Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

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1 Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations Loïc Foissy Laboratoire de Mathématiques - UMR6056, Université de Reims Moulin de la Housse - BP REIMS Cedex, France loicfoissy@univ-reimsfr ABSTRACT We consider the combinatorial Dyson-Schwinger equation X B + (P (X)) in the non-commutative Connes-Kreimer Hopf algebra of planar rooted trees H NCK, where B + is the operator of grafting on a root, and P a formal series The unique solution X of this equation generates a graded subalgebra A N,P of H NCK We describe all the formal series P such that A N,P is a Hopf subalgebra We obtain in this way a -parameters family of Hopf subalgebras of H NCK, organized into three isomorphism classes: a first one, restricted to a polynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Faà di Bruno Hopf algebra By taking the quotient, the last classe gives an infinite set of embeddings of the Faà di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees Moreover, we give an embedding of the free Faà di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, together with a non commutative version of this embedding Contents 1 Preliminaries 3 11 Valuation and n-adic topology 3 1 The commutative Connes-Kreimer Hopf algebra of trees 4 13 The non commutative Connes-Kreimer Hopf algebra of planar trees 5 14 The Faà di Bruno Hopf algebra 6 Subalgebras associated to a formal series 7 1 Construction 7 Main theorem 8 3 When is A P a Hopf subalgebra? 9 31 Preliminary results 9 3 Proof of Is A N,α,β a Hopf subalgebra? Generators of A N,α,β 1 4 Equalities of the subalgebras A N,P The A N,α,β s are Hopf subalgebras 14 5 Isomorphisms between the A N,α,β s Another system of generators of A N,1,β 16 5 Isomorphisms between the A N,α,β s 17 1

2 6 The case of the free Faà di Bruno algebra with D variables Construction 0 6 Subalgebras of H D NCK 0 63 Description of the Y i w s in the generic case Introduction The Connes-Kreimer Hopf algebra H CK of rooted trees is introduced in [1] It is commutative and not cocommutative A particular Hopf subalgebra of H CK, namely the Connes-Moscovici subalgebra, is introduced in [5] It is the subalgebra generated by the following elements: δ 1, δ, δ 3 +, δ δ , The appearing coefficients, called Connes-Moscovici coefficients, are studied in [4, 7] It is shown in [6] that the character group of this subalgebra is isomorphic to the group of formal diffeomorphisms, that is to say the group of formal series of the form h + a 1 h +, with composition In other terms, the Connes-Moscovici subalgebra is isomorphic to the Hopf algebra of functions on the group of formal diffeomorphisms, also called the Faà di Bruno Hopf algebra A non commutative version H NCK of the Connes-Kreimer Hopf algebra of trees is introduced in [9, 11] It contains a non commutative version of the Connes-Moscovici subalgebra, described in [10] Its abelianization can be identified with the subalgebra of H CK, here denoted by A 1,1, generated by the following elements of H CK : a 1, a, a 3 +, a a , This subalgebra is different from the Connes-Moscovici subalgebra, but is also isomorphic to the Faà di Bruno Hopf algebra In this paper, we consider a family of subalgebras of H NCK, which give a non commutative version of the Faà di Bruno algebra They are generated by a combinatorial Dyson-Schwinger equation [, 15, 16]: X P B + (P (X P )), where B + is the operator of grafting on a common root, and P p k h k is a formal series such that p 0 1 All this makes sense in a completion of H NCK, where this equation admits a unique solution X P a k, whose coefficients are inductively defined by: a 1, n a n+1 p k B + (a α1 a αk ), k1 α 1 ++α k n,,

3 For the usual Dyson-Schwinger equation, P α(1 h) 1 We characterise the formal series P such that the associated subalgebra is Hopf: we obtain a two-parameters family A N,α,β of Hopf subalgebras of H NCK and we explicitely describe the system of generator of these algebras We then characterise the equalities between the A N,α,β s and then their isomorphism classes We obtain three classes: 1 A N,0,1, equal to K[ ] A N,1, 1, the subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions 3 The A N,1,β s, with β 1, a non commutative version of the Faà di Bruno Hopf algebra By taking the quotient, we obtain three classes of Hopf subalgebras of H CK : 1 A 0,1, equal to K[ ] A 1, 1, the subalgebra of ladders, isomorphic to the Hopf algebra of symmetric functions 3 The A 1,β s, with β 1, isomorphic to the Faà di Bruno Hopf algebra We finally give an embedding of a non commutative version of the free Faà di Bruno on D variables (see [1]) in a Hopf algebra of planar rooted trees decorated by the set {1,, D} 3 By taking the quotient, the free Faà di Bruno algebra appears as a subalgebra of a Hopf algebra of decorated rooted trees This text is organized as follows The first section gives some recalls about the Hopf algebras of trees and the Faà di Bruno algebra We define the subalgebras of H CK and H NCK associated to a formal series P in section and also give here the main theorem (theorem 4), which characterizes the P s such that the associated subalgebras are Hopf In section 3, we prove 3 of theorem 4 In section 4, we prove 4 1 of theorem 4 We also describe there the system of generators, and the case of equalities of the subalgebras We describe the isomorphism classes of these subalgebras in the following section In the last one, we consider the multivariable case Notations 1 K is any field of characteristic zero Let λ K We put: g λ (h) (1 h) λ 1 Preliminaries 11 Valuation and n-adic topology λ(λ + 1) (λ + k 1) h k k! Q k (λ)h k K[[h]] In this paragraph, let us consider a graded Hopf algebra A Let A n be the homogeneous component of degree n of A For all a A, we put: val(a) max n N / a A k N {+ } k n For all a, b A, we also put d(a, b) val(a b), with the convention 0 Then d is a distance on A The induced topology over A will be called the n-adic topology 3

