The symmetric operation in a free pre-Lie algebra is magmatic

A pre-Lie product is a binary operation whose associator is symmetric in the last two variables. As a consequence its antisymmetrization is a Lie bracket. In this paper we study the symmetrization of the pre-Lie product. We show that it does not satisfy any other universal relation than commutativity. It means that the map from the free commutative-magmatic algebra to the free pre-Lie algebra induced by the symmetrization of the pre-Lie product is injective. This result is in contrast with the associative case, where the symmetrization gives rise to the notion of Jordan algebra. We first give a self-contained proof. Then we give a proof which uses the properties of dendriform and duplicial algebras.


Introduction
A pre-Lie algebra is a vector space equipped with a binary operation x * y whose associator is right-symmetric. Its name comes from the fact that the antisymmetrization [x, y] := x * y − y * x of this binary operation is a Lie bracket. In this paper we investigate the symmetrization x#y := x * y + y * x of the pre-Lie product. We show that, contrarily to the bracket, it does not satisfy any universal relation (but commutativity of course). In other words the map of operads ComM ag → preLie induced by the symmetrization is injective. Here ComM ag stands for the operad encoding the algebras equipped with a commutative binary operation (sometimes called commutative nonassociative algebras in the literature). We give two different proofs of this result. The first one is self-contained and rely on the combinatorics of trees. The second one uses the dendriform algebras. More precisley we prove that the map of operads M ag → Dend is injective by using the theory of generalized bialgebras for duplicial algebras. Vladimir Dotsenko informed us that he found an alternative proof by using Groebner basis [3]. As a byproduct of our proof we show that the symmetrization of the binary operation generating the operad N AP induces an injective morphism ComM ag → N AP.

Prerequisite on operads
Let Fin be the category of finite sets, and Vect be the category of vector spaces over a field K. Let n = {1, . . . , n} be the standard finite set of cardinality n. We will use sometimes the fact that it is equipped with a total order. Any functor We recall that a type of algebras gives rise to an algebraic operad by taking the free algebra functor (cf. for instance [11,8]).

Pre-Lie algebra
A pre-Lie algebra is a vector space L together with a binary operation * satisfying the relation The operad P = preLie has been described in terms of trees by Chapoton and Livernet in [2] as follows. Given a finite set S of cardinality n let P[S] be the vector space spanned by the labelled rooted trees on n vertices with distinct labels chosen in S. For example the space P[{a, b, c}] is the linear span of the following trees: The pre-Lie product is obtained by the following process. Given two disjoint sets I, J and two labelled trees T ∈ P[I] and Y ∈ P[J] one defines Any pre-Lie algebra L gives rise to a Lie algebra whose bracket is defined by the antisymmetric product [x, y] := x * y − y * x. This gives a morphism of operads Lie −→ preLie. A FREE PRE-LIE ALGEBRA IS MAGMATIC   3 This morphism is injective and has been well studied in the literature, cf. for instance [2] and [9].

THE SYMMETRIC OPERATION IN
In this paper we consider the symmetric product This new binary operation is related to the commutative-magmatic operad ComM ag, where a ComM ag-algebra is a vector space A together with a binary operation · satisfying the relation x · y = y · x, ∀x, y ∈ A.
The symmetric product # in preLie gives a morphism of operads This has been much less studied and our main theorem is to show that this morphism is injective. Let us recall that ComM ag(n) is of dimension (2n − 3)!! = 1 × 3 × · · · × (2n − 3) and that preLie(n) is of dimension n n−1 .
An element in ComM ag[S] is a product x · y where x ∈ ComM ag[I], y ∈ ComM ag[J] for some decomposition S = I ∪ J (disjoint union). Let us suppose that S is equipped with a total order. Thanks to the commutativity of the product we can assume that the label max(S) is in J. In fact, there is a unique way to write an element in ComM ag[S] such that max(S) is on the right side and the two factors x and y satisfy also this property. We call it the normalized writing. For instance the elements of ComM ag(3) are written: We can now define Φ : ComM ag → P explicitly. If S = {a} is of cardinality one, then Φ(a) = a. If |S| > 1, then any Y ∈ ComM ag[S] decomposes uniquely as Y = V · W . We let Φ(Y ) := Φ(V )#Φ(W ) and extend linearly to ComM ag[S].

