = ( S Γ ) | Int (Γ) . • T g (Γ) = ( T Γ ) | Int (Γ) .Roughly speaking, one deletes the external edges of Γ to obtain g (Γ) . Note that if G ∈ G ( A ) and I ⊆ A , non-empty, then G/I → a is also a graph, whereas G | I isnot always a graph (external edges may appear). Theorem 73 We define two operadic composition on G : if G ∈ G ( A ) , G ′ ∈ G ( B ) and b ∈ A , G ∇ a G ′ = X G ′′ ∈G ( A ⊔ B \{ b } ) ,g ( G ′′| B )= G ′ , G ′′ /B → b = G G ′′ , G ◦ a G ′ = X G ′′ ∈G ( A ⊔ B \{ b } ) ,g ( G ′′| B )= G ′ , G ′′ /B → b = G,B is G -convex G ′′ . The unit is: I = X G ∈G ( { } ) G. Moreover, the following map is an injective morphism from ( G , ∇ ) to ( F , ∇ ) and from ( G , ◦ ) to ( F , ◦ ) : φ : G ( A ) −→ F ( A ) G −→ X Γ ∈F ( A ) , g (Γ)= G Γ . Proof. The map φ is clearly injective. Let G ∈ G ( A ) , G ′ ∈ G ( B ) , b ∈ A . Note that φ ( G ) ∇ b φ ( G ′ ) is a sum of Feynman graphs with multiplicity , more precisely: φ ( G ) ∇ b φ ( G ′ ) = X Γ ∈F ( A ⊔ B \{ b } ) ,g (Γ | B )= G, g (Γ /B → b )= G ′ Γ . Let us assume that Γ appears in φ ( G ) ∇ b φ ( G ) and that g (Γ) = g (Γ ′ ) . Then, obviously, g (Γ ′| B ) = g (Γ | B ) = G ′ , and g (Γ ′ /B → b ) = g (Γ /B → b ) = G , so Γ ′ also appears in φ ( G ) ∇ b φ ( G ′ ) .Hence: φ ( G ) ∇ b φ ( G ′ ) = φ X G ′′ ∈G ( A ⊔ B \{ b } ) ,g ( G ′′| B )= G ′ , G ′′ /B → b = G G ′′ . φ is injective, this defines an operadic composition on G , making φ an operad morphism.The proof is similar for ◦ , observing that if Γ , Γ ′ ∈ F ( A ⊔ B \ { b } ) are such that g (Γ) = g (Γ ′ ) ,then B is Γ -convex if, and only if, B is Γ ′ -convex. (cid:3) Remark. The unit of G is: I = X k ≥ ?>=<89:; k (cid:10) (cid:10) . Definition 74 Let A be a finite set.1. We denote by N c G ( A ) the set of graphs G with no oriented cycle such that V ( G ) = A . Wealso put: NcG ( A ) = M G ∈N c G ( A ) K G, NcG ( A ) = Y G ∈N c G ( A ) K G. 2. We denote by SG ( A ) the set of simple graphs G such that V ( G ) = A . We also put: SG ( A ) = M G ∈SG ( A ) K G. This is a finite-dimensional space, of dimension | A | ( | A |− .3. We denote by N c SG ( A ) the set of simple graphs G with no oriented cycle such that V ( G ) = A . We also put: NcSG ( A ) = M G ∈N c SG ( A ) K G. This is a finite-dimensional space. By restriction of φ , we obtain: Corollary 75 NcG , SG and NcSG are operads for ∇ and ◦ . Remark. The unit of SG is I = ?>=<89:; .We obtain two commutative diagrams of operads. For ∇ : FSF -(cid:13) ; ; ✇✇✇✇✇✇✇✇✇ NcF d d ■■■■■■■■■ G k k ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ NcSF ,(cid:12) : : ✉✉✉✉✉✉✉✉✉ c c ●●●●●●●● SG ,(cid:12) ; ; ✈✈✈✈✈✈✈✈✈❲ ❲ ❲ ❲ ❲ ❲ ❲ k k ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ NcG d d ■■■■■■■■■❲ ❲ ❲ ❲ ❲ ❲ ❲ k k ❲ ❲ ❲ ❲ ❲ ❲ ❲ NcSG ,(cid:12) : : tttttttttt d d ❍❍❍❍❍❍❍❍❍ Y9 k k ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ◦ : F / / / / NcFSF / / / / -(cid:13) ; ; ✇✇✇✇✇✇✇✇✇ NcSF NcF d d ■■■■■■■■■ Id O O G / / / / ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ k k ❲ ❲ ❲ ❲ ❲ ❲ NcGNcSF ,(cid:12) : : ✉✉✉✉✉✉✉✉✉ c c ●●●●●●●● Id O O SG / / / / ,(cid:12) ; ; ✈✈✈✈✈✈✈✈✈❲ ❲ ❲ ❲ ❲ ❲ ❲❲ ❲ ❲ k k ❲ ❲ ❲ ❲ ❲ NcSG NcG d d ■■■■■■■■■ Id O O ❲ ❲ ❲ ❲ ❲ ❲ ❲ k k ❲ ❲ ❲ ❲ ❲ ❲ ❲ NcSG ,(cid:12) : : tttttttttt d d ❍❍❍❍❍❍❍❍❍ Id O O Y9 k k ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ A quasi-order on a set A is a transitive, reflexive relation on A . By Alexandroff’s theorem, if A is finite, there is a bijective correspondence relation between quasi-orders on A and topologieson A [1, 10]. Definition 76 Let A be a finite set.1. We denote by q O ( A ) the set of quasi-orders on A and we denote by qO ( A ) the spacegenerated by q O ( A ) .2. We denote by O ( A ) the set of orders on A and we denote by O ( A ) the space generated by O ( A ) . The elements of q O ( A ) will be denoted under the form P = ( A, ≤ P ) , where ≤ P the consideredquasi-order defined on the set A . Definition 77 Let Γ ∈ F ( A ) . We define a quasi-order on A by: ∀ x, y ∈ A, x ≤ Γ y if there exists an oriented path from x to y in Γ . Remark. The quasi-order ≤ Γ is an order if, and only if, Γ has no oriented cycle. Definition 78 Let P ∈ q O ( A ) and I ⊆ A , non-empty.1. We shall say that I is P -convex if, and only if, for all x, y, z ∈ A : x, z ∈ I and x ≤ P y ≤ P z = ⇒ y ∈ I. 2. The quasi-order ≤ P/I → a is defined on A ⊔ { a } \ I : if x, y ∈ A \ I , • x ≤ P/I → a y if x ≤ P y or if there exists x ′ , y ′ ∈ I , x ≤ P y ′ and x ≤ P y ′ . • x ≤ P/I → a a if there exists y ′ ∈ I such that x ≤ P y ′ . • a ≤ P/I → a y if there exists x ′ ∈ I , such that x ′ ≤ P y . Remark. If Γ ∈ F ( A ) and I ⊆ A , then I is Γ -convex if, and only if, I is ≤ Γ -convex. More-over, ≤ Γ /I → a = ( ≤ Γ ) /I → a .The following operad is described in [11]: 70 heorem 79 We define an operadic composition ◦ on qO : if P ∈ q O ( A ) , Q ∈ q O ( B ) and b ∈ A , P ◦ a Q = X R ∈ q O ( A ⊔ B \{ b } ) ,R | B = Q, R/I → a = P,B R -convex R. The following map is an injective operad morphism from ( qO , ◦ ) to ( SG , ◦ ) : κ : qO ( A ) −→ SG ( A ) P −→ X G ∈SG ( A ) , ≤ G = P G. Proof. Let P ∈ q O ( A ) , Q ∈ q O ( B ) and b ∈ B . Then: κ ( P ) ◦ b κ ( Q ) = X G ∈ SG ( A ⊔ B \{ b } ) , ≤ G | B = Q, ≤ G/ → b = P,B G -convex G. Let us assume that G appears in κ ( P ) ◦ b κ ( Q ) and that ≤ G ′ = ≤ G . Then, as B is G -convex, it is ≤ G -convex, hence ≤ G ′ -convex, hence G ′ -convex. Moreover: ≤ G ′| B = ( ≤ G ′ ) | B = ( ≤ G ) | B = ≤ G | B = P, ≤ G ′ /B → b = ( ≤ G ′ ) /B → b = ( ≤ G ) /B → b = ≤ G/B → b = Q, so G ′ appears in κ ( G ) ◦ b κ ( G ′ ) . Therefore: κ ( P ) ◦ a κ ( Q ) = κ X R ∈ q O ( A ⊔ B \{ b } ) ,R | B = Q, R/I → a = P,B R -convex R . Moreover, κ is injective: indeed, if P ∈ q O ( A ) , the arrow graph G P of this quasi-order satisfies ≤ G P = ≤ P . Therefore, this defines an operadic composition on quasi-orders. (cid:3) The following operad O is described