4 Let A be the completion of A for this distance In other terms: A The elements of A are written in the form + + A n a n, with a n A n for all n Moreover, A is naturally given a structure of associative algebra, by continuously extending the product of A The coproduct of A can also be extended in the following way: For all p + Indeed, for all n, m N, val : A A ˆ A i,j N A i A j a n h n K[[h]], for all a A such that val(a) 1, we put: ( n+m kn p(a) + p n a n A p n a n ) n, so this series is Cauchy, and converges It is an easy exercise to prove that for all p, q K[[h]], such that q has no constant term, for all a A, with val(a) 1, (p q)(a) p(q(a)) 1 The commutative Connes-Kreimer Hopf algebra of trees This Hopf algebra is introduced by Kreimer in [1] and studied for example in [3, 5, 7, 8, 13, 14] Definition 1 1 A rooted tree t is a finite, connected graph without loops, with a special vertex called root The set of rooted trees will be denoted by T CK The weight of a rooted tree is the number of its vertices 3 A planar rooted tree is a tree which is given an imbedding in the plane The set of planar rooted trees will be denoted by T NCK Examples 1 Rooted trees of weight 5:,,,,,,,,,,,,,,,, Planar rooted trees of weight 5:,,,,,,,,,,,,,,,,,,,,,, 4

5 The Connes-Kreimer Hopf algebra of rooted trees H CK is the free associative commutative algebra freely generated over K by the elements of T CK A linear basis of H CK is given by rooted forests, that is to say monomials in rooted trees The set of rooted forests will be denoted by F CK The weight of a rooted forest F t 1 t n is the sum of the weights of the t i s Examples Rooted forests of weight 4: 1,,,,,,,,,,,,,,,, We now recall the Hopf algebra structure of H CK An admissible cut of t is a non empty cut such that every path in the tree meets at most one cut edge The set of admissible cuts of t is denoted by Adm(t) If c is an admissible cut of t, one of the trees obtained after the application of c contains the root of t: we shall denote it by R c (t) The product of the other trees will be denoted by P c (t) The coproduct of t is then given by : (t) t t + P c (t) R c (t) c Adm(t) This coproduct is extended as an algebra morphism Then H CK becomes a Hopf algebra Note that H CK is given a gradation of Hopf algebra by the weight Examples ( ) ( ) ( ) , , + + +, ( ) We define the operator B + : H CK H CK, that associates to a forest F F CK the tree obtained by grafting the roots of the trees of F on a common root For example, B + ( ) Then, for all x H CK : (B + (x)) B + (x) 1 + (Id B + ) (x) (1) This means that B + is a 1-cocycle for a certain cohomology of coalgebra, see [5] for more details Moreover, this operator B + is homogeneous of degree 1, so is continuous So it can be extended in an operator B + : H CK H CK 13 The non commutative Connes-Kreimer Hopf algebra of planar trees This algebra is introduced simultaneously in [9, 11] As an algebra, H NCK is the free associative algebra generated by the elements of T NCK A basis of H NCK is given by planar rooted forests, that is to say words in elements of T NCK The set of planar rooted forests will be denoted by F NCK Examples Planar rooted forests of weight 4: 1,,,,,,,,,,,,,,,,,,,,,, The coproduct of H NCK is defined, as for H CK, with the help of admissible cuts For example : 5

6 ( ) , ( ) Note that H NCK is a graded Hopf algebra, with a gradation given by the weight We define an operator, also denoted by B + : H NCK H NCK, as for H CK For example, B + ( ) and B + ( ) Then (1) is also satisfied on H NCK Moreover, this operator B + is homogeneous of degree 1, so is continuous In consequence, it can be extended in an operator B + : H NCK H NCK We proved in [9, 10] that H NCK is a self-dual Hopf algebra: it has a non degenerate pairing denoted by <, >, and a dual basis (e F ) F FNCK of the basis of planar decorated forests The product in the dual basis is given by graftings (in an extended sense) For example: e e e + e + e + e + e + e + e 14 The Faà di Bruno Hopf algebra Let K[[h]] be the ring of formal series in one variable over K We consider: G h + a n h n+1 K[[h]] n 1 This is a group for the composition of formal series The Faà di Bruno Hopf algebra H F db is the Hopf algebra of functions on the opposite of the group G More precisely, H F db is the polynomial ring in variables Y i, with i N, where Y i is the function on G defined by: G K Y i : h + a n h n+1 a i n 1 The coproduct is defined in the following way: for all f H F db, for all P, Q G, (f)(p Q) f(q P ) This Hopf algebra is commutative and not cocommutative It is also a graded, connected Hopf algebra, with Y i homogeneous of degree i for all i We put: Y 1 + Y n H F db n1 Then: (Y) Y n+1 Y n n1 6