Symmetrizing the pre-Lie product
The main result of this paper is the following 3.1. Theorem. The morphim of operads Φ : ComM ag → preLie induced by the symmetric product x#y is injective.
Proof. Plan of the proof. We introduce the auxiliary operation x * y on the labelled trees as follows. For any labelled trees T and Y (labelled on disjoint finite sets), T * Y is the unique tree obtained by connecting the root of Y to the root of T . The root of T * Y is the former root of T : T * Y := ?>=< 89:; Y ?>=< 89:; T r t The proof of the theorem is divided into three steps.
-step 1. We show that it is sufficient to prove injectivity for the symmetrization of the operation * (in place of * ).
-step 2. We identify a subset of P[S] which is in bijection with ComM ag[S] and we show that * realizes this bijection.
-step 3. We show that the symmetrization of the operation * induces an injective map ComM ag[S] → P[S].
Step 1. We need to show that the linear maps Φ S : ComM ag[S] → P[S] are injective for all finite sets S. We first introduce a filtration on P[S]. For a tree T ∈ P[S], we say that T is of degree d if the number of subtrees connected to the root of T is d. We write deg(T ) = d. This induces a filtration of P[S]. The pre-Lie product with respect to this filtration is where T * Y is the unique tree obtained by connecting the root of Y to the root of T . So, for the purpose of this proof, we may use the product * instead of * .
Step 2. We define a map Ψ : ComM ag[S] → P[S], we identify its image, and finally we prove that it is injective.
Let S be a totally ordered finite set. Let t = x · y be an element in ComM ag[S], whose writing has been normalized, that is max(S) is a label of y. We define inductively Any tree T ∈ P[S] is uniquely determined by its root r and a set of subtrees {T 1 , T 2 , . . . , T ℓ } attached to the root. We denote this decomposition as T = (r, For the factorization in Eq (1) to exist in A[S] we must have that r is not max(T ). That is r < b ℓ . We remark that recursively, X 1 will be factorizable if r > b ℓ−1 . As we repeat this we see that In particular, we get a characterization of the T ∈ A[S] as follows.
, which is characterized by the following properties: Proof of claim 1. The tree T i either comes from Ψ(x) or is Ψ(y). In both cases it is in A[S i ] by induction. The index b i either come from Ψ(x) or is max Ψ(y) = max(y). In both cases we have r < b i as expected.

Claim 2. Ψ is a bijection onto its image A[S].
Proof of claim 2. Starting with T ∈ Im Ψ, by claim 1 we have Step 3. For T ∈ A[S] and its factorisation T = X 1 * X 2 , for which max(T ) = max(T 2 ), we define d(T ) = |X 2 |, the size of X 2 (the number of vertices). We can now turn to the map Φ : ComM ag[S] → P[S] given by Our purpose is to show that it is injective.  We proceed by induction on |S|. If |S| = 1, then Φ is a linear isomorphism and the result is clear. Assume |S| > 1. Our induction hypothesis is that the image of Φ[S ′ ] is isomorphic to A[S ′ ] for all |S ′ | < |S|. More precisely, we assume that the image of Φ[S ′ ] is exactly A[S ′ ] modulo terms in B[S ′ ]. This is certainly true for |S| = 1.
To obtain this, we order the trees of A[S] (a basis) as follows. Let T, T ′ be trees in A[S]. We say that T < T ′ whenever d(T ) > d(T ′ ). When |S| > 1 we have that  (1), we have that there are unique S = S 1 + S 2 and trees X 1 ∈ A[S 1 ] and X 2 ∈ A[S 2 ] such that T = X 1 * X 2 . By induction hypothesis, there are basis elements X 1 + W 1 in the image of Φ[S 1 ] and X 2 + W 2 in the image of Φ[S 2 ]. By construction we have that We study this equation. In the following let b = max(T ). Remark that b is necessarily a label in X 2 Let T ′ = X 2 * X 1 . If T ′ ∈ A[S], then b must be in a proper subtree V 2 of T ′ and the decomposition as in Eq (1) for T ′ is T ′ = V 1 * V 2 . We remark that V 2 must be a proper subtree of X 2 (since the root of T ′ is the root of X 2 ), hence |X 2 | > |V 2 |. This implies that T < T ′ . It may happen that T ′ ∈ B[S] but this does not cause any problem. We need to inspect the other terms in Eq (2). Let Z i be a tree in the support of W i , in particular Z i ∈ B[S i ]. For any tree Y , the tree Z i will eventually be a factor as in Eq (1) for the tree Y * Z i (this may take more than one recursive step). This shows that (X 1 + W 1 ) * W 2 and (X 2 + W 2 ) * W 1 are both in B[S]. For elements of the form Z i * Y we have to be more careful. For Z = Z 1 * X 2 , we have that max(Z) = b is in X 2 , this implies that Z = Z 1 * X 2 is the unique decomposition of Z as in Eq (1) , then b must be in a proper subtree V 2 of Z and the decomposition as in Eq (1) for Z is Z = V 1 * V 2 . We remark that V 2 must be a proper subtree of Z 2 , hence |X 2 | = |Z 2 | > |V 2 |. This implies that T < Z. For all T ∈ A[S], we have shown that there is an element in the image of Φ[S] given by Eq (2) which takes the form (x * y) * z = (x * z) * y for any labelled trees x, y, z. Algebras with one binary operation satisfying this identity already occured in the literature in the work of M. Livernet [4], and were called N AP -algebras (for NonAssociative Perm). She showed that the labelled trees do describe the associated operad and that the product is precisely given by the operation * on trees described in the proof of Theorem 3.1. Therefore, as a byproduct of the proof of the main theorem we obtain the following:

Theorem. The symmetrization of the operation * induces an injection of operads
ComM ag → N AP.

4.3.
Jordan algebras. Let x · y := xy + yx where xy is an associative operation. It is easy to check that the 3 operations (x · y) · z, (x · z) · y, (y · z) · x are linearly independent in the free associative algebra. However the 15 operations in arity 4 are not linearly independent: they satisfy the formula: Since this formula is invariant under any permutation of the three variables x, y, z it is equivalent, when 3 is invertible, to a formula in two variables only. It leads to the definition of a Jordan algebra which is a vector space equipped with a symmetric binary operation a · b satisfying the relation: Let us denote by Jord the operad governing Jordan algebras. Since any symmetrized associative product satisfies the Jordan relation there is a morphism of operads Jord → As . Contrarily to the Lie-As case and the ComMag-preLie case, this morphism is not injective. In other words there are relations satisfied by the symmetrized associative product which are not consequences of the Jordan relation. The first examples appear in arity 8: the Glennie relations, see p. 79 of [13].

5.
Conjecture on the factorization of preLie 5.1. On the injectivity of Lie → As. There are several ways of proving the injectivity of the morphism of operads Lie → As. One of them is to take advantage of the existence of the notion of "cocommutative bialgebra". Indeed, the Poincaré-Birkhoff-Witt theorem implies that there is an isomorphism of endofunctors of V ect Therefore it is natural to look for a similar statement when replacing Lie-As by ComMag-preLie.

5.2.
Conjecture: splitting of preLie. There exists an S-module X such that preLie = X • ComM ag.

Black and red trees.
Since the conjecture is only about the structure of S-module of preLie and since N AP has the same underlying S-module we could as well put N AP in the statement. For the rest of the section we denote by P the underlying S-module of both preLie and N AP .
Here are some arguments in favor of the conjecture. In this section we construct a linear isomorphism where φ ⊢ S denotes a set partition of S. Unfortunately, the family of spaces X that we construct is not defining an S-module. We believe that our construction can be modified to realize this, but we have not yet found the right way to do so. In Step 2 of Section 3 we constructed an injection Ψ : ComM ag → P. We say that a tree of P that is in the image of Ψ is a tree of type A. In Claim 1 of Step 2 we have a characterization of the tree of type A. To obtain the linear isomorphism in (4), we decompose (uniquely) any tree of P into special trees (of type X ) labelled with trees of type A.
To get this decomposition, we work recursively from the leaves down to the root. We will mark some edges red (double lines) and others will remain black. The black connected components will represent the trees of type A and the double red edges will connect them to form a tree. We also have that a double red edge is always between the root of a black tree to the maximum in the black tree below connected to it by that double red edge.
If T is a single leaf, then there are no edges to colour. Now assume we have T = (r, {T 1 , T 2 , . . . , T ℓ }) and all the trees T i for i ∈ {1, 2, . . . , ℓ} have been colored. Let T ′ i be the black connected component at the root of We assume (by construction up to that point) that in T i , all the double red edges connected to T ′ i are connected to b ′ i . If r > b ′ 1 or if r < b ′ 1 and T ′ i = T i for 1 ≤ i < ℓ, then we mark double red all edges out of the root r otherwise, we mark them black. We proceed to the next step. This will mark the edges of T double red or black. If we cut all marked edges, thanks to Claim 1 we get a forest of trees of type A. They are connected by marked edges and this structure forms a tree. This decomposition is unique. Remark that the condition ensures that the bicolored tree T that we obtain is black at r only if all double red edges in the black connected component of r are connected to the maximum vertex of that component. Otherwise, the tree is double red at r.
The tree below show how the special double red marking works. Let

