The Hopf algebra of Fliess operators and its dual pre-Lie algebra

Loïc Foissy Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville Université du Littoral Côte d'Opale, Centre Universitaire de la Mi-Voix 50, rue Ferdinand Buisson, CS 80699 62228 Calais Cedex - France

e-mail : [email protected]

ABSTRACT. We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, nite-dimensional, connected grading. Dually, the vector space Rhx0 , x1 i is both a pre-Lie algebra for the pre-Lie product dual of the coproduct of H , and an associative, commutative algebra for the shue product. These two structures admit a compatibility which makes Rhx0 , x1 i a Com-Pre-Lie algebra. We give a presentation of this object as a pre-Lie algebra. KEYWORDS. Fliess operators; pre-Lie algebras; Hopf algebras. AMS CLASSIFICATION. 16T05, 17B60, 93B25, 05C05.

Contents

1 Construction of the Hopf algebra 1.1 1.2 1.3

3

Denition of the composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual Hopf algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Pre-Lie structure on Khx0 , x1 i 2.1 2.2

7

Pre-Lie coproduct on V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dual pre-Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Presentation of the pre-Lie algebra 3.1 3.2 3.3 3.4

Khx0 , x1 i Free Com-Pre-Lie algebras . . . . . . . . . . . . . From partitioned trees to Khx0 , x1 i . . . . . . . . From trees to Khx0 , x1 i . . . . . . . . . . . . . . Pre-Lie product in the basis of admissible words .

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4 Appendix 4.1 4.2 4.3

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18

Enumeration of partitioned trees . . . . . . . . . . . . . . . . . . . . . . . . . . . Study of the dendriform structure on admissible words . . . . . . . . . . . . . . . Freeness of the pre-Lie algebra gPT (D) . . . . . . . . . . . . . . . . . . . . . . . . 1

18 19 20

Introduction

Right pre-Lie algebras, or shortly pre-Lie algebras [4, 1], are vector spaces with a bilinear product • satisfying the following axiom:

(x • y) • z − x • (y • z) = (x • z) • y − x • (z • y). Consequently, the antisymmetrization of • is a Lie bracket. These objects are also called rightsymmetric algebras or Vinberg algebra [12, 17]. If A is a pre-Lie algebra, the symmetric algebra S(A) inherits a product ? making it a Hopf algebra, isomorphic to the enveloping algebra of the Lie algebra A [13, 14]. Whenever it is possible, we can consider the dual Hopf algebra S(A)∗ and its group of characters G, which is the exponentiation, in some sense, of the Lie algebra A. We here consider an inverse construction, starting from a group used in Control Theory, namely the group of Fliess operators [3, 5, 6]; this group is used to study the feedback product. We limit ourselves here to the one-dimensional case. We work with the alphabet X = {x0 , x1 }; the set of words with letters in X is denoted by X ∗ . We inductively dene a family of operators Fw : L1 ([t0 , t1 ], R) −→ C([t0 , t1 ]) indexed by elements of X ∗ by:

Z F∅ [u](t) = 1,

t

Z Fc [u](s)ds,

Fx0 c [u](t) = X

u(s)Fc [u](s)ds. t0

t0

For any formal series f =

t

Fx1 c [u](t) =

aw w ∈ Rhhx0 , x1 ii, we put:

w∈X ∗

Ff [u] =

X

aw Fw [u].

w∈X ∗

Under a condition on the coecients aw , this converges; the set {Id + Ff } is a group for the composition. By transport of structure, we obtain a group product ◦ on Rhhx0 , x1 ii. A way of computing the composition of two words of X ∗ , recalled here in denition 1, is given in [5], where it is also proved that it this composition respects the grading by the length of words; note that this grading is not connected and not nite-dimensional. We rst give a way to describe the composition in the group Rhhx0 , x1 ii and the coproduct of H by induction on the length of words (lemma 2 and proposition 3). We prove that H admits a second gradation, which is connected; the dimensions of this gradation are given by the Fibonacci sequence (proposition 8). As the product of Rhhx0 , x1 ii is left-linear, H is a commutative, right-sided combinatorial Hopf algebra [10], so, dually, Rhx0 , x1 i inherits a pre-Lie product •, which is inductively dened in proposition 11. We prove that the words xn1 , n ≥ 0, form a minimal subset of generators of this pre-Lie algebra (theorem 12). The pre-Lie algebra Rhx0 , x1 i has also an associative, commutative product, namely the shue product [15]. We prove that the following axiom is satised (proposition 13):

(x

y) • z = (x • z)

y+x

(y • z).

So Rhx0 , x1 i is a Com-Pre-Lie algebra [11] (denition 14). We give a presentation of this pre-Lie algebra in theorem 24. We use for this a description of free Com-Pre-Lie algebras in terms of partitioned trees (denition 16), which generalizes the construction of pre-Lie algebras in terms of rooted trees of [1]. This presentation induces a new basis of Rhx0 , x1 i in terms of words with letters in N∗ , see corollary 25. The pre-Lie product of two elements of this basis uses a dendriform structure [2, 9] on the algebra of words with letters in N∗ (theorem 28). The study of this dendriform structure is postponed to the appendix, as well as the enumeration of partitioned trees; we also prove that free Com-Pre-Lie algebras are free as pre-Lie algebras, using Livernet's 2

rigidity theorem [7].

Aknowledgment. The research leading these results was partially supported by the French National Research Agency under the reference ANR-12-BS01-0017. Notation. We denote by K a commutative eld of characteristic zero. All the objects (algebra, coalgebras, pre-Lie algebras. . .) in this text will be taken over K. 1

Construction of the Hopf algebra

1.1 Denition of the composition Let us consider an alphabet of two letters x0 and x1 . We denote by Khhx0 , x1 ii the completion of the free algebra generated by this alphabet, that is to say the set of noncommutative formal series in x0 and x1 . Note that Khhx0 , x1 ii is an algebra for the concatenation product and for the shue product, which we denote by . The unit for both products is the empty word, which we denote by ∅. The algebra Khhx0 , x1 ii is given its usual ultrametric topology.

Denition 1 [5]. 1. For any d ∈ Khhx0 , x1 ii, we dene a continuous algebra map ϕd from Khhx0 , x1 ii to End(Khhx0 , x1 ii) in the following way: for all X ∈ Khhx0 , x1 ii, ϕd (x0 )(X) = x0 X,

ϕd (x1 )(X) = x1 X + x0 (d

X).

2. We dene a composition ◦ on Khhx0 , x1 ii in the following way: for all c, d ∈ Khhx0 , x1 ii, c ◦ d = ϕd (c)(∅) + d. It is proved in [5] that this composition is associative.

Notation. Remark.

For all c, d ∈ Khhx0 , x1 ii, we put c˜ ◦d = c ◦ d − d = ϕd (c)(∅). If c1 , c2 , d ∈ Khhx0 , x1 ii, λ ∈ K:

(c1 + λc2 )˜◦d = ϕd (c1 + λc2 )(∅) = (ϕd (c1 ) + λϕd (c2 ))(∅) = ϕd (c1 )(∅) + λϕd (c2 )(∅) = c1 ˜◦d + λc2 ˜ ◦d. So the composition ˜ ◦ is linear on the left. As ϕd is continuous, the map c −→ c˜◦d is continuous for any d ∈ Khhx0 , x1 ii. Hence, it is enough to know how to compute c˜ ◦d for any word c, which is made possible by the next lemma, using an induction on the length:

Lemma 2 For any word c, for any d ∈ Khhx0 , x1 ii: 1. ∅˜◦d = ∅. 2. (x0 c)˜◦d = x0 (c˜◦d). 3. (x1 c)˜◦d = x1 (c˜◦d) + x0 (d (c˜◦d)).

Proof.

1. ∅˜ ◦d = ϕd (∅)(∅) = Id(∅) = ∅.