in [12]: Corollary 80 O is a suboperad of ( qO , ◦ ) . Moreover, if, for any finite set A , we denote by J ( A ) the space generated by the elements of q O ( A ) \ O ( A ) , then J is an ideal of ( qO , ◦ ) and thequotient qO /I is isomorphic to O . Proof. This is implied by O = κ − ( κ ( qO ) ∩ NcSG ) . (cid:3) xamples. In O : qq ◦ qq = qqq + q qq ∨ qq ◦ qq = qqq + q ∧ qq 31 2 qq ◦ qq = qqq + q qq ∨ qq ◦ qq = qqq + q ∧ qq 21 3 qq ◦ q q = q ∧ qq 31 2 + qq q + q qq qq ◦ q q = q qq ∨ + qq q + qq q qq ◦ q , = qq , qq ◦ q , = qq , qq ◦ qq = q ∧ qq 21 3 + qqq qq ◦ qq = qqq + q qq ∨ qq ◦ qq = q ∧ qq 12 3 + qqq qq ◦ qq = qqq + q qq ∨ qq ◦ q q = q qq ∨ + qq q + q qq qq ◦ q q = q ∧ qq 12 3 + qq q + qq q qq ◦ q , = qq , qq ◦ q , = qq , q q ◦ qq = qq q q q ◦ qq = q qq q q ◦ qq = qq q q q ◦ qq = q qq q q ◦ q q = q q q q q ◦ q q = q q q q q ◦ q , = q , q q q ◦ q , = q q , q , ◦ qq = 0 q , ◦ qq = 0 q , ◦ qq = 0 q , ◦ qq = 0 q , ◦ q , = 0 q , ◦ q , = 0 q , ◦ q , = 0 q , ◦ q , = 0 We obtain a diagram of operads (for ◦ ): F / / / / NcFSF / / / / -(cid:13) ; ; ✇✇✇✇✇✇✇✇✇ NcSF NcF d d ■■■■■■■■■ Id O O G / / / / ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ k k ❲ ❲ ❲ ❲ ❲ ❲ NcGNcSF ,(cid:12) : : ✉✉✉✉✉✉✉✉✉ c c ●●●●●●●● Id O O SG / / / / ,(cid:12) ; ; ✈✈✈✈✈✈✈✈✈❲ ❲ ❲ ❲ ❲ ❲ ❲❲ ❲ ❲ k k ❲ ❲ ❲ ❲ ❲ NcSG NcG d d ■■■■■■■■■ Id O O ❲ ❲ ❲ ❲ ❲ ❲ ❲ k k ❲ ❲ ❲ ❲ ❲ ❲ ❲ NcSG ,(cid:12) : : ✉✉✉✉✉✉✉✉✉✉ c c ❍❍❍❍❍❍❍❍❍ Id O O Y9 k k ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ qO / / / / ❲ ❲ ❲ ❲ ❲ ❲ ❲❲ ❲ ❲ k k ❲ ❲ ❲ ❲ ❲ OO a a ❇❇❇❇❇❇❇❇ Id O O Y9 k k ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ (4.1)By composition, we obtain morphism from qO to several other operads; for example: qO ( A ) −→ G ( A ) P −→ X G ∈G ( A ) , ≤ G = P G, qO ( A ) −→ SF ( A ) P −→ X G ∈SF ( A ) , ≤ G = P G, qO ( A ) −→ F ( A ) P −→ X G ∈F ( A ) , ≤ G = P G. emark . The image of κ is not a suboperad of ( SG , ∇ ) . For example, let us take P and Q be the quasi-orders associated to the following graphs: ?>=<89:; ) ) ?>=<89:; b j j ; ?>=<89:; ?>=<89:; . Then G appears in κ ( P ) ∇ b κ ( Q ) , and G ′ not: G = ?>=<89:; / / ?>=<89:; * * ?>=<89:; j j , G ′ = ?>=<89:; / / ! ! ?>=<89:; * * ?>=<89:; j j , although ≤ G = ≤ G ′ . b ∞ structures Theorem 81 There exists a unique operad morphism: AsCom −→ O m −→ q q ,⋆ −→ q q + qq . Proof. In O : q q (12) = q q , q q ◦ q q = q q ◦ q q = q q q , ( qq + q q ) ◦ ( qq + q q ) = ( qq + q q ) ◦ ( qq + q q )= qqq + q qq ∨ + q ∧ qq 31 2 + q qq + qq q + qq q + q q q . So these elements define a morphism from AsCom to qO . (cid:3) By composition, we obtain morphisms from AsCom to any operad of the commutativediagram (4.1). We always denote by m and ⋆ the image of the two products of AsCom in allthese operads. Definition 82 Let A a finite set and I ⊆ A .1. Let Γ ∈ F ( A ) . We shall say that I is an ideal of Γ if for all x, y ∈ A : x ≤ Γ y and x ∈ I = ⇒ y ∈ I. 2. Let ≤∈ q O ( A ) . We shall say