The structure group for quasi-linear equations via universal enveloping algebras

We consider the approach of replacing trees by multi-indices as an index set of the abstract model space $\mathsf{T}$ introduced by Otto, Sauer, Smith and Weber to tackle quasi-linear singular SPDEs. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group $\mathsf{G}$. In particular, $\mathsf{G}\subset{\rm Aut}(\mathsf{T})$ arises from a Hopf algebra $\mathsf{T}^+$ and a comodule $\Delta\colon\mathsf{T}\rightarrow \mathsf{T}^+\otimes\mathsf{T}$. In fact, this approach, where the dual $\mathsf{T}^*$ of the abstract model space $\mathsf{T}$ naturally embeds into a formal power series algebra, allows to interpret $\mathsf{G}^*\subset{\rm Aut}(\mathsf{T}^*)$ as a Lie group arising from a Lie algebra $\mathsf{L} \subset{\rm End}(\mathsf{T}^*)$ consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (nonlinearities, functions of space-time mod constants). These actions are shift of space-time and tilt by space-time polynomials. The Hopf algebra $\mathsf{T}^+$ arises from a coordinate representation of the universal enveloping algebra ${\rm U}(\mathsf{L})$ of the Lie algebra $\mathsf{L}$. The coordinates are determined by an underlying pre-Lie algebra structure of the derived algebra of $\mathsf{L}$. Strong finiteness properties, which are enforced by gradedness and the restrictive definition of $\mathsf{T}$, allow for this purely algebraic construction of $\mathsf{G}$. We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the generalized parabolic Anderson model.

In this article, we connect the regularity structure (A, T, G) introduced by the second author, Sauer, Smith and Weber in [27] for a simple class of quasi-linear equations to the general Hopf-algebraic framework formulated by Hairer [16] and later expanded in [6,10,4]. The main difference between [27] on the one hand, and the output of the general strategy in [16] applied to this class of equations on the other hand, lies in the effectively smaller abstract model space T: The basis elements in [27] amount to specific linear combinations of the basis in [16], which is indexed by trees. Trees do not play any role in the contribution of this paper; thus, the Hopf algebras underlying rough paths [24,14] as worked out in [15,18], and regularity structures [16,6] are not at our disposal. The goal of this paper is to unveil this Hopf structure in the tree-free set-up of [27]; loosely speaking, this amounts to replacing combinatorics by Lie geometry. For an introduction to our framework, we refer to the series of lectures [26] and the notes [22]; for an introduction to classical regularity structures, we refer to the review article [17].
In our approach to the regularity structure (A, T, G), we start from the space of tuples (a, p) of (polynomial) nonlinearities a and space-time polynomials p, which we think of parameterizing the entire manifold of solutions 1 u. We consider the actions of shift by a space-time vector h ∈ R d+1 and of tilt by a space-time polynomial q on (a, p)-space, where, crucially, the tilt by a constant is encoded as a shift of the (one-dimensional) u-space because we think of p as p mod constants. We consider the infinitesimal generators of these actions, and pull them back as derivations on the algebra of formal power series R[[z k , z n ]] in the natural coordinates {z k } k≥0 and {z n } N d+1 0 \{0} of (a, p)-space 2 , which give rise to an index set of multi-indices. These derivations define a Lie algebra L; the corresponding Lie group coincides with the pointwise dual G * of the structure group G. However, we take a completely algebraic route to construct G, which passes via the universal envelope U(L) of L, and a module structure which identifies U(L) with a space of endomorphisms of T * , the algebraic dual of the model space T.
The algebraic construction of the present paper is similar in spirit to the recent work by Bruned and Manchon [8], who construct Hopf algebras starting from a (multi) pre-Lie algebra that encodes grafting of decorated trees, following the general theory developed by Guin and Oudom [28]. Also our Lie structure comes from a natural pre-Lie product on L, which however is not closed (see Subsection 3.8 for details); more recently, and motivated by the present paper, Bruned and Katsetsiadis [7] have interpreted this structure as a post-Lie algebra. Like in [28] we use it to canonically identify the enveloping algebra U(L) with the symmetric algebra S(L), which we implement through the choice of a specific basis 3 for U(L). This basis is crucial for recovering the intertwining property that relates the coproduct and coaction from regularity structures [17, (4.14)].
While in [27] the regularity structure (A, T, G) was introduced for quasilinear equations of the form the structures defined in this paper cover other (semi-linear) equations relying on a single scalar nonlinearity a(u). In fact, the algebraic structure we build is oblivious to the form of the equation and just relies on the solution to the linearized problem being of positive regularity, as will become apparent in Section 6 and, more importantly, Section 7. 4 In Section 6 we consider a driven ODE of the form (1.2) du dx 2 = a(u)ξ, provide a dictionary φ between our index set of multi-indices and linear combinations of trees in the Connes-Kreimer Hopf algebra (which is at the basis of branched rough paths), and prove that φ generates a Hopf algebra morphism. Section 7 is devoted to the stochastic heat equation (SHE) where now the morphism property is established with respect to the Hopf algebra in regularity structures [17]. While the morphism φ between our model space and the one based on decorated trees changes from one equation to another, the consistency between our geometric definition and the combinatorial definitions persists, and we expect it to hold as well for the class (1.1).
Working with our more parsimonious regularity structure (A, T, G) and model (Π x , Γ xy ) has the potential advantage of reducing the number of counter-terms in renormalization. In joint work with P. Tsatsoulis [23], we show that algebraic renormalization of (1.1) combines well with our greedier setting: we show that under a natural symmetry condition on the noise ξ, a BPHZ-type choice of renormalization can be performed, leading to a renormalized equation of the form ∂ ∂x 2 − ∂ 2 ∂x 2 1 u = a(u) ∂ 2 u ∂x 2 1 + h(u) + ξ with a deterministic and local counter-term h, as postulated in [27,Theorem 1], and -most importantly -we show that the greedy model (Π x , Γ xy ) can be naturally estimated without resorting to trees. We believe this to be a general principle, namely that multi-indices provide a more efficient bookkeeping of the renormalization constants; in Subsection 6.6 we show this for (1.2), connecting to translation of rough paths [5].
Our Ansatz has two invariances built-in, with beneficial effects for renormalization. The first invariance is the independence on the choice of an origin in u-space, which is ensured by the prominent role of the infinitesimal generator of u-shifts. The second invariance relates to the more specific class of quasi-linear equations (1.1). Namely, our theory is not affected by interpreting (1.1) as a perturbation around ( ∂ ∂x 2 − 2 ∂ 2 ∂x 2 1 )u = ξ, i. e. by 4 Our structure would also work, for example, for a generalized KPZ equation with only one nonlinearity, i. e. u + ξ. In other words, our approach is invariant under changing the reference value in a-space. Insisting on such (collective) in-or rather covariances in renormalization has been implemented in a much broader way in [21] 5 .

Motivation and interpretation of the main result
2.1. Modding out constants. We take the perspective Butcher [9] introduced on the level of ODEs, and which was extended in [15] to driven ODEs of the form 6 (1.2), of viewing the solution of the homogeneous initial value problem, i. e. with u(x 2 = 0) = 0, as a function(al) of the nonlinearity a, i. e. u = u[a](x 2 ). Obviously, the solutionũ for an (inhomogeneous) initial datum u 0 can then be recovered by a u-shift:ũ = u[a(· + u 0 )] + u 0 . (2.1) In particular, re-centering in the sense of imposing homogeneous initial conditions at some other time instance, say u 1 (x 2 = 1) = 0, can be recovered by a suitable variable u-shift π = π[a] in the form of the Ansatz u 1 [a] = u[a(· + π[a])] + π[a].
The extension to a driven PDE, e. g. (1.3), is more subtle, since even for fixed a, the solution manifold is infinite-dimensional. Relaxing the equation to hold only modulo space-time polynomials, one expects that the solution manifold can be (locally) parameterized by all space-time polynomials p. It is therefore natural to think in terms of u = u[a, p](x), like is implicitly done in 7 [4, p.879]. However, this is an over-parameterization in the sense that it does not take advantage of u-shifts, cf. (2.1). A key feature of our approach, which will be spelled out in the upcoming Subsections 2.2 and 2.3 is to consider p only modulo constants (and to keep track of u only modulo constants). In Subsection 3.4 we argue that this greedy approach to the regularity structure is actually truthful.
2.2. The (a, p)-space. At the basis of our construction is the space R[u] × (R[x 1 , x 2 ]/R), which is the set 8 of pairs (a, p), where a is a polynomial in a single variable u, and p is a polynomial in two variables (x 1 , x 2 ) = x. As indicated by the quotient, we consider p's only up to additive constants. Note that R[u]×(R[x 1 , x 2 ]/R) is the direct sum indexed by the disjoint union of N 0 and N 2 0 \ {0}; we often write {k ≥ 0} ∪ {n = 0}.
We recall that the polynomial a plays the role of the nonlinearity in case of the quasi-linear class (1.1), its argument u is a placeholder for the solution. The polynomial p plays the role of a (local) parameterization of the manifold of solutions; the values of u and p are thus thought to be in the same space, i. e. the real line, whereas the argument x of p is in space-time.

Actions of shift and tilt.
There are two natural actions on the (a, p)-space R[u]×(R[x 1 , x 2 ]/R), which we shall call "shift" and "tilt". We start by introducing the shift, by which we think of shifts of space x 1 and time x 2 . We seek an action 9 of the additive group R 2 ∋ h on (a, p)-space. We (momentarily) identify (2.2) which in particular allows to define the composition a • p ∈ R[x 1 , x 2 ] on (a, p)-space R[u] × (R[x 1 , x 2 ]/R). Then for h ∈ R 2 , the transformation (a, p) → a · +p(h) , p(· + h) − p(h) is well-defined. The action on the p-component is such that it corresponds to shift projected onto (2.2). The action on the a-component is made such that the composition a • p is mapped onto its (unprojected) shift x → (a • p)(x + h). Thus under the lens of a, this action corresponds to the plain shift by h. It is easy to check that (2.3) is indeed an action. The presence of the composition a • p connects to the Faà di Bruno formula, cf. [13], which expresses composition in terms of coefficients and thus encodes the chain rule. It explicitly enters in the exponential formula (5.16) via (5.17) and (A.1).
We now turn to tilt. By this we momentarily 10 think of an action on (a, p)space of the polynomial space R[x 1 , x 2 ] (now including the constants). It is defined by writing R[x 1 , x 2 ] ∋ q = n π (n) x n , where 11 x n = x n 1 1 x n 2 2 , and considering (a, p) → a · +π (0) , p + n =0 π (n) x n . (2.4) This treatment of the p-component ensures that the transformation (2.4) is well-defined in view of (2.2). The treatment of the a-component is such that the composition a • p is mapped onto a • (p + q) under (2.4). So once more, under the lens of a, this action corresponds to the tilt of p by q. Note that the addition of a polynomial is involved in recentering, by analogy with the addition of constants in the ODE case at the beginning of Subsection 2.1.