2. (x0 c)˜ ◦d = ϕd (x0 c)(∅) = ϕd (x0 ) ◦ ϕd (c)(∅) = ϕd (x0 )(c˜◦d) = x0 (c˜◦d). 3. (x1 c)˜ ◦d = ϕd (x1 c)(∅) = ϕd (x1 ) ◦ ϕd (c)(∅) = ϕd (x1 )(c˜◦d) = x1 (c˜◦d) + x0 (d (c˜◦d)). 3



1.2 Dual Hopf algebra We here give a recursive description of the Hopf algebra of the coordinates of the group Khhx0 , x1 ii of [5]. For any word c, let us consider the map Xc ∈ Khhx0 , x1 ii∗ , such that for any d ∈ Khhx0 , x1 ii, Xc (d) is the coecient of c in d. We denote by V the subspace of A∗ generated by these maps. Let H = S(V ), or equivalently the free associative, commutative algebra generated by the Xc 's. The elements of H are seen as polynomial functions on Khhx0 , x1 ii; the elements of H ⊗ H are seen as polynomial functions on Khhx0 , x1 ii × Khhx0 , x1 ii. Then H is given a multiplicative coproduct dened in the following way: for any word c, for any f, g ∈ Khhx0 , x1 ii,

∆(Xc )(f, g) = Xc (f ◦ g). As ◦ is associative, ∆ is coassociative, so H is a bialgebra.

Notations. 1. The space of words is a commutative algebra for the shue product . Dually, the space V inherits a coassociative, cocommutative coproduct, denoted by ∆ . For example, if a, b, c ∈ {x0 , x1 }:

∆ (Xabc ) = Xabc ⊗ X∅ + Xa ⊗ Xbc + Xb ⊗ Xac + Xc ⊗ Xab + Xab ⊗ Xc + Xac ⊗ Xb + Xbc ⊗ Xa + X∅ ⊗ Xabc . 2. We dene two linear endomorphisms θ0 , θ1 of V by θi (Xc ) = Xxi c for any word c. The following proposition allows to compute ∆(Xc ) for any word c by induction on the length.

˜ Proposition 3 For all x ∈ V , we put ∆(x) = ∆(x) − 1 ⊗ x. ˜ ∅ ) = X∅ ⊗ 1. 1. ∆(X ˜ ◦ θ0 = (θ0 ⊗ Id) ◦ ∆ ˜ + (θ1 ⊗ m) ◦ (∆ ˜ ⊗ Id) ◦ ∆ . 2. ∆ ˜ ◦ θ1 = (θ1 ⊗ Id) ◦ ∆ ˜. 3. ∆

Proof.

For any word c, for any f, g ∈ Khhx0 , x1 ii:

˜ c )(f, g) = ∆(Xc )(f, g) − (1 ⊗ Xc )(f, g) = Xc (f ◦ g) − Xc (g) = Xc (f ⊗ g − g) = Xc (f ˜◦g). ∆(X ˜ c ) ∈ V ⊗ H , so formulas in points 2 and 3 make sense. As ˜ ◦ is linear on the left, ∆(X Let f ∈ Khhx0 , x1 ii. It can be uniquely written as f = x0 f0 + x1 f1 + λ∅, with f0 , f1 ∈ Khhx0 , x1 ii, λ ∈ K . For all g ∈ Khhx0 , x1 ii:

f ˜◦g = (x0 f0 )˜ ◦g + (x1 f1 )˜ ◦g + λ∅˜◦g = x0 (f0 ˜◦g + g

(f1 ˜◦g)) + x1 (f1 ˜◦g) + λ∅.

1. We obtain:

˜ ∅ )(f, g) = X∅ (x0 (f0 ˜ ∆(X ◦g + g

(f1 ˜◦g)) + x1 (f1 ˜◦g) + λ∅) = 0 + 0 + λ = (X∅ ⊗ 1)(f, g),

so ∆(X∅ ) = X∅ ⊗ 1.

4

2. Let c be a word.

˜ ◦ θ0 (Xc )(f, g) = ∆(X ˜ x c )(f, g) ∆ 0 = Xx0 c (x0 (f0 ˜ ◦g + g = Xc (f0 ˜ ◦g + g

(f1 ˜◦g)) + x1 (f1 ˜◦g) + λ∅)

(f1 ˜ ◦g)) + 0 + 0

˜g + (f1 ◦ ˜g) g) + 0 + 0 = Xc (f0 ◦ ˜ c )(f0 , g) + (∆ ˜ ⊗ Id) ◦ ∆ (Xc )(f1 , g, g) = ∆(X ˜ ⊗ Id) ◦ ∆ (Xc )(f1 , g) ˜ c )(f0 , g) + (Id ⊗ m) ◦ (∆ = ∆(X ˜ c )(f, g) + (θ1 ⊗ Id) ◦ (Id ⊗ m) ◦ (∆ ˜ ⊗ Id) ◦ ∆ (Xc )(f, g), = (θ0 ⊗ Id) ◦ ∆(X ˜ ◦ θ0 (Xc ) = (θ0 ⊗ Id) ◦ ∆(X ˜ c ) + (θ1 ⊗ Id) ◦ (Id ⊗ m) ◦ (∆ ˜ ⊗ Id) ◦ ∆ (Xc ). so ∆ 3. This is proved by a similar computation.



Examples. ∆(Xx0 ) = Xx0 ⊗ 1 + 1 ⊗ Xx0 + Xx1 ⊗ X∅ , ∆(Xx20 ) = Xx20 ⊗ 1 + 1 ⊗ Xx20 + Xx0 x1 ⊗ X∅ + Xx1 x0 ⊗ X∅ + Xx1 x1 ⊗ X∅2 + Xx1 ⊗ Xx0 , ∆(Xx0 x1 ) = Xx0 x1 ⊗ 1 + 1 ⊗ Xx0 x1 + Xx1 x1 ⊗ X∅ + Xx1 ⊗ Xx1 , ∆(Xx1 x0 ) = Xx1 x0 ⊗ 1 + 1 ⊗ Xx1 x0 + Xx1 x1 ⊗ X∅ . By an easy induction on n:

˜ xn ) = Xxn ⊗ 1 and ∆(Xxn ) = Xxn ⊗ 1 + 1 ⊗ Xxn . Corollary 4 For all n ≥ 1, ∆(X 1 1 1 1 1

1.3 Grading It is proved in [5] that the Hopf algebra H is graded by the length of words, but this grading is not connected, that is to say that the homogeneous component of degree 0 is not (0), as it contains X∅ . Moreover, the homogeneous components of H are not nite-dimensional, as for example X∅n Xxk is homogeneous of degree k for all n ≥ 0. We now dene another grading on H , 0 which is connected and nite-dimensional.

Denition 5

1. Let c = c1 . . . cn be a word. We put: deg(c) = n + 1 + ] {i ∈ {1, . . . , n} | ci = x0 } .

2. For all k ≥ 1, we put Vk = V ect(Xc | deg(x) = k). This dene a connected grading of V , that is to say: M V =

Vk .

k≥1

3. This grading induces a connected grading of the algebra H : H=

M

Hk , and H0 = K.

k≥0

˜ c) ∈ Lemma 6 If c is a word of degree n, then ∆(X

M i+j=n

Proof.

Let us start by the following observations: 5

Vi ⊗ Hj .