7 Indeed, with the convention a 0 b 0 1: (Y) a i h i+1 b i h i+1 Y i 0 i 0 i 0 i 0 b i b i a j h j+1 j 0 a j j 0 i+1, i+1 ( ) Y n+1 Y n a i h i+1 b i h i+1 n1 i 0 i 0 a i i 0 n1 n+1 b n The graded dual HF db is an enveloping algebra, by the Cartier-Quillen-Milnor-Moore theorem A basis of P rim(hf db ) is given by (Z i) i N, where: Z i : H F db K Y α 1 1 Y α k k 0 if α α k 1, Y j δ i,j By homogeneity, for all i, j N, there exists a coefficient λ i,j K such that [Z i, Z j ] λ i,j Z i+j Moreover: [Z i, Z j ](Y) λ i,j (Z i Z j Z j Z i ) (Y) ( ) (Z i Z j Z j Z i ) Y n+1 Y n Z i (Y j+1 ) Z j (Y i+1 ) (j + 1) (i + 1) j i So the bracket of P rim(hf db ) is given by: [Z i, Z j ] (j i)z i+j n1 Subalgebras associated to a formal series 1 Construction We denote by K[[h]] 1 the set of formal series of K[[h]] with constant term equal to 1 Proposition Let P K[[h]] 1 1 There exists a unique element X P a n H CK, such that X P B + (P (X P )) k 1 There exists a unique element X P k 1 a n H NCK, such that X P B + (P (X P )) Proof 7

8 1 Unicity We put X P n 1 a n, with a n homogeneous of degree n for all n Then the a n s satisfy the following equations: a 1, n a n+1 Hence, the a n s are uniquely defined k1 α 1 ++α k n p k B + (a α1 a αk ) Existence The a n s defined inductively by () satisfy the required condition () We put X P n 1 a n, a n homogeneous of degree n for all n Then the a n s satisfy the following equations: a 1, n a n+1 k1 The end of the proof is similar α 1 ++α k n p k B + (a α1 a αk ) (3) Definition 3 Let P K[[h]] 1 1 The subalgebra A P of H CK is the subalgebra generated by the a n s The subalgebra A N,P of H NCK is the subalgebra generated by the a n s Remarks 1 A P is a graded subalgebra of H CK, and A N,P is a graded subalgebra of H NCK For all n N, a n is an element of V ect(t CK ) Hence: The same holds in the non commutative case Main theorem V ect(t CK ) A P V ect(a n, n N ) (4) One of the aim of this paper is to prove the following theorem: Theorem 4 Let P K[[h]] 1 The following assertions are equivalent: 1 A N,P is a Hopf subalgebra of H NCK A P is a Hopf subalgebra of H CK 3 There exists (α, β) K, such that P satisfies the following differential system: { (1 αβh)p S α,β : (h) αp (h) P (0) 1 4 There exists (α, β) K, such that: (a) P (h) 1 if α 0 (b) P (h) e αh if β 0 (c) P (h) (1 αβh) 1 β if αβ 0 An easy computation proves the equivalence between assertions 3 and 4 Moreover, using the Hopf algebra morphism ϖ : H NCK H CK, defined by forgetting the planar data, it is clear that A P ϖ(a N,P ) So, assertion 1 implies assertion 8

9 3 When is A P a Hopf subalgebra? 31 Preliminary results The aim of this section is to show 3 in theorem 4 Lemma 5 Suppose that A P is a Hopf subalgebra Then two cases are possible: 1 P 1 In this case, X P and A P K[ ] p 1 0 In this case, a n 0 for all n 1 Proof Suppose first that p 1 0, and suppose that there exists n such that p n 0 Let us choose n minimal Then, by (), a a n 0 and a n+1 p n B + ( n ) Then: (B + ( n )) B + ( n ) 1 + n ( ) n k B + ( n k ) A P A P, k which implies that B + ( ) A P V ect(t CK ) V ect(a n ), so a 0: contradiction So P 1 and X P Suppose p 1 0 By (), the canonical projection of a n+1 on Im((B + ) ) (vector space of trees such that the root has only one child) is p 1 B + (a n ) for all n 1 Hence, for all n 1, a n+1 0 a n 0 So for all n 1, a n 0 We put: { Z : H CK K F F CK δ,f Note that Z is an element of the graded dual H CK Moreover, Z can be extended to H CK, and satisfies, for all a, b H CK : Z(ab) Z(a)ε(b) + ε(a)z(b) Lemma 6 Let P K[[h]] 1 If A P is a Hopf subalgebra of H CK, then: (Z Id) (X P ) A P Proof As A P is a Hopf subalgebra, for all n N, (a n ) A P A P Hence, (Z Id) (a n ) A P Lemma 7 We consider the following continuous applications: { Z Z ˆ Id : H CK ˆ H CK H CK, ε ε ˆ Id : H CK ˆ H CK H CK Then ε is an algebra morphism and Z is a ε-derivation, ie satisfies: Z(ab) Z(a) ε(b) + ε(a) Z(b) Proof Immediate 9