Seeking a representation as algebra endomorphisms.
We are interested in the group G of transformations on (a, p)-space generated by the two actions (2.3) and (2.4). We seek a representation of G as a matrix group, i. e. as a subgroup of Aut(T) for a suitable linear space T. The natural approach is to lift (2.3) and (2.4) to an action on a space 9 By action one means that addition in the group R 2 is compatible with composition of the transformations of (a, p)-space. 10 We need an extension later on. 11 With the implicit understanding that n ∈ N 2 0 if not stated otherwise.
of nonlinear functionals π on (a, p)-space by "pull-back". Indeed, it is tautological that (2.3) defines an algebra endomorphism Γ * of the algebra of functions π on (a, p)-space via For the moment, the notation Γ * is just suggestive; it will become meaningful when we identify this object with the dual of an element of G. The same remark applies to the forthcoming T * and G * .
This pull-back also suggests to naturally extend (2.6) from constant tilt π (n) ∈ R to variable tilt, meaning that π (n) itself is a function on (a, p)space: Note that also (2.5) has this form.
We use the notation π for a generic function on (a, p)-space, since it acts as a placeholder for the model Π = Π[a, p](x), which indeed can be considered as a parameterization of the solution manifold by p and depending on a (next to depending on space-time x).
While according to (2.8), the set of Γ * 's defined through (2.7), where π (n) runs through all functions on (a, p)-space, is closed under composition, there is in general no inverse. For this, we will have to pass to a more restricted space for the π (n) 's. Incidentally, while (2.5) is contained in (2.7) when π (n) runs through all functions on (a, p)-space, this will not be the case for the restricted space. 12 Here, Γ * , Γ ′ * andΓ * are defined through (2.7) by {π (n) } n , {π ′(n) } n and {π (n) } n , respectively. 13 We will provide a rigorous proof in the context of Proposition 5.1.

Seeking a matrix representation.
A reason for not only restricting the space of π (n) 's but also the one of π's in (2.7) is that the algebra of all functions on (a, p) is too large for a representation in terms of countably many coordinates. Let us therefore start from the following coordinates on (a, p)-space: (2.9) In (2.9) we use the standard abbreviation n! = n 1 ! n 2 ! and d n dx n = d n 1 dx n 1 1 . Note that {z k , z n } k,n can be considered as the dual basis to the standard monomial basis {u k , x n } k,n of (a, p)-space. In particular, these coordinates arbitrarily fix an origin of the affine u-space and the affine x-space. The effect of changing these origins is considered in Subsection 3.2. Clearly, every polynomial expression in (2.9) can be identified with a function on (a, p)-space. This allows us to identify the polynomial algebra R[z k , z n ] with a sub-algebra of the algebra of functions on (a, p)-space. Note that R[z k , z n ] is the direct sum over the index set of multi-indices 14 γ. In particular, the monomials form a countable basis of R[z k , z n ]. We will denote by e k and e n the multiindices such that z e k = z k and z en = z n , respectively.
However, R[z k , z n ] is not preserved by the Γ * defined through (2.6): Taking π (0) = v ∈ R \ {0} and π (n) = 0 for n = 0, and considering the function π = z 0 , we have Γ * π[a, p] = a(v). Now a(v) cannot be expressed as a finite linear combination of z γ 's. Actually, it follows from Taylor's formula that a(v) can be written as (2.11) so that the function Γ * z 0 can be identified with a formal power series in the variables (2.9), that is, Hence in coordinates, we a priori only know that (2.7) defines an algebra morphism from R[z k , z n ] into the larger R[[z k , z n ]]. Loosing the endomorphism property of course obscures the group structure. We thus seek an extension of the above Γ * 's to endomorphisms of R[[z k , z n ]]. This will require restricting Γ * to a (linear) subspace T * of R[[z k , z n ]], which amounts to the restriction of the space of π's mentioned at the beginning of this subsection.

