Algebraic deformation for (S)PDEs
March 24, 2021
Yvain Bruned1, Dominique Manchon2
1
University of Edinburgh 2
LMBP, CNRS - Université Clermont-Auvergne,
Email: [email protected], [email protected].
Abstract
We introduce a new algebraic framework based on the deformation of pre-Lie products. This allows us to provide a new construction of the algebraic objects at play in Regularity Structures in the work [10] and in [13] for deriving a general scheme for dispersive PDEs at low regularity. This construction also explains how the algebraic structure in [10] can be viewed as a deformation of the Butcher-Connes-Kreimer and the extraction-contraction Hopf algebras. We start by deforming various pre-Lie products via a Taylor deformation and then we apply the Guin-Oudom procedure which gives us an associative product whose adjoint can be compared with known coproducts. This work reveals that pre-Lie products and their deformation can be a central object in the study of (S)PDEs.
Contents
1 Introduction 2
2 Algebraic deformation of multi-pre-Lie algebras 6
2.1 Multi-pre-Lie algebras . . . 6
2.2 Taylor deformation of free multi-pre-Lie algebras . . . 9
3 Algebraic deformations of various products 13 3.1 Reminder on the Chu-Vandermonde identity . . . 14
3.2 Deformation of grafting product . . . 14
3.3 Deformation of a plugging product . . . 21
3.4 Deformation of the insertion product . . . 30
3.5 Cointeraction between deformed products . . . 33
4 Applications 35 4.1 Numerical Analysis . . . 35
4.2 Regularity Structures . . . 37
A Symbolic index 41
Introduction 2
1 Introduction
Decorated trees and their various Hopf algebra structures have revealed themselves to be very efficient for solving (stochastic) partial differential equations ((S)PDEs). Indeed, the theory of singular SPDEs has developed very fast in the last years. This is due to the theory of Regularity Structures introduced by Martin Hairer in [32]. Many equations whose well-posedness was open are now covered by this theory like the KPZ equation, ϕ43 (see [32]), ϕ44−δ (see [5]), sine-Gordon in the full subcritical regime (see [18]), generalised KPZ equation (see [8]), Yang-Mills in dimension two (see [15]). For surveys on this developement one can look at [11, 9] or the textbook [26]. The idea of Regularity Structures takes its root in rough paths [34, 30, 31]; one wants to approximate the solution of a singular SPDE by a Taylor expansion whose monomials are iterated integrals. This type of expansion is truncated to a sufficiently high order but it could be arbitrarily large when we consider a model close to subcriticality like ϕ4−δ4 . This is where algebraic structures enter the game for dealing with the computational challenge of such an expansion.
The core of this structure resides in decorated trees which are suitable combi-natorial objects for encoding iterated integrals. Indeed, decorations on the edges encode noises, convolution with kernels and derivatives on the previous objects, whereas decorations on the vertices correspond to monomials over d + 1 variables for the time and the spatial components. Recently, decorated trees have been also used in the context of PDEs for writing a resonance based scheme for dispersive equations in [13]. They allow to push an efficient numerical scheme from the first to the higher orders of approximation. It also covers a large class of equations. This reveals that more applications to PDEs could be expected in the future.
Such a structure was sketched in [32] and formalised in [10]. Two Hopf algebras act on the solution expansion: the first one recenters iterated integrals around a base point, the second renormalises these integrals which can be ill-defined in many situations due to the singularity of the noises. The convergence of the renormalised iterated integrals is proved in [17]. These two structures can be understood as a generalisation of the Butcher-Connes-Kreimer Hopf algebra [14, 21, 22] and the extraction-contraction Hopf algebra [19, 16] which appear both in Numerical analysis and Quantum field theory. We want to derive a rigorous mathematical statement describing this analogy. The idea is to construct a map that will transport all the properties from the original structures to the ones used for singular SPDEs.
Theorem 1.1 The structures defined in [10] and [13] are deformations of the
Butcher-Connes-Kreimer and the extraction-contraction Hopf algebras.
For proving this theorem, one has to build on the specificity given by (S)PDEs. Indeed, one uses in the various applications Taylor expansions which are encoded at the algebraic level in [10]. The main idea is therefore to deform all classical structures by means of "algebraic" Taylor expansions. Derivatives and monomials appear
Introduction 3
as decorations on decorated trees and by changing them along with the transfor-mation one gets a formal Taylor series with a leading term and terms of lower degrees.
Behind Theorem 1.1, one can think about a larger programme. Indeed, many algebraic structures use trees and graphs and they have prescribed rules on how to carry some operations as extraction-contraction and cutting edges. The main difficulty is to untangle all these operations in order to get the basic operations that will generate all the structure.
Starting with the grafting product which is a pre-Lie product and applying an algebraic procedure, D. Guin and J. M. Oudom recovered in [28, 29] the Grossman-Larson product [27], which is an associative product obtained from the Butcher-Connes-Kreimer coproduct by dualisation and suitable normalisation of the dual basis by symmetry factors. Their procedure works for any pre-Lie product and gives an alternative way of presenting the Hopf algebras at hand. A pre-Lie perspective on renormalisation has already been proposed in rough paths (see [6]) which was the inspiration for the work [5]. For edge-decorated trees, one needs to use Multi-pre-Lie algebras introduced in [5]. Multi-pre-Lie algebras are vector spaces (on some field k of characteristic zero) endowed with a family of pre-Lie products such that any finite linear combination of those remains a pre-Lie product. For decorated trees, it will be a family of grafting products indexed by the edge decorations. In fact, one can see the whole family as a pre-Lie product by considering only planted trees. In the diagram below we provide a partial answer to theorem 1.1 and it reveals the approach followed in this paper
y Guin-Oudom // Θ ˜ ?0 Φ Dual //∆ CK Ψ c y Guin-Oudom //?0 Dual //∆DCK (1.1)
The upper part of the diagram corresponds to the application of the Guin-Oudom procedure to the pre-Lie product y (collection of grafting products). This step has been performed in [25] and leads to the construction of a Butcher-Connes-Kreimer coproduct on decorated trees. The lower part of the diagram is the same procedure applied to the deformation cy of y introduced in [5]. This is the content of Theorem 3.4 and leads to the construction of ∆DCK. It is more involved due to the fact that one has to introduce infinite sums and use bigraded spaces as in [10]. Then, one wants a way to move from the upper part to the lower part. The maps Φ and Ψ are not easy to define. Instead, we construct the isomorphism Θ in Theorem 2.7 which can be extended naturally to decorated planted trees such that:
Θ(¯σ y ¯τ ) = Θ(¯σ)cy Θ(¯τ ) (1.2) where ¯σ, ¯τ are decorated planted trees. Then, the pre-Lie structure of y is transported tocy via the isomorphism Θ. The construction of Θ relies on the freeness of y
Introduction 4
and the fact that: ¯
σcy ¯τ = ¯σ y ¯τ + lower grading terms (1.3) where one measures the grading of a decorated trees as the sum of its edge decorations. In this way,cy is an algebraic deformation of y. Therefore, the coproduct ∆DCKin the diagram 1.1 can be understood as a deformation of the Butcher-Connes-Kreimer coproduct ∆CK. The coproduct ∆DCKhas an important application in Numerical Analysis (see [13]). Yet, coproducts in [10] are different from ∆DCK. This is due to derivations compatible with the pre-Lie product that needs to be added. In order to repeat the diagram 1.1, one has to work with another pre-Lie product defined by means of plugging denoted by. We obtain a similar diagram when we construct two coproducts ∆P(resp. ∆DP) from (resp. a deformation of ):b
Guin-Oudom //? Dual //∆ P b Guin-Oudom //? 2 Dual //∆DP (1.4)
Then, after performing an identification one can get from ∆DPthe coproduct ∆2 which has been introduced in [10] for recentering iterated integrals. The lower part of the diagram is the subject of Theorem 3.16. One cannot find an isomorphism that will preserve the pre-Lie structure like in (1.2), sending to (see Proposition 3.11).b One still has the deformation property (1.3) namely:
σ τ = σ τ + lower grading termsb (1.5) where σ, τ are decorated trees. For proving that b is a pre-Lie product one needs to use the Chu-Vandermonde identity, which we recall in Paragraph 3.1, which is also crucial for deriving the coassociativity in [10, Prop. 3.11] for the coproduct denoted by ∆2 therein. The pre-Lie product given by Θ is e satisfying
Θ(σ τ) = Θ(σ) Θ(τ)e
Then, one is able to give an expression for b in Proposition 3.13: σbvτ =↑
nσ
v Πσevτ
(1.6) where bvis the deformed plugging at the node v ∈ Nτ(Nτ nodes of τ ), Π removes the node decoration at the root of σ denoted by nσ and ↑nvσ adds the decoration nσ to the node v. We explain in Proposition 3.17 how the product ?2can be connected tocy:
Ib(σ ?2τ ) = ˜↑ nσ
Nτ(ΠσcyIb(τ )) (1.7)
where˜↑nNστ splits the decoration nσ among the nodes of τ andIb(τ ) grafts the tree τ
onto a new root via an edge decorated by b. The two formulae (1.6) and (1.7) give a different perspective with the use of derivations ↑nvσ and˜↑
nσ
Introduction 5
For the extraction-contraction coproduct ∆EC, we use the Chapoton-Livernet insertion product I of a tree into another. It is a pre-Lie product and can be directly defined via the product ?. This definition seems new in comparison with the literature. Then, the deformation bI is obtained by replacing ? by ?2. One can run the Guin-Oudom procedure and get the following diagram:
? Def //I Guin-Oudom //˜?1 Dual //∆EC
?2 Def //Ib
Guin-Oudom //?