10 + Let us fix t T CK We put P (h) p n h n As X P B + (P (X P )), we have: Z (X P ) + + p n (Z Id) B + (X n P ) p n Z(B + (X n P ))1 + + ( + ) Z(X P )1 + B + p n Z( (XP ) n ) p n (Z B + )( (X P )) n ( + ) Z(X P )1 + B + np n ε( (X P )) n 1 Z( (XP )) Z(X P )1 + B + ( + We consider the following linear application: np n X n 1 P Z( (X P )) Z(X P )1 + B + ( P (X P )(Z Id) (X P )) ) ) L P : { HCK H CK a B + (P (X P )a) Then, immediately, for all a H CK, val(l P (a)) val(a)+1, so Id L P is invertible Moreover, by the preceding computation: Z (X P ) Z(X P )1 + L P ((Z Id) (X P )) (Id L P )((Z Id) (X P )) Z(X P )1 Z (X P ) Z(X P )(Id L P ) 1 (1) Hence, as Z(X P ) 1, lemma 6 induces the following result: Proposition 8 Let P K[[h]] 1 If A P is a Hopf subalgebra of H CK, then: (Id L P ) 1 (1) A P 3 Proof of 3 We put Y way: + In particular, b 1 p 1 b n (Id L P ) 1 (1) Then b n can be inductively computed in the following b 0 1, b n+1 n (k + 1)p k+1 B + (a α1 a αk ) k1 α 1 ++α k n + n k1 α 1 ++α k n kp k B + (b α1 a α a αk ) (5) 10

11 Suppose that A P is a Hopf subalgebra Then b n (A P V ect(t CK )) n V ect(a n ) for all n 1, so there exists α n K, such that b n α n a n Let us compare the projection on Im((B + ) ) of a n+1 and b n+1 : { p 1 B + (a n ) for a n+1, p B + (a n ) + p 1 B + (b n ) (p + p 1 α n )B + (a n ) for b n+1 Suppose that p 1 0 Then the a n s are all non zero by lemma 5, so α n is uniquely determined for all n N We then obtain, by comparing the projections of a n+1 and b n+1 over Im((B + ) ): Hence, for all n N, α n p 1 + p p 1 (n 1) { α1 p 1, α n+1 p p 1 + α n Let us compare the coefficient of B + ( n ) in a n+1 and in b n+1 with () and (5) we obtain: { p n for a n+1, (n + 1)p n+1 + np n p 1 for b n+1 Hence, α n+1 p n (n + 1)p n+1 + np n p 1 for all n 1 As a consequence: ( (n + 1)p n+1 + p 1 p ) np n p 1 p n p 1 This property is still true for n 0, as p 0 1 By multiplying by h n and taking the sum: We then put α p 1 and β p p 1 Hence: 1 ( P (h) + p 1 p ) hp (h) p 1 P (h) p 1 (1 αβh)p (h) αp (h) This equality is still true if p 1 0, with α 0 and any β Hence, we have shown: Proposition 9 If A P is a Hopf subalgebra of H CK, then there exists (α, β) K, such that P satisfies the following differential system: This implies: 1 P (h) 1 if α 0 P (h) e αh if β 0 S α,β : 3 P (h) (1 αβh) 1 β if αβ 0 { (1 αβh)p (h) αp (h) P (0) 1 4 Is A N,α,β a Hopf subalgebra? The aim of this section is to prove in 4 1 in theorem 4 11

12 41 Generators of A N,α,β We denote by P α,β the solution of S α,β and we put A α,β A Pα,β and A N,α,β A N,Pα,β, in order to simplify the notations We also put X Pα,β n N a n (α, β) and P α,β system S α,β is equivalent to: p 0 (α, β) 1, S α,β : p n+1 (α, β) α 1 + nβ n + 1 p n(α, β) for all n N Definition 10 1 For all i N, we put [i] β (1 + β(i 1)) In particular, [i] 1 i et [i] p n (α, β)h n The We put [i] β! [1] β [i] β In particular, [i] 1! i! et [i] 0! 1 We also put [0] β! 1 Immediately, for all n N: p n (α, β) α n [n] β! n! For all F F CK, we define the coefficient F! by: F! (fertility of s)! s vertex of F Note that these are not the coefficients F! defined in [4, 7, 19] They can be inductively defined by:! 1, (t 1 t k )! t 1! t k!, B + (F )! k!f! In a similar way, we define the following coefficients: [F ] β! [fertility of s] β! s vertex of F They can also be inductively defined: [ ] β! 1, [t 1 t k ] β! [t 1 ] β! [t k ] β!, [B + (F )] β! [k] β![f ] β! In particular, for all forest F, [F ] 1! F! and [F ] 0! 1 Finally, for all n N, we put a n (α, β) Theorem 11 For any tree t, t T NCK, t n a t (α, β) α t 1 [t] β! t! a t (α, β)t In particular, a t (1, 0) 1 t!, a t(1, 1) 1 and a t (0, β) δ t, for all β Moreover: a t (1, 1) { 0 if t is not a ladder, 1 if t is a ladder 1