1. Let c be a word of degree k . Then x0 c is a word of degree k + 2. Hence, θ0 is homogeneous of degree 2 on V . 2. Let c be a word of degree k . Then x1 c is a word of degree k + 1. Hence, θ1 is homogeneous of degree 1 on V . 3. Let c and d be two words of respective degrees k and l. Then any word obtained by shuing c and d is of degree k + l − 1: its length is the sum of the length of c and d, and the number of x0 in it is the sum of the numbers of x0 in c and d. Hence, dually, the coproduct ∆ is homogeneous of degree 1 from V to V ⊗ V . Let us prove the result by induction on the length k of c. If k = 0, then c = ∅ so n = 1, and ˜ ∆(Xc ) = Xc ⊗ 1 ∈ V1 ⊗ H0 . Let us assume the result for all words of length < k − 1. Two cases can occur. P 0 xi ⊗ x00i . By the preceding third 1. If c = x0 d, then deg(d) = n − 2. we put ∆ (Xd ) = observation, we can assume that for all i, x0i , x00i are homogeneous elements of V , with deg(x0i ) + deg(x0i ) = n − 2 + 1 = n − 1. Then: X ˜ c ) = (θ0 ⊗ Id) ◦ ∆(X ˜ d) + ˜ 0 ) ⊗ x00 ). ∆(X (θ1 ⊗ m) ◦ (∆(x i i i

˜ d ) ∈ (V ⊗ H)n−1 . By the second observation, (θ0 ⊗ Id) ◦ By the induction hypothesis, ∆(X ˜ ˜ 0 )⊗x00 ) ∈ (V ⊗ ∆(Xd ) ∈ (V ⊗H)n . By the induction hypothesis applied to x0i , for all i, (∆(x i i 0 00 ˜ H ⊗V )n−1 , so by the rst observation, (θ1 ⊗m)◦(∆(xi )⊗xi ) ∈ (V ⊗H)n−1+1 ⊆ (V ⊗H)n . So ∆(Xc ) ∈ (V ⊗ H)n . ˜ c ) = (θ1 ⊗ Id) ◦ ∆(X ˜ d ). By the induction 2. c = x1 d, then deg(d) = n − 1. Moreover, ∆(X ˜ d ) ∈ (V ⊗ H)n−1 . By the second observation, ∆(X ˜ c ) ∈ (V ⊗ H)n . hypothesis, ∆(X So the result holds for any word c.



The multiplicativity of ∆ implies:

Proposition 7 With this grading, H is a graded, connected Hopf algebra. Let us now study the formal series of V and H .

Proposition 8

1. For all k, let us put pk = dim(Vk ) and FV =

∞ X

pk X k . Then:

k=1

X , 1 − X − X2   √ !k √ !k 1 1+ 5 1− 5  . This is the Fibonacci sequence and for all k ≥ 1, pk = √  − 2 2 5 FV =

(A000045 in [16]). 2. We put FH =

∞ X

dim(Hk )X . Then FH = k

k=0

k=1

is A166861 of [16].

Proof.

∞ Y

1 . The sequence (dim(Hk ))k≥0 (1 − X k )pk

Let us consider the formal series: X F (X0 , X1 ) = ]{words in x0 , x1 with i x0 and j x1 }X0i X0j . i,j≥0

6

Then F (X0 , X1 ) =

1 . Moreover, by denition of the degree of a word: 1 − X0 − X1 FV = XF (X 2 , X) =

X . 1 − X − X2

As H is the symmetric algebra generated by V , its formal series is given by the second point. 

Remark.

Consequently, the space V inherits a bigrading:

Vk,n = V ect(Xc | deg(c) = k and lg(c) = n).   n It is not dicult to prove that dim(Vk,n ) = and the formal series of this bigrading k−n−1 is: X X dim(Vk,n )X k Y n = . 1 − XY − X 2 Y k,n≥0

2

Pre-Lie structure on

Khx0 , x1 i

2.1 Pre-Lie coproduct on V ˜ ) ⊆ V ⊗ H , so H As the composition ◦ is linear on the left, the dual coproduct satises ∆(V is a commutative right-sided Hopf algebra in the sense of [10], and V inherits a right pre-Lie coproduct: if π is the canonical projection from H = S(V ) onto V , ˜ δ = (π ⊗ π) ◦ ∆ = (Id ⊗ π) ◦ ∆. It satises the right pre-Lie coalgebra axiom:

(23).((δ ⊗ Id) ◦ δ − (Id ⊗ δ) ◦ δ) = 0. The following proposition allows to compute δ(Xc ) by induction on the length of c.

Proposition 9

1. δ(X∅ ) = 0.

2. δ ◦ θ0 = (θ0 ⊗ Id) ◦ δ + (θ1 ⊗ Id) ◦ ∆ . 3. δ ◦ θ1 = (θ1 ⊗ Id) ◦ δ .

Proof. ∆ (x) =

x0

The rst point comes from ∆(X∅ ) = X∅ ⊗ 1 + 1 ⊗ X∅ . Let x ∈ V . We put ˜ ⊗ x00 ∈ V ⊗ V . For any y ∈ V , we put ∆(y) − y ⊗ 1 = y (1) ⊗ y (2) ∈ V ⊗ H+ . Then:

˜ ⊗ Id) ◦ ∆ (x) = (θ1 ⊗ m)(x0 ⊗ 1 ⊗ x00 + x0(1) ⊗ x0(2) ⊗ x00 ) (θ1 ⊗ m) ◦ (∆ 0(2) 00 = θ1 (x0 ) ⊗ |{z} x00 +x0(1) ⊗ x | {zx} . ∈V

∈Ker(π)

Applying Id ⊗ π , it remains:

˜ ⊗ Id) ◦ ∆ (x) = (θ1 ⊗ Id) ◦ ∆ (x). (Id ⊗ π) ◦ (θ1 ⊗ m) ◦ (∆ Let i = 0 or 1. Then:

˜ = (θi ⊗ Id) ◦ (Id ⊗ π) ◦ ∆ ˜ = (θi ⊗ Id) ◦ δ. (Id ⊗ π) ◦ (θi ⊗ Id) ◦ ∆ The result is induced by these remarks, combined with proposition 3. 7



Proposition 10 Proof.

Ker(δ) = V ect(Xxn1 , n ≥ 0).

The inclusion ⊇ is trivial by corollary 4. Let us prove the other inclusion.

First step. Let us prove the following property: if x ∈ Vk is such that δ(x) = λ

X i+j=k−2

(k − 2)! Xxi ⊗ Xxj , 1 1 i!j!

then there exists µ ∈ K such that x = µxk−1 1 . It is obvious if k = 1, as then x = µ∅. Let us assume the result at all ranks < k . We put x = xα 1 (x0 f0 + x1 f1 ), where α ≥ 0, f0 is homogeneous of degree k − 2 − α and f1 is homogeneous of degree k − 1 − α.

δ(x) = (θ1α ⊗ Id) ((θ0 ⊗ Id) ◦ δ(f0 ) + (θ1 ⊗ Id) ◦ δ(f1 ) + (θ1 ⊗ Id) ◦ ∆ (f0 )) . Let us consider the terms in this expression of the form X∅ ⊗ Xc , with c a word. This gives:

λX∅ ⊗ Xxk−2 = 0, 1

so λ = 0 and δ(x) = 0. Let us now consider the terms of the form Xxα1 x0 c ⊗ Xd , with c, d words. We obtain: (θ1α ◦ θ0 ⊗ Id) ◦ δ(f0 ) = 0. As both θ0 and θ1 are injective, we obtain δ(f0 ) = 0. By the induction hypothesis, f0 = νX1 xl1 , with l = k − 2 − α < k . Hence:

∆ (f0 ) = ν

X l! X i ⊗ Xxj , 1 i!j! x1

i+j=l

and:



 X l! X i ⊗ Xxj  = 0. (θ1α+1 ⊗ Id) δ(f1 ) + ν 1 i!j! x1 i+j=l

As θ1 is injective, we obtain with the induction hypothesis that f1 = µXxk−2−α , so: 1

x = µXxk−1 + νXxα x0 xk−α−2 . 1

1

1

This gives:

 δ(x) = ν(θ1α+1 ⊗ Id) 

X

i+j=k−α−2

X

=ν

i+j=k−α−2

 (k − α − 2)! Xxi ⊗ Xxj  1 1 i!j!

(k − α − 2)! Xxi+α ⊗ Xxj 1 1 i!j!