1 Dual //∆1
(1.8)
The fact that ?1 is the dual of ∆1 is given in Theorem 3.24 wich concludes the proof of Theorem 1.1. We are also able to write the cointeraction between ∆2and ∆1obtained in [10] at the level of the deformed products, between ?2 and bI (see Theorem 3.25). This cointeraction has been previously observed in [19, 16] for the non-deformed structures.
In the end, we have proposed one single procedure allowing to recover the structure of [13] and [10] from the standard Butcher-Connes-Kreimer coproduct. Indeed, the isomorphism Θ transports the grafting product used for constructing the Butcher-Connes-Kreimer coproduct tocy which allows us to recover [13]. Then the formulae (1.6) and (1.7) shows how the construction in [10] is connected to the isomorphism Θ andcy. In this paper, we view the extraction-contraction coproduct as constructed from the pre-Lie insertion that depends on the product dual of the Butcher-Connes-Kreimer coproduct. Therefore, the deformed extraction-contraction coproduct follows from this property. Let us mention that the formalism developed in this work has already some impact on the study of singular SPDEs. The work proposes [4] a shorter proof of the renormalised equations first established in [5]. The spirit of our construction is also used in [35] for constructing a structure group for quasi-linear equations.
Let us outline the paper by summarising the content of its sections. In Section 2, we recall basics on multi-pre-Lie algebras and explore derivation for these structures. We see how they could be encoded directly into the grafting product for decorated trees. Then, we present the Taylor deformation considered in this paper. It is constructed via the isomorphism Θ in Theorem 2.7 which transports the pre-Lie structure of yatocy
a .
In Section 3, we consider various products and their deformations. This is where we provide the main results for making Theorem 1.1 precise. We start by the product y on decorated planted trees which groups the products yatogether. It is deformed via Θ intocy. By applying the Guin-Oudom procedure, we get an associative product ?0. In Theorem 3.4, we identify the dual of ?0 that is the
Algebraic deformation of multi-pre-Lie algebras 6
deformed Butcher-Connes-Kreimer coproduct ∆DCK. This explicit formula is very
useful in Section 4. Then, we move to another product, the plugging of a decorated tree into another. We obtain a deformed product b not coming directly from Θ. The Guin-Oudom approach gives again an associative product ?2whose adjoint is ∆2 (see Theorem 3.16). The last product is the insertion product I whose deformation
b
I is constructed from ?2. We then define an associative product ?1from bI whose adjoint ∆1is a deformed extraction-contraction coproduct (see Theorem 3.24). The section ends on studying a cointeraction property between deformed products.
In Section 4, we consider the two main applications of these algebraic structures which are a numerical scheme for dispersive PDEs and the theory of Regularity Structures for singular SPDEs. In subsection 4.1, we focus on the numerical applications and show that the coproduct used for the local error analysis is actually very close to the one obtained in Theorem 3.16. In subsection 4.2, we show that the pre-Lie approach developped in Section 4 gives an immediate and elegant construction of the algebraic objects at play in the theory of Regularity Structures. One gets the recentering coproduct from Theorem 4.2 and the extraction-contraction coproduct from Theorem 4.3 (the coproduct which is used for implementing the BPHZ algorithm [12, 33, 37]) Then, the cointeraction between these two maps is related to the pre-Lie cointeraction given in Section 4.
Acknowledgements
First discussions on this work were initiated while the authors participated in the workshop "Algebraic and geometric aspects of numerical methods for differential equations" held at the Institut Mittag-Leffler in July 2018. The authors thank the organisers of this workshop for putting together a stimulating program bringing different communities together, and the members of the institute for providing a friendly working atmosphere.
2 Algebraic deformation of multi-pre-Lie algebras
In this section, we present the notion of Multi-pre-Lie algebra introduced in [5]. One of the main examples are decorated trees equipped with a family of grafting products. We consider derivations for this structure and define a Taylor deformation. The latter is the deformation used for constructing various products in Section 3.
2.1 Multi-pre-Lie algebras
Definition 2.1 Let E be any set. A multi-pre-Lie algebra indexed by E is a vector
spacePover a field k of characteristic zero, endowed with a family (ya)a∈E of bilinear products such that
x ya(y yb z) − (x yay) yb z = y yb (x yaz) − (y ybx) yaz (2.1) for any a, b ∈ E and for any x, y, z ∈P.
Algebraic deformation of multi-pre-Lie algebras 7
In particular, each product yais left pre-Lie as well as any finite linear combination of those. The corresponding brace elements are recursively defined by
(x1· · · xn) ya1···an z := x1 ya1 (x2· · · xn) ya2···an z − n X i=2 x2· · · xi−1(x1 ya1 xi)xi+1· · · xn) ya2···an z
for any x1, . . . , xn, z ∈P. They are invariant under any permutation of the pairs (x1, an), . . . , (xn, an). Let us give a few important examples:
Example 1 Let E and V be two sets, let TEV be the set of rooted trees with vertices decorated by V and edges decorated by E, and letTEV be the linear k-span of TEV. The family of pre-Lie products is given by grafting by means of decorated edges, namely:
σ yaτ := X v∈Nτ
σ yav τ, (2.2)
where σ and τ are two decorated rooted trees, Nτ is the set of vertices of τ and where σ yav τ is obtained by grafting the tree σ on the tree τ at vertex v by means of a new edge decorated by a ∈ E. The proof of (2.1) is similar to the proof of the fact that grafting is a pre-Lie operation, and can be visualized by the following picture, symmetric with respect to the two branches:
τ σ
w a b
where σ, τ and w are decorated trees. Below we present an example of grafting:
•αya γ β b = γ β α a b + γ β α b a (2.3)
Proposition 2.2 For any set E, the E-multiple pre-Lie operad is given by thek-span
of the labellised rooted trees with E-decorated edges, where the labellisation is understood with respect to the vertices.
Proof. The proof (see [25]) is a straightforward adaptation of the description of the
pre-Lie operad by F. Chapoton and M. Livernet [20].
Corollary 2.3 For any pair of sets (E, V ), the free E-multiple pre-Lie algebra
generated by V is given by thek-span of rooted trees with E-decorated edges and V -decorated vertices.
Algebraic deformation of multi-pre-Lie algebras 8
Example 2 We keep the notations of the previous example, but we suppose that a
commutative monoid Ω acts on V . For any a ∈ E and ω ∈ Ω, the binary product ya,ω is defined as follows:
σ ya,ωτ := X v∈Nτ
↑ωv (σ yav τ ), (2.4)
where the operator ↑ωv stands for changing the decoration nvof the vertex v into ωnv. As ↑ωv and σ yav − commute, we also have:
σ ya,ω τ = X v∈Nτ
σ yav (↑ωv τ ). (2.5)
One can notice that the operators ↑ωdefined by
↑ω τ := X v∈Nτ
↑ωv τ. (2.6)
form a family of commuting derivations for all pre-Lie products ya,ω0.
Proposition 2.4 The vector spaceTEV endowed with the binary products ya,ωis a
E × Ω-multiple pre-Lie algebra.
Proof. In the particular case when the three trees are made of a single vertex,
Equation (2.1) comes from the fact that
•αya,ω (•β yb,ω0 •γ) − (•α ya,ω •β) yb,ω0 •γ) = ωω0γ
β α
a b
is manifestly symmetric in (a, ω) and (b, ω0), due to the commutativity of the monoid Ω. The general case is straightforward and left to the reader.
We abbreviate ya,1with yawhere 1 is the unit of Ω.
Proposition 2.5 Suppose that V = S × Ω, and that the action of Ω on V is given
by ω.(s, ω0) := (s, ωω0). ThenTEV, (ya)a∈E, (↑ω)ω∈Ω
is the free S-generated
E-multiple pre-Lie algebra endowed with an Ω-indexed family of commuting
derivations. Proof. Let
A, (a)a∈E
be an E-multiple pre-Lie algebra endowed with an Ω-indexed family of commuting derivations (Dω)ω∈Ω. Fix a collection (αs)s∈S of elements of A. By Corollary 2.3, there is a unique morphism Φα of E-multiple
Algebraic deformation of multi-pre-Lie algebras 9
pre-Lie algebras fromTESto A such that Φα(•s) = αsfor any s ∈ S. We want to extend Φαto a morphism Ψα: TEV, (ya)a∈E, (↑ω)ω∈Ω −→A, (a)a∈E, (Dω)Ω∈E .
From Ψ◦ ↑ω= Dω◦ Ψαwe immediately get: Ψα(•s,ω) = Dω(αs).
By Corollary 2.3 again, there is a unique morphism χα of E-multiple pre-Lie algebras fromTEV to A such that χα(•s,ω) = Dω(αs) for any ω ∈ Ω and s ∈ S. To prove that Ψαand χαcoincide, it only remains to prove χα◦ ↑ω (τ ) = Dω◦ χα(τ ) for any ω ∈ Ω and for any τ ∈ TEV.