Main result.
Our main results are: i) The goals outlined in Subsections 2.4, 2.5, and 2.6 can be achieved, provided we restrict to a suitable subspace T * ⊂ R[[z k , z n ]] and restrict the admissible π (n) 's. ii) The objects are dual to a regularity structure. iii) They arise from a natural Hopf algebra structure based on a pre-Lie algebra structure.
For arbitrary yet fixed α > 0 introduce the homogeneity 15 of a multi-index γ by and let 16 N 2 0 ∋ n → |n| ∈ R + be additive and satisfy |(1, 0)|, Then Γ * defined through (2.7) extends to an automorphism of T * , which respects the algebra structure of the ambient R[[z k , z n ]], cf. (5.19). The same holds true for the Γ * defined through (2.5). These two types of Γ * 's generate a group G * ⊂ End(T * ) consistent with (2.8). As a set, G * is parameterized by a shift h ∈ R 2 and a tilt {π (n) } n through an exponential formula 17 , cf. (5.16).
ii) There exists a linear space T of which T * is the algebraic dual; for every Γ * ∈ G * there exists a Γ ∈ End(T) of which Γ * is the dual, thereby defining a group G ⊂ End(T). Letting A := (αN 0 + N 0 ) \ {0}, the triple (A, T, G) forms a regularity structure.
iii) Consider the Lie algebra L ⊂ End(T * ) spanned by the infinitesimal generators of shift and tilt. Consider its universal enveloping algebra U(L) with its canonical algebra morphism U(L) → End(T * ). There exists a nondegenerate pairing between U(L) and a linear space T + such that the Hopf algebra structure on U(L) defines a Hopf algebra structure on T + . Likewise, the pairing allows to lift the action given through the algebra morphism U(L) → End(T * ) to a coaction ∆ : T → T + ⊗ T. In line with regularity structures, the group G ⊂ End(T) then arises from the Hopf algebra structure of T + together with ∆. The exponential formula arises from choosing a specific basis in U(L), which is based on a pre-Lie algebra structure on L.
This basis determines the pairing and ensures the intertwining of ∆ + and ∆ modulo the re-centering maps J n , cf. (4.49).
2.8. Outline of the paper. Section 3 introduces and motivates the main objects. More precisely, in Subsection 3.2, we will introduce the infinitesimal generators of shift . In Subsection 3.3, the polynomial sector 18T will be defined; in Subsection 3.6, we define the space T * ⊂ R[[z k , z n ]], which turns out to be dual to the abstract model space T. The corresponding mapping properties of {∂ i } i , {D (n) } n , and their respective transposed versions are characterized. In Subsection 3.4, we point out that the commutators of {∂ i } i and {D (n) } n behave in the same way shift and tilt operators would act on polynomials including the constants 19 . In Subsection 3.7, we extend from constant to variable tilt parameters π (n) , in form of monomials z γ , by introducing the infinitesimal generator z γ D (n) of variable tilt. In Subsection 3.8, we explore the natural pre-Lie algebra structure of the set of generators and a bigrading. In Subsection 3.9, the homogeneity |γ| of a multi-index γ and thus the set of homogeneities A and the ensuing grading of T will be introduced. In Subsection 3.10, we define the Lie algebra L as the subspace of End(T * ) spanned by While Section 3 is mostly about definitions and elementary properties, Section 4 states the main, partially technical, results that require a proof. In Subsection 4.1, we appeal to the general theory of Hopf algebras: We consider the universal enveloping algebra U(L) of the Lie algebra L, which is obtained from the tensor algebra factorized by the ideal generated by the Lie bracket, and which naturally is a Hopf algebra. Moreover, since L ⊂ End(T * ), there is a canonical algebra morphism ρ : U(L) → End(T * ) and the concatenation product on U(L) coincides with the composition in End(T * ). This action naturally defines a (left) module structure U(L) ⊗ T * → T * . In Subsection 4.2, the pre-Lie product of Subsection 3.8 is extended to an operation of {z γ D (n) } |γ|>|n| on U(L), cf. (4.6), which is shown to be consistent with the Hopf algebra structure. This operation will allow us, in Subsection 4.3, to select a basis that is natural, but different from the typical bases considered in the Poincaré-Birkhoff-Witt theorem, cf. (4.15). Such a basis also provides a non-degenerate pairing between U(L) and a space T + , see (4.42), which is introduced in Subsection 4.5. Under this pairing, the coproduct on U(L) turns into a product on T + that allows to identify T + with the polynomial algebra in variables indexed by the index set of L, cf. (4.19).
Next, we embark on the more subtle part of the dualization. This heavily relies on finiteness properties stated in (4.35) and (4.41), which in turn are an outcome of extending the bigrading of L to U(L); this is carried out in Subsection 4.4. In order to obtain these finiteness properties, it is 18 in the jargon of regularity structures 19 This subsection is logically not needed, but provides a key intuition. crucial to pass from R[[z k , z n ]] to T * . As a consequence, the action and the product of U(L), in terms of their coordinate representation with respect to our basis, turn into coaction and coproduct, respectively, for the couple T and T + , see Proposition 4.11. More precisely, the action U(L) ⊗ T * → T * gives rise to a coaction ∆ : T → T + ⊗ T, and the concatenation product U(L) ⊗ U(L) → U(L) gives rise to a coproduct ∆ + : T + → T + ⊗ T + . In particular, T + carries the structure of a (graded connected) Hopf algebra. In Subsection 4.6 we argue that ∆ and ∆ + intertwine as postulated by regularity structures.
Section 5 deals with the group structure and connects to the goals of Theorem 2.1. In Subsection 5.1, we apply general Hopf algebra theory to T + . This allows to endow the space of multiplicative linear forms Alg(T + , R) ⊂ (T + ) * with a group structure, with help of the (convolution) product coming from the coproduct ∆ + . Together with the coaction ∆, this gives rise to our G ⊂ End(T), establishing part iii) of Theorem 2.1. We also state that G is consistent with the requirements of regularity structures with respect to the gradedness of T, cf. (5.10), and the polynomial sectorT, cf. (5.11). This concludes part ii) of Theorem 2.1. In Subsection 5.2, we connect back to Section 2 by establishing part i) of Theorem 2.1. Namely, we show that the Γ * 's extend the definition (2.7) (Proposition 5.1 ii)), that they respect the algebra structure of R[[z k , z n ]] (Proposition 5.1 v)), and that they respect the group structure (2.8) (Proposition 5.1 vi)). In Subsection 5.3 we enlarge our regularity structure to meet exactly 20 Hairer's axioms, showing that our smaller structure contains all the relevant information.
At this stage, the reader may wonder why the passage from the Lie algebra L ⊂ End(T * ) to the corresponding group G * ⊂ Aut(T * ), which both live on the dual side, has to pass via the primal side in form of T and T + . The reason resides in our purely Hopf-algebraic approach, which prevents us from appealing to the matrix exponential in End(T * ) that analytically links the Lie algebra L to its (Lie) group G * , even if, as in our case, the exponential sum is effectively 21 finite because of gradedness. Indeed, the universal enveloping algebra U(L), as finite linear combinations of products of elements of L, is obviously too small to contain matrix exponentials of elements of L, even if T * were finite dimensional. First passing to T + , which as a linear space is isomorphic to U(L), and then to its algebraic dual (T + ) * , which as a linear space is much larger than U(L), is an algebraic way of extending U(L). It turns out to contain the matrix exponentials of 22 {D † | D ∈ L ⊂ End(T * )} ⊂ End(T), namely in form of Alg(T + , R) ⊂ (T + ) * as seen through the coaction ∆. Note that the primal G is more valuable than its dual G * , since one may always pass from Γ ∈ End(T) to Γ * ∈ End(T * ), while the opposite is only possible in finite dimensions. 20 not just up to the constant part, and including an abstract integration map 21 meaning that it is finite for a given matrix element 22 The elements D † are well defined by (3.3), since we have the stronger (3.42).
Sections 6 and 7 are logically independent of the rest of the paper, but connect the combinatorial structures to our Lie-geometric construction. More precisely, we make this connection in the well-studied cases of branched rough paths (1.2) and the stochastic heat equation (1.3).
In Section 6, we consider the driven ODE example (1.2). Here at a fixed a the solution manifold is parameterized by R, the space of initial conditions. This allows us to restrict the (a, p)-space to the space of nonlinearities a, hence the index set to multi-indices γ over {k ≥ 0}, and thus the Lie algebra to {z γ D (0) }. We show that the construction of Sections 3 and 4 is compatible with the Connes-Kreimer Hopf algebra underlying branched rough paths [15,18]. More specifically, we associate our multi-indices (2.10) with linear combinations of (undecorated) trees in the Connes-Kreimer framework via a map φ, and show that it is a Hopf algebra morphism. To do so, Lemma 6.2 establishes a pre-Lie algebra morphism property of the transpose of φ with respect to the grafting pre-Lie product (after a suitable normalization), which is at the core of the construction of Connes and Kreimer [12,20]. This morphism property is shown to be related to the one involving the map Υ in [2] in Subsection 6.5. In addition, in Subsection 6.6 we discuss how renormalization fits our setting and show that the transpose of φ intertwines with the translation maps of rough paths defined in [5], cf. Lemma 6.5.
In Section 7, we connect to tree-based regularity structures [16,17,4] for SHE. This example is simple in the sense that it only involves one integration kernel so that no edge decorations appear in the tree-based approach. We adopt the two perspectives in [4] when it comes to the treatment of the polynomial decorations. In Subsections 7.1 to 7.3, we consider a detailed description of the polynomials, in line with the space B in [4,Subsection 4.1], and show that our Lie algebra L reflects the grafting operations in [4,Definition 4.7]; as in Section 6, a pre-Lie morphism property of our dictionary is shown, connecting to the morphism properties of Υ in [4]. In Subsection 7.4, we relate to the coarser description which contracts all polynomials by multiplication giving rise to the model space T H ; the morphism φ between T and T H will no longer be one-to-one. In Proposition 7.5 we establish that φ induces a morphism Φ between T + and the Hopf algebra T + H in [17] (without appealing to a pre-Lie structure).
we may consider the components of the sequence Dz γ as a matrix representation {D γ β } β, γ . Since the D's we construct below will have the finiteness property we may write Dz γ = β D γ β z β . Moreover, they also will have the dual finiteness property This second finiteness property is just the one needed to have a unique D † ∈ End(R[[z k , z n ]] † ) such that (D † ) * = D; on the level of the matrix representation this just means Passing from the polynomial space R[z k , z n ] to the formal power series space R[[z k , z n ]] serves us well in the actual application, cf. [23,22], where we think of (2.9) as coordinates on the manifold of all solutions u. As a consequence, the general solution u can be seen as a function of the variables z k , z n (with values in the space of functions of x), or rather as a formal power series in these variables, described by its coefficients Π β indexed by our multi-indices β. Formally, Π β is, up to the combinatorial factor β!, a partial derivative of the general solution u w. r. t. the above variables. By Leibniz' rule, (1.1) gives rise to the following family of linear equations indexed by β: It turns out that this {Π β } β can be inductively rigorously constructed 24 , and thus naturally gives rise to the formal power series In particular, the algebra structure of R[[z k , z n ]], which will contain the dual T * of the abstract model space T, is inherent to our approach. This dual perspective is consistent with the definition of the model as a distribution with values in T * [17, Definition 3.3].
In the solution theory of regularity structures, one considers truncations of this formal power series as approximations of the actual solution; such truncations, in turn, shall be seen as (coherent) modelled distributions, in the language of [16,4], after the application of the model 25 Π.
We start with D (0) , which is to capture the action of R onto (a, p)-space by tilt by constants, which in view of (2.4) amounts to a shift of u-space Here, the action on the p-component, which is made such that a • p stays invariant, is immaterial because we "mod out" constants. As for the shift (2.3) of x, this action lifts by pull-back to functions π of (a, p). We formally define D (0) to be the infinitesimal generator of this action 27 This can be given sense for π ∈ {z k , z n }, cf. (2.9), and yields In addition, (3.7) suggests that D (0) is a derivation, which we postulate. This and (3.8) yield that, on the space R[z k , z n ], D (0) assumes the form the sum is obviously effectively finite on R[z k , z n ]. From (3.9) we infer the matrix representation with respect to the monomial basis: Note that the finiteness property (3.3) is satisfied so that D (0) is well defined as a derivation of R[[z k , z n ]]. We also see that the finiteness property (3.2) holds, so that (3.4) defines an endomorphism ( After this representation (3.10) of infinitesimal shifts of u, we turn to the shifts of space x 1 and time x 2 , that is, the action (2.3) of R 2 on (a, p)-space. Again, this action extends by pull-back to functions π on (a, p), see (2.5). We formally consider its infinitesimal generators 28 π a · +p(y 1 , 0) , p · +(y 1 , 0) − p(y 1 , 0) , π a · +p(0, y 2 ) , p · +(0, y 2 ) − p(0, y 2 ) . (3.11) 26 i.e. satisfying Dππ ′ = (Dπ)π ′ + πDπ ′ for π, π ′ ∈ R[[z k , z n ]]. 27 Here is yet another characterization of D (0) : For arbitrary u ∈ R consider π, π ′ ∈ R[[z k , z n ]] specified through π[a, p] = a(u) and π ′ [a, p] = da dv (u); they are related by π ′ = D (0) π. 28 For arbitrary x ∈ R 2 consider π, π ′ ∈ R[[z k , z n ]] characterized through π[a, p] = p(x) and π ′ [a, p] = ∂p ∂x1 (x); they are related by π ′ = ∂ 1 π.
By the chain rule and (3.8) for z k , and using the same argument (with p playing the role of a) that led to (3.8) for z n , we formally derive which we now postulate. Together with the postulate that ∂ 1 be a derivation, this implies that on the sub-algebra R[z k , z n ] we have The notation D (n) = ∂ zn is redundant, but very convenient; we obviously have the matrix representation Incidentally, still for n = 0, we have (3.14) which can be given a sense as an endomorphism on both R[[z k , z n ]] and R[z k , z n ].
We now have defined the building blocks, which are derivations on R[[z k , z n ]] (3.16) satisfying the finiteness properties (3.2) and (3.3).

The polynomial sectorT.
In view of the second item of (2.9), which identifies the coordinate z n = z en of R[[z k , z n ]] with the derivative 1 with R[x 1 , x 2 ]/R, the space of polynomials in the variables x 1 , x 2 quotiented by the constants. Following [17, Assumption 3.20], we callT the polynomial sector. We note that the transposed endomorphisms of (3.16) preserve this polynomial sector 29T which on the level of the matrix representation amounts to and can be inferred from (3.10), (3.13), and (3.15). We note that ∂ † 1 , ∂ † 2 almost act as partial derivatives on the polynomial sector 30T , which (in In terms of the matrix representation, this means which in turn can be read off from (3.15). The reason why the case n = (1, 0) (and analogously for ∂ 2 the case n = (0, 1)) is excluded is that we modded out constants in the polynomial p, cf. (2.2); see however the upcoming Subsection 3.4.

Commutators of {D
We now make a connection between {D (n) } n ∪ {∂ i } i and the classical Lie algebra of tilt and shift on polynomials. We start noting that which is obvious in case of n = 0 and n ′ = 0, and can be easily inferred from (3.9) for n = 0 and n ′ = 0. We next argue that (3.11) implies Indeed, by the finiteness property (3.2), the monomial z γ is mapped by ∂ 1 , ∂ 2 onto finite linear combinations of monomials. Hence we may indeed appeal to (3.11) when computing (∂ 1 ∂ 2 − ∂ 2 ∂ 1 )z γ , which shows that this expression vanishes by the symmetry of second derivatives. Turning to the commutator between D (n) 's and ∂ i 's, we first observe that by the characterization (3.12) and the commutation relation (3.21) we have [D (0) , ∂ 1 ] = 0 by the second item in (3.8). Likewise, for n = 0, we have [D (n) , ∂ 1 ] = n 1 D (n−(1,0)) (with the understanding that this expression vanishes if n 1 = 0) by the second item in (3.12). We retain that , when it comes to their commutators, precisely behave like certain endomorphisms on R[x 1 , x 2 ]. The important fact here is that this is the full space R[x 1 , x 2 ], not just the space R[x 1 , x 2 ]/R with the constants factored out, which was our starting point in Section 2. The corresponding endomorphisms on R[x 1 , x 2 ] are just the infinitesimal generators of shift and tilt. In particular, the subtle D (0) has a simple analogue in the infinitesimal generator of the "tilt" by a constant polynomial. This shows that the incorporation of constants into the a-part in (2.3) and (2.4) did not lead to a loss of information.
We now point out that the building blocks {D (n) } n ∪ {∂ i } i are strictly triangular with respect to the following two additive functionals on multiindices γ Here N 2 0 ∋ n → |n| ∈ N 0 denotes a scaled length of n. In applications to parabolic equations of the form (1.1), one considers |n| = n 1 + 2n 2 ; in general, |n| is an additive and coercive map which is determined by the scaling of the differential operator. We note that the combination of k≥0 kγ(k) and n =0 γ(n) in (3.24) is natural: Like we identified z en with p(x) = x n at the beginning of Subsection 3.3, we may identify z e k with a(u) = u k ; hence while k≥0 kγ(k) measures the homogeneity in the uvariable, n =0 γ(n) measures the homogeneity in the polynomial p; u and p-values have the same "physical" dimension. Considering the difference [γ] is forced upon us by the following. .

