2 Post-Lie algebras

In this section, we briefly recall the definition of a post-Lie algebra and its various properties.

Definition 2.1. A post-Lie algebra is a Lie algebra $ (\mathfrak {g}, [.,.]) $ equipped with a bilinear product $ \triangleright $ satisfying the following identities:

(3) $$ \begin{align} x \triangleright [y,z] = [x \triangleright y,z] + [y, x \triangleright z] \end{align} $$

and

(4) $$ \begin{align} [x,y] \triangleright z = a_{\triangleright}(x,y,z) - a_{\triangleright}(y,x,z) \end{align} $$

with $ x,y,x \in \mathfrak {g} $ , and the associator $ a_{\triangleright }(x,y,z) $ is given by:

$$ \begin{align*} a_{\triangleright}(x,y,z) = x \triangleright ( y \triangleright z ) - (x \triangleright y) \triangleright z. \end{align*} $$

When $ \mathfrak {g} $ is an abelian Lie algebra, (3) is void and (4) is the classical definition of the pre-Lie relation. The equation gives then a (in general) nontrivial Lie bracket. One can define a new Lie bracket $[[ .,.]]$ given by

$$ \begin{align*} [[x,y]] = [x,y] + x \triangleright y - y \triangleright x. \end{align*} $$

The post-Lie product $ \triangleright $ can be extended to a product on the universal enveloping algebra $ U(\mathfrak {g}) $ by first defining it on $ \mathfrak {g} \otimes U(\mathfrak {g})$ :

$$ \begin{align*} x \triangleright {\mathbf{1}} = 0, \quad x \triangleright y_1 ... y_n = \sum_{i=1}^n y_1 ... (x \triangleright y_i) ... y_n \end{align*} $$

and then extending it to $ U(\mathfrak {g}) \otimes U(\mathfrak {g}) $ by defining

$$ \begin{align*} {\mathbf{1}} \triangleright A & = A, \quad \nonumber x A \triangleright y = x \triangleright (A \triangleright y) -(x \triangleright A) \triangleright y, \\ A \triangleright B C & = \sum_{(A)} (A^{(1)} \triangleright B)(A^{(2)} \triangleright C), \end{align*} $$

where $ A, B, C \in U(\mathfrak {g}) $ and $ x, y \in \mathfrak {g} $ . Here, we have used Sweedler’s notation for the coproduct $ \Delta $ : $ \Delta A =\sum _{(A)} A^{(1)}\otimes A^{(2)} $ . This coproduct is defined for $ x \in \mathfrak {g} $ by

$$ \begin{align*} \Delta x = x \otimes {\mathbf{1}} + {\mathbf{1}} \otimes x \end{align*} $$

and then extended multiplicatively with respect to the concatenation product. Finally, one is able to define an associative product $*$ on $U(\mathfrak {g})$ :

(5) $$ \begin{align} A * B = \sum_{(A)} A^{(1)} (A^{(2)} \triangleright B). \end{align} $$

Then, one of the main results in [Reference Ebrahimi-Fard, Lundervold and Munthe-Kaas24, Thm 3.4], which depends on the above construction, allows us to exploit the underlying post-Lie structure on $\mathfrak {g}$ in order to obtain information about the structure of $U(\mathfrak {g})$ .

Furthermore, one has the following proposition which shows that a post-Lie structure underlying a Lie algebra produces a representation-theoretic datum (see [Reference Mencattini, Quesney and Silva37, Sec. 5]):

Proposition 2.4. The mapping

$$ \begin{align*} \rho : \mathfrak{\bar{g}} \rightarrow End_{\mathbb{R}}(U(\mathfrak{g})), \quad x \mapsto ( y \mapsto x \triangleright y + x y) \end{align*} $$

is a linear representation of the Lie algebra $\mathfrak {\bar {g}}$ . Furthermore, by the universal property of the enveloping algebra, it extends to a linear representation

$$ \begin{align*} \rho:U(\mathfrak{\bar{g}}) \rightarrow End_{\mathbb{R}} (U(\mathfrak{g})) \end{align*} $$

of the algebra $U(\mathfrak {\bar {g}})$ . We call this the representation induced by the post-Lie algebra $(\mathfrak {g}, [ .,.], \triangleright )$ or, more shortly, the induced representation.