13 Proof Induction on t If t 1, then t and a t (α, β) 1 Suppose the result true for all tree of weight strictly smaller than t Then, with t B + (t 1 t k ), by (3): a t (α, β) α t t k 1 [t 1] β! [t k ] β! α k [k] β! t 1! t k! k! α t 1 ++ t k [t] β! t! α t 1 [t] β! t! The formulas for (α, β) (1, 0), (1, 1) and (0, β) are easily deduced Finally, for (α, β) (1, 1), it is enough to observe that [1] β! 1 and [k] β! 0 if k Examples a 1 (α, β) a (α, β) α a 3 (α, β) α ( (1 + β) + ) ( a 4 (α, β) α 3 (1 + β)(1 + β) 6 a 5 (α, β) α 4 (1+3β)(1+β)(1+β) 4 + (1+β) 4 (1 + β) + + (1+β) 4 + (1+β) + (1+β)(1+β) 6 (1 + β) + + (1+β) + (1+β) + (1+β) (1 + β) + + (1+β)(1+β) 6 + (1+β) ) + + (1+β)(1+β) 6 + (1+β)(1+β) 6 + (1+β) + In particular, a(1, 1) is the sum of all planar trees of weight n, so A N,1,1 is the subalgebra of formal diffeomorphisms described in [10] Moreover, a(1, 1) is the ladder of weight n, so A N,1, 1 is the subalgebra of ladders of H NCK 4 Equalities of the subalgebras A N,P Lemma 1 Let P, Q K[[h]] 1 Suppose that Q(h) P (γh) for a certain γ We denote X P a n Then X Q γ n 1 a n In particular, if γ 0, A N,P A N,Q n 1 n 1 Proof We put Y n 1 γ n 1 a n Then: B + (Q(Y)) n N n N n N n N n γ k p k B + (γ n1 1 a n1 γ nk 1 a nk ) k1 n 1 ++n k n n k1 n 1 ++n k n γ n n γ k+n k p k B + (a n1 a nk ) k1 n 1 ++n k n γ n a n+1 p k B + (a n1 a nk ) Y By unicity, Y X Q 13

14 Theorem 13 Let (α, β) and (α, β ) K The following assertions are equivalent: 1 A N,α,β A N,α,β A α,β A α,β 3 (β β and αα 0) or (α α 0) Proof 1 Obvious 3 By theorem 11: a 1, a α, a 3 α + α (1+β) a 1, a α, a 3 α + α (1+β ) As A α,β A α,β, there exists γ 0, sucht that α γα Hence, α γα In particular, if α 0, then α 0 Suppose that α 0 As a 3 and a 3 are colinear, the following determinant is zero: α As α and α are non zero, β β α (1+β) α α (1+β ) 1 α α (β β) 0 ; 3 1 Suppose first α α 0 Then P α,β P α,β 1, so A N,α,β A N,α,β Suppose β β and αα 0 Then there exists γ K {0}, such that α γα Then, immediately, P α,β (γh) P α,β (h) By the preceding lemma, A N,α,β A N,α,β 43 The A N,α,β s are Hopf subalgebras We now prove 4 1 in theorem 4 If α 0, then A N,α,β K[ ] and it is obvious We take α 0 By theorem 13, we can suppose that α 1 Lemma 14 Let k, n N We consider the following element of K[X 1,, X n ]: P k (X 1,, X n ) α 1 ++α nk X 1 (X 1 + 1) (X 1 + α 1 1) α 1! X n(x n + 1) (X n + α n 1) α n! By putting S X X n : P k (X 1,, X n ) S(S + 1) (S + k 1) k! 14

15 Proof Induction on k This is obvious for k 1 Suppose the result true at rank k Then: P k (X 1,, X n )(X X n + k) α 1 ++α nk α 1 ++α nk+1 (k + 1) n i1 n i1 X 1 (X 1 +1)(X 1 +α 1 1) α 1! X i (X i +1)(X i +α i ) α i! X n(x n+1)(x n+α n 1) α n! α i α 1 ++α n k+1 (k + 1)P k+1 (X 1,, X n ) This implies the announced result X 1 (X 1 +1)(X 1 +α 1 1) α 1! X i (X i +1)(X i +α i 1) α i! X n(x n+1)(x n+α n 1) α n! X 1 (X 1 +1)(X 1 +α 1 1) α 1! X i (X i +1)(X i +α i 1) α i! X n(x n+1)(x n+α n 1) α n! Let F t 1 t k be a forest and t be a tree Using the dual basis (e F ) F FNCK : coefficient of F t in (X 1,β ) < e F e t, (X 1,β ) > < e F e t, X 1,β > < e s, X 1,β > s grafting of F on t [s] β! s! s tree, grafting of F on t Let n be the weight of t and s 1,, s n its vertices Let f i be the fertility of s i Let (α 1,, α n ) such that α α n k and consider the graftings of F on t such that α i trees of F are grafted on s i for all i Then: 1 If s is such a grafting, we have: [s] β! [t] β![t 1 ] β! [t k ] β! [f 1 + α 1 ] β! [f 1 ] β! s! t!t 1! t k! (f 1 + α 1 )! f 1! The number of such graftings is: ( ) f1 + α 1 α 1 (f n + α n )! f n! ( fn + α n α n ) [f n + α n ] β!, [f n ] β! 15