We proceed by induction on the number |τ | of vertices of τ , the case |τ | = 1 being obvious. Any τ with at least two vertices can be written as
τ = k X j=1
τ1j yaj τ2j,
where the components τ1j and τ2jare strictly smaller with respect to the number of vertices. We can then compute:
χα◦ ↑ω (τ ) = χα◦ ↑ω k X j=1 τ1j yaj τ2j = χα k X j=1 ↑ω τ1j yaj τ2j+ τ1j yaj↑ωτ2j = k X j=1 Dωχα(τ1j)aj χα(τ2j) + χα(τ1j)ajDωχα(τ2j) = Dω k X j=1 χα(τ1j)ajχα(τ2j) + χα(τ1j)aj χα(τ2j) = Dωχα(τ ).
2.2 Taylor deformation of free multi-pre-Lie algebras
We suppose here that Ω is the monoid Nd+1endowed with componentwise addition. A grading is given by
Algebraic deformation of multi-pre-Lie algebras 10
where s := (s1, . . . , sd+1) ∈ Nd+1>0 is fixed. We suppose that V = S × Nd+1and E = S0× Nd+1where S and S0 are two finite sets. Then Ω acts freely on both E and V in a graded way. We denote by + the addition in Ω as well as both actions of Ω on E and V . A family of deformed grafting products onTEV is defined as follows:
σcyaτ := X v∈Nτ X `∈Nd+1 nv ` σ ya−`v (↑−`v τ ). (2.7) Here nv ∈ Nd+1denotes the second component of the decoration at the vertex v. The generic term is self-explanatory if there exists a (unique) pair (b, α) ∈ E × V such that a = ` + b and nv = ` + α. It vanishes by convention if this condition is not satisfied. We define the grading of a tree inTEV by the sum of the gradings of its edges given by | · |grad:
|τ |grad := X e∈Eτ
|e(e)|s. (2.8)
where Eτ are the edges of τ and e(e) is the decoration of the edge e. Then,cy a
is a deformation of yain the sense that:
σcyaτ = σ yaτ + lower grading terms. We repeat the example in (2.3) but now with the deformation:
•αcya γ β b = γ β α a b + γ β α b a (2.9) +X ` γ ` γ− ` β α a− ` b +X ` γ ` γ β− ` α b a− `
The second line of (2.9) contains the lower grading terms.
Remark 2.6 The deformation (2.7) has been first introduced for singular SPDEs in
[5]. The leading term corresponds to the higher order of a Taylor jet. Indeed, if we consider τ to be •k, the tree composed of a single node decorated by k, and σ to be •, we get: •cya•k= X `∈Nd+1 k ` • ya−`•k−` (2.10)
We interpret the tree •kas a monomial of degree k and the edge yais associated to the derivatives of some smooth function f : Rd+1→ R. Indeed, if we multiply a decorated tree by the product of the factors n1v!where v is any vertex, one has
•cya 1 k!•k = X `∈Nd+1 1 `!• y a−` 1 (k − `)! •k−`. (2.11)
Algebraic deformation of multi-pre-Lie algebras 11
We suppose that a ≥ k and we can rewrite (2.11) as a Taylor jet of f(a−k)when the bound is given by k: X `≤k xk−` (k − `)!f (a−`)(0).
where we have made the following identifications 1 (k − `)!•k−` ≡ xk−` (k − `)!, f (a−`)(0) ≡ 1 `!• y a−` .
Theorem 2.7 There exists a unique linear isomorphism Θ :TEV → TEV such that
Θ(•α) = •α for any α ∈ V
and
Θ(σyaτ ) = Θ(σ)cyaΘ(τ ) (2.12)
for any σ, τ ∈TEV and for any a ∈ E.
Proof. We introduce the set P TEV of planar rooted trees with edges decorated by E and vertices decorated by V , andPTVE will stand for the linear span of P TEV. A collection of left graftings is defined by
σyyaτ := X v∈Nτ
σyyavτ, (2.13)
for any σ, τ ∈ P TEV, where σyyavτ is obtained by grafting σ on τ at vertex v on the left of the other branches, by means of a new edge decorated by a.
Lemma 2.8 (PTVE, (yya)a∈E) is the free algebra endowed with a family of mag-matic products indexed by E, generated by V .
Proof. The free magma endowed with a family of magmatic products indexed by E
generated by V is obtained as the set of planar binary trees with leaves decorated by V and internal vertices decorated by E. The magmatic product ∨aof two such trees S and T , with a ∈ E, is obtained by plugging S and T on the left (resp. right) branch of the Y -shaped tree with a unique internal node decorated by a.
D. Knuth’s rotation correspondence maps bijectively this set of planar binary trees onto P TEV. It transforms leaves into vertices and internal nodes into edges. The magmatic product ∨ais transformed into the left Butcher product indexed by a defined by
Algebraic deformation of multi-pre-Lie algebras 12
where %τ is the root of τ . By freeness property, there is a unique linear map Ψ :PTVE →PTVE such that Ψ(•α) = •αfor any α ∈ V , and such that
Ψ(σ◦→aτ ) = Ψ(σ)yyaΨ(τ ) (2.15)
for any σ, τ ∈ P TEV. It is obvious that
σyyaτ = σ◦→aτ + higher potential energy terms,
where the potential energy of a tree is the sum of the heights of its vertices. From this we can infer, by induction on the number of vertices, that
Ψ(σ) = σ + higher potential energy terms
for any σ ∈ P TEV. This in turns implies that Ψ is a linear isomorphism, the matrix of which is triangular with 1’s on the diagonal in a suitable basis. Details in the particular case E = {∗} can be found in [24] and in [1], adaptation to any finite set E is straightforward.
Let us now finish up the proof of Theorem 2.7: by universal property of the free magmatic algebra, there exists a unique linear map
Θ :PTVE →PTVE
such that Θ(•α) = •αfor any α ∈ V , and such that Θ(σyyaτ ) = Θ(σ)ycy aΘ(τ ) (2.16) with σycyaτ := X v∈Nτ X `∈Nd+1 nv ` σyya−`v (↑↑−`v τ ) (2.17) for any a ∈ E. Here ↑↑−`v is the obvious analogue of ↑−`v in the planar setting. From (2.16) and (2.17) one easily shows, by induction on the number of vertices, that
Θ(σ) = σ + lower grading terms
for any σ ∈ P TEV, hence Θ is a linear isomorphism. Moreover, one can also easily see, by induction on the number of vertices, that Θ(σ) is a sum of planar trees all of which coincide with σ modulo a change of decoration. It therefore gives rise to a linear isomorphism Θ :TEV → TEV such that the following diagram commutes (where π stands for the canonical projection which forgets the planarity):
PTVE Θ // π PTVE π TEV Θ //TEV
Algebraic deformations of various products 13
Corollary 2.9 The algebra (TEV, (cya)a∈E) is E-multiple pre-Lie.
Proof. This is immediate from the fact that (TEV, (ya)a∈E) is E-multiple pre-Lie, and from
σcyaτ = Θ(Θ−1(σ) yaΘ−1(τ )).
It is natural to introduce the associated Ω-indexed family of modified derivations:
b
↑ω := Θ−1↑ω Θ, (2.18)
which obviously form a family of commuting derivations with respect to the new family of pre-Lie products (cya)a∈E. As an obvious consequence of Proposition 2.5 and Theorem 2.7,
Corollary 2.10 The space TEV endowed with(ya)a∈E and (b↑ ω
)ω∈Ω is the free S-generated E-multiple pre-Lie algebra endowed with an Ω-indexed family of
commuting derivations. The isomorphism with the original structure
TEV, (ya )a∈E, (↑ω)ω∈Ω
is given by Θ.
Let us remark that b↑ ω
does not coincide with ↑ω: for example,
↑ω Θ ω00 ω0 a =X `∈Ω ω00 ` ω00− ` ω + ω0 a− ` +X `∈Ω ω00 ` ω + ω00− ` ω0 a− ` 6= Θ ↑ω ω00 ω0 a =X `∈Ω ω00 ` ω00− ` ω + ω0 a− ` +X `∈Ω ω + ω00 ` ω + ω00− ` ω0 a− `
Finally, from Proposition 2.4, we get that the vector spaceTEV endowed with the binary productscy
a,ω
is a E × Ω-multiple pre-Lie algebra, with σcy
a,ωτ = Θ(Θ−1(σ) ya,ω Θ−1(τ )).
(2.19)
3 Algebraic deformations of various products
We consider three products on decorated trees: grafting, plugging and insertion. The grafting product is deformed by using the isomorphism Θ and we construct an associative product via the Guin-Oudom procedure associated to it. Then, the coproduct dual of this product is identified as the deformed Butcher-Connes-Kreimer coproduct on decorated trees. For the plugging, the deformation is not exactly given by Θ but follows the same spirit. The associated product obtained by applying
Algebraic deformations of various products 14
Guin-Oudom is the dual of a Butcher-Connes-Kreimer coproduct where derivations have been added in the definition. Then, we also define the deformed insertion from this deformed product. The last part of this section focuses on the cointeraction between deformed products.
3.1 Reminder on the Chu-Vandermonde identity
Let S be any finite set, and let k, ` : S → N with N := {0, 1, 2, . . .}. Factorials and binomial coefficients are defined by:
`! := Y x∈S `(x)!, k ` := Y x∈S k(x) `(x) , (3.1) where a b := a!
b!(a − b)! if a, b ∈ N and a ≥ b, with the convention a
b
= 0
if a, b ∈ N and a < b. now let eS be another finite set, and let π : S → eS be a map (not necessarily surjective nor injective). For any map ` : S → N we define π∗` : eS → N by: π∗`(x) =
0 if x is not in the image of π, X
y∈S, π(y)=x
`(y) if x is in the image of π. (3.2)
Proposition 3.1 (Chu-Vandermonde identity) For any e` : eS → N and for any k : S → N the following holds:
π∗k e ` = X `, π∗`=e` k ` . (3.3)
Proof. Consider the obvious equality between polynomials with variables indexed
by eS: Y s∈S (1 + Xπ(s))k(s)= Y e s∈ eS (1 + X e s)π∗k(es), (3.4)
and pick the coefficient of the monomial Y e s∈ eS X`(ees) e s on both sides.