We finish this section by presenting standard examples of pre- and post-Lie algebras. We first consider the set of non-planar rooted trees denoted by ${\mathcal {T}}$ . We define a product $ \bar {\curvearrowright } $ called grafting product defined for two rooted trees $ \tau , \sigma $ by

$$ \begin{align*} \sigma \bar{\curvearrowright} \tau:=\sum_{v\in N_{\tau} } \sigma \bar{\curvearrowright}_v \tau, \end{align*} $$

where $ N_\tau $ is the set of vertices of $ \tau $ and $\sigma \bar {\curvearrowright }_v \tau $ is obtained by grafting the tree $\sigma $ on the tree $\tau $ at vertex v by means of a new edge. Below, we provide an example of computation for $ \bar {\curvearrowright } $ :

The space $ {\mathcal {T}} $ equipped with $ \bar {\curvearrowright } $ is the free pre-Lie algebra over one generator $ \bullet $ (see [Reference Chapoton and Livernet23]). We consider the linear span of planar rooted trees denoted by $ \mathcal {P}\mathcal {T} $ . Now, one has

which is not the case in the non-planar setting. One can define a magmatic operation $ \bar {\curvearrowright }_{l} $ called left grafting. It is given for $ \tau , \sigma $ non-planar rooted trees by

$$ \begin{align*} \sigma \bar{\curvearrowright}_{l} \tau:=\sum_{v\in N_{\tau} } \sigma \bar{\curvearrowright}_{l,v} \tau, \end{align*} $$

where $\sigma \bar {\curvearrowright }_{l,v} \tau $ is obtained by grafting the tree $\sigma $ on the tree $\tau $ at vertex v by means of a new edge at the leftmost location. Below, we provide an example of computation for $ \bar {\curvearrowright }_{l} $ :

If we consider the free Lie algebra generated by $ \mathcal {P} \mathcal {T} $ equipped with the leftmost grafting, one obtains the free post-Lie algebra (see [Reference Munthe-Kaas and Lundervold39, Thm. 3.2]).

3 Decorated trees, multi-indices and non-commuting operators

In this section, we recall two different combinatorial structures that are both used in the context of Regularity Structures with the aim of solving singular SPDEs. The first one is that of decorated trees and the second one is that of multi-indices. We shall stress the non-commutative nature of some key operators defined on these structures and proceed to show that their non-commutative properties can be elegantly captured by certain post-Lie algebraic structures.

Decorated trees as introduced in [Reference Bruned, Hairer and Zambotti9] are described in the following way. Pick two symbols I and $\Xi $ and let $ \mathcal {D} := \lbrace I,\Xi \rbrace \times \mathbb {N}^{d+1}$ define the set of edge decorations. These two symbols represent a convolution with a kernel $ I $ and a noise term $ \Xi $ . One may add more symbols if one works with a system of SPDEs with more than one noise and one kernel. Decorated trees over $ \mathcal {D} $ are of the form $T_{\mathfrak {e}}^{{\mathfrak {n}}} = (T,{\mathfrak {n}},{\mathfrak {e}}) $ , where T is a non-planar rooted tree with node set $N_T$ and edge set $E_T$ . The maps ${\mathfrak {n}} : N_T \rightarrow \mathbb {N}^{d+1}$ and ${\mathfrak {e}} : E_T \rightarrow \mathcal {D}$ are node, respectively edge, decorations. We denote the set of decorated trees by $ {\mathfrak {T}} $ . The tree product is defined by

$$ \begin{align*} (T,{\mathfrak{n}},{\mathfrak{e}}) \cdot (T',{\mathfrak{n}}',{\mathfrak{e}}') = (T \cdot T',{\mathfrak{n}} + {\mathfrak{n}}', {\mathfrak{e}} + {\mathfrak{e}}')\;, \end{align*} $$

where $T \cdot T'$ is the rooted tree obtained by identifying the roots of $ T$ and $T'$ . The sums $ {\mathfrak {n}} + {\mathfrak {n}}'$ mean that decorations are added at the root and extended to the disjoint union by setting them to vanish on the other tree. We make the connection with symbolic notation introduced in the previous part.

• – An edge decorated by $ (I,a) \in \mathcal {D} $ is denoted by $ I_{a} $ . The symbol $ I_{a} $ is also viewed as the operation that grafts a tree onto a new root via a new edge with edge decoration $ a $ . The new root at hand remains decorated with $0$ .