16 Hence, by putting x i f i + 1/β and s x x n, by lemma 14: coefficient of F t in (X 1,β ) [t] β! [t 1 ] β! [t k] β! [f 1 + α 1 ] β! [f n + α n ] β! t! t 1! t k! [f 1 ] β!α 1! [f n ] β!α n! α 1 ++α nk Moreover, as t is a tree: [t] β! [t 1 ] β! [t k] β! t! t 1! t k! [t] β! [t 1 ] β! [t k] β! t! t 1! t k! α 1 ++α nk α 1 ++α nk n i1 n i1 [t] β! [t 1 ] β! [t k] β! β k P k (x 1,, x n ) t! t 1! t k! [t] β! [t 1 ] β! [t k] β! k s(s + 1) (s + k 1) β t! t 1! t k! k! (1 + f i β) (1 + (f i + α i 1)β) α i! β α i x i(x i + 1) (x i + α i 1) α i s f f n + n/β number of edges of t + n/β n 1 + n/β n(1 + 1/β) 1 S(S + 1) (S + k 1) So, as Q k (S) : k! (X 1,β ) X 1,β 1 + X 1,β 1 + X 1,β 1 + F t 1 t k, t n1 [t] β! [t 1 ] β! t! t 1! [t k] β! β k Q k ( t (1 + 1/β) 1)F t t k! Q k (n(1 + 1/β) 1)β k X k 1,β a n(1, β) (1 βx 1,β ) n(1/β+1)+1 a n (1, β) n1 Proposition 15 The coproduct of the a n (1, β) s is given by: (X 1,β ) X 1,β 1 + (1 βx 1,β ) n(1/β+1)+1 a n (1, β) n1 As a consequence, A N,1,β is a Hopf subalgebra of H NCK Remark By taking the abelianization of A N,1,β, the same holds in A 1,β : (X 1,β ) X 1,β 1 + (1 βx 1,β ) n(1/β+1)+1 a n (1, β) n1 5 Isomorphisms between the A N,α,β s 51 Another system of generators of A N,1,β Notation We denote by B the inverse of B + : H NCK V ect(t NCK ), that is to say the application defined on a tree by deleting the root We define b n (α, β) B (a n+1 (α, β)) for all n N, and: Y(α, β) b(α, β) 16

17 We have: Y(α, β) α B+(F ) 1 [B+ (F )] β! B + F (F )! F F CK α t1 ++ tk [k] β![t 1 ] β! [t k ] β! t 1 t k k!t 1! t k! t 1,,t k T CK [k] β! X k α,β k! 1(1 + β) (1 + β(k 1)) X k α,β k! 1/β(1/β + 1) (1/β + k 1)) β k X k α,β k! Q k (1/β)β k X k α,β (1 βx α,β ) 1/β By the last equality, b n (α, β) A N,α,β for all n N Moreover, by the second equality, if n 1: b n (α, β) t T CK, t n α n [t] β! t + forests with more than two trees t! αa n (α, β) + forests with more than two trees So (b n (α, β)) n 1 is a set of generators of A N,α,β if α 0 Proposition 16 Suppose α 1 Then: (Y(1, β)) Y(1, β) n(β+1)+1 b n (1, β) Proof As Y(1, β) B (X(α, β)), by (1): (Y(1, β)) (Id B ) ( (X(1, β)) X(1, β) 1) (1 βx 1,β ) n(1/β+1)+1 B (a n (1, β)) n1 n1 5 Isomorphisms between the A N,α,β s Y n(1+β) β 1,β b n 1 (1, β) Y (n+1)(1+β) β 1,β b n (1, β) Y n(1+β)+1 1,β b n (1, β) Proposition 17 If β 1 and β 1, then A N,1,β and A N,1,β are isomorphic Proof Let γ K {0} We put: Z(1, β) Y(1, β) γ 17 c n (1, β),