that I is an ideal of ≤ if for all x, y ∈ A : x ≤ y and x ∈ I = ⇒ y ∈ I. Proposition 83 Let A and B be disjoint finite sets. If P ∈ {F , SF , G , SG , q O} , for all Γ ∈ P ( A ) , Γ ′ ∈ P ( B ) : m ◦ (Γ , Γ ′ ) = ΓΓ ′ , ⋆ ◦ (Γ , Γ ′ ) = X Υ ∈P ( A ⊔ B ) , Υ | A =Γ , Υ | B =Γ ′ ,B ideal of Υ Υ . Proof. We prove it for P = F ; the other cases are proves similarly In F , m is the sum overall Feynman graphs Γ on { , } , with no internal edge between and and no internal edgebetween and . Then m ◦ Γ is the sum over all Feynman graphs Υ over A ⊔ { } , with Υ | A = Γ and no edge between any vertex of A and and no edge between and any vertex of A . Inother words, m ◦ Γ is the sum over all Feynman graphs Υ = ΓΥ ′ , where Υ ′ is a Feynman graphon { } . Consequently, m ◦ (Γ , Γ ′ ) = ( m ◦ Γ) ◦ Γ ′ is the sum over all Feynman graphs Υ on A ⊔ B , with Υ | A = Γ and Υ | B = Γ ′ , and no internal edge between A and B an no internal edgebetween B and A , that is to say Υ = ΓΓ ′ . The proof is similar for ⋆ : ⋆ ◦ (Γ , Γ ′ ) is the sum overall Feynman graphs Υ on A ⊔ B , such that Υ | A = Γ and Υ | B = Γ ′ , and no internal edge between B and A . (cid:3) .4.2 Associated Hopf algebras Corollary 84 The vector space CF generated by connected Feynman graph is a suboperadof ( F , ∇ ) and ( F , ◦ ) . Proof. Let Γ ∈ F ( A ) and I ⊆ A , non-empty. Let us prove that if Γ | I and Γ /I → a areconnected, then Γ is connected. For all x ∈ A , let us denote by CC ( x ) the connected componentof x in Γ . As Γ | I is connected, for all x ∈ I , I ⊆ CC ( x ) . Let x / ∈ I . As Γ /I → a is connected,there exists an non oriented path from x to a in Γ /I → a , so there exists a non-oriented pathfrom x to a vertex y ∈ I in Γ : we obtain that I ⊆ CC ( y ) = CC ( x ) for all x / ∈ I . So Γ has onlyone connected component. (cid:3) Following corollary 80, it is possible to define suboperad of connected objects for all theoperads in the commutative diagram. As the product m is, in all cases, the disjoint union,the morphism from b ∞ to any of these operads obtained by restriction of the morphism from AsCom takes its image in the suboperad of connected objects. For example: θ O ( ⌊− , −⌋ , ) = qq , θ O ( ⌊− , −⌋ , ) = q qq ∨ ,θ O ( ⌊− , −⌋ , ) = q ∧ qq 12 3 , θ O ( ⌊− , −⌋ , ) = q qq q (cid:30)(cid:31) + q qq q (cid:31) + q qq q (cid:31) + q qq q (cid:30) + q qq q (cid:30) . For all k, l ≥ , θ O ( ⌊− , −⌋ k,l ) is the sum of all connected bipartite graphs with blocks { , . . . , k } and { k + 1 , . . . , k + l } . The number of such graphs is given by sequence A227322 of the OEIS.For any vector space V , for any ( P , CP ) in the following set: (cid:26) ( F , CF ) , ( NcF , CNcF ) , ( SF , CSF ) , ( NcSF , CNcSF ) , ( G , CG ) , ( NcG , CNcG ) , ( SG , CSG ) , ( NcSG , CNcSG ) , ( qO , CqO ) , ( O , CO ) (cid:27) , we have: F P ( V ) = S ( F Cp ( V )) . Let us describe the product ⋆ induced by the b ∞ structure on all these