• – An edge decorated by $ (\Xi ,0) \in \mathcal {D} $ is denoted by $ \Xi $ . We suppose that these edges are terminal edges with zero node decoration at their leaves.

• – A factor $ X^{\ell }$ encodes a single node $ \bullet ^{\ell } $ decorated by $ \ell \in \mathbb {N}^{d+1}$ . We write $ X_i$ , $ i \in \lbrace 0,1,\ldots ,d\rbrace $ to denote $ X^{e_i}$ . Here, we have denoted by $ e_i $ the vector of $ \mathbb {N}^{d+1} $ with $ 1 $ in ith position and $ 0 $ otherwise. The element $ X^0 $ is identified with the empty tree .

Using this symbolic notation, given a decorated tree $ \tau $ , there exist decorated trees $ \tau _1, ..., \tau _r $ such that

$$ \begin{align*} \tau = X^{\ell} \Xi^m \prod_{i=1}^r I_{a_i}(\tau_i), \end{align*} $$

where $ \prod _i $ is the tree product, $ \ell \in \mathbb {N}^{d+1} $ , $ m, r \in \mathbb {N} $ . In relevant applications, a product of noises is not allowed and one can only consider the case $ m \in \lbrace 0,1 \rbrace $ . A tree of the form $ I_a(\tau ) $ is called a planted tree as there is only one edge connecting the root to the rest of the tree. Below, we present an example of decorated trees:

where we have put $ \Xi $ on the edge to specify that it is decorated by $ (\Xi ,0) $ . For an edge decorated by $ (I,a) $ , we have just put $ a $ . Nodes without decoration mean that their decoration is equal to zero. We define a product called grafting product:

$$ \begin{align*} \sigma \curvearrowright^a \tau:=\sum_{v\in N_{\tau} } \sigma \curvearrowright^a_v \tau, \end{align*} $$

where $\sigma $ and $\tau $ are two decorated rooted trees, $ N_\tau $ is the set of vertices of $ \tau $ and $\sigma \curvearrowright ^a_v \tau $ is obtained by grafting the tree $\sigma $ on the tree $\tau $ at vertex v by means of a new edge decorated by $a\in \mathbb {N}^{d+1}$ . Grafting onto noise-type edges, that is, edges decorated by $ (\Xi ,0) $ are forbidden. Therefore, there is a bijection between noises as decorated edges and noises as node decorations. Below, we provide an example of this grafting product:

One should notice that we did not graft onto the noise edge. The family of grafting products $ (\curvearrowright ^b )_{b \in \mathbb {N}^{d+1}} $ forms a multi-pre-Lie algebra, in the sense that they satisfy the following identities:

$$ \begin{align*} \left( \tau_1 \curvearrowright^a \tau_2 \right) \curvearrowright^b \tau_3 - \tau_1 \curvearrowright^a ( \tau_2 \curvearrowright^b \tau_3 ) = ( \tau_2 \curvearrowright^b \tau_1 ) \curvearrowright^a \tau_3 - \tau_2 \curvearrowright^b ( \tau_1 \curvearrowright^a \tau_3 ), \end{align*} $$

where the $ \tau _i $ are decorated trees and $ a,b $ belong to $ \mathbb {N}^{d+1} $ . This is an extension of the notion of a pre-Lie product (which is recovered when the family is reduced to one element) and was first introduced in [Reference Bruned, Chandra, Chevyrev and Hairer5, Prop. 4.21]. This multi-pre-Lie algebra can be summarised into a single pre-Lie structure on the space of planted trees, given by the product:

$$ \begin{align*} I_{a}(\sigma) \curvearrowright I_{b}(\tau):= I_{b}(\sigma \curvearrowright^a \tau). \end{align*} $$

This was first noticed in [Reference Foissy28] (see also [Reference Bruned and Manchon11, Prop. 3.2]). Below, we provide an example that illustrates the grafting operation:

Notice that we do not graft at the root, which has its decoration set to zero.

Furthermore, the products $ \curvearrowright ^a $ can be deformed via a pre-Lie isomorphism described in [Reference Bruned and Manchon11, Sec. 2.2]. The deformed products are given by

$$ \begin{align*} \sigma \widehat{\curvearrowright}^a \tau:=\sum_{v\in N_{\tau}}\sum_{\ell\in\mathbb{N}^{d+1}}{{\mathfrak{n}}_v \choose \ell} \sigma \curvearrowright_v^{a-\ell}(\uparrow_v^{-\ell} \tau). \end{align*} $$

Intuitively, one should think of the operators $\uparrow _v^{-\ell }$ as encoding the operation of partial differentiation at a purely algebraic level. Formally, $ {\mathfrak {n}}_v \in \mathbb {N}^{d+1}$ denotes the decoration at the vertex $ v $ and the operator $ \uparrow _v^{-\ell } $ is defined as subtracting $ \ell $ from the node decoration of $ v $ .