18 with c n (1, β) A(1, β), homogeneous of degree n This makes sense, because b 0 (1, β) 1 Moreover, for all n 1: c n (1, β) Q 1 (γ)b n (1, γ) + forests with more than two trees γb n (1, γ) + forests with more than two trees, so (c n (1, β)) n 1 is a set of generators of A(1, β) Moreover: (Z(1, β)) ( Q k (γ) Y(1, β) n(β+1)+1 b n (1, β) Q k (γ) l0 n1 Q k (γ) a 1,,a k 1 a 1 ++a k l Y(1, β) l(β+1) l0 ) k Y(1, β) (a 1++a k )(β+1)+k b a1 (1, β) b ak (1, β) Y(1, β) l(β+1)+k b a1 (1, β) b ak (1, β) a 1 ++a k l Y(1, β) l(β+1) Y(1, β) γ c l (1, β) l0 l0 ( ) Z(1, β) l β+1 +1 γ cl (1, β) Q k (γ)y(1, β) b a1 (1, β) Y(1, β) b ak (1, β) We now chose γ β+1 β +1 As β 1, this is well defined; as β 1, this is non zero Then: (Z(1, β)) Z(1, β) l(β +1)+1 c l (1, β) So the unique isomorphism of algebras defined by: { A 1,β A 1,β b n (1, β ) c n (1, β) is a Hopf algebra isomorphism l0 In the non commutative case, the following result holds: Corollary 18 There are three isomorphism classes of A N,α,β s: 1 the A N,1,β s, with β 1 These are not commutative and not cocommutative A N,1, 1, isomorphic to QSym, the Hopf algebra of quasi-symmetric functions ([17, 0]) This one is not commutative and cocommutative 3 A N,0,1 K[ ] This one is commutative and cocommutative Consequently, in the commutative case: Corollary 19 There are three isomorphism classes of A α,β s: 1 the A 1,β s, with β 1 These are isomorphic to the Faà di Bruno algebra on one variable A 1, 1, isomorphic to Sym, the Hopf algebra of symmetric functions This one is commutative and cocommutative 18

19 3 A 0,1 K[ ] Proof As A α,β is the abelianization of A N,α,β, if A N,α,β A N,α,β, then A α,β A α,β Moreover, A 1,β is not cocommutative if β 1, whereas A 1, 1 is So A 1,β and A 1, 1 are not isomorphic if β 1 It remains to show that A 1,β is isomorphic to the Faà di Bruno Hopf algebra on one variable if β 1 Let us consider the dual Hopf algebra of A 1,β By Cartier- Quillen-Milnor-Moore s theorem ([18]), this is an enveloping algebra U(L 1,β ) Moreover, L 1,β has for basis (T n ) n N defined by: T n : L 1,β K a α 1 1 aα k k 0 if α α k 1, a m δ m,n Moreover, T n is homogeneous of degree n By proposition 15, for all i, j 1: (T i T j ) (X 1,β ) T i (X 1,β )T j (1) + n1 0 + Q 1 (j(1/β + 1) 1)βT i (X 1,β ) j(1 + β) β Q k (n(1/β + 1) 1)β k T i (X1,β k )T j(a n (1, β)) By homogeneity, there exists λ i,j K such that [T i, T j ] λ i,j T i+j Then: λ i,j [T i, T j ](X 1,β ) (T i T j ) (X 1,β ) (T j T i ) (X 1,β ) j(1 + β) β i(1 + β) + β (i j)(1 + β) Then, there exists a Lie algebra morphism: { Lα,β P rim(h F db ) T n (1 + β)z n In particular, if β 1, this is an isomorphism Hence, A 1,β is isomorphic to H F db, so A 1,β is isomorphic to the Faà di Bruno Hopf algebra on one variable Remark The Connes-Moscovici subalgebra H CM of H CK (see [5, 6]) does not appear here: as it is generated by,, +,, it would be A (1,1) The fourth generator of A (1,1) is: whereas the fourth generator of H CM is: So they are different + + +, The case of the free Faà di Bruno algebra with D variables We here fix an integer D 1 We denote by W the set of non empty words in letters {1,, D} 19

20 61 Construction We now recall the construction of the free Faà di Bruno algebra in D variables (see [1]) Consider the ring of non commutative formal series K h 1,, h D on D variables We consider: ( ) G D a (i) w h w / a (i) j δ i,j w W 1 i D We use the following convention: if u 1 u k W, then h w h u1 h uk In other terms, G D is the set of formal diffeomorphisms on K D which are tangent to the identity at the origin This is a group for the composition of formal series Then H F db,d is the Hopf algebra of functions on the opposite of the group G D Hence, it is the polynomial ring in variable Yw, i 1 i D, with the convention that if w has only one letter j, then Yj i δ i,j The coproduct is given in the following way: for all f H F db,d, for all P, Q G D, (f)(p Q) f(q P ) In particular, if: then: P ( w W a (i) w h w ) (Yw)(P i Q) Yw(Q i P ) Yw i 1 i D and Q ( w W b j u 1 u k u 1 u k W w 1,,w k W u 1 u k W w 1 w k w b j u 1 u k a u 1 w 1 a u k w k b (i) w h w ) a u 1 w 1 a u k 1 i D, w k h w 1w k 1 j D Hence: ( Yw i ) n k1 For D 1, we recover H F db 1 u i D w 1,,w k W, w 1 w k w Y u 1 w 1 Y u k w k Y i u 1 u k 6 Subalgebras of H D NCK We now put D {1,, D} 3 The elements of D will be denoted in the following way: i, (u 1, u ) In the same way, it is possible to construct a commutative Hopf algebra HCK D of rooted trees decorated by D, and a non commutative Hopf algebra HNCK D of planar rooted trees decorated by D In both cases, we define, for all i, (u 1, u ) D, a linear endomorphism B + i,(u 1,u ), which sends forest F on the tree obtained by grafting all the trees of F on a common root decorated by i, (u 1, u ) Definition 0 Let i {1,, D} and w u 1 u n W We define an element Yw i inductively on n in the following way: H D NCK Yw i δ i,w if n 1, Yw i 1 α,β D w 1,w W, w 1 w w B + i,(α,β) (Y α w 1 Y β w ) if n 0