Hopf algebras. We restrictourselves to F F ( V ) , the other cases are similar. We fix V = V ect ( X , . . . , X N ) . • As a vector space, A CF ( V ) = F F ( V ) is generated by isoclasses b Γ of Feynman graphs Γ whose vertices are decorated by elements of [ N ] . • For any Feynman graph Γ whose vertices are decorated by [ N ] : ∆( b Γ) = X Γ=Γ Γ c Γ ⊗ c Γ . • For any Feynman graphs Γ ∈ F ( A ) , Γ ′ ∈ F ( B ) , whose vertices are decorated by [ N ] : b Γ ∗ b Γ ′ = X Γ ′′ ∈F ( A ⊔ B ) , Γ ′′| A =Γ , Γ ′′| B =Γ ′ ,B ideal of Γ ′′ c Γ ′′ . The dual Hopf algebra A ∗ CF ( V ) has the same basis, up to an identification. Moreover: • For any Feynman graph Γ ∈ F ( A ) , whose vertices are decorated by [ N ] : ∆ ∗ ( b Γ) = X I ideal of Γ \ Γ | V (Γ) \ I ⊗ c Γ | I . For any Feynman graphs Γ , Γ ′ , whose vertices are decorated by [ N ] : b Γ . b Γ ′ = d ΓΓ ′ . By functoriality, we obtain a diagram of Hopf algebra morphisms: A ∗ CF y y y y ttttttttt % % % % ❑❑❑❑❑❑❑❑❑❑ ❳❳❳❳❳❳ , , , , ❳❳❳❳❳❳❳❳❳❳❳❳ A ∗ CNcF Id (cid:15) (cid:15) ? _ o o A ∗ CSF % % % % ❏❏❏❏❏❏❏❏❏ ❳❳❳❳❳ ❳❳❳ , , , , ❳❳❳❳❳❳❳❳ A ∗ CNcSF Id (cid:15) (cid:15) ? _ o o A ∗ CNcF y y y y ssssssssss ❳❳❳❳❳❳❳❳ , , , , ❳❳❳❳❳❳❳❳ A ∗ CG y y y y sssssssss % % % % ▲▲▲▲▲▲▲▲▲▲ A ∗ CNcG Id (cid:15) (cid:15) ? _ o o A ∗ CNcSF , , , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ A ∗ CSG % % % % ❑❑❑❑❑❑❑❑❑ (cid:15) (cid:15) (cid:15) (cid:15) A ∗ CNcSG Id (cid:15) (cid:15) ? _ o o A ∗ CNcG y y y y rrrrrrrrrr A ∗ CNcSG (cid:15) (cid:15) (cid:15) (cid:15) A ∗ CqO % % % % ❏❏❏❏❏❏❏❏❏ A ∗ CO Id (cid:15) (cid:15) ? _ o o A ∗ CO Here are examples of morphisms in this diagram: ( A ∗ CF ( V ) −→ A ∗ CSF ( V ) b Γ −→ d s (Γ) , A ∗ CF ( V ) −→ A ∗ CNcF ( V ) b Γ −→ (b Γ if Γ has no oriented cycle , otherwise , ( A ∗ CF ( V ) −→ A ∗ CG ( V ) b Γ −→ d g (Γ) , (cid:26) A ∗ CG ( V ) −→ A ∗ CqO ( V ) b G −→ c ≤ G . Let us describe the bialgebra D ∗ F ( V ) . It is generated by pairs ( b Γ , d ) , where Γ is a connectedFeynman graph decorated by [ N ] , and d ∈ [ N ] . We shall need the following notions: Definition 85 Let Γ be a Feynman graph, with V (Γ) = A , and let { A , . . . , A k } be a partitionof A .1. We shall say that this partition is Γ -admissible if: • For all i , Γ | A i is connected. • – A is Γ -convex. – A is (Γ /A → -convex. – ... – A k is (( . . . (Γ /A → / . . . ) /A k − → ( k − -convex.By lemma 62-3, this does not depend on the choice of the order on A , . . . , A k .2. Let us assume that { A , . . . , A k } is Γ -admissible. Let D : [ k ] −→ [ N ] .