(3.27)
Note that (3.25) and (3.27) are of similar character: If the matrix element does not vanish, both functionals (3.24) are ordered and one of them is strictly ordered. However, (3.26) is of a different character, and we will get back to this in (3.43).
3.6. The abstract model space T. Now is a good moment to introduce the model space 32 T and its dual T * . We define T * ⊂ R[[z k , z n ]] to be the direct product over the multi-indices γ with This restriction of R[[z k , z n ]] to T * is motivated by the fact that the model component Π γ is only non-vanishing when [γ] ≥ 0 or γ ∈ {e n } n =0 . This can be read off from (3.5); the same holds for (1.2) and (1.3), see (6.3) and (7.2), respectively 33 . We denote byT * the subspace of elements of the dual space T * that vanish on the spaceT introduced in (3.17); in particularT * is the direct product over the multi-indices satisfying [γ] ≥ 0. Then T * =T * ⊕T * , whereT * is the direct product over the multi-indices γ ∈ {e n } n =0 . Thus we can identify the model space T with the direct sum of the polynomial sector T introduced in (3.17) and the spaceT spanned by all monomials z γ with [γ] ≥ 0. Since [γ] ≥ 0 is closed under addition of multi-indices, and thus under multiplication,T * is a sub-algebra. We note that the derivations (3.16) mapT * intoT * : which on the level of the coordinate representation means

The infinitesimal generators of variable tilt {z
After introducing the building blocks (3.16), we now specify the full collection of derivations on R[[z k , z n ]] that will act as the basis of L. Again, we start with a motivation: The purpose of the structure group G ⊂ End(T), or rather its pointwise dual G * ⊂ End(T * ), is to provide the transformations of the T * -valued model Π x when passing from one base-point x to another, see [17,Definition 3.3]. This re-centering involves subtracting a Taylor 32 It differs from a standard model space in regularity structures, see the discussion in Subsection 5.3.
33 Actually, such statements may be shown inductively under some uniqueness assumption for the corresponding equation, which guarantees that the only solution to the homogeneous problem is 0. This more analytic remark is outside the scope of this paper; we refer to [22,23] for a full argument.
polynomial. Denoting the coefficients of such a polynomial by {π (n) } n , and treating the constant part (i. e. the part with n = 0) differently in line with (3.6) and (2.3), this corresponds to the action (2.4).
In the inductive construction of theT * -valued centered model Π x , the coefficients π (n) depend on theT * -valued Π x itself, which for us means . We pass from π (n) ∈T * to a finite linear combination 34 of monomials z γ with [γ] ≥ 0. Hence on an infinitesimal level, in view of the characterization (3.7) of D (0) and the definition (3.12) of D (n) for n = 0, transformations of the type (2.4) give rise to the derivations SinceT * is closed under multiplication, it follows that (3.29) is preserved: However, even for [γ] ≥ 0, multiplication with z γ does not mapT * into T * . Luckily, the composition z γ D (n) does; we have Indeed, this is an immediate consequence of (3.9) and (3.12) together with (3.32).
On the level of the matrix representation, this implies that for all these operators, and not just for ∂ 1 and ∂ 2 , the finiteness property (3.3) holds. Moreover, when passing from D = D (n) to D = z γ D (n) , (3.19) is preserved so that (3.18) can be upgraded to as an affine space is flat, the Lie bracket [·, ·] arises from the pre-Lie product ⊲ that is given by the covariant derivative of one vector field along another vector field, see e. g. [25]; the relation between the bracket and the product is given by [D, and an analogous formula with ∂ 1 replaced by ∂ 2 . However, ∂ 1 ⊲ ∂ 1 cannot be expressed in terms of a linear combination of {∂ i } i ∪ {z γ D (n) } γ,n , so that the span of the latter is not closed under ⊲. Note that it is not possible to fix this by postulating ∂ 1 ⊲ ∂ 1 = 0, since then the (left) pre-Lie identity is not satisfied. 35 Nevertheless, it follows from (3.36) and (3.22) that the span , which will be used in Subsection 3.10.
The presence of a pre-Lie structure connects to the pre-Lie algebras in rough paths [5] and regularity structures [4]. Indeed, as we shall see in Section 6 in the specific case of driven ODEs, ⊲ is related to the grafting pre-Lie product (up to combinatorial factors, see Subsection 6.3 for a detailed discussion).
We now come to an important observation: There is a bigrading 36 on the index set {1, 2} ∪ {(γ, n) | [γ] ≥ 0, n = 0} of our (linearly independent) family of derivations that is compatible with the pre-Lie product ⊲. Indeed, we associate a pair of integers to every index by the following map bi:  By compatibility we mean that for any two elements D, D ′ of our family, provided not both are of the form ∂ i , the product D ⊲ D ′ is a linear combination of elements of our family that only correspond to indices such that their bigrading is the sum of the bigradings of the index for D and for D ′ . This is obvious for the second item in (3.36). Expanding the first item in (3.36) as and appealing to definition (3.37) we see that our claim amounts to and to (3.27). The second part of this implication also follows from Lemma 3.1.
Bigraded spaces appear in the context of regularity structures in [6]. In the tree-based setting, one chooses a bigrading [6, (2.4)] which encodes the size of the tree, on the one hand, and the decorations, on the other. The same guiding principle is present in (3.37): the quantity 1 + [γ] is the number of edges of the trees represented by the multi-index γ, whereas the second component is, roughly speaking, counting the polynomial decorations. We refer to Sections 6 and 7 for more details.
We now return to the strict triangular structure with respect to (3.24) and in particular the deficiency of (3.26). The choices we make now are guided by the application to the quasi-linear equation (1.1) with a driver ξ of regularity α − 2. Inspired by (3.37) we choose an α > 0 and define the homogeneity of a multi-index γ as where the normalization |0| = α, which destroys additivity, is made such that, in line with [17,Assumption 3.20], satisfies the assumptions of [17, Definition 3.1] of being bounded from below (namely by min{α, 1}) and locally finite. Note that in our setting, the entire index set of T has positive homogeneity, and only corresponds to the "integrated" part of the model; for a detailed connection to Hairer's set of homogeneities see Subsection 5.3.
Property (3.43) results in the following gradedness: For κ ∈ A let T κ ⊂ T denote the subspace corresponding to the indices γ with |γ| = κ; we obviously have in line with [17, Definition 3.1]. Then (3.43) can be reformulated as with the implicit understanding that κ, κ ′ ∈ A. We note that because of the presence of the z 0 -variable, and thus the γ(k = 0)-component on which |γ| is not coercive, T κ is not finite dimensional. However, in the practice of (1.1), this is of no concern since the model Π( , which a priori is a formal power series, actually is analytic in z 0 , which plays the role of a constant coefficient in where both sums are finite due to (3.2). By (3.30), we learn that because (3.43) they restrict to |β| > |γ| and |β ′ | > |γ ′ |. Moreover, by assumption (3.41) we have |γ| > |n| and |γ ′ | > |n ′ |; hence as desired, the sums in (3.46) involve only multi-indices with |β ′ | > |n ′ | and |β| > |n|.
It remains to show that L is bigraded; this is clear from (3.27) and (3.38), which show that the pre-Lie product (and thus the Lie bracket) is compatible with (3.37), together with the commutation relation [∂ 1 , ∂ 2 ] = 0.
4. The Hopf algebra structure 4.1. The universal enveloping algebra U(L), T * as a module over U(L). We now adopt a more abstract point of view and consider the elements of the Lie algebra L as mere symbols rather than endomorphisms, and we interpret (3.22), (3.46) and (3.47) as a coordinate representation of the Lie bracket in terms of the basis (3.45). We denote by U(L) the corresponding universal enveloping algebra [1, p. 28], an algebra which is based on the tensor algebra formed by L and quotiented through the ideal generated by the relations defining the Lie bracket. We may think of the tensor algebra as the direct sum indexed by words.
Due to the mapping properties (3.33), the canonical Lie algebra morphism ρ : L → End(T * ), which replaces every abstract symbol D ∈ L with its corresponding endomorphism, is well defined, and as a consequence of Subsection 3.10, ρ is a Lie algebra morphism. By the universality property [1, (U), p.29], such ρ extends in a unique way to an algebra morphism ρ : U(L) → End(T * ); in particular, concatenation of words turns into composition of endomorphisms. However, this representation is not faithful 37 . In a canonical way, we may rewrite ρ as a map U(L) ⊗ T * → T * , so that T * as a linear space becomes a left module over U(L).
The universal enveloping algebra U(L) is naturally a Hopf algebra, cf. [1, Examples 2.5, 2.8]; the product is given by the concatenation of words, whereas the coproduct is characterized by its action on the elements D ∈ L (which we call primitive elements), namely and in general by the compatibility with the product, meaning that for all U, U ′ ∈ U(L) The derived algebraL and the pre-Lie structure ⊲ revisited. As mentioned in Subsection 3.8, the Lie algebra L is not closed under the pre-Lie product ⊲. However, the only failure, namely SinceL is also an ideal, the quotient Lie algebra L/L is Abelian, see [19,Lemma 1.2.5], and thus is isomorphic to {∂ 1 , ∂ 2 }. Moreover, the [1, p. 29]. This algebra morphism in turn induces the decomposition By definition, U 0 is canonically isomorphic to U(L). SinceL is closed under ⊲, the pre-Lie structure provides a canonical isomorphism, as cocommutative coalgebras, between U(L) and the symmetric tensor algebra S(L), see [28,Theorem 2.12]. Via the definition (4.5), the pre-Lie structure ⊲ : L ×L → L provides a natural isomorphism between the linear spaces U m and U(L), as will become apparent in Subsection 4.3. These natural isomorphisms, of which we will make no explicit use, will guide our construction of a basis in Subsection 4.3.
We now will be more precise on how we salvage the pre-Lie structure ⊲. We use ⊲ in terms of the product The map (4.6) is inductively defined in the length of U by anchoring through z γ 1D (n) = z γ D (n) and postulating for any D ∈ L ⊂ U(L) Let us comment on (4.7): First of all, the identity (4.7) is consistent with the map ρ : U(L) → End(T * ) in the sense of since D as an element of End(T * ) is a derivation. As an identity in U(L), it is to be read as follows: On the l. h. s., we first multiply U by D via concatenation, and then apply (4.6). For the first r. h. s. term, we reverse this order. The second r. h. s. term is a linear combination of several versions of (4.6) (with γ replaced by β); the coefficients are given by identifying D ∈ L with D ∈ End(T * ), and (3.43) shows that (β, n) is in the index set of L. Hence (4.7) indeed provides an inductive definition of (4.6).
A first crucial observation is that the maps (4.6) commute 38 : Proof. We argue by induction. The base case of U = 1 follows from using (4.7) twice (once for D = z γ ′ D (n ′ ) and U = 1) and connecting the outcomes via (3.46). We now assume that (4.9) is satisfied for some U and give ourselves an element D ∈ L. Applying (4.7) twice, we obtain and the analogous expression in case of z γ z γ ′ DUD (n ′ ) D (n) ; by the induction hypothesis, both are equal.
A second crucial observation is that the maps (4.6) commute with the coproduct on U(L) in the following sense: Proof. Once more we argue by induction. The base case of U = 1 is included in (4.1) and our definition z γ 1D (n) = z γ D (n) . We now assume that (4.11) is satisfied for some U and give ourselves an element D ∈ L. Using the inductive definition (4.7) we may feed in the induction hypothesis, leading to which by the inductive definition (4.7) compactifies to Since by (4.1) and (4.2) the proof is complete.
A third crucial observation is that the maps (4.6) connect product and coproduct in the following sense: Under the assumption (4.10), Proof. Again we argue by induction. For U = 1 the identity follows by noting that cop 1 = 1 ⊗ 1. Assume now that (4.13) is satisfied for some U, then for D ∈ L by the induction hypothesis which by (4.7) leads to Using β D β β ′ (U (1) ) γ β = (DU (1) ) γ β ′ together with (4.12) finishes the proof.
A final observation is an intertwining of the maps (4.6) with ∂ 1 : and an analogous statement holds for ∂ 1 replaced by ∂ 2 .
We now assume that (4.14) is true for a given U and aim to prove it for DU, where D ∈ L. Then by (4.7) and appealing to the induction hypothesis yields which by a second application of (4.7) takes the desired form.
Here J denotes a multi-index on tuples (γ, n) with [γ] ≥ 0 and |γ| > |n|; we set J! := (γ,n) J(γ, n)!, so that the normalization constant J!m! may be seen as the multi-index factorial (J, m)!. This normalization is chosen such that the basis representation of the coproduct is standard, see (4.19) below. The collateral damage of this normalization is that the basis representation of (4.6) acquires a combinatorial factor: where J = e (γ 1 ,n 1 ) + ... + e (γ k ,n k ) and (γ 1 , n 1 ) ... (γ k , n k ) is a basis of U(L). Applying (4.7) iteratively, one can show the representation Since {B (J,m) } (J,m) is a basis, it is easy to deduce from this identity that also {D (J,m) } (J,m) is a basis.
The advantage of the basis (4.15) over a Poincaré-Birkhoff-Witt basis of the form (4.18) is that the former does not rely on the choice of an order in L, cf. (4.9), whereas the latter crucially does. The only choice to be made is the order of the three symbols: having first the z γ 's, then the ∂'s and last the D (n) 's generates the only basis for which the analogue of [17, (4.14)], namely (4.49), is true. In addition, with the basis (4.15) we obtain the most direct identification of our group elements as exponentials of shift and tilt parameters, cf. Proposition 5.1.
The coproduct has the following simple structure in the basis (4.15), which is reminiscent of the Hopf algebra of constant-coefficient differential operators over the algebra of smooth functions, cf. [3].
We now address the case J = 0. We take a pair (γ, n) such that J(γ, n) = 0 and use (4. Then by (4.11) and the induction hypothesis, By (4.15) and Lemma A.1, this as desired reduces to (J(γ, n) + 1) Recall that by construction, the concatenation product on U(L) is an abstract lifting of the composition product on End(T * ), as can be seen by applying ρ. The upcoming lemma is a projection of this fact ontoL, see This identity should be seen as the dual of the forthcoming intertwining relation of ∆ + and ∆ via J n , cf. (4.49).