= 0, so necessarily ν = 0 and x = µXxk−1 . 1

Second step. Let x ∈ Ker(δ). As δ is homogeneous of degree 0, the homogeneous components

of x are in Ker(δ). By the rst step, with λ = 0, these homogeneous components, hence x, belong to V ect(Xxk , k ≥ 0).  1

8

2.2 Dual pre-Lie algebra As V is a graded pre-Lie coalgebra, its graded dual is a pre-Lie algebra. We identify this graded dual with Khx0 , x1 i ⊆ Khhx0 , x1 ii; for any words c, d, Xc (d) = δc,d . The pre-Lie product of Khx0 , x1 i is denoted by •. Dualizing proposition 9, we obtain:

Proposition 11

1. For all word c, ∅ • c = 0.

2. For all words c, d, (x0 c) • d = x0 (c • d). 3. For all words c, d, (x1 c) • d = x1 (c • d) + x0 (c d).

Proof.

Let u, v, w be words. Then Xw (u • v) = δ(Xw )(u ⊗ v). Hence, if d is a word:

X∅ (u • v) = 0, Xx0 d (u • v) = (θ0 ⊗ Id) ◦ δ(Xd )(u ⊗ v) + (θ1 ⊗ Id) ◦ ∆ (Xd )(u ⊗ v) = Xd (θ0∗ (u) • v + θ1∗ (u)

v),

Xx1 d (u • v) = (θ1 ⊗ Id) ⊗ δ(Xd )(u ⊗ v) = Xd (θ1∗ (u) • v). Moreover, for all word c:

θ0∗ (∅) = 0,

θ0∗ (x0 c) = c,

θ0∗ (x1 c) = 0,

θ1∗ (∅) = 0,

θ1∗ (x0 c) = 0,

θ1∗ (x1 c) = c.

Hence, for any words c, d:

Xx0 d (x0 c • v) = Xd (c • v)

Xx0 d (x1 c • v) = Xd (c

= Xx0 d (x0 (x • v)), Xx1 d (x0 c • v) = 0

v)

= Xx0 d (x1 (c • v) + x0 (c

v)),

Xx1 d (x1 c • v) = Xd (c • v)

= Xx1 d (x0 (x • v)),

= Xx1 d (x1 (c • v) + x0 (c

v)).

Hence, for any w, Xw (x0 c • v) = Xw (x0 (x • v)) and Xw (x1 c • v) = Xw ((x1 (c • v) + x0 (c v)).  Dualizing proposition 10:

Theorem 12

Khx0 , x1 i = V ect(xn1 , n ≥ 0) ⊕ (Khx0 , x1 i • Khx0 , x1 i). Hence, (xn1 )n≥0 is a minimal system of generators of the pre-Lie algebra Khx0 , x1 i.

Proof.

As • = δ ∗ , Im(•) = Ker(δ)⊥ = V ect(Xxn1 , n ≥ 0)⊥ .

Proposition 13 For all x, y, z ∈ Khx0 , x1 i, (x Proof.



y) • z = (x • z) y + x (y • z).

We prove it if x, y, z are words. If x = ∅, then:

(∅

y) • z = y • z = (∅ • z)

y+∅

(u • z).

If y = ∅, the result is also true, using the commutativity of . We can now consider that x, y are nonempty words. Let us proceed by induction on k = lg(x) + lg(y). If k = 0 or 1, there is nothing to prove. Let us assume the result at all rank < k . Four cases can occur.

9

First case. x = x0 a and y = x0 b. Then: y) • z = (x0 (a

(x

x0 b) • z + (x0 (x0 a

b)) • z

x0 b) • z) + x0 ((x0 a

= x0 ((a

b) • z)

= x0 ((a • z)

x0 b) + x0 (a

((x0 b) • z)) + x0 (((x0 a) • z)

= x0 ((a • z)

x0 b) + x0 (a

(x0 (b • z)) + x0 ((x0 (a • z))

= x0 (a • z) = (x • z)

x0 b + x0 a y+x

(b • z))

b) + x0 (x0 a b) + x0 (x0 a

(b • z))

b) + x0 (x1 a

(b • z)),

x0 (b • z)

(y • z).

Second case. x = x1 a and y = x0 b. This gives: (x

y) • z = (x1 (a

x0 b)) • z + (x0 (x1 a

= x1 ((a • z)

(x • z)

z) + x0 (((x1 a) • z)

= x1 ((a • z)

x0 b) + x1 (a

+ x0 (a

z) + x0 ((x1 (a • z))

x0 b

y = (x1 (a • z)) = x1 ((a • z) + x0 (a

(y • z) = x1 a

x

x0 (b • z))

x0 b) + x1 (a

x0 b

+ x0 (a

b)) • z

z

(b • z))

b) + x0 (x1 a

x0 (b • z))

x0 b + (x0 (a

z))

b) + x0 ((x0 (a (x0 b)

(x0 b)) + x0 (x1 (a • z) x0 b) + x0 ((x0 (a

z))

z))

b) b),

x0 (b • z)

= x1 (a

x0 (b • z)) + x0 (x1 a

(b • z)).

These computations imply the required equality.

Third case. x = x0 a and y = x1 b. This is a consequence of the second case, using the

commutativity of

.

Last case. x = x1 a and y = x1 b. Similar computations give: (x

y) • z = x1 ((a • z) + x1 (x1 a

(x • z) x

x1 (b • w)) + x1 (a

(b • z)) + x1 ((x1 (a • z))

y = x1 ((a • z)

(y • z) = x1 (a

x1 b) + x1 (a

x1 b) + x1 ((x1 (a • z))

x1 (b • w)) + x1 (a

x0 (b

x0 (b

b) + x1 ((x0 (a b) + x0 (a

z))

x1 b

z)) + x1 (x1 a

z)) + x0 (a

b) + x0 (a

(b • z)) + x0 (a

z)

x1 b

z),

z))

b),

z) + x1 ((x0 (a

So the result holds in all cases. 3

x1 b

x1 b

z). 

Presentation of the pre-Lie algebra

Khx0 , x1 i

Proposition 13 motivates the following denition:

Denition 14 [11] A Com-Pre-Lie algebra is a triple (V, •,

), such that:

1. (V, •) is a pre-Lie algebra. 2. (V,

) is a commutative, associative algebra (non necessarily unitary).

3. For all a, b, c ∈ V , (a b) • c = (a • c) b + a (b • c). For example, Khx0 , x1 i is a Com-Pre-Lie algebra. See [11] for an example of Com-Pre-Lie algebra based on rooted trees. 10

3.1 Free Com-Pre-Lie algebras Denition 15

1. A partitioned forest is a pair (F, I) such that:

(a) F is a rooted forest (the edges of F being oriented from the leaves to the roots). (b) I is a partition of the vertices of F with the following condition: if x, y are two vertices of F which are in the same part of I , then either they are both roots, or they have the same direct descendant. 2. We shall say that a partitioned forest is a partitioned tree if all the roots are in the same part of the partition. 3. Let D be a set. A partitioned tree decorated by D is a pair (t, d), where t is a partitioned tree and d is a map from the set of vertices of t into D. For any vertex x of t, d(x) is called the decoration of x. 4. The set of isoclasses of partitioned trees will be denoted by PT . For any set D, the set of isoclasses of partitioned trees decorated by D will be denoted by PT (D).

Examples. We represent partitioned trees by the Hasse graph of the underlying rooted forest, the partition being represented by horizontal edges, of dierent colors. Here are all the partitioned trees with ≤ 4 vertices: q qq qq q q , = q q, q q q; q qq qq q ∨q q = q ∨q , qq q = q qq ,

qq qq q ; qq , q q ; ∨q , ∨q ,

q

q q

q

qq qq q

q qq qqq qqq qqq qqq q q qq qq ∨q , ∨q = ∨q , ∨q , ∨q = ∨q , ∨q = ∨q , ∨qq , ∨qq , qq , qq qq ∨q q = q ∨q , qq qq , qq q q = q qq q = q q qq , q q q q .