The generic term is self-explanatory if there exists a (unique) pair $(b,\alpha )\in \mathbb {N}^{d+1} \times \mathbb {N}^{d+1}$ such that $a=\ell +b$ and ${\mathfrak {n}}_v =\ell +\alpha $ . It vanishes by convention if this condition is not satisfied. Given a scaling ${\mathfrak {s}} \in \mathbb {N}_0^{d+1} = \mathbb {N}^{d+1} \setminus \lbrace 0 \rbrace $ , we define the grading of a tree as the sum of the gradings of its edges and denote it by $ |\cdot |_{\text {grad}} $ :

$$ \begin{align*} |\tau|_{\text{grad}}:=\sum_{e\in E_{\tau}}\big|{\mathfrak{e}}(e) \big|_{{\mathfrak{s}}}, \end{align*} $$

where $ E_{\tau } $ are the edges of $ \tau $ , $ {\mathfrak {e}}(e) $ is the decoration of the edge $ e $ and for a given $ \mathbf {n} \in \mathbb {N}^{d+1} $ , one has

$$ \begin{align*} |\mathbf n|_{{\mathfrak{s}}}:= \sum_{i=0}^d s_i n_i. \end{align*} $$

A scaling ${\mathfrak {s}}$ is necessary as these decorated trees are used in the context of singular SPDEs where different variables come with different degrees of homogeneity. A good example to keep in mind is that of parabolic equations, where the linear operator is given by $ \partial _t - \Delta $ , with the Laplacian acting on the spatial variable. It is then natural to make powers of t ‘count double’ and take the parabolic scaling $ (2,1,...,1) $ .

The deformed product $\widehat {\curvearrowright }^a$ has been first introduced in [Reference Bruned, Chandra, Chevyrev and Hairer5, Rem. 4.12]. One notices that $\widehat {\curvearrowright }^a$ is a deformation of $\curvearrowright ^a$ in the sense that

$$ \begin{align*} \sigma \widehat{\curvearrowright}^a \tau = \sigma \curvearrowright^a \tau + \text{lower grading terms}. \end{align*} $$

Again, one may summarise the above family of multi-pre-Lie products into a single pre-Lie product. More specifically, the deformed pre-Lie product $\widehat {\curvearrowright }$ is given by

$$ \begin{align*} I_{a}(\sigma) \, \widehat{\curvearrowright} \, I_{b}(\tau):= I_{b}(\sigma \, \widehat{\curvearrowright}^{a} \, \tau). \end{align*} $$

Below, we provide an example illustrating the product $ \widehat {\curvearrowright } $ :

Another important operation we will need to define on the space of trees is $ \uparrow ^{i} $ :

$$ \begin{align*} \uparrow^{i} \tau = \sum_{v \in N_{\tau}} \uparrow^{e_i}_v \tau. \end{align*} $$

Here, the insertion does not happen on top of noise-type edges. We provide below an example of computation:

This operator is a derivation for the grafting product $ \curvearrowright ^a $ in the sense that

$$ \begin{align*} \uparrow^{i} \left( \sigma \curvearrowright^a \tau \right) = (\uparrow^{i} \sigma) \curvearrowright^a \tau + \sigma \curvearrowright^a \, ( \uparrow^{i} \tau) \end{align*} $$

and one has the following right derivation property:

$$ \begin{align*} \uparrow^{i}_{N_\tau} \left( \sigma \curvearrowright^a \tau \right) = \sigma \curvearrowright^a \, ( \uparrow^{i} \tau), \end{align*} $$

where $\uparrow ^{i}_{N_\tau } $ is defined as $\uparrow ^{i} $ but with $ N_{\tau } \sqcup N_{\sigma } $ replaced by $ N_{\tau } $ . These two properties are clearly not true for the deformed pre-Lie products $ \widehat {\curvearrowright }^{a} $ . They are, however, true up to some deformation. This is made formal by the following proposition:

Proposition 3.1. One has, for all decorated trees $ \sigma , \tau $ and $ a \in \mathbb {N}^{d+1} $ , $ i \in \lbrace 0,...,d \rbrace $ ,

(6) $$ \begin{align} \uparrow^{i}_{N_\tau} \left( \sigma \widehat{\curvearrowright}^a \tau \right) = \sigma \widehat{\curvearrowright}^a \, ( \uparrow^{i} \tau) - \sigma \widehat{\curvearrowright}^{a-e_i} \tau. \end{align} $$

Proof. Identity (6) is a consequence of the following:

$$ \begin{align*} \sigma \widehat{\curvearrowright}^a \, ( \uparrow^{i} \tau) & = \sum_{v\in N_{\tau}}\sum_{\ell\in\mathbb{N}^{d+1}}{{\mathfrak{n}}_v + e_i \choose \ell} \sigma \curvearrowright_v^{a-\ell}(\uparrow_v^{-\ell + e_i} \tau) \\ \sigma \widehat{\curvearrowright}^{a-e_i} \, \tau & = \sum_{v\in N_{\tau}}\sum_{\ell\in\mathbb{N}^{d+1}}{{\mathfrak{n}}_v \choose \ell} \sigma \curvearrowright_v^{a-\ell - e_i}(\uparrow_v^{-\ell } \tau) \\ & = \sum_{v\in N_{\tau}}\sum_{\ell\in\mathbb{N}^{d+1}}{{\mathfrak{n}}_v \choose \ell-e_i} \sigma \curvearrowright_v^{a-\ell }(\uparrow_v^{-\ell + e_i } \tau) \\ \uparrow^{i}_{N_\tau} \left( \sigma \widehat{\curvearrowright}^a \tau \right) & = \sum_{v\in N_{\tau}}\sum_{\ell\in\mathbb{N}^{d+1}}{{\mathfrak{n}}_v \choose \ell} \sigma \curvearrowright_v^{a-\ell }(\uparrow_v^{-\ell + e_i } \tau). \end{align*} $$

Then, we conclude by the fact that

$$ \begin{align*} {{\mathfrak{n}}_v \choose \ell-e_i} + {{\mathfrak{n}}_v \choose \ell} = {{\mathfrak{n}}_v + e_i \choose \ell}.\\[-34pt] \end{align*} $$

Recently, a different encoding of iterated integrals has been introduced in [Reference Linares, Otto and Tempelmayr34] based on multi-indices. Let’s briefly recall the definitions given by the authors. We suppose we are given two sets of abstract variables $ (z_k)_{k \in \mathbb {N}} $ and $ (z_{n})_{n \in \mathbb {N}^{d+1}} $ . The $ z_k $ encode nodes of arity $ k $ and the $ z_n $ are monomials. Multi-indices $ \beta $ over $ \mathbb {N} $ and $ \mathbb {N}^{d+1} $ measure the frequency of the variables $ z_k, z_n $ , so that we can write monomials

$$ \begin{align*} z^{\beta} : = \prod_{k \in \mathbb{N}, \, n \in \mathbb{N}^{d+1}} z_k^{\beta(k)} z_{n}^{\beta(n)}. \end{align*} $$

One can write $ \beta $ according to the canonical basis of $ \bar {e}_k, \bar {e}_n $ encoding the $ z_k, z_{n} $ :

$$ \begin{align*} \beta = \sum_{k \in \mathbb{N}} \beta(k) \bar{e}_k + \sum_{n \in \mathbb{N}^{d+1}} \beta(n) \bar{e}_n. \end{align*} $$

We introduce a family of derivations on these multi-indices: $ D^{(0)}, (D^{(n)})_{n \neq 0} $ and $ \partial _i, \, i \in \lbrace 0,...,d\rbrace $ . These are defined in the following way:

$$ \begin{align*} D^{(0)} = \sum_{k \in \mathbb{N}} (k+1) z_{k+1} \partial_{z_k}, \quad D^{(n)} = \partial_{z_n}, \, n \neq 0, \end{align*} $$

where $ \partial _{z_k} $ is the derivative in the coordinates $ z_k $ . The action of this derivative operator corresponds to increasing the arity of a node by one. Then, the derivatives $ \partial _i $ are given by

$$ \begin{align*} \partial_i = \sum_{n} (n_i +1 ) z_{n + e_i} D^{(n)}. \end{align*} $$