(4.26)
We now note that (4.20) implies the following: In the case of J 2 = 0, and thus ε m D (J 2 ,m 2 ) = 0, once more we choose a pair (γ ′ , n ′ ) such that J 2 (γ ′ , n ′ ) = 0 and use (4.15) to write On the first r. h. s. term we apply once more (4.13) is a linear combination of basis elements (4.15) with strictly positive length, as may be seen by an iterative application of (4.13). We thus appeal to (4.27) to the effect of (4.28) . For the second r. h. s. term, we note that if (J ′ 2 , m ′ 2 ) = (0, 0) the sum is empty, so that (4.22) follows from (4.28). If (J ′ 2 , m ′ 2 ) = (0, 0) , the length is strictly smaller than that of D (J 2 ,m 2 ) , so that by the induction hypothesis the second r. h. s. term is given by Note that by (4.27), By definition of the algebra morphism ρ, this equals 2 ) D (n ′ ) = 0, this shows that (4.22) holds.

The bigrading revisited and finiteness properties.
Recall that L is a bigraded Lie algebra with respect to (3.37), and thus U(L) becomes a bigraded Hopf algebra. This means that there exists a decomposition U(L) = b∈N 0 ×Z U b such that the concatenation product maps Note that this decomposition is different from (4.4). It turns out that our basis elements (4.15) are homogeneous:  We adopt the same notation bi as in (3.37), without any risk of confusion.
Proof. Let us fix a pair (γ, n). We will show that this clearly proves the lemma, since D (J,m) is built starting from 1 m! ∂ m ∈ U (0,|m|) and applying (4.6) iteratively. We argue in favor of (4.31) by induction. The case U = 1 holds since by construction 1 ∈ U (0,0) . We now assume (4.31) to be true for a given U and give ourselves a D ∈ L such that D ∈ U b ′ . We express z γ DUD (n) using (4.7). Since DU ∈ U b+b ′ , our goal is to prove z γ DUD (n) ∈ U b+b ′ +bi(γ,n) . Indeed, for the first r. h. s. term of (4.7), by the induction hypothesis, For the second r. h. s. term, we note that by the compatibility of bi and ⊲, see Subsection 3.8, D γ β = 0 implies bi(β, n) = b ′ + bi(γ, n), which combined with the induction hypothesis yields With help of the bigrading (4.30) and the grading (4.32) we will now establish finiteness properties of the action and the product. For this, we first write the basis representations of both maps. It is tautological that the basis representation of the action U(L) ⊗ T * → T * with respect to (4.15) and the monomial "basis" 41 , i. e. where (D (J,m) ) γ β is the matrix representation of ρD (J,m) ∈ End(T * ). We choose the notation ∆ since it will give rise to a coaction, cf.    ∆ γ β (J,m) = 0 =⇒ |γ| < |β|. We stress that the restriction of γ is crucial in our approach, as will become apparent in the proof. Incidentally, by (3.33), it implies the same restriction for β.  The base case |(J, m)| = 0 is trivial, since this implies β = γ. In the induction step, we fix (J, m) and distinguish two cases: if J = 0, then the claim follows from (∂ m ) γ β = γ ′ (∂ m−(1,0) ) γ ′ β (∂ 1 ) γ γ ′ via (3.27) and the induction hypothesis in form of |n ′ |γ ′ (n ′ ) + (0, |m − (1, 0)|).
If J = 0, the claim likewise follows via (4.13), which we may use thanks to (4.19), into which we insert (3.38).
We shall now give a characterization of the basis representation of the concatenation product, i. e.  0), (0, 1)} and J = 0 or m = 0 and the multi-index J having just one non-trivial entry -equal to one -at (γ, n); for this, we write J = e (γ,n) . The former case is easy; indeed, by (4.7) and (4.14), we see that J = 0 implies J ′ = J ′′ = 0, which reduces all possible situations to formula (4.17). In particular, this yields   Proof. Once more by (4.2) and (4.19) it is enough to show (4.41) for |(J, m)| = 1, see the discussion after (4.39). The case J = 0 and |m| = 1 is trivial from (4.40). For |J| = 1 and m = 0, we write J = e (β,n) and claim that (4.41) follows from (4.22). For this, we apply (4.22) with U 1 = D (J ′ ,m ′ ) and U 2 = D (J ′′ ,m ′′ ) , and consider the coefficient of the z β -term, which is nonvanishing by assumption. By (4.27), the first r. h. s. term in (4.22) is nonvanishing only if U 2 = z γ D (n) for some γ; applying (4.35) to the first factor ρU 1 , we see that there are only finitely many γ's and (J ′ , m ′ )'s which give a non-vanishing contribution to the z β -coefficient. Turning to the second r. h. s. term of (4.22), its z β -coefficient is non-zero unless U 1 = z β D (n+m) and U 2 = D (0,m) for some m. The constraint |n + m| < |β| only allows for finitely many m's. (J ′ ,m ′ ) . As a consequence of (4.42), U(L) canonically is a subspace of (T + ) * ; note that the latter is much larger, since it is the direct product over the index set of all (J, m)'s, whereas U(L) is just the direct sum.
Our next goal is to provide a structure for T + by dualization of the Hopf algebra and the module structures of U(L). First, we note that the basis representation of a coproduct has the algebraic properties of a product, and thus (4.19) defines a product 43 · in T + given by This way (T + , ·) becomes the (commutative) polynomial algebra over variables indexed by the index set of L.
In a similar way, we want to transpose the action and the coproduct mentioned in the previous subsection. The transposition in these two cases is possible thanks to the finiteness properties which were stated in Lemmas 4.9 and 4.10. Starting with the action, analogously to (3.4), from the basis representation (4.34) we define a map ∆ : T → T + ⊗ T by The sum is finite due to (4.35), and hence ∆ is well-defined. We stress that the restriction to T is crucial for our argument; it does not seem possible We now turn to the product; by the basis representation (4.39), we define a map ∆ + : Such a map has the algebraic properties of a coproduct in T + . The fact that this map is well-defined is a consequence of the finiteness property (4.41).
The only missing ingredient to make T + a Hopf algebra is an antipode 44 S: since (4.32) and (4.33) make T + a connected 45 graded bialgebra, this is guaranteed by general theory, see [19, Proposition 3.8.8].
These observations are collected in the following result.  Note that (4.46) is the dualization of the morphism property of ρ.
We now have introduced all the objects required to construct a structure group G ⊂ End(T) according to [17,Section 4.2]. A minor difference is that, in our case, (T, ∆) is a left comodule while in [17, (4.15)] it is a right comodule, a fact that transfers to (4.49) and more upcoming identities. This does not affect the construction. In fact, with a similar (though more cumbersome) definition of the Lie algebra L, working at the level of the transposed endomorphisms from the beginning, we would have been able to recover the same structure, but paying the price of blurring the connection to the actions on (a, p)-space that served as a motivation in Section 2.