21 Examples For i, u 1, u, u 3 and u 4 elements of {1,, D}: Y i u 1 δ i,u1, Yu i 1 u i, (u1, u ), Yu i 1 u u 3 ( α, (u 1, u ) i, (α, u 3 ) + α, (u, u 3) i, (u 1, α) ), Y i u 1 u u 3 u 4 1 α D 1 α,β D ( α, (u 1, u ) i, (α, β) β, (u 3, u 4 ) + β, (u, u 3 ) α, (u 1, β) i, (α, u 4 ) + β, (u 1, u ) α, (β, u 3 ) i, (α, u 4 ) + β, (u 3, u 4 ) α, (u, β) i, (u 1, α) + β, (u, u 3 ) ) α, (β, u 4 ) i, (u 1, α) An easy induction shows that Y i u 1 u n is homogeneous of degree n 1 Theorem 1 For all i {1,, D}, w W of length n: ( Yw i ) n k1 1 α i D w 1,,w k W, w 1 w k w Y α 1 w 1 Y α k w k Y i α 1 α k Proof By induction on n It is obvious if n 1 or Suppose it is true for all rank < n Then: (Y i w) α,β w 1 w w Yw i 1 + α,β α 1,,α k Y i w 1 + k Yw i 1 + k k 1 w 1 w k w B + i,(α,β) (Y α w 1 Y β w ) w 1 w w w 1,1 w 1,k w 1 w,1 w,l w w 1 w k w w 1 w k w α 1,,α k Y α1 w 1 β 1,,β l α 1,,α k Y α1 w 1 α 1,,α k Y α1 w 1 Y α k w k Y α1 w 1,1 Y αk w 1,k Y β1 w,1 Y βl w,l B + i,(α,β) Y α k w k Y α k w k Y i α 1 α k α,β Y i α 1 α k w 1 w α 1α k B + i,(α,β) ( Y α α 1 α k Y β β 1 β l ) ( Y α w 1 Y β w ) Hence, the subalgebra of H D NCK generated by the Y i w s is a Hopf subalgebra Its abelianization can be seen as a subalgebra of H D CK, and is isomorphic to H F db,d Remark In the case where D 1, we put Y 1 1 } {{ 1 Y n Then, by definition: } n+1 times Y 0 1, Y 1, Y n n 1 n B + (Y k Y n 1 k ) B + (Y n 1 ) + B + (Y k Y n 1 k ) if n Hence, by (4), this is the subalgebra associated to 1 + h + h (1 + h) P 4, 1 A D,4, 1 1 k1 (h), namely

22 63 Description of the Y i w s in the generic case Definition A tree t T D CK is admissible if: 1 Every vertex is of fertility less than For each vertex of fertility 1, the decorations are set in this way: a, (c, d) i, (a, b) or b, (c, d) i, (a, b) 3 For each vertex of fertility, the decorations are set in this way: a, (c, d) i, (a, b) b, (e, f) Let t be an admissible tree We associate to it a word in W in the following inductive way: 1 w( i, (a, b)) ab If the root of t has fertility 1, with decorations set as a, (c, d) i, (a, b), then, if we denote t B (t), w(t) w(t )b 3 If the root of t has fertility 1, with decorations set as b, (c, d) i, (a, b), then, if we denote t B (t), w(t) aw(t ) 4 If the root of t has fertility, then, if we denote t t B (t), w(t) w(t )w(t ) Remark The cases 1 and are not incompatible, so w(t) is not well defined For example, for t a, (c, d) i, (a, a), two results are possible: cda and acd An easy induction shows that: Proposition 3 Suppose that w W is generic, that is to say all his letters are distinct Then Y i w is the sum of admissible trees t such that: 1 w(t) w The decoration of the root of t is of the form i, (a, b), with 1, a, b D If the word is not generic, we obtain Y i w by specializing the generic case For example, if w aaa, we have: Y i abc Y i aaa 1 α D α, (a, b) + i, (α, c) In particular, a, (a, a) i, (a, a) appears with multiplicity 1 α D 1 α D α, (a, a) i, (α, a) + 1 α D α, (b, c) i, (a, α) α, (a, a) i, (a, α) References [1] Michael Anshelevich, Edward G Effros, and Mihai Popa, Zimmermann type cancellation in the free Fa di Bruno algebra, J Funct Anal 37 (006), no 1, , math/ [] Christoph Bergbauer and Dirk Kreimer, Hopf algebras in renormalization theory: locality and Dyson-Schwinger equations from Hochschild cohomology, IRMA Lect Math Theor Phys 10 (006), , math/

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