(a) We obtain a Feynman graph on [ k ] : Γ / { A , . . . , A k } = ( . . . (Γ /A → / . . . ) /A k → k. Its isoclasse does not depend on the order chosen on the partition A ⊔ . . . ⊔ A k by lemma62. If Γ is connected, then Γ / { A , . . . , A k } is connected. Moreover, (Γ / { A , . . . , A k } , D ) is a Feynman graph decorated by [ N ] . b) Γ | A . . . Γ | A k is a decorated Feynman graph, with k connected components, namely A , . . . , A k . For all connected Feynman graphs Γ and d ∈ [ N ] , in D ∗ F ( V ) : ∆ ∗ (( b Γ , d )) = X { A , . . . , A k } Γ -admissible ,D :[ k ] −→ N ( \ (Γ / { A , . . . , A k } , D ) , d ) ⊗ ( [ (Γ | A , D (1)) . . . ( d Γ | A k , D ( k )) . For any Feynman graph decorated by [ N ] : ρ ( b Γ) = X { A , . . . , A k } Γ -admissible ,D :[ k ] −→ N ( \ Γ / { A , . . . , A k } , D ) ⊗ ( d Γ | A , D (1)) . . . ( d Γ | A k , D ( k )) . Similar formulas can be given for the other operads. For example, if a, b, c, d ∈ [ N ] , in D ∗ qO ( V ) : ∆ ∗ (( q a , d )) = N X p =1 ( q p , d ) ⊗ ( q a , p ) , ∆ ∗ (( qq ab , d )) = N X p,q =1 ( qq pq , d ) ⊗ ( q a , p )( q b , q ) + N X p =1 ( q p , d ) ⊗ ( qq ab , p ) , ∆ ∗ (( q qq ∨ acb , d )) = N X p,q,r =1 ( q qq ∨ prq , d ) ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 ( qq pq , d ) ⊗ ( qq ab , p )( q c , q )+ N X p,q =1 ( qq pq , d ) ⊗ ( qq ac , p )( q b , q ) + N X p =1 ( q p , d ) ⊗ ( q qq ∨ acb , p ) , ∆ ∗ (( q ∧ qq ab c , d )) = N X p,q,r =1 ( q ∧ qq pq r , d ) ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 ( qq pq , d ) ⊗ ( q c , p )( qq ba , q )+ N X p,q =1 ( qq pq , d ) ⊗ ( q b , p )( qq ca , q ) + N X p =1 ( q p , d ) ⊗ ( q ∧ qq ab c , p ) , ∆ ∗ (( qqq abc , d )) = N X p,q,r =1 ( qqq pqr , d ) ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 ( qq pq , d ) ⊗ ( qq ab , p )( q c , q )+ N X p,q =1 ( qq pq , d ) ⊗ ( q a , q )( qq bc , p ) + N X p =1 ( q p , d ) ⊗ ( qqq abc , p ) . Note that the subalgebra of D ∗ CqO ( V ) generated by rooted trees is a subbialgebra. This comesfrom the sujective operad morphism from CqO to PreLie , sending any rooted tree to itself andthe other quasi-order to . 76or the coaction: ρ ( q a ) = N X p =1 q p ⊗ ( q a , p ) ,ρ ( qq ab ) = N X p,q =1 qq pq ⊗ ( q a , p )( q b , q ) + N X p =1 q p ⊗ ( qq ab , p ) ,ρ ( q qq ∨ acb ) = N X p,q,r =1 q qq ∨ prq ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 qq pq ⊗ ( qq ab , p )( q c , q )+ N X p,q =1 qq pq ⊗ ( qq ac , p )( q b , q ) + N X p =1 q p ⊗ ( q qq ∨ acb , p ) ,ρ ( q ∧ qq ab c ) = N X p,q,r =1 q ∧ qq pq r ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 qq pq ⊗ ( q c , p )( qq ba , q )+ N X p,q =1 qq pq ⊗ ( q b , p )( qq ca , q ) + N X p =1 q p ⊗ ( q ∧ qq ab c , p ) ,ρ ( qqq abc ) = N X p,q,r =1 qqq pqr ⊗ ( q a , p )( q b , q )( q c , r ) + N X p,q =1 qq pq ⊗ ( qq ab , p )( q c , q )+ N X p,q =1 qq pq ⊗ ( q a , q )( qq bc , p ) + N X p =1 q p ⊗ ( qqq abc , p ) . In order to obtain the coproduct of B ∗ qO ( V ) , let us quotient by relations ( q i , j ) = δ i,j . In B ∗ CqO ( V ) : ∆ ∗ (( qq ab , d )) = ( qq ab , d ) ⊗ ⊗ ( qq ab , d ) , ∆ ∗ (( q qq ∨ acb , d )) = ( q qq ∨ abc , d ) ⊗ N X p =1 ( qq pc , d ) ⊗ ( qq ab , p ) + N X p =1 ( qq pb , d ) ⊗ ( qq ac , p ) + 1 ⊗ ( q qq ∨ acb , d ) , ∆ ∗ (( q ∧ qq ab c , d )) = ( q ∧ qq ab c , d ) ⊗ N X q =1 ( qq cq , d ) ⊗ ( qq ba , q ) + N X q =1 ( qq bq , d ) ⊗ ( qq ca , q ) + 1 ⊗ ( q ∧ qq ab c , d ) , ∆ ∗ (( qqq abc , d )) = ( qqq abc , d ) ⊗ N X p =1 ( qq pc , d ) ⊗ ( qq ab , p ) + N X q =1 ( qq aq , d ) ⊗ ( qq bc , p ) + 1 ⊗ ( qqq abc , d ) . 