Denition 16 Let t = (t, I) and t0 = (t0 , J) ∈ PT . 1. Let s be a vertex of t0 . The partitioned tree t •s t0 is dened as follows: (a) As a rooted forest, t •s t0 is obtained by grafting all the roots of t0 on the vertex s of t. (b) We put I = {I1 , . . . , Ik } and J = {J1 , . . . , Jl }. The partition of the vertices of this rooted forest is I t J = {I1 , . . . , Ik , J1 , . . . , Jl }. 2. The partitioned tree t t0 is dened as follows: (a) As a rooted forest, t t0 is tt0 . (b) We put I = {I1 , . . . , Ik } and J = {J1 , . . . , Jl } and we assume that the set of roots of t is I1 and the set of roots of t0 is J1 . The partition of the vertices of t t0 {I1 t J1 , I2 , . . . , Ik , J2 , . . . , Jl }. q q qq qqq q q qq Examples. There are three possible graftings ∨q •s q : ∨q , ∨q and ∨q . There are two possible qq qqq ∨q q graftings q •s q q : ∨q and q .

These operations can also be dened for decorated partitioned trees. A short combinatorial study proves the next result:

Proposition 17 Let D be a set. We denote by gPT (D) the vector space generated by PT (D). We extend by bilinearity on gPT (D) and we dene a second product • on gPT (D) in the following way: if t, t0 ∈ PT (D), X t • t0 =

t •s t0 .

s∈V (t)

Then (gPT (D) , •,

) is a Com-Pre-Lie algebra. 11

In particular, gPT (D) is pre-Lie. Let us use the extension of the pre-Lie product • to S(gPT (D) ) dened by Oudom and Guin [13, 14]: 1. If t1 , . . . , tk ∈ gPT (D) , t1 . . . tk • 1 = t1 . . . tk . 2. If t, t1 , . . . , tk ∈ gPT (D) , t • t1 . . . tk = (t • t1 . . . tk−1 ) • tk − t • (t1 . . . tk−1 • tk ). 3. If a, b, c ∈ S(gPT (D) ), ab • c = (a • c(1) )(b • c(2) ), where ∆(c) = c(1) ⊗ c(2) is the usual coproduct of S(gPT (D) ). In particular, if t1 , . . . , tk , t ∈ PT (D):

t1 . . . tk • t =

k X

t1 . . . (ti • t) . . . tk .

i=1

Lemma 18 Let t = (t, I), t1 = (t1 , I (1) ), . . . , tk

= (tk , I (k) ) be partitioned trees (k ≥ 1). Let s1 , . . . , sk ∈ V (t). The partitioned tree t •s1 ,...,sk (t1 , . . . , tk ) is obtained by grafting the roots of ti on si for all i, the partition being I t I (1) t . . . t I (k) . Then: X t •s1 ,...,sk (t1 , . . . , tk ). t • t1 . . . tk = s1 ,...,sk ∈V (t)

Proof.

By induction on k . This is obvious if k = 1. Let us assume the result at rank k .

t • t1 . . . tk+1 = (t • t1 . . . tk ) • tk+1 −

k X

t • (t1 . . . (ti • tk+1 ) . . . tk )

i=1

X

=

(t •s1 ,...,sk (t1 , . . . , tk )) • tk+1 −

=

t • (t1 . . . (ti •s tk+1 ) . . . ti )

i=1 s∈V (ti )

s1 ,...,sk ∈V (t)

X

k X X

(t •s1 ,...,sk (t1 , . . . , tk )) •sk+1 tk+1

s1 ,...,sk+1 ∈V (t)

+

k X X

(t •s1 ,...,sk (t1 , . . . , tk )) •s tk+1

i=1 s∈V (ti )

−

k X

X

X

t •s1 ,...,sk (t1 , . . . , ti •s tk+1 , . . . , ti )

i=1 s1 ,...,sk ∈V (t) s∈V (ti )

X

=

t •s1 ,...,sk+1 (t1 , . . . , tk+1 ).

s1 ,...,sk+1 ∈V (t)

Hence, the result holds for all k .



Theorem 19 Let D be a set, let A be a Com-Pre-Lie algebra, and let ad ∈ A for all d ∈ D.

There exists a unique morphism of Com-Pre-Lie algebra φ : gPT (D) −→ A, such that φ( q d ) = ad for all d ∈ D. In other words, gPT (D) is the free Com-Pre-Lie algebra generated by D.

Proof. Unicity. ti,1 , . . . , ti,ki

Let t ∈ T d . We denote by r1 , . . . , rn its roots. For all 1 ≤ i ≤ n, let be the partitioned trees born from ri and let di be the decoration of ri . Then:

t = ( q d1• t1,1 . . . t1,k1 )

...

( q dn• tn,1 . . . tn,kn ).

So φ is inductively dened by:

φ(t) = (ad1 • φ(t1,1 ) . . . φ(t1,k1 ))

... 12

(adn • φ(tn,1 ) . . . φ(tn,kn )).

(1)

Existence. As the product of A is commutative and associative, (1) denes inductively a morphism φ from gPT (D) to A. By denition, it is compatible with the product . Let us prove the compatibility with the product •. Let t, t0 be two partitioned trees, let us prove that φ(t • t0 ) = φ(t) • φ(t0 ) by induction on the number N of vertices of t. If N = 1, then t = q d and: φ(t • t0 ) = ad • φ(t0 ) = φ(t) • φ(t0 ), by denition of t0 . If N > 1, two cases are possible. First case. If t has only one root, then t = q d • t1 . . . tk , and:

t • t0 = q d • t1 . . . tk t0 +

k X

q d • t1 . . . ti ◦ t0 • tk .

i=1

Using the induction hypothesis on t1 , . . . , tk : k X

φ(t • t0 ) = ad • φ(t1 ) . . . φ(tk )φ(t0 ) +

i=1 k X

= ad • φ(t1 ) . . . φ(tk )φ(t0 ) +

ad • φ(t1 ) . . . φ(t1 ◦ t0 ) . . . φ(tk ) ad • (φ(t1 ) . . . φ(t1 ) ◦ φ(t0 ) . . . φ(tk ))

i=1 0

= (ad • φ(t1 ) . . . φ(tk )) • φ(t ) = φ(t) • φ(t0 ).

Second case. If t has k > 1 roots, we put t = t1

tk . The induction hypothesis holds

...

for t1 , . . . , tk , so:

φ(t • t0 ) =

=

=

k X i=1 k X i=1 k X

ti • t0

φ(t1

...

tk )

φ(t1 )

φ(ti • t0 )

φ(t1 )

φ(ti ) • φ(t0 )

...

φ(tk )

...

φ(tk )

i=1

= (φ(t1 )

φ(tk )) • φ(t0 )

...

= φ(t) • φ(t0 ). Hence, φ is a morphism of Com-Pre-Lie algebras.



3.2 From partitioned trees to Khx0 , x1 i Proposition 20 As a Com-Pre-Lie algebra, Khx0 , x1 i is generated by ∅ and x1 . Proof.

Let A be the Com-Pre-Lie subalgebra of Khx0 , x1 i generated by ∅ and x1 . For all n ≥ 1, it contains x1 n = n!xn1 , so it contains xn1 for all n ≥ 0. As Khx0 , x1 i is generated by these elements as a pre-Lie algebra, A = Khx0 , x1 i.  We denote by φCP L : gPT ({1,2}) −→ Khx0 , x1 i the unique morphism of Com-Pre-Lie algebras which sends q 1 to ∅ and q 2 to x1 . By proposition 20, it is surjective.

Lemma 21 Let t1 , . . . , tk ∈ PT ({1, 2}). 1. φCP L ( q 1 • t1 . . . tk ) = 0 if k ≥ 1. 13

2. φCP L ( q 2 • t1 . . . tk ) = 0 if k ≥ 2. 3. If t ∈ PT ({1, 2}), φCP L ( q 2 • t) = x0 φCP L (t).

Proof.