For all these derivatives, we will use a matrix representation $ (D^{(0)})^{\gamma }_{\beta }, (D^{(n)})^{\gamma }_{\beta } $ and $ (\partial _i)^{\gamma }_\beta $ , where $ \gamma $ and $ \beta $ are multi-indices.

$$ \begin{align*} (D^{({ 0})})_\beta^\gamma&=\sum_{k\ge 0}\left\{\begin{array}{cl} (k+1)\gamma(k)&\mbox{if}\;\gamma+\bar{e}_{k+1}=\beta+\bar{e}_k\\ 0&\mbox{otherwise}\end{array}\right\}\\(D^{({ n})})_\beta^\gamma&=\left\{\begin{array}{cl} \gamma({ n})&\mbox{if}\;\gamma=\beta+\bar{e}_{ n}\\ 0&\mbox{otherwise}\end{array}\right\}\quad\mbox{for}\;{ n}\not={ 0}\\(\partial_i)_\beta^\gamma&=\sum_{k\ge 0}\left\{\begin{array}{cl} (k+1)\gamma(k)&\mbox{if}\;\gamma+\bar{e}_{k+1}+\bar{e}_{e_i}=\beta+e_k \\ 0&\mbox{otherwise}\end{array}\right\} \\& \quad +\sum_{{ n}\not=0}\left\{\begin{array}{cl} (n_i+1)\gamma({ n})&\mbox{if}\;\gamma+\bar{e}_{{ n}+ e_i}=\beta+\bar{e}_{ n}\\ 0&\mbox{otherwise}\end{array}\right\}. \end{align*} $$

The non-commutative property (6) appears at the level of the derivations. Indeed, one has

(7) $$ \begin{align} \partial_i D^{(n)} = D^{(n)} \partial_i + n_i D^{(n-e_i)}. \end{align} $$

This motivates the introduction of a Lie bracket in the next section for taking into account that non-commutative property. We note that the encoding given by multi-indices does not precisely correspond to the original encoding via decorated trees. Indeed, the decorated trees associated to them contain more node decorations, in the spirit of [Reference Bruned, Chandra, Chevyrev and Hairer5] where a new class of trees has been introduced for proving a pre-Lie morphism property on some elementary differentials. The new decorated trees are of this form:

(8) $$ \begin{align} \tau = \Xi \prod_{j=1}^{k_1} X^{\ell_j} \prod_{i=1}^{k_2} I(\tau_i), \end{align} $$

where now the noise $ \Xi $ systematically appears at every node and can be omitted in the notation. All the edges have the same decoration $ (I,0) $ . The main novelty are the new decorations given by the product of the $ X^{m_j} $ and this time, we have

$$ \begin{align*} \prod_{j=1}^{k_1} X^{\ell_j} \neq X^{\sum_{j=1}^{k_1} \ell_j}. \end{align*} $$

We denote the set of these decorated trees by $ \mathfrak {T}_0 $ . Then, the mapping to a multi-index can be performed recursively via the following map $ \Psi $ :

(9) $$ \begin{align} \Psi(\tau) = (k_1 + k_2)! \, z_{k_1 + k_2} \prod_{j=1}^{k_1 } (\ell_j !) \, z_{\ell_j} \prod_{i=1}^{k_2} \Psi(\tau_i), \quad \Psi(\Xi) = {\mathbf{1}}. \end{align} $$

The deformed pre-Lie product in this context takes the form

(10) $$ \begin{align} \begin{aligned} \sigma \widehat{\curvearrowright}^a \tau & :=\sum_{v\in N_{\tau}}\sum_{\ell\in \lbrace 0, a \rbrace}{{\mathfrak{n}}_v \choose \ell} \sigma \widehat{\curvearrowright}_v^{a-\ell}(\hat{\uparrow}_v^{-\ell} \tau) \\ & = \sigma \curvearrowright^a \tau + \sum_{v\in N_{\tau}} \sigma \widehat{\curvearrowright}_v^{0}(\hat{\uparrow}_v^{-a} \tau), \end{aligned} \end{align} $$

where $ \hat {\uparrow }_v^{-a} $ removes one $ X^{a} $ at the node $ v $ ; otherwise, it is equal to zero. Then, we need to put an extra restriction as decorated trees described in (8) have only zero edge-decorations. Therefore, when $ a = 0 $ , one has

$$ \begin{align*} \sigma \widehat{\curvearrowright}^a \tau = \sigma \curvearrowright^a \tau. \end{align*} $$