3.2 Deformation of grafting product
We denote by PEV the set of planted trees and byPEV their linear span. A planted tree is of the formIa(τ ) where τ ∈TEV andIa(τ ) denotes the grafting of the tree τ onto a new root via an edge decorated by a ∈ E. Remark that, according to our conventions, the root of a planted tree is not decorated. We define two pre-Lie products on the planted trees by:
Ia(σ) yIb(τ ) = Ib(σ yaτ ), Ia(σ)cyIb(τ ) = Ib(σcy aτ ).
(3.5) We have the following property
Algebraic deformations of various products 15
Proposition 3.2 The spacePEV endowed with y orcy is a pre-Lie algebra.
Proof. This comes from the obvious identification ofPEV withTEV ⊗ kE given by
Ia(τ ) 7−→ τ ⊗ a.
It is easily seen that the structure yaresp. (cya)a∈EonTEis E-multiple pre-Lie if and only if the bilinear product y resp.cy onT
V
E ⊗ kE given by (σ ⊗ a) y (τ ⊗ b) := (σ yaτ ) ⊗ b resp. (σ ⊗ a)cy (τ ⊗ b) := (σcy
aτ ) ⊗ b
is pre-Lie [25].
Below we present an example of computation with y:
α a y β b = β α b a (3.6)
The main difference with (2.3) is that we do not have decorations at the root and one cannot graft at the root: we get one term less. If we extend Θ defined in Theroem (2.7) toPEV by:
Θ(Ia(τ )) :=Ia(Θ(τ )), (3.7)
we obtain thatcy is a deformation of y. For every τ, σ ∈ T V E , one has Θ(Ia(σ) yIb(τ )) = Θ(Ia(σ))cy Θ(Ib(τ )). (3.8) Indeed, ΘIa(σ) yIb(τ ) = ΘIb(σ yaτ ) =Ib Θ(σ ya τ ) =Ib Θ(σ)cyaΘ(τ ) =Ia Θ(σ)cyIb Θ(τ ) = Θ Ia(σ) c y Θ Ib(τ )
where we have used (3.8), (3.5) and (3.7). The deformation of (3.6) is given by
α a c y β b = β α b a + X |`|s<|β|s∧|a|s β ` β− ` α b a− ` (3.9)
Algebraic deformations of various products 16
We want to construct a Hopf algebra from the pre-Lie algebra (PEV,cy). We follow the path initiated by Guin and Oudom in [28, 29]. It has also been used in [25] for the pre-Lie algebra (PEV, y) for constructing a Butcher-Connes-Kreimer type coproduct. We want to see how this perspective is robust toward the deformation of the pre-Lie product. We denote by g the Lie algebra (PEV, [·]) where the bracket is given for ¯σ, ¯τ ∈ TEV by:
[¯σ, ¯τ ] = σcy ¯τ − ¯τcy ¯σ.
We first consider the symmetric algebra S(PEV) with its usual multiplicative coproduct ∆ defined for every ¯τ ∈PEV by:
∆¯τ = ¯τ ⊗ 1 + 1 ⊗ ¯τ . (3.10)
A natural basis of this symmetric algebra is given by trees with undecorated root. Identifying the monomial Xkwith the decorated one-vertex tree •k. We extend the map ∆ multiplicatively to trees with decorated root by
∆Xk= Xk⊗ 1 + 1 ⊗ Xk. (3.11)
Multiplicativity is not intended for the tree product but for the natural action of S(PEV) onTEV obtained by identifying the roots of the planted trees with the one of the tree inTEV. Using Sweedler’s notation it reads for w ∈ S(PEV): ∆w =
P
(w)w(1)⊗w(2).
We are now in position to apply the Guin-Oudom construction to the pre-Lie algebra (PEV,cy). One can define a product on S(P
V E) for every u, v, w ∈ S(PEV), x, y ∈PEV by 1 • w = w, u • 1 = 1?(u), w • uv =X (w) (w(1)• u)(w(2)• v), xv • y = xcy (v • y) − (xcy v) • y (3.12)
where 1?stands for the counit, and wherecy is extended toP V E ⊗ S(PEV) in the following way: xcy x1. . . xk= k X i=1 x1. . . (xcy xi) . . . xk,
with xi∈PEV. Then, we define the associative product ?0as:
w ?0v = X
(w)
(w(1)• v)w(2)
(3.13)
Algebraic deformations of various products 17
Proposition 3.3 The space (S(PEV), ?0, ∆) is a Hopf algebra isomorphic to the enveloping algebra U(g) where the Lie algebra g isPEV endowed with the antisym-metrization of the pre-Lie productcy.
In the sequel, we want to identify the dual of the product ?0. We suppose that the set V of vertex decorations coincides with the commutative monoid Ω. It will be given by the following maps ∆DCK : S(P
V
E) → S(PEV) ˆ⊗ S(PEV) and ¯
∆DCK :T
V
E → S(PEV) ˆ⊗TEV defined recursively by: ∆DCKIa(τ ) = id ˆ⊗Ia ¯ ∆DCKτ +Ia(τ ) ˆ⊗ 1, ∆¯DCKX k= 1 ˆ⊗ Xk ¯ ∆DCKIa(τ ) = id ˆ⊗Ia ¯ ∆DCKτ + X `∈Nd+1 1 `!Ia+`(τ ) ˆ⊗ X `. (3.14)
The map ∆DCKis extended using the product of S(T
V
E ) (denoted by e Q
). We use the tree product (denoted byQ) for extending the map∆¯DCK. We also define the map
¯ ∆rootDCKby ¯ ∆rootDCKIa(τ ) = X `∈Nd+1 1 `!Ia+`(τ ) ˆ⊗ X `, ∆¯root DCKX k= 1 ˆ⊗ Xk
and extend it multiplicatively with respect to the tree product. We have replaced the standard tensor product ⊗ by ˆ⊗ for making sense of the infinite sum over `. Indeed, the maps ∆DCKand∆¯DCKare triangular maps for the bigrading given in [10] by
(|Ten|bi) = (|Ten|grad, |NT \ {%T}| + |ET|),
where T is a rooted tree, e : ET → E are the edge decorations and n : NT → V are the node decorations, Ten is the decorated tree with such decorations. We have used the grading | · |graddefined in (2.8).
We recall some basics on bigraded spaces extracted from [10, Sec. 2.3]. A bigraded space V is given by
V = n∈N2
Vn,
where each Vnis a vector space and any element v of V is given as a formal sum v =P
n∈N2vnwith vn∈ Vnand such that there exists k ∈ N, vn= 0 when n2> k.
For two bigraded spaces V and W , we define V ˆ⊗ W as the bigraded space V ˆ⊗ W def = n∈N2 " M m+`=n (Vm⊗ W`) # . (3.15)
We consider the following partial order on N2
Algebraic deformations of various products 18
A family {Am,n}m,n∈N2 of linear maps Am,n : Vn → ¯Vm between two bigraded
spaces V andV is called triangular if A¯ m,n = 0 unless m ≥ n.
Following [10, Remark 2.16], one has a canonical inclusion V∗ ⊗ W∗ ⊂ (V ˆ⊗ W )∗and the dual V∗consists of formal sumsPnv∗nwith vn∗ ∈ Vn∗, and for every k ∈ N there exists f (k) such that vn∗ = 0 for every n ∈ N2 with n1≥ f (n2). From this definition of the dual, we clearly see from (2.7) thatcy is a map defined fromPEV
∗ ⊗ˆ
PEV∗ into PEV ∗
. Indeed, the product cy gives a finite sum which is clearly an element of the dual. We can derive a recursive formulation ofcy in order to match the one for ∆DCK. We also associate a grafting product to∆¯DCK
denoted by ¯y. This product plays a central role in the recursive expression ofcy: ¯ σcyIb(τ ) =Ib(¯σ ¯y τ ), Ia(τ ) ¯y Xkτ0 = X `∈Nd+1 k ` Xk−`Ia−`(τ )τ0+ Xk(Ia(τ )cy τ0) (3.16)
where σ ∈ S(PEV), τ ∈ TEV. Here τ0 is a tree with root decoration zero, and can therefore be identified as an element of S(PEV). The identities (3.16) separate the grafting at the root from the grafting on the other nodes. For the dual, we consider an inner product on the decorated trees h·, ·i such that for every σ, τ ∈TEV:
hσ, τ i = δσ,τS(τ )
where the algebraic symmetry factor S is defined inductively as follows: for any tree τ written as τ = Xk Qm
j=1Iaj(τj)
βj
, where we group terms (uniquely) in such a way that (ai, τi) 6= (aj, τj) for i 6= j, we inductively set
S(τ ) = k! m Y j=1 S(τj)βjβj! . (3.17)
The presence of k! can be seen as an internal hidden symmetry at each vertex, as if each vertex, seen under the microscope, behaves as a list of d + 1 packets of smaller points. We adopt the same definition on S(PEV) where the product
Q
is replaced by ˜
Q
. We extend naturally the inner product (3.17) to the spaceTEV ⊗TEV by setting for σ1, σ2, τ1, τ2 ∈TEV
hσ1⊗ σ2, τ1⊗ τ2i = hσ1, τ1ihσ2, τ2i.