We prove 1.-3. by induction on k . If k = 1:

φCP L ( q 1 • t) = ∅ • φCP L (t) = 0,

φCP L ( q 2 • t) = x1 • φCP L (t) = x0 φCP L (t).

Let us assume the results at rank k − 1 ≥ 1. Then:

φCP L ( q 1 • t1 . . . tk ) = ∅ • φCP L (t1 ) . . . φCP L (tk ) = (∅ • φCP L (t1 ) . . . φCP L (tk−1 )) • φCP L (tk ) −

k X

∅ • φCP L (t1 ) . . . φCP L (ti • tk ) . . . φCP L (tk−1 )

i=1

= 0, φCP L ( q 2 • t1 . . . tk ) = x1 • φCP L (t1 ) . . . φCP L (tk ) = (x1 • φCP L (t1 ) . . . φCP L (tk−1 )) • φCP L (tk ) −

k X

x1 • φCP L (t1 ) . . . φCP L (ti • tk ) . . . φCP L (tk−1 ).

i=1

If k ≥ 3, the induction hypothesis immediately allows to conclude that φCP L ( q 2 • t1 . . . tk ) = 0 − 0 = 0. If k = 2, this gives:

φCP L ( q 2 • t1 t2 ) = (x1 • φCP L (t1 )) • φCP L (t2 ) − x1 • φCP L (t1 • t2 ) = (x0 φCP L (t1 )) • φCP L (t2 ) − x0 φCP L (t1 • t2 ) = x0 (φCP L (t1 ) • φCP L (t2 ))φCP L (t1 • t2 )) = 0. Hence, the result holds for all k ≥ 1.



3.3 From trees to Khx0 , x1 i Let gT (N∗ ) be the free pre-Lie algebra generated by N∗ , as described in [1]. It can be seen as the subspace of gPT (N∗ ) generated by rooted trees (which are seen as partitioned trees such that any part of the partition is a singleton), with the restriction of the pre-Lie product • dened by graftings. This pre-Lie algebra is graded, the degree of a tree being the sum of its decorations. By theorem 12, there exists a unique surjective map of pre-Lie algebras ΦP L : gT (N∗ ) −→ Khx0 , x1 i, sending q n to xn−1 for all n ≥ 1. As xi−1 is homogeneous of degree i for all i, this 1 1 morphism is homogeneous of degree 0.

Notation.

If t1 . . . tk ∈ T (N∗ ) and n ∈ N∗ , we put:

Bn (t1 . . . tk ) = q n • t1 . . . tk . This is the tree obtained by grafting t1 , . . . , tk on a common root decorated by n.

Proposition 22 Let

Then:

t = Bn (t1 . . . tk ) ∈ T (N∗ ). We put φP L (ti ) = wi for all 1 ≤ i ≤ k . ( x0 w1 . . . x0 wk x1n−1−k if k < n, φP L (t) = 0 otherwise. 14

Proof.

As gPT ({1,2}) is pre-Lie, there exists a unique morphism of pre-Lie algebras:

ψ:

  gT (N∗ )

−→ gPT ({1,2})

  qn

−→

1 q2 (n − 1)!

Then φCP L ◦ψ is a pre-Lie algebra morphism sending q n to so φCP L ◦ ψ = φP L . We obtain, by lemma 18:

ψ( q n • t1 . . . tk ) =

1 q2 (n − 1)!

1 = (n − 1)!

(n−1)

.

1 x (n − 1)! 1

(n−1)

= x1n−1 for all n ≥ 1,

(n−1)

• (ψ(t1 ) . . . ψ(tk ))   Y X q2 •  ti 

 ...

q2 • 

i∈I1

I1 t...tIn−1 ={1,...,k}

 Y

ti 

i∈In−1

Let us apply φCP L to this expression. If |Ij | ≥ 2, by lemma 21:





φCP L  q 2 • 

 Y

ti  = 0.

i∈Ij

Consequently, if k ≥ n, at least one of the Ij contains two elements, so φCP L ◦ψ(t) = φP L (t) = 0. Let us assume that k < n. Hence, using the commutativity of :

 1 φP L ( q n • t1 . . . tk ) = (n − 1)! 1 = (n − 1)! 1 = (n − 1)!

X

x1 • 

I1 t...tIn−1 ={1,...,k}, |Ij |≤1

X

wi 

...

x1 • 

i∈I1

x 1 • w1





 Y

Y

wi 

i∈Ik

. . . x1 • wk

x1

(n−1−k)

ι:{1,...,k}−→{1,...,n−1}, injective

X

x 0 w1

. . . x 0 wk

x1

(n−1−k)

ι:{1,...,k}−→{1,...,n−1}, injective

(n − 1) . . . (n − k) (n−1−k) x0 w1 . . . x0 wk x1 (n − 1)! (n − 1) . . . (n − k)(n − 1 − k)! = x0 w1 . . . x0 wk xn−1−k 1 (n − 1)! =

= x 0 w1

. . . x 0 wk

x1n−1−k ,

which is the announced result.



Corollary 23 Let s1 , . . . , sm , t1 , . . . , tn ∈ T (N∗ ), m, n ≥ 0. For all i, j, k ≥ 1: φP L (Bk+1 ((Bi (s1 . . . sm )Bj (t1 . . . tn ))) = φP L (Bk (Bi+1 (s1 . . . sm Bj (t1 . . . tn ))) + φP L (Bk (Bj+1 (Bi (s1 . . . sm )t1 . . . tn )) .

Proof.

We note:

T1 = Bk+1 ((Bi (s1 . . . sm )Bj (t1 . . . tn )) = q k + 1 • (( q i • s1 . . . sm )( q j • t1 . . . tn )), T2 = Bk (Bi+1 (s1 . . . sm Bj (t1 . . . tn ))) = q k • ( q i + 1 • (s1 . . . sm ( q j • t1 . . . tn ))), T3 = Bk (Bj+1 (Bi (s1 . . . sm )t1 . . . tn )) = q k • ( q j + 1• (( q i • s1 . . . sm )t1 . . . tn )). 15

If m ≥ i, or n ≥ j , or k = 1, all these elements are sent to zero by φP L by proposition 22. Let us assume now that m < i, n < j , k > 1. We put vi = φP L (si ) and wi = φP L (ti ). Then:

φP L (T1 ) = x0 (x0 v1 |

...

X

φP L (T2 ) = x0 (x0 v1 = x0 (X

...

φP L (T3 ) = x0 (x0 (x0 v1

As x0 X

vm

x0 Y = x0 (X

x0 (x0 w1 |

...

x0 wn {z Y

xj−1−n ) 1 }

xk−2 1

...

x 0 wn

xj−1−n ) 1

x1i−1−m )

x 0 w1

x 0 wn

x0 (x0 w1

xi−1−m ) 1

xk−2 1

xk−2 1 ,

x0 Y )

= x0 (x0 X

xi−1−m ) 1 }

x1k−2 ,

x0 Y

= x0 X

x0 vm {z

...

Y)

x0 vm

xj−1−n ) 1

xk−2 1

xk−2 1 .

x0 Y ) + x0 (x0 X

Y ), we obtain the result.



Theorem 24 The kernel of φP L is the pre-Lie ideal generated by: 1. B1 (t1 . . . tk ), where k ≥ 1, t1 , . . . , tk ∈ T (N∗ ). 2. Bk+1 (Bi (s1 . . . sm )Bj (t1 . . . tn ))−Bk (Bi+1 (s1 . . . sm Bj (t1 . . . tn ))−Bj+1 (Bi (s1 . . . sm )t1 . . . tn )), where i, j, k ∈ N∗ , m, n ≥ 0, s1 , . . . , sm , t1 , . . . , tn ∈ T (N∗ ).