Otherwise, one has

$$ \begin{align*} \sigma \widehat{\curvearrowright}^a \tau = \sum_{v\in N_{\tau}} \sigma \widehat{\curvearrowright}_v^{0}(\hat{\uparrow}_v^{-a} \tau). \end{align*} $$

Such deformation is reminiscent of the one used in the context of numerical analysis in [Reference Bruned and Schratz13, Reference Bruned and Manchon11], where there is only one term of lower order. In [Reference Otto, Sauer, Smith and Weber40, Reference Linares, Otto and Tempelmayr34], the authors consider only a subclass of multi-indices; that is,

$$ \begin{align*} \{ z^{\gamma}D^{(n)} \}_{[\gamma] \geq 0 }, \end{align*} $$

where $ [\gamma ] $ is given by

$$ \begin{align*} [\gamma] = \sum_{k \geq 0} k \gamma(k) - \sum_{n \neq 0} \gamma(n). \end{align*} $$

The condition $ [\gamma ] \geq 0 $ corresponds to the fact that we have fewer monomial multi-indices than branches coming from the $ \bar {e}_k $ . The authors are also more restrictive by projecting according to the homogeneity of the multi-indices given by a map $ \gamma \mapsto |\gamma |_{{\mathfrak {s}}} $ depending on the chosen scaling $ {\mathfrak {s}} $ . The extra condition is given by: $ |\gamma |_{{\mathfrak {s}}} - |n|_{{\mathfrak {s}}}> 0 $ . In the context of decorated trees, it corresponds to planted trees with positive homogeneity.

4 A post-Lie algebra for decorated trees

In this section, we explain how the previous formalism can be expressed directly on decorated trees in order to obtain the coproduct $\Delta _{2}$ appearing in [Reference Bruned and Manchon11] which has been introduced in [Reference Hairer32, Reference Bruned, Hairer and Zambotti9]. This coproduct is crucial for defining the Hopf algebra governing the recentering procedure in Regularity Structures. It is used for constructing recentered iterated integrals that are the building blocks of solutions for singular SPDEs.

We give analogues to the spaces introduced in the previous section by using the letter $\mathcal {V}$ . We define the following spaces:

$$ \begin{align*} \mathcal{V} & = \Big \langle \{ I_a(\tau), \, a \in \mathbb{N}^{d+1}, \, \tau \in \mathfrak{T} \} \cup \{ X_{i} \}_{i = 0,..., d} \Big \rangle_{\mathbb{R}}, \\ \tilde{\mathcal{V}} & = \Big \langle \{ I_a(\tau), \, a \in \mathbb{N}^{d+1}, \, \tau \in \mathfrak{T} \} \Big \rangle_{\mathbb{R}}. \end{align*} $$

The space $ \tilde {\mathcal {V}} $ is the linear span of planted decorated trees, and $ \mathcal {V}$ is the linear span of planted trees with the monomials $ X_i $ . We introduce a Lie bracket and a product on the space $\mathcal {V}$ that are compatible with one another and will give us a post-Lie algebraic structure.

Definition 4.1. We define a product $ \widehat {\triangleright } $ on $ \mathcal {V} $ for every $ a,b \in \mathbb {N}^{d+1}, \, i,j \in \lbrace 0,\ldots ,d\rbrace $ as

(11) $$ \begin{align} X_i \, \widehat{\triangleright} \, I_{a}(\tau) = \uparrow^i I_{a}(\tau), \quad I_{a}(\tau) \, \widehat{\triangleright} \, X_{i} = 0, \quad X_i \, \widehat{\triangleright} \, X_{j} = 0, \end{align} $$

and

$$ \begin{align*} I_{a}(\sigma) \, \widehat{\triangleright} \, I_{b}(\tau) = I_{a}(\sigma) \, \widehat{\curvearrowright} \,I_{b}(\tau). \end{align*} $$

Definition 4.2. We define the Lie bracket on $\mathcal {V}$ as $[x, y]_{0} = 0$ for $x, y \in \tilde {\mathcal {V}}$ , $[x, y]_{0} = 0$ for $x, y \in \langle \ X_{i} \ \rangle _{\mathbb {R}}$ and as