In the proof below and also in the rest of the paper, we will use multinomial coefficients for k, `1, ...`n∈ Nd+1: k `1, ..., `n = k (`i)i∈{1,...,n} = k! (k −P i`i)! Q i(`i!) .
Let I be a finite set and let A be a commutative algebra. We consider an element τ = ⊗i∈Iτi in the tensor algebra ⊗i∈IA. For every partition π of I, we define the
Algebraic deformations of various products 19 productMπ by: Mπτ =O J∈π Y j∈J τj, τ ∈ ⊗i∈IA
where the productQis the product in A, and where J ∈ π means that J is a block of the partition π. We will also use this definition when the algebras in the tensor product are different. From the context, one can deduce the products used in every element of the partition.
Theorem 3.4 The product ?0is the dual of the deformed Butcher-Connes-Kreimer coproduct ∆DCK. One has
hτ1cy ¯τ2, ¯τ3i = hτ1⊗ ¯ˆ τ2, ∆DCKτ¯3i, hτ1y τ¯ 2, τ3i = hτ1⊗ τˆ 2, ¯∆DCKτ3i(3.18) where τ1 ∈ S(PEV), ¯τ2, ¯τ3 ∈PEV and τ2, τ3 ∈TEV.
Proof. First, one can easily check that for τ1, τ2 ∈ S(PEV), τ3∈ TEV
hXkτ1τ2, τ3i = hτ1⊗ Xkτ2, ∆τ3i (3.19)
where ∆ is the extension given in (3.11). Such an identity results from manipulating the definition of the symmetry factor S. We proceed by induction on the number of edges for proving (3.18). For τ1 = eQi∈IIai(σi) ∈ S(PEV) and τ2, τ3 ∈ TEV, one has
hτ1cyIb(τ2),Ib(τ3)i = hτ1y τ¯ 2, τ3i
Then by applying the induction hypothesis on ¯y, we get hτ1y τ¯ 2, τ3i = hτ1⊗ τˆ 2, ¯∆DCKτ3i
= hτ1⊗ˆ Ib(τ2), id ˆ⊗Ib ¯
∆DCKτ3i
= hτ1⊗ˆ Ib(τ2), ∆DCKIb(τ3)i
where we have used (3.14) in the last line. For τ2= Xkτ0 where τ0has zero node decoration at the root, one gets from (3.16)
f Y i∈IIai(σi) ¯y τ2 = X J⊂I X `i∈Nd+1, k (`i)i∈J Xk−Pi∈J`i Y i∈J Iai−`i(σi) ! f Y i∈I\JIai(σi)cy τ0
Then by using the recursive definition of the map∆¯DCKgiven in (3.14), one has
¯ ∆DCK X kY i∈I Iai(σi) ! =1 ˆ⊗ Xk Y i∈I (∆DCKIai(σi)
Algebraic deformations of various products 20 + X `i∈Nd+1 1 `i! Iai+`i(σi) ˆ⊗ X `i) =X J⊂I X `i∈Nd+1 f Y i∈J 1 `i! Iai+`i(σi) ˆ⊗ X k+P i∈J`i ∆DCK f Y i∈I\JIai(σi)
One can match in the dual the factors of the previous two identities by using (3.19). Indeed, one has
hτ1y τ¯ 2, τ3i = X J⊂I h X `i∈Nd+1, k (`i)i∈J Xk−Pi∈J`i Y i∈J Iai−`i(σi) ! ˆ ⊗ f Y i∈I\JIai(σi)cy τ0 , ∆τ3i =X J⊂I h Y i∈J Iai(σi) ˆ⊗ X k ! ˆ ⊗ f Y i∈I\JIai(σi) ˆ⊗ τ0 , ¯∆rootDCK⊗ ∆ˆ DCK∆τ3i =X J⊂I h Y i∈J Iai(σi) ˆ⊗ f Y i∈I\JIai(σi) ˆ⊗ X kτ 0 ! ,M(1)(3)(24) ∆¯rootDCK⊗ ∆ˆ DCK∆¯τ3i = h ∆Y i∈I Iai(σi) ˆ⊗ X kτ 0 ! ,M(1)(3)(24) ∆¯rootDCK⊗ ∆ˆ DCK∆¯τ3i = h Y i∈I Iai(σi) ˆ⊗ X kτ 0 ! ,M(13)(24) ∆¯rootDCK⊗ ∆ˆ DCK∆τ3i = hτ1⊗ τˆ 2, ¯∆DCKτ3i
where we have also used the identity:
M(13)(24) ∆¯rootDCK⊗ ∆ˆ DCK∆ = ¯∆DCK.
Moreover, we have applied the induction hypothesis oncy and used that
h X `i∈Nd+1, k (`i)i∈J Xk−Pi∈J`i Y i∈J Iai−`i(σi) ! , τ3i (3.20) = h fY i∈JIai(σi) ˆ⊗ X k, ¯∆root DCKτ3i.
For proving (3.20), one has to check that the combinatorial coefficients provided by the polynomials are the same:
k (`i)i∈J hXk−Pi∈J`i, Xk−Pi∈J`ii = k! (k −P i∈J`i)!Qi∈J`i! (k −X i∈J `i)! = Y i∈J 1 `i! ! k! = Y i∈J 1 `i! ! hXk, Xki.
Algebraic deformations of various products 21
3.3 Deformation of a plugging product
Suppose here that the set V of vertex decorations is equal to the commutative monoid Ω. We define a plugging operator onTEV as the following. For every σ, τ ∈TEV, we set
σ τ = X
v∈Nτ
σvτ
where σvτ is the tree obtained by the identification of the root of σ with the node v. Both decorations of the root of sigma σ and v are simply added in this process. Below, we give an example of this operation:
ω β α a b δ γ c = ω + δ γ β α c a b + δ ω + γ β α c b a (3.21)
This operation is different from (3.6) where only planted trees are considered with no decorations at the root. It also does not coincide with ya,ω defined on planted trees in (2.4). For later use we decompose the plugging product into
:= root+non−root, (3.22)
where σrootτ := σ%(τ )τ is the tree obtained by plugging at the root of τ .
Proposition 3.5 The spaceTEV endowed with is a pre-Lie algebra but not free. Proof. The proof of the pre-Lie relation is straightforward and left to the reader.
The non-freeness comes from the fact that any brace element (σ1· · · σn) τ is obtained by summing up all possibilities of merging the roots of the components σj with distinct vertices of τ . It therefore vanishes whenever n exceeds the number of vertices of τ .
In the sequel, we will use the shorthand notation ` = (`1, ..., `n) ∈ (Nd+1)nand |`| =P
i`i. On can define a deformed version of this plugging operator in the same spirit as before. But we graft simultaneously several edges. We set for a tree σ of the form XkQni=1Iai(σi)
σ τ :=b X v∈Nτ σbvτ (3.23) where σ bvτ is given by σbvτ := X `∈(Nd+1)n nv ` Xk n Y i=1 I(ai−`i)(σi ! v(↑−|`|v τ ) (3.24)
Algebraic deformations of various products 22
As an example, the deformation of (3.21) is given by
ω β α a b b δ γ c =X ` δ ` ω + δ− |`| γ β α c a− `1 b− `2 +X ` γ ` δ ω + γ− |`| β α c b− `2 a− `1 (3.25)
We decompose the previous bilinear operation as:
b =b root +b non−root (3.26)
where broot(resp. bnon−root) is defined the same way as b except that we consider only the root (resp. we sum over the nodes except the root).
Proposition 3.6 The spaceTEV endowed with is a pre-Lie algebra but not free.b
Proof. Let σ = XkQn
i=1Iai(σi), τ = X
¯ kQm
i=1I¯ai(τi) and w elements ofT
V E . We want to prove that
σ (τb w) − (σb τ)b w = τb (σb w) − (τb σ)b w.b (3.27) The only difficult part is when σ and τ are plugged on the same node v ∈ Nw decorated by nv. In terms of the binomial coefficient, we get from the left hand side of (3.27) X `∈(Nd+1)n X ¯ `∈(Nd+1)m nv ¯ ` nv− |¯`| + ¯k ` − X `,r∈(Nd+1)n X ¯ `∈(Nd+1)m nv ¯ `, r ¯ k `
On the right hand side, we get
X `∈(Nd+1)n X ¯ `∈(Nd+1)m nv ` nv− |`| + k ¯ ` − X `∈(Nd+1)n X ¯ `,¯r∈(Nd+1)m nv `, ¯r k ¯ ` .
We first notice that
nv ¯ `, r =n¯v ` nv− |¯`| r .
By using Chu-Vandermonde identity, we get
X `,r∈(Nd+1)n nv− |¯`| r ¯ k ` = X `∈(Nd+1)n nv− |¯`| + ¯k `
which allows us to conclude. The non-freeness argument is similar to the one invoked in the proof of Proposition 3.5.
Algebraic deformations of various products 23
Remark 3.7 One cannot repeat the proof given in theorem 2.7 because the product
is not free. Instead, we have used the Chu-Vandermonde identity.