Proof. Let I be the ideal generated by these elements. By proposition 22 and corollary 23, I ⊆ Ker(φP L ). We put h = gT (N∗ ) /I . Applying repeatedly the relation given by elements of the second form, it is not dicult to prove that for any t ∈ T (N∗ ), there exists a linear span of ladders t0 such that t = t0 in h. Moreover, by the relation given by elements 1., if one of the vertices of a ladder t which is not the leaf is decorated by 1, then t = 0. Let us denote by L(n) the set of ladders decorated by N∗ , of weight n, such that all the vertices which are not S the leaf are decorated by integers > 1. It turns out that h is generated by the elements t, t ∈ L = L(n). Let φP L be the morphism form h to Khx0 , x1 i induced by φP L . By homogeneity, as φP L is surjective, for all n ≥ 1: φP L (V ect(L(n))) = V ect(words of degree n). In order to prove that I = Ker(φP L ), it is enough to prove that φP L is injective. By homogeneity, it is enough to prove that φ|V ect(L(n)) is injective for all n ≥ 1. Hence, it is enough to prove that for all n ≥ 1, |L(n)| = dim(V ect(words of degree n)) = pn , where the pn are the integers dened in proposition 8. Let ln = |L(n)| and qn be the number of t ∈ L(n) with no vertex decorated by 1. Then for all n ≥ 2, ln = qn + qn−1 , and l1 = 1. We put:

L=

∞ X

ln X n ,

Q=

n=1

∞ X

qn X n .

n=1

We obtain P = X + Q + XQ. Moreover:

Q=

1−

1 X

Xi

−1=

1 1−

X2 1−X

−1=

X2 , 1 − X − X2

i≥2

Finally, L =

X = F . So, for all n ≥ 1, |L(n)| = pn . 1 − X − X2



As an immediate corollary, a basis of h is given by the classes of the elements of L. Turning to Khx0 , x1 i, we obtain: 16

Corollary 25 Let w = a1 . . . ak be a word with letters in N∗ . 1. We put:

a

mw = xa11 −1 • (xa11 −1 • (. . . (x1k−1

−1

• xa1k ) . . .).

2. We shall say that w is admissible if a1 , . . . , ak−1 > 1. The set of admissible words is denoted by Adm. Then (mw )w∈Adm is a basis of Khx0 , x1 i.

Remark.

If w is not admissible, that is to say if there exists 1 ≤ i < k , such that ai = 1, then mw = 0 by proposition 22. We extend the map w −→ mw by linearity.

3.4 Pre-Lie product in the basis of admissible words Notations. 1. For all k, l, we denote by Sh(k, l) the set of (k, l)- shues, that is to say permutations ζ ∈ Sk+l such that ζ(1) < . . . < ζ(k), ζ(k + 1) < . . . < ζ(k + l). For all k, l we denote by Sh≺ (k, l) the set of (k, l)-shues ζ such that ζ −1 (k + l) = k . For all k, l we denote by Sh (k, l) the set of (k, l)-shues ζ such that ζ −1 (k + l) = k + l. 2. The symmetric group Sn acts on the set of words with letters in N∗ of length n by permutation of the letters: σ.(a1 . . . an ) = aσ−1 (1) . . . aσ−1 (n) .

Proposition 26 Let KhN∗ i be the space generated by words with letters in N∗ . We dene a

dendriform structure on this space by:

(a1 . . . ak ) ≺ (b1 . . . bl ) =

X

ζ.a1 . . . ak b1 . . . bk−1 (bk + 1)

ζ∈Sh≺ (k,l)

(a1 . . . ak )  (b1 . . . bl ) =

X

ζ.a1 . . . ak−1 (ak + 1)b1 . . . bk .

ζ∈Sh (k,l)

The associative product ≺ +  is denoted by ?.

Proof.

Direct verications.



We postpone the study of this dendriform algebra to section 4.2.

Notations.

For all a1 , . . . , ak ∈ N∗ , we denote by l(a1 . . . ak ) = Ba1 ◦ . . . ◦ Bak (1) the ladder decorated from the root to the leaf by a1 , . . . , ak . Note that ma1 ...ak = φP L (l(a1 . . . ak )).

Lemma 27 Let k, l ≥ 1 and let a1 , . . . , al , b1 , . . . , bl ∈ N∗ . Then: φP L (Ba1 +1 (l(a2 . . . ak )l(b1 . . . bl )) + Bb1 +1 (l(a1 . . . ak )l(b2 . . . bl )) = ma1 ...ak ?b1 ...bl .

Proof.

By induction on k + l. If k = l = 1, then: q q φP L ( q ba11 + 1 + q ab11 + 1 ) = m(a1 +1)b1 +(b1 +1)a1 = ma1 ?b1 .

17

Let us assume the result at all ranks < k + l. If k = 1, then:

φP L (Ba1 +1 (l(b2 . . . bl )) + Bb1 +1 (l(a1 )l(b2 . . . bl )) = φP L ( q a1 + 1 • l(b2 . . . bl ) + q b1 + 1 • (l(a1 )l(b2 . . . bl ))) = φP L (l((a1 + 1)b2 . . . bl )) + φP L ( q b1 • (l((a1 + 1)b2 . . . bl ) + q b2 + 1 • (l(a1 )l(b3 . . . bl ))) = m(a1 +1)b2 ...bl + mb1 (a1 ?b2 ...bl ) = m(a1 +1)b2 ...bl +

l−1 X

mb1 ...bi (a1 +1)...bl + mb1 ...(bl +1)a1

i=1

= ma1 ?b1 ...bl . If l = 1, a similar computation, permuting the ai 's and the bj 's, proves the result. If k, l > 1, then:

φP L (Ba1 +1 (l(a2 . . . ak )l(b1 . . . bl )) + Bb1 +1 (l(a1 . . . ak )l(b2 . . . bl )) = φP L ( q a1• ( q a2 + 1 • l(a3 . . . ak )l(b1 . . . bl )) + q b1 + 1 • l(a1 . . . ak )l(b2 . . . bl ))) + φP L ( q b1 • ( q a1 + 1 • l(a2 . . . ak )l(b2 . . . bl )) + q b2 + 1 • l(a1 . . . ak )l(b3 . . . bl ))) = ma1 (a2 ...ak ?b1 ...bl )+b1 (a1 ...ak ?b2 ...bl ) = ma1 ...ak ?b1 ...bl . Hence, the result holds for all k, l ≥ 1.



Theorem 28 For all a1 , . . . , ak , b1 , . . . , bl ∈ N∗ : ma1 ...ak • mb1 ...bl =

k−1 X

ma1 ...ai−1 (ai −1)(ai+1 ...ak ?b1 ...bl ) + ma1 ...ak b1 ...bl .

i=1

Proof. By denition of ma1 b1 ...bl , if k = 1, ma1 • mb1 ...bl = ma1 b1 ...bl . So the result holds if k = 1. Let us assume that k ≥ 2. In gT (N∗ ) , we have: l(a1 . . . ak ) • l(b1 . . . bl ) = q a1• (l(a2 . . . ak ) • l(b1 . . . bl )) + q a1• l(a2 . . . ak )l(b1 . . . bl ). Applying φP L :

ma1 ...ak • mb1 ...bl = ma1 (a2 ...ak )•(b1 ...bl ) + φP L ( q a1 − 1 • ( q a2 + 1 l(a3 . . . ak )l(b1 . . . bl )) + q b1 + 1 • l(a1 . . . ak )l(b2 . . . bl ))) = ma1 (a2 ...ak )•(b1 ...bl ) + m(a1 −1)(a2 ...ak ?b1 ...bl ) , by the preceding lemma. The result follows from an easy induction. 4



Appendix

4.1 Enumeration of partitioned trees Let d ≥ 1. For all n ≥ 1, let fn be the number of partitioned trees decorated by {1, . . . , d} with n vertices and let tn be the number of partitioned trees decorated by {1, . . . , d} with n vertices and one root. By convention, f0 = 1. We put:

T =

∞ X

n

tn X ,

F =

n=1

∞ X n=0

18

fn X n .