4.6.
Intertwining of ∆ and ∆ + through J n . Let us define for every n a map J n : T → T + in coordinates by Note that in view of (4.20) J n is the transposition of ι n up to a combinatorial factor: The normalization with n! is made such that the dualization of (4.22) takes the form of the following intertwining relation between the coaction ∆ and the coproduct ∆ + , which is an identity in T + ⊗ T + : Indeed, (4.49) amounts to (4.22) once tested with U 1 ⊗ U 2 , where we use the pairings (3.1) and (4.42), the definitions (4.39) and (4.34), and the fact that U, Z (0,m) = ε m U in view of (4.21). Combined with (4.50) ∆ + Z (0,(1,0)) = Z (0,(1,0)) ⊗ 1 + 1 ⊗ Z (0,(1,0)) , which follows from (4.40), we see that ∆ + is determined by ∆ through J n in agreement with regularity structures, cf. [17, (4.14)]. Let us also mention that the coaction applied to the polynomial sectorT is in agreement 46 with [17, p. 23], This may be seen by (4.44) and (4.34). First note that ρD (J,n ′ ) preserves T * , as a consequence of the same property of L, see (3.32), and that ρD (J,n ′ ) maps toT * only if J = 0 as can be read off (4.15). Therefore, (4.51) follows from D (0,n ′ ) z n ′′ = n ′ +n ′′ n ′ z n ′ +n ′′ , which is a consequence of (4.15) and (3.12). for f ∈ (T + ) * , by (4.43) the space Alg(T + , R) is characterized by Due to this property, the elements f ∈ Alg(T + , R) are parameterized by 47 where h ∈ R 2 and {π (n) } n ⊂T * is constrained by From the Hopf algebra structure of T + , the space Alg(T + , R) inherits a natural group structure, namely the convolution product of functionals: The neutral element e of this group, which is the counit of T + , maps Z (0,0) to 1 and every other basis element of T + to 0, and the inverse elements are given by f −1 = f S, cf. [1, Theorem 2.1.5].
Following [17,Subsection 4.2], we now define a map Γ : (T + ) * → End(T) by Then the set Applying definition (5.6) to z β , plugging in (4.44), (5.1) and (4.34), we obtain from (3.4) the representation 48 and note that this sum is effectively finite because of (4.35). Moreover, as a consequence of (4.36), this may be rewritten more in line with the corresponding requirement in [17, Definition 3.1]: The elements of the structure group behave nicely with the polynomial sectorT, see Subsection 3.3, in the sense that for f ∈ Alg(T + , R) with h ∈ R 2 being its parameter according to (5.3), and for all n = 0, note that the order in the composition rule is reversed as a consequence of transposition.
We gather all our results in the following proposition.
Proof. We first show (5.19); indeed, it is a direct consequence of (5.2) and the following generalized Leibniz rule: For all π 1 , ..., π l ∈ R[[z k , z n ]] where the sum runs over all (J 1 , m 1 ), ..., (J l , m l ) with (J 1 , m 1 ) + . . . + (J l , m l ) = (J, m). It is easy to see that by induction, (5.22) for general l ∈ N follows from the case l = 2. In view of (4.19), this case reduces to for U ∈ U(L), where we do not distinguish between ρU ∈ End(T * ) and U. Formula (5.23) is trivial for U = 1; it is obvious for U ∈ L ⊂ Der(T * ). It remains to pass from U to UD for some D ∈ L. For the l. h. s. of (5.23) we note that by induction hypothesis (and base case) we have For the r. h. s. of (5.23) we have by the compatibility of the coproduct with concatenation (composition) and (4.1) We now turn to the proof of (5.16). We first argue that the r. h. s. of (5.16), when interpreted as an endomorphism of T * , is effectively finite (note that we already know that the l. h. s. is effectively finite from (4.35)). For this, we note that the r. h. s. of (5.16) is an infinite sum of terms of the form where either [γ i ] ≥ 0 or γ i ∈ {e n } n =0 for i = 1, ..., k. We extend the family of derivationsL by incorporating purely polynomial multi-indices, so that we consider the set {z γ ′ D (n) } [γ ′ ]≥0,|γ ′ |>|n| ∪{z m D (n) } m>n . It can be easily checked that this family is closed under the standard pre-Lie product ⊲ given by the first item in (3.36): For the mixed terms this follows from (z m D (0) )z γ = k≥0 (k + 1)γ(k)z γ−e k +e k+1 +em with |γ − e k + e k+1 + e m | = |γ| + α + |m| > |γ|, cf. (3.10); from (z m D (n) )z γ ′ = γ ′ (n)z em+γ−en with |e m + γ − e n | = |γ ′ | + |m| − |n| > |γ| > |n ′ |, cf. (3.12); and from (z γ ′ D (n ′ ) )z m = δ n ′ m z γ ′ . With this extension at hand, we follow the strategy of Lemma 4.9. For this we need the two following properties: • extension of (3.38) to purely polynomial multi-indices, i. e.
Hence, to show (5.16) it is enough to apply both sides to a monomial z γ with [γ] ≥ 0 or γ ∈ {e n } n =0 . Note that both sides are multiplicative 49 , so it is enough to consider γ's of length one. Thus, in view of (3.9) and (3.12), showing (5.16) amounts to showing (5.17) and (5.18). We start with (5.17). Using (5.12) and applying (A.1) (with the roles of l and k flipped), we see that it is enough to show that Here we interpret ι 0 + n =0 ε n ⊗ z n as a linear map from U(L), a space endowed with coproduct, into the algebra R[[z k , z n ]], so that powers make sense. By the binomial formula 50 applied to (ι 0 + n =0 ε n ⊗z n ) k and (4.19), the l. h. s. equals (ε n D (J k ,m k ) )z n .

By (5.3), (4.20) and (4.21), this equals
which by the binomial formula and (5.14) is seen to coincide with (π (0) ) k . To show (5.18), we appeal to (5.12) to see Indeed, this holds for any effectively finite expression of the form of the r. h. s. of (5.16) with D (n) 's being commuting derivations and π (n) 's being multiplication operators. 50 Note that the coproduct in U(L) is co-commutative, see (4.19), therefore the product of linear maps from U(L) to any commutative algebra is Abelian. since D (J,m) with J = 0 and m = 0 would annihilate z n , cf. (4.15). The first sum has contributions only from J = e (γ,n) , therefore by (3.12) and which together with (5.14) yields (5.18).
For k ≥ 0 we apply (5.17) twice and use multiplicativity (5.19) to obtain A re-summation together with the binomial formula and applying once more (5.17) yield which finishes the proof of (2.8).
We now argue in favor of v). By (5.13), we have to show that for f of the form the product (5.5) amounts to addition of h ∈ R 2 . Indeed, and we conclude by the binomial formula. For f of the form (5.24), and using (2.9), (5.14) assumes the form n π (n) [a, p]x n = p(x + h), so that (5.20) turns into (5.21).
We finally turn to the proof of vi). By (5.13), it suffices to show that the set of f ∈ Alg(T + , R) such that f (J,m) = 0 for m = 0 is a subgroup; this is a direct consequence of U(L), cf. (4.3), being a sub-Hopf algebra of U(L).
In this subsection, while keeping G as an abstract group, we enlarge the abstract model space T on which it acts. We do so in order to draw a closer connection to [17]. We will proceed in two steps, first enlarging the abstract model space by a placeholder for the omitted constants, and then by a placeholder for the right-hand side of the equation.
While thinking of p only modulo constants was an important guiding principle in uncovering the algebraic structure, see Section 2, we will now reintroduce constants into the polynomial sectorT, cf. Subsection 3.3, by augmenting its basis {x n } n =0 by the element x 0 . As a consequence, we pass from T to R ⊕ T. We now argue that the action of G naturally extends to R ⊕ T, where we first adopt the point of view of Subsection 5.1: Indeed, given h ∈ R d and {π (n) } n ⊂T * , which gives rise to Γ ∈ End(T), the extension to an endomorphism of R ⊕ T is visualized by the block structure 51 This form of extension completes the action (5.11) on the (extended) polynomial sector R ⊕T in the sense of [17,Assumption 3.20]: Since an element ofT * , like π (0) , is characterized by vanishing onT, (5.25) maps the basis element x n onto m n n h m x n−m , which formally can be written as (x + h) n . We will motivate the presence of π (0) in (5.25) below.
For (5.25) to define an action, we need to check that the composition of two endomorphisms of the form of (5.25) preserves this form. This is more easily seen for the induced dual action on (R ⊕ T) * ∼ = R ⊕ T * , which is of the block form id 0 It is now convenient to adopt the point of view of Subsection 5.2, which amounts to viewing the lower left entry of (5.26) as a single object, as done in (5.14) and (5.15), still labelled by π (0) . The desired statement then follows from (2.8), which was rigorously established in part iii) of Proposition 5.1. It is also on the level of this extended definition (5.14) of π (0) that we may motivate (5.26): The purpose of G * is to contain elements Γ * xy that "algebrize" the re-centering of the model, which is a T * -valued function 52 of space-time, from its version Π y centered at one base point y to its version Π x centered at another base point x, see [17,Definition 3.3]. In the application 51 the sum in the upper right entry is effectively finite and thus defines an element of T * , in line with the meaning of this block 52 or distribution, depending on the application [23, (2.41)] of our setting, this holds only up to a space-time constant π (0) xy ∈ T * , that is, Π x = Γ * xy Π y + π (0) xy , (5.27) which however is tied to Γ * xy via (5.17). Now (5.26), with Γ * and π (0) specified to Γ * xy and π x as aT *valued Schwartz distribution; recall thatT * is canonically characterized as the space of all linear functionals π ∈ T * that vanish on the x n 's. In order to capture this on the level of our abstract model space, or rather its dual, we pass from (R ⊕ T) * to (R ⊕ T) * ⊕T * .
The motivation for the extension (5.28) is again given by the application [23, (2.40)] of our abstract structure. Indeed, the new model components transform according to Γ * xy up to aT * -valued (formal) power series in spacetime where again the coefficients are tied to Γ * xy via (5.18); note that the terms m = (0, 0), (1, 0) do not contribute. We note that by [23, (2. here we denote by z the active variable of Π x Hence we see that (5.29) assumes the axiomatic form [17,Definition 3.3], which is free of polynomial corrections, under the extension (5.28).
In order to establish that (5.28) provides indeed a representation of G, we now argue that it is compatible with composition. By the compatibility of (5.26), and of the bottom right block of (5.28), we are left with the bottom left block of the product, which consists of the summands where (5.25) ′ stands for (5.25) with Γ and π (0) replaced by Γ ′ and π ′ (0) , respectively. We note that p m := (∂ 2 − ∂ 2 1 ) † x m is an element of the (extended) polynomial sector R ⊕T; hence applying (5.25) ′ to it, we obtain 54 p m (· + h ′ ), as we have shown above. Hence (5.30) assumes the form We need to re-express (5.31) in terms of the extended definition (5.14) in order to (eventually) apply (2.8). To this purpose we introduce which since {z n } n =0 is dual to {x n } n =0 defines a projection P from T * ontoT * that allows to pass from the extended definition (5.14) of π (m) to the original one used in (5.31). Clearly, the second summand in (5.31) calls for the commutator of Γ * and P , of which we need a representation: Using once more that Γ * preservesT * , which can be written as Γ * P = P (Γ * − Γ * (id − P )), we obtain by (5.18) In view of (5.31), we apply this to π ′ (m) , which by (5.15) yields Hence in terms of the extended definition (5.14), (5.31) assumes the form According to (2.8), the term in line (5.32) is the desired output. Hence it remains to argue that the term in (5.33) vanishes when summed over m, which by resummation and relabelling amounts to Recalling that p n = (∂ 2 − ∂ 2 1 ) † x n , and by (3.20), this follows from the same formula with p n replaced by x n , which amounts to Leibniz' rule.
We note thatT ⊂ T is canonically defined as consisting of those elements that are annihilated by {z n } n =0 ⊂ T * ; it is a natural complement ofT in T. Moreover, (R ⊕ T) * ⊕T * is the dual of (R ⊕ T) ⊕T. Passing to this primal side (R ⊕ T) ⊕T, (5.28) turns into since the bottom r. h. s. of (5.28) can be rewritten as Γ * P , and since P † defined through is the dual of P , and a projection from T ontoT. Hairer's integration map [17,Assumption 3.21] has the block form where ι is the injectionT ⊂ T. By definition (5.25), the commutator ΓI − I Γ has the block form so that by (5.35), the image of ΓI − I Γ is contained inT, in line with the axiom [17, (3.9)].
We close this subsection by arguing, in line with the axiom [17, Definition 3.1], that the matrix representation of Γ−id is strictly triangular, provided we extend the basis {z β } {[β]≥0}∪{β=en} of T to a basis of R ⊕ T ⊕T, and extend the homogeneity of its index set as follows: On the R-component, we take the dual basis vector to x 0 , and endow this single index, which we (momentarily) denote by 0, with homogeneity 0; on theT-component, we take the basis {z β } [β]≥0 , endowing the index β with the homogeneity |β|−2.