We want to know if the plugging operator b can be derived using the Θ defined in Theorem (2.7) before and the operator. The answer is yes but it does not come from a direct transport of structure. Considering the transported plugging operator
e
defined as
Θ(σ τ) = Θ(σ) Θ(τ)e
we will see that it does not coincide with b. Let us first give an explicit expression of the linear isomorphism Θ:
Proposition 3.8 Let τ = XkQn i=1Iai(τi) ∈ T V E , one has: Θ(τ ) = X `∈(Nd+1)n k ` Xk−|`| n Y i=1 I(ai−`i)(Θ(τi)).
Proof. One has
τ = (τ1· · · τn) ya1···an Xk, hence we easily get
Θ(τ ) = (Θ(τ1) · · · Θ(τn))cy a1···anΘ(Xk) = (Θ(τ1) · · · Θ(τn))cy a1···anXk = X `∈(Nd+1)n k ` Xk−|`| n Y i=1 I(ai−`i) Θ(τi) by (3.8) and (3.7).
In the next proposition, we give an explicit expression of the product eroot defined by
Θ(σrootτ ) = Θ(σ)e root
Θ(τ ).
Similarly to (3.22) one has then:
e =e root +e non−root. Proposition 3.9 Let σ = XkQn i=1Iai(σi) and τ = X ¯ kQ¯n i=1I¯ai(τi) ∈T V E , one has σe root τ = X `∈(Nd+1)n X ¯ `∈(Nd+1)n¯ ¯ k ` k ¯ ` (3.28) Xk+¯k−|`|−|¯`| n Y i=1 I(ai−`i)(σi) ¯ n Y i=1 I(¯ai−¯`i)(τi).
Algebraic deformations of various products 24
Proof. The proof relies on the recursive definition of Θ given in Proposition 3.8.
Indeed, one has:
Θ(σrootτ ) =X `,¯` ¯ k + k `, ¯` Xk+¯k−|`|−|¯`| n Y i=1 I(ai−`i) Θ(σi) Yn¯ i=1 I(¯ai−¯`i)Θ(τi) .
On the other hand, from (3.28) one has
Θ(σ)e root Θ(τ ) = X `,¯`,m, ¯m k ` ¯ k ¯ ` k − |`| ¯ m ¯ k − |¯`| m Xk+¯k−|`|−|¯`|−|m|−| ¯m| n Y i=1 I(ai−`i−mi) Θ(σi) Yn¯ i=1 I(¯ai−¯`i− ¯mi)Θ(τi)
Now, we perform the change of variable ri = `i+ miand ¯ri = ¯`i+ ¯mi:
Θ(σ)e root Θ(τ ) = X `,¯`,r,¯r k ` ¯ k ¯ ` k − |`| ¯ r − ¯` ¯ k − |¯`| r − ` Xk+¯k−|r|−|¯r| n Y i=1 I(ai−ri) Θ(σi) Yn¯ i=1 I(¯ai−¯ri) Θ(τi)
We fix r and ¯r and by the Chu-Vandermonde identity, we get: X `,¯` k ` ¯ k ¯ ` k − |`| ¯ r − ¯` ¯ k − |¯`| r − ` =X `,¯` k ¯ r − ¯`, ` ¯ k ¯ `, r − ` = ¯ k + k ¯ r, r .
Then we conclude from the previous identity that: Θ(σ root τ ) = Θ(σ)e
root Θ(τ ).
Remark 3.10 One can derive a formula for σevτ when v is not the root of τ . It will be similar to (3.23) and it will be symmetric like eroot when it turns out to modify the edge decoration.
The next propostion reveals the main difference between b and which can bee observed at the root.
Proposition 3.11 The operators,,b satisfy the following properties:e σ rootτ = τrootσ, σe root τ = τe root σ, σb root τ 6= τb root σ
Proof. For root
, the property follows from its definition. Then Θ transports this structure to eroot. But with an explicit computation, one can check that σb
root
τ 6= τb root
Algebraic deformations of various products 25
Remark 3.12 The main consequence of Proposition 3.11 is thatb root
is not directly obtained from Θ which makes its construction more subtle. Proposition 3.13 shows that one needs to take into account insertion of polynomials after the plugging.
Proposition 3.13 Let σ, τ ∈TEV, one has for every v ∈ Nτ
σbvτ =↑ nσ
v Πσevτ
(3.29)
where Π sets to zero the node decoration equal to nσat the root of σ.
Proof. When σ has zero node decoration from (3.28), one has
σbvτ = σevτ Then, we conclude from (3.23) which gives
σbvτ =↑ nσ
v Πσbvτ.
Remark 3.14 The identity (3.29) allows us to write:
σbvτ =↑ nσ v Πσevτ =↑ nσ v Θ Θ−1(Πσ) v Θ−1(τ ) = Θ ˆ↑nvσ Θ−1(Πσ) v Θ−1(τ ) whereˆ↑ nσ v is given as in (2.18).
As for cy, we apply the Guin-Oudom construction to the pre-Lie product b on the spaceTEV. We now consider the symmetric algebra S(TEV) with its usual coproduct ∆ defined for every τ ∈TEV by:
∆τ = τ ⊗ 1 + 1 ⊗ τ. (3.30)
On S(TEV), we identify single node with no decorations as the empty forest. We obtain a product •2 on S(TEV) defined as in (3.12). One can notice that for x ∈ S(TEV), y ∈TEV:
x •2y = 0
when the number of trees in x is larger than the number of nodes in y. Then, we define the following product ?2as:
w ?2v = X
(w)
(w(1)•2v)w(2).
We denote by g2 the Lie algebra (TEV, [·]) where the bracket is given for σ, τ ∈TEV by:
Algebraic deformations of various products 26
Proposition 3.15 The space (S(TEV), ?2, ∆) is a Hopf algebra isomorphic to the enveloping algebra U(g2) where the Lie algebra g2 is TEV endowed with the antisymmetrization of the pre-Lie product.b
Let us recall from [2, Sec. 4.2] a symbolic notation for decorated forests with one distinguished tree. This notation was crucial in that paper for deriving a recursive formula for an extraction-contraction coproduct. Here it plays another important role in identifying the dual of the product ?2. The linear span TEV ⊗ S(TEV) of forests with one distinguished tree is denoted byS(ˆ TEV). The map
τ ⊗ τ1· · · τn7−→ τ τ1· · · τn
fromS(ˆ TEV) into S(TEV) is denoted by C. Adopting the notation τC(τ1) · · ·C(τn) for the left-hand side, the map Csatisfies
C(τC(τ1) · · · C(τn)) = C(τ )C(τ1) · · ·C(τn).
We endowS(ˆ TEV) with the following product: σ fY iC(σi) τ fY jC(τj) = στ fY iC(σi) f Y jC(τj)
where στ is now the tree product between σ and τ . We extend ∆ onS(ˆ TEV) by defining it on the distinguished tree as multiplicative for the tree product. Now for monomials (identified with one-vertex trees), one has:
∆X = X ⊗ 1 + 1 ⊗ X (3.31)
which gives rise to the following identity
∆Xk=X ` k ` X`⊗ Xk−`.
We consider the following maps ∆DP:T
V
E → ˆS(TEV) ˆ⊗TEV and∆¯DP :T
V E → S(TEV) ˆ⊗TEV defined recursively by:
∆DPX = X ˆ⊗ 1 + 1 ˆ⊗ X, ∆¯DPX = X ˆ⊗ 1, ∆¯DPIa(τ ) = C⊗ˆ Ia∆DPτ ∆DPIa(τ ) = ¯∆DPIa(τ ) + X `∈Nd+1 1 `!Ia+`(τ ) ˆ⊗ X `.
Then, they are extended multiplicatively.
Theorem 3.16 The product ?2 is the dual of the deformed coproduct ∆2 = (C⊗ id)∆DP. One has
hˆτ1⊗ τ2, ∆2τ3i = hˆτ1 τb 2, τ3i, hˆτ1⊗ τ2, ¯∆DPτ3i = hˆτ1b
non−rootτ 2, τ3i where ˆτ1 ∈ S(TEV) and τ2, τ3 ∈TEV.
Algebraic deformations of various products 27
Proof. The proof follows essentialy the same steps as in Theorem 3.4. First, one
can easily check that for τ1, τ2 ∈TEV, τ3∈TEV
hXkτ1τ2, τ3i = hXnτ1⊗ Xˆ n−kτ2, ∆τ3i
where here we view an element of TEV as an element of S(ˆ TEV) having one distinguished tree and an empty forest. For τ2 = Xkτ0 = XkQnj=1Iaj(σj) ∈T
V E and ˆτ1= eQi∈IC XkiQ j∈JiIaij(τij) ∈ S(TEV), one has ˆ τ1 τb 2 = X i∈I X `j∈Nd+1 k (`j)j∈Ji Xk− P j∈Ji`j+ki Y j∈Ji Iaij−`j(τij) f Y r∈I\{i}C Xkr Y j∈Jr Iarj(τrj) b non−rootYn j=1 Iaj(σj) + X kτˆ 1b non−rootτ 0
Then by using the recursive definition of the coproduct ∆DP, one has
∆DP XmY j∈J Iaj(τj) = ∆DPX mY j∈J ¯ ∆DPIaj(τj) + X `j∈Nd+1 1 `j! Iaj+`j(τj) ˆ⊗ X `j) =X I⊂J X k,`i∈Rd+1 m k Xm−kY i∈I Iai+`i(τi) `i! ˆ ⊗ Xk+Pi∈J`i ! Y i∈J\I ¯ ∆DPIai(τi)
Then the proof follows the same lines as in Theorem 3.4, where now one uses ∆ defined onTEV seperating the polynomial decorations, see (3.31). The only property missing is to match the combinatorial coefficient in the dual when we plug at the root similarly as (3.20). Indeed, one has
n +P i∈J`i (`i)i∈J hXn+k, Xn+ki =n + P i∈J`i (`i)i∈J (n + k)!