5.4.
Bimodule structure. Comment from Vladimir Dotsenko: the behavior of the double red trees suggests that the S-module X itself can be decomposed as X = ComM ag • Z for some S-module Z. Computation of the power series agrees with this statement. That would make preLie into a free bimodule over ComM ag.

5.5.
Comment on the splitting of preLie and N AP . Since preLie (resp. N AP ) and ComM ag are operads it is natural to ask oneself if there could be an operad structure on X which would be compatible with the other operad structures under the isomorphism of S-modules. Analogously, it is natural to ask for "good triples of operads" (X , preLie, ComM ag) (resp. (X , N AP, ComM ag)) analogous to the good triple (Com, As, Lie) , cf. [7].

Comparison with the operad of dendriform algebras
We give another proof of Theorem 3.1 based on the comparison between the operad preLie and the operads Dend and Dup. First, we recall results about dendriform algebras and their parametrized version. Second, we recall results of [7] Chapter 5 about the operad Dup encoding duplicial algebras. Finally we give an alternative proof to the injectivity of the map ComM ag → preLie .
6.1. Parametrized dendriform algebras. Let λ ∈ K be a parameter. We define a parametrized dendriform algebra as a vector space A equipped with two binary operations ≺: A ⊗ A → A and ≻: A ⊗ A → A called the left operation and the right operation respectively, satisfying the following three relations For λ = 1 it is the notion of dendriform algebra introduced in [5,6] and for λ = 0 it is the notion of duplicial algebra introduced in [7,12]. The relevant operads are denoted by Dend and Dup respectively.
We introduce the operation {x, y} given by and the operation x y given by In the dendriform case {x, y} is known to be a pre-Lie product. Proof. The commutativity of the diagram follows from the following equalities: 6.3. On duplicial algebras. As mentioned above a duplicial algebra is defined by two binary operations x ≺ y and x ≻ y satisfying three relations: There is a notion of generalized bialgebra which is duplicial as an algebra and coassociative as a coalgebra. The primitive part is an algebra encoded by the operad M ag. The generating operation of the magmatic operad is given by x y = x ≺ y − x ≻ y in terms of the duplicial operations. Since the "triple of operads" (Com, Dup, M ag) is good (Corollary 5.2.6 of [7]), there is an isomorphism Dup ∼ = As • M ag, and therefore the map M ag → Dup is injective.
6.4. Lemma. The map of ns operads M ag → Dend induced by the operation x y := x ≺ y − x ≻ y is injective.
Proof. The free λ-Dend-algebra on a generator x, that is λ-Dend(x) ∼ = n≥1 λ-Dend n , is isomorphic to n≥1 K[P BT n+1 ]. The proof is the same as the proof for λ = 1 given in [6]. The inverse of this isomorphism, which we denote by ϕ, is obtained as follows:
We consider, in Dend(x) and Dup(x) respectively, the subspace generated by x under the operation . It give rise to the following commutative diagram (of graded vector spaces) M ag Since the map M ag → Dup is injective so is the map M ag → Dend.
6.5. Alternative proof of Theorem 3.1. Since the map of operads ComM ag → M ag is obviously injective, Lemma 6.4 implies that the composite ComM ag → M ag → Dend is injective. Since this is also the composite ComM ag → P reLie → Dend by Lemma 6.2, the map ComM ag → P reLie is injective. So, we are done with the second proof.
Observe that P relie → Dend is also injective, it has been proved by M. Ronco in [10]. 6.6. Splitting of Dend. Since there is an injection ComM ag → Dend, it is natural to ask the same question as in section 5: does there exist an S-module Y such that Dend = Y • ComM ag ?
More precisely its generating series is f Y (t) = 1 1−3t−t 3 . Since, as an S-module, Dend = Dup, and since by [7] we know that Dup = As • M ag, it would suffice to find Y such that M ag = Y • ComM ag.