Homomorphism to the Connes-Kreimer Hopf algebra
In this logically independent and rather combinatorial section, we specify to the particularly simple case of scalar 55 branched rough paths. We will argue that our coproduct ∆ + on T + , when suitable restricted, arises from the Connes-Kreimer coproduct. More precisely, there is a linear subspace T RP ⊂ T and a pre-Lie subalgebra L RP ⊂ L (which as a linear space is isomorphic to T RP ) of derivations D (which are such that D † preserves T RP ) such that the corresponding restriction of ∆ + intertwines with the Connes-Kreimer coproduct on forests, see Subsection 6.4. The intertwining is provided by the linear one-to-one map φ that relates our model, which is indexed by multi-indices, to branched rough paths indexed by trees, see Subsection 6.2.
6.1. Relating the model Π to branched rough paths. Since it does not affect the algebraic insight of this section, we consider a qualitatively smooth driver ξ to avoid renormalization. Following our initial discussion in Subsection 2.1, we consider the solution of the initial value problem du dx 2 = a(u)ξ, u(x 2 = 0) = 0 (6.1) as a functional u = u[a](x 2 ) of the (polynomial) nonlinearity a. It lifts to a function of the coordinates {z k } k≥0 introduced in (2.9). Hence we may take derivatives with respect to these coordinates evaluated at z k = 0; these partial derivatives are indexed by multi-indices 56 β. It is easy to (formally) verify that the resulting partial derivatives Π β satisfy with the understanding that Π e 0 (x 2 ) = x 2 0 ξ. These components combine to the centered 57 model Π = {Π β } β . Incidentally, interpreting (6.2) can be compactly written as dΠ dx 2 = k≥0 z k Π k ξ. While this derivation of (6.2) is formal, Π = {Π β } β can be, inductively in the length of β, constructed rigorously for sufficiently regular ξ. 55 i. e. in a one-dimensional state space 56 which are maps N 0 ∋ k → β(k) ∈ N 0 with finitely many non-zero values 57 Centered at time x 2 = 0, which however we suppress in our notation.
Based on (6.2) we may read off that not all the multi-indices are populated. More precisely, we claim that Π β = 0 implies We establish (6.3) in its negated form by induction in k≥0 kβ(k). In the base case k≥0 kβ(k) = 0, which is equivalent to β ∈ N 0 e 0 , in which case the r. h. s. of (6.2) reduces to k = 0 and thus β = e 0 , which satisfies (6.3). Turning to the induction step, we note that the r. h. s. of (6.1) restricts to k ≥ 1 so that the induction hypothesis can be applied to β 1 , . . . , β k . Hence the induction step follows from the fact that (6.3) is preserved when passing from β 1 , . . . , β k to β = e k + β 1 + · · · + β k .
We now compare (6.2) to the standard definition of branched rough paths, which is based on (if not otherwise stated: rooted and thus non-empty and undecorated) trees τ instead of multi-indices β. We recall that for a collection τ 1 , . . . , τ k , τ of such trees, the notation τ = B + (τ 1 · · · τ k ) (6.4) means that τ is the tree that is obtained from attaching an edge to each of the trees τ 1 , . . . , τ k and merging them in a common root, with the understanding that B + (∅) gives the tree with a single node 58 , denoted by . We recall from [15,Section 4] that the branched rough path 59 {X τ } τ is, inductively in the number of edges, defined through dX τ dx 2 = X τ 1 · · · X τ k ξ, X τ (x 2 = 0) = 0 provided (6.4) holds, (6.5) which includes X (x 2 ) = x 2 0 ξ. It is clear from (6.2) and (6.5) that every Π β is a linear combination of the X τ 's. Lemma 6.1. For every multi-index β, Here T β is the set of trees that have β(k) nodes with k children 60 , and where σ(β) and σ(τ ) are symmetry factors defined as follows: is the size of the group of all transformations of a tree τ ∈ T β that are obtained by permuting the children (with their descendants attached) at every node; σ(τ ) is the size of the subgroup that leaves a particular tree 58 which is the root 59 the canonical lift of ξ 60 Note that thanks to the restriction (6.3), this set is not empty. τ ∈ T β invariant 61 ; hence σ(β) σ(τ ) is the size of the orbit of τ under all above transformations 62 .
that comes with the intuition of grafting the tree τ 1 onto the tree τ 2 ; it can be extended to a pre-Lie product on B by linearity. While (6.15) shows that the two pre-Lie structures on L 1 and on B are isomorphic it is helpful to distinguish them here. This is related to the fact that we consider the standard pairing between B and L 1 , i. e. we think of z τ ∈ B and Z τ ∈ L 1 as dual bases 66 . These pre-Lie products have been evoked in branched rough paths [5,Subsection 3.2.2] and in regularity structures [4,Remark 4.1].
According to the definition (6.7) of σ(β) this reduces to This last identity holds because also the l. h. s. is the number of nodes of τ 2 with k children on which τ 1 can be attached (via a new edge).
The fact that φ † is not one-to-one reflects that our (pre-)Lie algebra is not free, as opposed to L 1 (cf. [11,Theorem 1.9]). Moreover, L RP is isomorphic to L 1 quotiented by an ideal. 66 Alternatively, one could work with but impose the pairing Z τ .z τ ′ = σ(τ )δ τ ′ τ , see more in [4,Subsection 3.3] on the choice of pairings viz. inner products. Corollary 6.3. As pre-Lie algebras, (L RP , ⊲) and (L 1 /R, ) are isomorphic, where (6.20) Proof. By Lemma 6.2, φ † : L 1 → L RP is a morphism. Thanks to the restriction (6.3), it is onto. It only remains to show that kerφ † = R. By (6.17), φ † vanishes on the generating set (6.20), hence R ⊂ kerφ † . To show the opposite inclusion, we fix τ c τ Z τ ∈ kerφ † , and by linearity it is enough to show that τ ∈T β c τ Z τ ∈ R for every β. By the representation (6.17), τ c τ Z τ is in the kernel of φ † if and only if τ ∈T β cτ σ(τ ) = 0 for all β. Remark 6.4. Note that, although freeness is lost, the generation property is preserved; in this setting, z 0 D (0) is the generator of L RP .

6.4.
Relating the coproduct ∆ + RP to Butcher's. The pre-Lie algebra morphism property (6.18) of φ † obviously implies that it is also a Lie-algebra morphism between L 1 and L RP . By the characterizing property of universal envelopes, φ † lifts to a morphism between the Hopf algebras U(L 1 ) and U(L RP ). According to [12,Theorem 3 b)] the standard pairing [12, (105)] between the Hopf algebra U(L 1 ) and the Connes-Kreimer Hopf algebra H respects the Hopf algebra structures. We recall that as an algebra, H is the polynomial algebra R[τ ] over trees τ , and the coproduct ∆ B is defined according to Butcher via cutting-off sub-trees ("pruning"), see e. g. [3,Section 3]. Defining T + RP and ∆ + RP , based on the Lie algebra L RP , in analogy to T + and ∆ + , see Subsection 4.5, we thus obtain that φ : T + RP → H is a Hopf algebra morphism, in particular (φ ⊗ φ)∆ + RP = ∆ B φ. Here, we used that on T + RP ⊗T + RP , which is naturally a subspace of (U(L RP )⊗ U(L RP )) * , we have that (φ † ⊗ φ † ) * = φ ⊗ φ.
6.5. Relating φ † to Υ. The morphism property (6.18) is closely related to the ones that appear in regularity structures [4,Corollary 4.15] and branched rough paths [2, Lemma 3.7], as we shall explain in this subsection for the latter: In view of its canonical pairing (6.13) with T RP , L RP can be canonically identified with a subspace 67 of T * RP , so that we may think of φ † as mapping into T * RP and then interpret (6.18) as the following identity in We note that the image of φ † is actually contained in the polynomial subspace R[z k ], and thus in view of (2.9) in the space of functions on a-space. Hence we may apply φ † Z τ to a polynomial 68 a, and thus also to a(· + u) for 67 Both have the same index set, but while L RP is a direct sum, T * RP is a direct product.
68 even to a formal power series some shift u ∈ R. We also note that D (0) preserves R[z k ], see (3.9). Hence we may "test" (6.21) with a(· + u) and obtain by definition (3.7) of D (0) which states that for fixed a, Υ a is a pre-Lie algebra morphism from (B, ) into the pre-Lie algebra of functions of u ∈ R.