On the other hand, one gets: 1 Q i∈J`i! n + k k hXk⊗ Xn+Pi∈J`i, Xk⊗ Xn+Pi∈J`ii = k!(n + P i∈J`i)! Q i∈J`i! n + k k
Algebraic deformations of various products 28
We consider the unital algebra morphismK: S(TEV) → TEV given byK(1) = •0and K f Y iC(τi) =Y i τi
where τi∈ TEV and the product Q
iτiis the tree product. The mapKmerges all the trees into one by identifying the roots. The decorations at the roots are added to the new root. One easily checks that
∆2K= K⊗ˆ K∆2
Then the coproduct ∆2 : S(TEV) → S(TEV) ˆ⊗ S(TEV) can be viewed as a coproduct fromTEV intoTEV ⊗ˆ TEV and it will be recursively defined by:
∆2X = X ˆ⊗ 1 + 1 ˆ⊗ X, ∆2Ia(τ ) = id ˆ⊗Ia∆2τ + X `∈Nd+1 1 `!Ia+`(τ ) ˆ⊗ X `.
We still denote the product associated to ∆2 by ?2. The dual map K∗ of K corresponds to identifying all the ways to split a tree into blocks:
K∗ XkY i∈I Iai(τi) ! = X k1+···+kn=k X J1t···tJn−m=I f Ym j=1C(X kj) (3.32) f Yn `=m+1C Xk` Y j∈J`−m Iaj(τj)
We finish this section by establishing a connection between the two productscy and ?2.
Proposition 3.17 One has for σ = XkQ
i∈IIai(σi), τ ∈TEV Ib(σ ?2τ ) = ˜↑ k Nτ Y i∈I Iai(σi)cyIb(τ ) ! (3.33) where ˜↑kN τ is defined by ˜ ↑kNτ = X k=P v∈Nτkv ↑kv v . (3.34)
Proof. From the definition of ?2. one has:
Algebraic deformations of various products 29
Due to the property of the product b, one can have only |Nτ| blocks containing polynomials decorations at the root in (3.32). Therefore, one gets
σ ?2τ = ˜↑ k Nτ K ∗Y i∈I Iai(σi) ! b τ. Then, we can conclude with the following identity
Ib K∗Y i∈I Iai(σi) b τ ! =Y i∈I Iai(σi)cyIb(τ ).
Indeed,K∗groups branches that will be grafted onto the same node in τ .
Remark 3.18 We could have used the right hand side of (3.33) for defining the
product ?2. Then, it is not trivial to see that the product ?2is associative. Moreover, the identity (3.34) corresponds to the identificationKon the polynomials. We find natural to apply this splitting to trees.
Remark 3.19 When k = 0, one recovers the product ?0given in (3.13).
We finish this subsection by presenting an example of computation for ?2. We consider the following trees:
σ = δ γ β b c , τ = ω α a , K∗σ = X δ=δ1+δ2 (•δ1 δ2 γ β b c + δ1 β b δ2 γ c + · · · ) Therefore, σ ?2τ is given by σ ?2τ = X δ=δ1+δ2 (•δ1 δ2 γ β b c + δ1 β b δ2 γ c )b ω α a . (3.35) = X δ=δ1+δ2 X ` ω ` ω + δ2− |`| γ β α + δ1 c− `2 a b− `1 +X ` α ` ω + δ1 α + δ2− |`| γ β a c− `2 b− `1 +X ` ω `1 α `2 ω + δ1− `1 α + δ2− `2 γ β a c− `2 b− `1 +X ` α `1 ω `2 ω + δ2− `2 α + δ1− `1 β γ a b− `1 c− `2 .
Note that τ has two vertices, therefore one only keeps in (3.35) at most two blocks decomposition fromK∗σ.
Algebraic deformations of various products 30
3.4 Deformation of the insertion product
The insertion of a tree σ into another τ is defined by
σ I τ = X v∈Nτ σ Iv τ where σ Iv τ is given as σ Iv τ = (Pv(τ ) ? σ)vTv(τ ) (3.36) where
• ? is the product associated to constructed via the Guin-Oudom procedure and the identificationK.
• Pv(τ ) corresponds to the subtree attached to v in τ .
• Tv(τ ) is the trunk obtained by removing all the branches attached to v.
• The decoration nvat the node v is set to be zero in Tv(τ ) and it is put in the tree Pv(τ ).
Remark 3.20 The identity (3.36) is new in comparison to the litterature on the
insertion product, which was defined for the first time in [20]. The idea is to construct I directly from ? which now becomes the central object. On can then transfer properties of the grafting operator to the insertion as the pre-Lie property.
We provide an example of computation:
σ = ω α a , τ = % δ γ β d c b Pv(τ ) = δ γ β b c Tv(τ ) = % d .
where v is the vertex of τ decorated by δ. On gets Pv(τ ) ?2σ from (3.35). We proceed as in the previous section by deforming the product I. A good candidate is given by σI τ =b X v∈Nτ σIbvτ where σ bIvτ is defined as σIbvτ = (Pv(τ ) ?2σ)vTv(τ ) (3.37) The deformation is performed by replacing ? by ?2. The next proposition shows that as I, the product bI is a pre-Lie product.
Algebraic deformations of various products 31
Proof. The proof relies on the fact that ?2 is an associative product. One has from (3.37) ηIbuσ b Ivτ = Pv(τ ) ?2 ηIbuσ vTv(τ ). (3.38) Then Pv(τ ) ?2 ηIbuσ = Pv(τ ) ?2((Pu(σ) ?2η)uTu(σ)). On the other hand
ηIbu σIbvτ = (Pu σIbvτ ?2η)uTu σIbvτ
(3.39)
Then if we assume that u ∈ Nσ Pu σIbvτ = X Pv,1(τ )Pv,2(τ )=Pv(τ ) (Pv,1(τ ) ?2Pu(σ)) Tu σIbvτ = Tu(Pv(τ ) ?2s)vTv(τ ) = X Pv,1(τ )Pv,2(τ )=Pv(τ ) (Pv,2(τ ) ?u2 Tu(σ))vTv(τ )
where ?u2 behaves like ?2except that insertions on the node u are not allowed. The product Pv,1(τ )Pv,2(τ ) under the sum has to be understood as the tree product. This yields ηIbu σIbvτ = X Pv,1(τ )Pv,2(τ )=Pv(τ ) ((Pv,1(τ ) ?2Pu(σ)) ?2η) u((Pv,2(τ ) ?u2 Tu(σ))vTv(τ )) In order to conclude, we use the associativity of ?2
(Pv,1(τ ) ?2Pu(σ)) ?2η = (Pv,1(τ ) ?2(Pu(σ)) ?2η)
to match (3.38) with (3.39). If u /∈ Nσ then the insertions are disjoint and one has: ηIbu σIbvτ = σIbv ηIbuτ.
Remark 3.22 One can try to apply Θ and see how I is deformed. One has
Θ(σ Iv τ ) = Θ(Pv(τ ) ?2σ)vΘ(Tv(τ ))
The productvis not deformed because Θ(Tv(τ )) has zero decoration at the node v. Then by using
Algebraic deformations of various products 32
we obtain
Θ(σ Iv τ ) = (Pv(Θ(τ )) ˜?2Θ(σ))vTv(Θ(τ )) = Θ(σ)IevΘ(τ ) where ˜?2 is the product obtained from e and
σIevτ := (Pv(τ ) ˜?2σ)vTv(τ ).
The product bI is not directly obtained from a deformation of I by Θ. This is due to the construction of ?2 from ? which is also not only an application of the isomorphism Θ.
We denote by ?1the product obtained by applying the Guin-Oudom procedure on the product bI. We want to identify the dual map of ?1. A good candidate is ∆1 : S(TEV) → S(TEV) ˆ⊗ S(TEV) recursively defined as:
∆1 =M(13)(2) ∆◦⊗ id∆ˆ 2 (3.40) where ∆◦ :TEV → S(TEV) ˆ⊗TEV is defined multiplicatively for the tree product by ∆◦Xk= 1 ˆ⊗ Xk, ∆◦Ia(τ ) = id ˆ⊗Ia∆1τ. (3.41)
Remark 3.23 The recursive formulation (3.40) and (3.41) has been introduced in
[2]. It is a decomposition of the extraction-contration coproduct into two different actions: one is the extraction at the root given by ∆2, the other is the extraction outside the root which is iterated by the map ∆◦. This recursive construction can be generalised for designing renormalisation maps. One just needs to replace ∆2 by a linear map R :TEV → TEV which has some compatibility with ∆2that is:
(id ⊗ R)∆2 = ∆2R Then, one can define a renormalisation map
M = M◦R, M◦Xkτ = XkM◦τ, M◦Ia(τ ) =Ia(M τ ), (3.42)
where M◦is multiplicative for the tree product. It just corresponds to replace the extraction by another procedure wich is iterated deeper in the tree. These new renormalisation maps can be used either in Regularity Structures (see [2]) or Rough Paths (see [3]). This approach has also been used in [23] under the name of local product.