77e obtain a commutative diagram of Hopf algebra morphisms: D ∗ CF y y y y sssssssss % % % % ▲▲▲▲▲▲▲▲▲▲ ❳❳❳❳❳❳ , , , , ❳❳❳❳❳❳❳❳❳❳❳❳ D ∗ CNcF Id (cid:15) (cid:15) ? _ o o D ∗ CSF % % % % ❑❑❑❑❑❑❑❑❑ ❳❳❳❳❳ ❳❳❳ , , , , ❳❳❳❳❳❳❳❳ D ∗ CNcSF Id (cid:15) (cid:15) ? _ o o D ∗ CNcF y y y y rrrrrrrrrr ❳❳❳❳❳❳❳❳ , , , , ❳❳❳❳❳❳❳❳ D ∗ CG y y y y ssssssssss & & & & ▲▲▲▲▲▲▲▲▲▲ D ∗ CNcG Id (cid:15) (cid:15) ? _ o o D ∗ CNcSF , , , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ D ∗ CSG % % % % ❑❑❑❑❑❑❑❑❑❑ (cid:15) (cid:15) (cid:15) (cid:15) D ∗ CNcSG Id (cid:15) (cid:15) ? _ o o D ∗ CNcG x x x x rrrrrrrrrr D ∗ CNcSG (cid:15) (cid:15) (cid:15) (cid:15) D ∗ CqO % % % % ❑❑❑❑❑❑❑❑❑ D ∗ CO Id (cid:15) (cid:15) ? _ o o D ∗ CO Remarks . The components of CF are not finite-dimensional, in order to obtain D ∗ CF ( V ) ,we use the duality between CF and CF such that for every Feynman graphs Γ , Γ ′ : ≪ Γ , Γ ′ ≫ = δ Γ , Γ ′ . The formulas given in the finite-dimensional case also make sense for this duality. The sameremark holds for other operads here appearing, such as CG .78 hapter 5 Summary Let P be an operad, such that P (0) = (0) and, for all n ≥ , P ( n ) is finite dimensional.1. (a) P is a graded, non connected brace algebra, with a bracket denoted by h− , −i . More-over, P + is a graded and connected brace subalgebra of P .(b) This induces a graded, non connected pre-Lie algebra structure on P , which pre-Lieproduct is denoted by • . The following diagram of pre-Lie algebras is commutative: P & & & & ▼▼▼▼▼▼▼▼▼▼▼ P + $ $ $ $ ■■■■■■■■■ ,(cid:12) : : tttttttttt coinv P coinv P + +(cid:11) rrrrrrrrrrr (c) This induces a monoid product ♦ on P and a group product ♦ on P + . The followingdiagram of monoids is commutative: ( P , ♦ ) ) ) ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ( P + , ♦ ) ( ( ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ ) (cid:9) ♠♠♠♠♠♠♠♠♠♠♠♠♠ ( coinv P , ♦ ) = M D P ( coinv P + , ♦ ) = G D P ' (cid:7) ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ 2. (a) There exist products ∗ , induced by the operadic composition of P , making the follow-ing diagram of graded bialgebras commutative: D P = ( S ( coinv P ) , ∗ , ∆) ( S ( P ) , ∗ , ∆) (cid:31) (cid:127) / / o o o o D P = ( T ( P ) , ∗ , ∆ dec ) B P = ( S ( coinv P + ) , ∗ , ∆) ?(cid:31) O O ( S ( P + ) , ∗ , ∆) ?(cid:31) O O (cid:31) (cid:127) / / o o o o B P = ( T ( P + ) , ∗ , ∆ dec ) ?