Let VT be the vector space generated by the set of partitioned trees decorated by {1, , . . . , d} and VF be the vector space generated by the set of partitioned trees decorated by {1, , . . . , d} with only one root. There is a bijection: ( S(VT ) −→ VF t1 . . . tk −→ t1 . . . tk . Hence:

F =

∞ Y i=1

There is a bijection:      

d M

1 . (1 − X k )tk

(2)

−→ VT

S(VF )

i=1

d  X   q i • (Fi,1 . . . Fi,ki ).  (F . . . , F , . . . , F . . . F ) −→ 1,k1 d,1 d,kd  1,1 i=1

This gives:

T = dX Formulas (2) and (3) allow to   f1       f2      f3      f4       f5

∞ Y

1 . (1 − X k )fk−1 i=1

(3)

compute inductively fk and tk for all k ≥ 1. This gives:

=d d(3d + 1) = 2 d(19d2 + 9d + 2) = 6 d(63d2 + 34d2 + 13d + 2) = 8 d(644d4 + 400d3 + 175d2 + 35d + 6) = 30

Here are examples of fn for d = 1 or 2:

n 1 2 3 4 5 6 7 8 9 10 d = 1 1 2 5 14 42 134 444 1518 5318 18989 d = 2 2 7 32 167 952 5759 36340 236498 1576156 10702333 The row d = 1 is sequence A035052 of [16].

4.2 Study of the dendriform structure on admissible words We here study the dendriform algebra KhN∗ i of proposition 26. It is clearly commutative, via the bijection from Sh≺ (k, l) to Sh (l, k) given by the composition (on the left) by the permutation (l + 1 . . . l + k 1 . . . l): in other terms, it is a Zinbiel algebra [8]. Let V be a vector space. The shue dendriform algebra Sh(V ) is T+ (V ), with the products given by: X (a1 . . . ak ) ≺ (b1 . . . bl ) = ζ.a1 . . . ak b1 . . . bk−1 bk ζ∈Sh≺ (k,l)

(a1 . . . ak )  (b1 . . . bl ) =

X ζ∈Sh (k,l)

19

ζ.a1 . . . ak−1 ak b1 . . . bk .

Moreover, this is the free commutative dendriform algebra generated by V , that is to say if A is a commutative dendriform algebra and f : V −→ A is any linear map, there exists a morphism of dendriform algebras φ : Sh(V ) −→ A such that φ| V = f . As a1 . . . ak  b = a1 . . . ak b in Sh(V ) for all a1 , . . . , ak , b ∈ V , this morphism φ is dened by:

φ(a1 . . . ak ) = (. . . (a1  a2 )  a3 ) . . .)  ak .

Proposition 29 KhN∗ i

1. Let V be the space generated by the words 1k i, k ∈ N, i ≥ 1. Then is isomorphic, as a dendriform algebra, to Sh(V ).

2. Let A be the subspace of KhN∗ i generated by admissible words. Then it is a dendriform subalgebra of KhN∗ i. Moreover, if W is the space generated by the letters i, i ≥ 1, then A is isomorphic, as a dendriform algebra, to Sh(W ).

Proof.

Let w = a1 . . . ak be a word with letters in N∗ . We denote by o(w) the sequence of indices j ∈ {1, . . . , k − 1} such that aj 6= 1. This sequences are totally ordered in this way: 0 , . . ., j 0 (j1 , . . . , jk ) < (j10 , . . . , jl0 ) if there exists a p such that jk = jl0 , jk−1 = jl−1 k−p+1 = jl−p+1 , 0 , with the convention j = j 0 0 jk−p < jl−p 0 −1 = . . . = j0 = j−1 = . . . = 0. Let φ : Sh(V ) −→ KhN∗ i be the unique morphism of dendriform algebras which extends the identity of V . Then:

φ((1k1 −1 a1 ) . . . (1kn −1 an )) = 1k1 −1 (a1 + 1) . . . 1kn−1 −1 (an−1 + 1)1kn −1 an + words w0 such that o(w0 ) > (k1 , . . . , kn−1 ). By triangularity, φ is an isomorphism. Moreover, for all a1 , . . . , an ≥ 1:

φ(a1 . . . an ) = (a1 + 1) . . . (an−1 + 1)an . Consequently, φ(Sh(W )) = A, so A is a dendriform subalgebra of KhN∗ i and is isomorphic to Sh(W ). 

4.3 Freeness of the pre-Lie algebra gPT (D) Notations.

put:

Let k ≥ 1, d1 , . . . , dk ∈ D and let F1 , . . . , Fk be decorated partitioned forests. We

Bd1 ,...,dk (F1 , . . . , Fk ) = ( q d1• F1 )

...

( q dk• Fk ).

Note that any partitioned tree can be written under the form Bd1 ,...,dk (F1 , . . . Fk ). This writing is unique up to a common permutation of the di 's and the Fi 's.

Proposition 30 We dene a coproduct δ on gPT (D) in the following way: for any decorated partitioned tree t = Bd1 ,...,dk (t1,1 . . . t1,n1 , . . . , tk,1 . . . tk,nk ), δ(t) =

k ni 1 XX Bd1 ,...,dk (t1,1 . . . t1,n1 , . . . , ti,1 . . . ti,j−1 ti,j+1 . . . ti,ni , . . . , tk,1 . . . tk,nk ) ⊗ ti,j . k i=1 j=1

1. For all x ∈ gPT (D) , (δ ⊗ Id) ◦ δ(x) = (23)(δ ⊗ Id) ◦ δ(x). 2. For all x, y ∈ gPT (D) , δ(x • y) = x ⊗ y + δ(x) • y .

Proof.

1. Let t = Bd1 ,...,dk (t1,1 . . . t1,n1 , . . . , tk,1 . . . tk,nk ). For all i, j , we put:

t/ti,j = Bd1 ,...,dk (t1,1 . . . t1,n1 , . . . , ti,1 . . . ti,j−1 ti,j+1 . . . ti,ni , . . . , tk,1 . . . tk,nk ). 20

Then:

δ(t) =

1X t/ti,j ⊗ ti,j . k i,j

Hence:

(δ ⊗ Id) ◦ δ(t) =

X

(t/ti,j )/ti0 ,j 0 ⊗ ti0 ,j 0 ⊗ ti,j

(i,j)6=(i0 ,j 0 )

As (t/ti,j )/ti0 ,j 0 and (t/ti0 ,j 0 )/ti,j are both the partitioned tree obtained by cutting ti,j and ti0 ,j 0 in t, they are equal, so (δ ⊗ Id) ◦ δ(t) is invariant under the action of (23). 2. Let t0 be a decorated partitioned tree.

δ(t • t0 ) =

k X

δ(Bd1 ,...,dk (t1,1 . . . t1,n1 , . . . , ti,1 . . . ti,ni t0 , . . . , tk,1 . . . tk,nk ))

i=1

+

X

δ(Bd1 ,...,dk (t1,1 . . . t1,n1 , . . . , ti,1 . . . ti,j • t0 . . . ti,ni , . . . , tk,1 . . . tk,nk ))

i,j

1 XX 1 = kt ⊗ t0 + Bd1 ,...,dk (t1,1 . . . t1,n1 , . . . , ti,1 . . . ti,ni t0 , . . . , tk,1 . . . tk,nk )/ti0 ,j 0 ⊗ ti0 ,j 0 k k 0 0 i

+

1 k

X

i ,j

Bd1 ,...,dk (t1,1 . . . t1,n1 , . . . , ti,1 . . . ti,j • t0 . . . ti,ni , . . . , tk,1 . . . tk,nk )/ti0 ,j 0 ⊗ ti0 ,j 0

(i,j)6=(i0 ,j 0 )

1X t/ti,k ⊗ ti,j • t0 k i,j X X = t ⊗ t0 + t(1) ⊗ t(2) • t0 + t(1) ⊗ t(2) • t0 .

+

So δ(t • t0 ) = t ⊗ t0 + δ(t) • t0 .



By Livernet's pre-Lie rigidity theorem [7]:

Corollary 31 The pre-Lie algebra gPT (D) is freely generated by Ker(δ). References

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