For proving the next theorem, we will use a recursive formula for the deformed insertion bI given for ˆτ1 = eQi∈Iτ1i∈ S(TEV), τ2, τ3 = XkQmj=1Iaj(τ3j) ∈ T
V E by ˆ τ1I τb 2 = ˆτ1Ib non−rootτ 2+ X j∈I f Y i6=jτ1iIb non−root(τ 1j?2τ2) τ2Ib non−rootXk m Y j=1 Iaj(τ3j) = m X i=1 XkIai(τ2Ib non−rootτ 3i) Y j6=i Iaj(τ3j).
Algebraic deformations of various products 33
Theorem 3.24 The product ?1is the dual of ∆1. One has
hˆτ1I τb 2, τ3i = hˆτ1⊗ τˆ 2, ∆1τ3i, hˆτ1Ib
non−rootτ
2, τ3i = hˆτ1⊗ τ2, ∆◦τ3i where ˆτ1 ∈ S(TEV), τ2, τ3 ∈TEV.
Proof. We proceed by induction on the size of the trees. From the identity (3.40),
one has for ˆτ1 = eQi∈Iτ¯1i∈ S(TEV), τ2, τ3 ∈TEV
hˆτ1⊗ τ2, ∆1τ3i = hˆτ1⊗ τ2,M(13)(2)(∆◦⊗ id)∆2τ3i
We notice that the dual map (M(13)(2))∗ofM(13)(2)is given by:
(M(13)(2))∗(ˆτ1⊗ τ2) = X j∈I f Y i6=jτ1i⊗ τ2⊗ τ1j + ˆτ1⊗ τ2⊗ 1.
Therefore by using the induction hypothesis, one gets
hˆτ1⊗ τ2,M(13)(2)(∆◦⊗ id)∆2τ3i = X j∈J D f Y i6=jτ1iIb non−root(τ 1j?2τ2), τ3 E + hˆτ1Ib non−rootτ 2, τ3i.
It remains to match the part given by ∆◦τ¯3. For τ3 = XkQmj=1Iaj(τ3j), we get
∆◦τ3 = (1 ⊗ Xk) m Y j=1
id ⊗Iaj∆1τ3j (3.43)
On the other hand,
ˆ τ1Ib non−rootXk m Y j=1 Iaj(τ3j) (3.44) = X I=tm j=1Jj Xk m Y j=1 Iaj f Y i∈Jj τ1iIb non−rootτ 3j
One can match the two identities (3.43) and (3.44) in order to conclude.
3.5 Cointeraction between deformed products
The classical cointeraction between grafting and insertion states that for τ1, τ2 ∈TEV and τ ∈ S(TEV):
X (τ )
τ(1) I τ1ya τ(2) I τ2 = τ I (τ1 yaτ2) (3.45) where Sweedler’s notation corresponds to the coproduct ∆ introduced in (3.30). This identity has been enounced at the Hopf algebraic level [16] and appears in
Algebraic deformations of various products 34
B-series [19]. It describes the interaction between multiplication and composition of B-series. The pre-Lie cointeraction (3.45) is extracted from [36]. One can also write this identity for the pre-Lie product y:
X (τ )
τ(1) Inon−root τ1y τ(2) I τ2 = τ I (τ1 y τ2) (3.46)
where this time τ1 ∈ PEV and Inon−root is the insertion everywhere except at the root. This identity is not true if one replaces y by the plugging product. But if we use the product ? with the identificationK, one gets:
X (τ )
τ(1) Inon−root τ1 ? τ(2) I τ2 = τ I (τ1? τ2) (3.47) We want to check that these indentities are preserved by the deformation. Applying the linear isomorphism Θ on (3.45), we get
Θ(τ I (τ1 yaτ2)) = Θ(τ )I(Θ(τe 1)cyaΘ(τ2))
which is not the desired identity. Indeed, Θ makes appear eI instead of I. Oneb cannot get a deformation of the cointeraction using Θ but we are able to get the following statement
Theorem 3.25 One has for τ1, τ2∈ TEV and τ ∈ S(TEV)
X (τ ) τ(1)Ib non−rootτ 1 c y τ(2)I τb 2 = τI (τb 1cy τ2) (3.48) X (τ ) τ(1)Ib non−rootτ 1 ?2 τ(2)I τb 2 = τI (τb 1?2τ2) (3.49)
Proof. We perform the proof using the identity (3.37) and with τ ∈TEV. It turns out that (3.49) can be rewritten as
τIb non−rootτ 1 ?2τ2+ τ1?2 τI τb 2 = τI (τb 1?2τ2) Then τI (τb 1?2τ2) = X v∈Nτ1\{%τ1} τIbv(τ1?2τ2) + X v∈Nτ2 τIbv(τ1?2τ2)
It is straightforward to check that
X v∈Nτ1\{%τ1} τIbv(τ1?2τ2) = X v∈Nτ1\{%τ1} τIbvτ1 ?2τ2
Applications 35 = τIb non−rootτ 1 ?2τ2.
For v ∈ Nτ2, one has
τIbv(τ1?2τ2) = (Pv(τ1?2τ2) ?2τ )vTv(τ1?2τ2) =X (τ1) τ1(1)?2Pv(τ2) ?2τ )v τ1(2)?v2Tv(τ2) = τ1?2 (Pv(τ2)) ?2τ )v(Tv(τ2)) = τ1?2 τIbvτ2
where we have used the identity:
τ1?2(τ2 v τ3) = X (τ1) τ1(1)?2τ2 v τ1(2)?v2τ3
with τi ∈TEV, v ∈ Nτ3 and τ3has zero node decoration at the node v. The proof
for (3.48) follows the same lines.
Remark 3.26 The proof of Theorem 3.25 does not depend on the deformation. It
is the same proof if one replaces ?2 by ? and bI by I. The proof exploits the fact that bI (resp. I) is defined in terms of ?2(resp. ?) and.
Remark 3.27 The cointeraction described in Theorem 3.25 has a counterpart for
the map M defined in (3.42):
M◦⊗ Mˆ ∆2= ∆2M
This formulation appears in [2] for defining more general renormalisation maps that could interact with the model given by the theory of Regularity Structures.
4 Applications
In this section, we will see that the various deformed products introduced in the previous section have many applications among (S)PDEs. The first application is about a numerical scheme for dispersive PDEs. The second application focuses on the main coproducts which are used within singular SPDEs. The deformed grafting insertion has its origin from that field.
4.1 Numerical Analysis
In [13], the authors derived a resonance based numerical schemes for equations of the type i∂tu(t, x) +L ∇,1 ε
u(t, x) = |∇|αp(u(t, x), u(t, x))
u(0, x) = v(x), (t, x) ∈ R+× Td
Applications 36
whereLis a differential operator and p is a monomial of the form p(u, ¯u) = uNu¯M. One can rewrite Duhamel’s formulation of (4.1) in Fourier space:
ˆ
uk(t) = eitP (k)vˆk− i|∇|α(k) eitP (k) Z t
0
e−iξP (k)pk(u(ξ), ¯u(ξ))dξ (4.2)
where P (k) denotes the differential operatorLin Fourier space
P (k) =L ∇,1 ε (k), pk(u(t), ¯u(t)) = X k=P iki−Pjk¯j N Y i=1 ˆ uki(t) M Y j=1 ¯ ˆ uk¯j(t).
The numerical scheme iterates this formulation and produces integrals that can be viewed as a character Π defined on a set of planted treesPEV.
The decorations encode the k, eitP and polynomials in ξ. Decorated trees are of the form:
I(o,m)(Xk`T ), T ∈ S(PEV)
where V = Nd+1× N and E = L × N. The finite set L collects different types and allows us to know if one has eitP orR0te−iξP · · · dξ (integrals in time correspond to a subset L+⊂ L). The second decoration on the edges given by N is associated to derivatives in time that one has to take in some Taylor expansions approximating the previous operators. The decoration on the node denoted by Xk`has to be understood as an inert decoration k ∈ Nd+1for the frequency k and a decoration ` ∈ N which corresponds to the polynomial in time coming from approximations. Then, one defines maps∆¯NA :T V E → TEV ⊗ S(ˆ PEV) and ∆NA: S(P V E) → S(PEV) ˆ⊗ S(PEV) such that ¯ ∆NA= X `⊗ 1ˆ ¯ ∆NAI(o,m)(X ` kT ) = I(o,m)(Xk`·) ˆ⊗ id ¯∆NAT + X n∈N Xn n! ⊗ˆ I(o,m+n)(X ` kT ) ∆NAI(o,m)(X ` kT ) = I(o,m)(Xk`·) ˆ⊗ id ¯∆NAT + 1 ˆ⊗I(o,m)(X ` kT ) (4.3) The identities (4.3) can be obtained by applyingM(2)(1)to (3.14):
¯ ∆NA=M (2)(1)∆¯ DCK, ∆NA=M (2)(1)∆ DCK. (4.4)
As a consequence of (4.4) and Theorem 3.4, we obtain one of our main results
Theorem 4.1 The map ∆NAis a deformation of the Butcher-Connes-Kreimer
co-product. It is obtained by applying the Guin-Oudom procedure oncy which is a
deformation of y given by the map Θ.
The maps used in [13] are slightly different from (4.3). Indeed, for numerical schemes, we define the order r of the scheme a priori. This produces projections