(cid:31) O O (b) They are all graded; the three bialgebras on the bottow row are graded Hopf algebras.3. (a) There exist coproducts ∆ ∗ making the following diagram of graded bialgebras com-mutative: D ∗ P = ( S ( inv P ∗ ) , m, ∆ ∗ ) (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) / / ( S ( P ∗ ) , m, ∆ ∗ ) (cid:15) (cid:15) (cid:15) (cid:15) D ∗ P = ( T ( P ∗ ) , m conc , ∆ ∗ ) (cid:15) (cid:15) (cid:15) (cid:15) o o o o B ∗ P = ( S ( inv P ∗ + ) , m, ∆ ∗ ) (cid:31) (cid:127) / / ( S ( P ∗ + ) , m, ∆ ∗ ) B ∗ P = ( T ( P ∗ + ) , m conc , ∆ ∗ ) o o o o B ∗ P , ( S ( P ∗ + ) , m, ∆ ∗ ) and B ∗ P are graded, connected Hopf algebras, dual to B P , ( S ( P + ) , ∗ , ∆) and B P respectively.(b) Considering the monoids of characters of these bialgebras, we obtain a commutativediagram of monoids: M D P ( P , ♦ ) o o o o M DP G B P ?(cid:31) O O ( P + , ♦ ) ?(cid:31) O O o o o o G BP ?(cid:31) O O 4. Let us consider an operad morphism θ P : b ∞ −→ P . Let V be a finite-dimensional vectorspace. We denote by C V the operad of morphisms from V to V ⊗ n .(a) We put: B P ( V ) = B P ⊗ C V , B ∗ P ( V ) = B ∗ P ⊗ C V , D P ( V ) = D P ⊗ C V , D ∗ P ( V ) = D ∗ P ⊗ C V ,B P ( V ) = B P ⊗ C V , B ∗ P ( V ) = B ∗ P ⊗ C V , D P ( V ) = D P ⊗ C V , D ∗ P ( V ) = D ∗ P ⊗ C V . (b) The morphism θ P induces a product ⋆ on S ( F P ( V )) , making ( S ( F P ( V )) , ⋆, ∆) agraded, connected Hopf algebra, denoted by A P ( V ) . Its graded dual is denoted by A ∗ P ( V ) .(c) A P ( V ) is a Hopf algebra in the category of D P ( V ) -modules; dually, A ∗ P ( V ) is a Hopfalgebra in the category of D ∗ P ( V ) -comodules.(d) The monoid ( M D P ( V ) , ♦ ) of characters of D ∗ P ( V ) acts by endomorphisms on the group G A P ( V ) of characters of A ∗ P ( V ) , and is isomorphic to the monoid of continuous endo-morphisms of the P -algebra F P ( V ) . The group of characters ( G B P ( V ) , ♦ ) of B ∗ P ( V ) acts by group automorphisms on G A P ( V ) , and is isomorphic to the group of formaldiffeomorphisms of F P ( V ) which are tangent to the identity.Here are diagrams of the different functors which appear in this text (contravariant functorsare represented by dashed arrows). Operads P & & ▼▼▼▼▼▼▼▼▼▼▼▼ x x qqqqqqqqqqq (cid:17) (cid:17) ❭ ❬ ❩ ❳ ❲ ❯ ◗ ▼ ❍ ❇ ❀ ✺ ✵ ✱ ✰ ✮ ✭ ✫✪✩ (cid:14) (cid:14) ❜❝❡❢✐♠q✇⑥☎✡✎✓✖✗✙✚✛ -bounded brace ( P , h− , −i ) Env.dend.alg. (cid:15) (cid:15) exp. o o -bounded pre-Lie ( coinv P , • ) Env.alg. (cid:15) (cid:15) exp. DendriformHopf algebras D P Cocom.Hopf algebras D P Bialgebras D ∗ P Char. (cid:15) (cid:15) ✤✤✤✤ Cocom.bialgebras D ∗ P Char. (cid:15) (cid:15) ✤✤✤ Monoids M DP Monoids M D P P by its augmentation ideal P + :Non unitaryoperads P + & & ▼▼▼▼▼▼▼▼▼▼▼▼ y y rrrrrrrrrrr (cid:18) (cid:18) ❩ ❳ ❲ ❱ ❚ ❙ ◗ ❖ ▼ ❑ ■ ❋ ❅ ✿ ✼ ✺ ✷ ✵ ✴ ✲ ✰ ✯ ✭ ✬ ✫ (cid:13) (cid:13) ❡❢❣✐❥❧♥♣rt②⑧✆✡✌✎✑✒✔✕✗✘✙ connected brace ( P + , h− , −i ) Env.dend.alg. (cid:15) (cid:15) exp. p p connected pre-Lie ( coinv P + , • ) Env.alg. (cid:15) (cid:15) exp. / / ConnecteddendriformHopf algebras B P ; ; Dual { { ✇ ✇ ✇ ✇ ✇ Cocom.connectedHopf algebras B P c c Dual ❍❍❍❍❍ ConnectedcodenfriformHopf algebras B ∗ P Char. (cid:15) (cid:15) ✤✤✤ Cocom.connectedHopf algebras B ∗ P Char. 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