Renormalization of rough paths via multi-indices.
We now give some details on future directions of our research, namely that renormalization can be carried out within the multi-index description without passing via trees. From the analytic and stochastic viewpoint, this is carried out in the case of quasi-linear SPDEs in the work [23]. In this section, we reveal the algebraic structure that guides renormalization in the simple case of branched rough paths, in line with [5].
Let us consider the following generalized version of (6.1): (6.24) du dx 2 = a 0 (u) + a 1 (u)ξ, u(x 2 = 0) = 0. so that shift-covariance is built in. As in Section 2, we lift (6.25) to the space of functions of (a 0 , a 1 ) by means of an algebra morphism M c : In analogy with (2.9), we introduce coordinates on (a 0 , a 1 )-space , and multi-indices which now involve both families of variables, i. e.
We still denote by T * RP the dual of the model space, now characterized by the population condition (6.28) the argument is the same as that of (6.3).
We moreover define the infinitesimal generator of tilt by a constant in line with (3.7): Then (6.26) acts on the coordinate functionals (6.27) as which thanks to the algebra morphism property defines the action of M c on the polynomial algebra R[z 0 k , z 1 k ]. Since M c amounts to plugging a power series into a power series, cf. (6.26), this action extends to the full power series space R[[z k , z n ]] as long as we impose The following result gathers the properties of the map M c .
Proof. Since the condition (6.31) is contained in (6.28), M c is well-defined. We first argue that M c commutes with D (0) ; indeed, this follows from (3.9) and (6.30) via , and is tautological for z 1 k . It then extends to the general case by the algebra morphism property of M c and Leibniz rule for D (0) . As a consequence, (6.32) is satisfied: We now argue that M c T * RP ⊂ T * RP . It is enough to show it for the space of finite sums, which is isomorphic to L RP 71 . Since (6.32) implies that M c is a pre-Lie morphism, and since the elements z 0 0 , z 1 0 generate L RP , cf. Remark 6.4, it suffices to show M c z 0 0 , M c z 1 0 ∈ T * RP ; this follows from (6.30) under the assumption c ∈ T * RP . Finally, the composition rule (6.33) may be read off from (6.26).
Combining (6.30) and (6.32), the map M c may be seen as a shift of the form z 0 0 → z 0 0 + c which, in addition, is a pre-Lie morphism. This connects the approach to the translation of (branched) rough paths as described in [5,Definition 14]. In the specific setting of this section, given an element 72 v ∈ L 1 , its associated translation map, which we denote by M BCF P v , is defined as the unique pre-Lie morphism that extends We follow the construction of previous subsections and build a dictionary φ from our model space T RP to the linear space of trees decorated by 0 and 1, so that (6.11) still holds with T β given by the set of trees which contain β(i, k) nodes decorated by i and with k children. This dictionary 73 intertwines with the translation maps: 71 See the discussion at the beginning of Subsection 6.5. 72 The general setting of [5] allows v to be a Lie series in the Grossman-Larson Hopf algebra generated by two nodes distinguished by decorations corresponding to each nonlinearity (in the specific setting of transformations of the form (6.25) only one decoration matters, so it is legitimate to think of v as non-decorated). Although we can also work with (infinite) Lie series, for notational convenience we restrict to finite sums and write v ∈ L 1 . 73 This time we consider φ † : L 1 → T * RP instead of φ † : L 1 → L RP , in line with Subsection 6.5. Lemma 6.6.
Proof. Since both φ † M BCF P v and M φ † v φ † are pre-Lie morphisms, cf. (6.21) and (6.32), it is enough to show (6.35) for the generators Z 0 and Z 1 . The case Z 0 follows from The case Z 1 is trivial from (6.34).
The map M c induces a (purely algebraic) transformation of the model. More precisely, if Π is the model constructed inductively from the ODE in which arises from (6.24), then thanks to the morphism property and (6.30), Π := M c Π solves The form of the last r. h. s. term in (6.36) connects to the form of the counter-term in the model equations for quasi-linear SPDEs which was postulated in [27, Subsection 1.1] and systematically constructed in [23].
We now argue that any translation in the sense of [5], which is expressed in terms of trees, may be expressed in terms of multi-indices in the form of (6.37). For this purpose, we note that, in the notation of [5, p. 37], Indeed, this follows from (6.27) for v = Z 0 , Z 1 , and is extended by the morphism property (6.22). Therefore by [5, Theorem 38 (ii)] equation (6.24) assumes the form Comparing (6.37) and (6.38), we see that the greedier setting of multiindices loses no information with respect to the tree-based approach; on the contrary, it reduces the complexity by grouping trees which give rise to the same renormalization procedure into a single multi-index. We expect this to extend to SPDEs, reducing the size of the renormalization group.
for [β] ≥ 0, together with Π en (x) = x n . This recursive definition leads to the following population condition: Π β = 0 implies and its integrated version Iτ , where I is the placeholder for integration, i. e. application of the kernel of the solution operator, is the placeholder for the noise 78 and with the understanding that n i = 0 for all i ∈ I. Although the factor ( i∈I I X n i ) is considered in [4] as a node decoration, 74 Since in our setting we work not with kernels but directly with the PDE, in order to guarantee uniqueness of the model we need to impose some extra conditions. Such conditions, which are irrelevant in the algebraic context of this article, amount to growth bounds that in turn allow for the application of Liouville principles; cf. e. g. [23,Proposition 5.3] or [22,Lemma 4.9]. 75 The reason why we do not restrict the model space T to the populated subspace, as we did in the rough path case, is that even then, our dictionary φ will no longer be one-to-one, see Subsection 7.4.
76 As opposed to the space V of combinatorial decorated trees [4, (3.5)], which have simpler polynomial decorations, and contain the actual model space in the tree-based framework; see Subsection 7.4.

77
In order to connect to [17], we switch the notation in [4] and use I for the abstract integration and I for the polynomial labelling. Here, I serves as a reminder that we cannot multiply polynomials. It is understood that (7.3) is independent of the order. 78 denoted in [4] as Ξ we will instead think of I X n i as a subtree; for example, the graphical representation of (I X n 1 )(I X n 2 ) I is given by The symmetry factor is defined accordingly, so that, for example, σ( X n X n ) = 2.
As pointed out in Subsection 5.3, the reason why we need both mapsφ − andφ is that the traditional model space in regularity structures relates to ours viaT ⊕ R ⊕ T; in trees, this means that every non-purely polynomial multi-index encodes both a linear combination of rooted and of integrated (planted) trees. In line with branched rough paths in Section 6, when restricted to populated multi-indices, cf. (7.2), both mapsφ − andφ are one-to-one but not onto. Moreover we have the following characterization: Lemma 7.1. For every multi-index β, Here T β is the set of trees with β(k) nodes with k children and β(n) decorations I X n . Recall that we think of the latter as subtrees, so they count as children. For example, (7.4) belongs to T β for β = e 0 + e 3 + e n 1 + e n 2 . The number σ(β) is the same as (6.7); in particular, it does not depend on β(n) for any n. The proof of Lemma 7.1 is similar to that of Lemma 6.1, and thus we omit it. Identity (7.8) establishes the matrix representation ofφ − , namely (7.9) (φ − ) β τ = σ(β) σ(τ ) if τ ∈ T β 0 otherwise ; 79 Following [4], we adopt the notation • to refer to the setting of expanded polynomial decorations; we will drop it in the contracted setting of Subsection 7.4. this, in turn, defines the transposed mapφ † − .

7.2.
Relating L to grafting operators on B.
In particular, (7.11) defines a pre-Lie product which is isomorphic to (7.10).
By the characterization (3.11) of ∂ 1 and definition (7.22) ofΥ a , this turns into the l. h. s. of (7.24). 7.4. Relating (∆, ∆ + ) to (∆ H , ∆ + H ). Our final goal is to connect our Hopf algebra structure to that of [17]. A first strong hint that the structures are compatible is (4.49); however, the analogue (7.31) for the coaction ∆ is missing.
We will pass to a coarser tree-based description, which no longer distinguishes different polynomial decorations of a given node but contracts them by multiplication. We identify the space of linear combinations of trees with contracted decorations with the model space 85 in [17,Subsection 4.2], and denote it by T H . More precisely, the model space T H consists of linear combinations of trees of the inductive form τ = X n ( j∈J Iτ j ), 85 In the more general context of [4], this contracted space corresponds to V , but we will directly work with the restriction to relevant trees. as well as their integrated versions Iτ . The passage from the detailed to the contracted description is encoded, as in [4,Subsection 4.1], by a linear map Q : B → T H recursively given, for τ as in (7.3), by Qτ = X i∈I n i ( j∈J IQτ j ).
For example, (7.4) turns into Q X n 2 X n 1 = X n 1 +n 2 .
The analogue of the space T + of [17, p. 24] is denoted by T + H with its basis elements X m i J H n i τ i , where τ i 's are elements of T H . Recall that J H n τ 86 Here, we are implicitly extendingφ − to the whole model space T by projecting onto the complement of the polynomial sector, i. e.φ − z en = 0.
vanishes for τ = X n ′ for every n ′ , and for |n| ≥ |τ | H + 2. Moreover, we In view of (7.31) and the multiplicativity of ∆ H we have ∆ H X n = n ′ +n ′′ =n n n ′ X n ′ ⊗ X n ′′ .
with the understanding that the pairing acts on the T + Lie algebra of {z γ D (n) } (γ,n) 23 L 1 (pre-)Lie algebra of Connes-Kreimer in [