(12) $$ \begin{align} [I_a(\tau),X_i]_0 = I_{a-e_i}(\tau). \end{align} $$

Proof. One has to check

(13) $$ \begin{align} x \, \widehat{\triangleright} \, [y,z]_0 = [x \, \widehat{\triangleright} \, y,z]_0 + [y, x \, \widehat{\triangleright} \, z]_0 \end{align} $$

and

(14) $$ \begin{align} [x,y]_0 \, \widehat{\triangleright} \, z = a_{\widehat{\triangleright}}(x,y,z) - a_{\widehat{\triangleright}}(y,x,z). \end{align} $$

It is easy to check (13), for $ x= X_i $ . Then, if we consider $ x = I_{a}(\sigma ) $ by symmetry, we can restrict ourselves to the case $ y = I_a(\sigma ) $ and $ z= X_i $ which will be non-zero. We have

$$ \begin{align*} I_a(\sigma) \, \widehat{\triangleright} \, [I_b(\tau),X_i]_0 = I_a(\sigma) \, \widehat{\curvearrowright} \, I_{b-e_i}(\tau). \end{align*} $$

We conclude by the fact that

$$ \begin{align*} [ I_a(\sigma) \, \widehat{\triangleright} \, I_b(\tau),X_i ]_0 = I_{b-e_i}( I_a(\sigma) \, \widehat{\curvearrowright} \tau) = I_a(\sigma) \, \widehat{\curvearrowright} \, I_{b-e_i}(\tau). \end{align*} $$

It remains to show (14). If $ z =X_i $ , then it is zero on both sides. Let us consider $ z = I_{b}(\tau ) $ . If $ x $ and $ y $ are both planted trees, the fact that $ \widehat {\curvearrowright } $ is a pre-Lie product gives the answer. For symmetry reason, we can consider $ x = I_a(\sigma ) $ and $ y = X_i $ . From the left-hand side, we get

$$ \begin{align*} [x,y]_0 \, \widehat{\triangleright} \, z = I_{a-e_i}(\sigma) \, \widehat{\curvearrowright} \, I_b(\tau). \end{align*} $$

From the right-hand side, only two terms remain:

$$ \begin{align*} \begin{aligned} a_{\widehat{\triangleright}}(x,y,z) & = I_a(\sigma) \, \widehat{\triangleright} \, ( X_i \, \widehat{\triangleright} \, I_b(\tau)) - (I_a(\sigma) \, \widehat{\triangleright} \, X_i) \, \widehat{\triangleright} \, I_b(\tau) \\ & = I_{a}(\sigma) \, \widehat{\curvearrowright} \, I_b( \uparrow^i \tau) \end{aligned} \end{align*} $$

because $ I_a(\sigma ) \, \widehat {\triangleright } \, X_i =0 $ . Then

$$ \begin{align*} \begin{aligned} a_{\widehat{\triangleright}}(y,x,z) & = X_i \, \widehat{\triangleright} \, (I_a(\sigma) \, \widehat{\triangleright} \, I_b(\tau)) - (X_i \, \widehat{\triangleright} \, I_a(\sigma)) \, \widehat{\triangleright} \, I_b(\tau) \\ & = \ \uparrow^{i}(I_a(\sigma) \, \widehat{\curvearrowright} \, I_b(\tau) ) - (\uparrow^{i}I_{a}(\sigma)) \, \widehat{\curvearrowright} \, I_{b}(\tau) \\ &= I_b( \uparrow^i_{N_{\tau}} (I_{a}(\sigma) \, \widehat{\curvearrowright} \, \tau)), \end{aligned} \end{align*} $$

where we have used

$$ \begin{align*} I_{b} ( \uparrow^{i} (I_{a}(\sigma) \, \widehat{\curvearrowright} \, \tau)) - I_{b}(I_{a}(\uparrow^{i} \sigma) \, \widehat{\curvearrowright} \, \tau)) = I_b( \uparrow^i_{N_{\tau}} (I_{a}(\sigma) \, \widehat{\curvearrowright} \, \tau)). \end{align*} $$

Indeed by definition, one has

$$ \begin{align*} \uparrow^{i} (I_{a}(\sigma) \, \widehat{\curvearrowright} \, \tau) & = \sum_{v \in N_{\sigma} \sqcup N_{\tau}} \uparrow^{i}_v (I_{a}(\sigma) \, \widehat{\curvearrowright} \, \tau) \\ & = \sum_{v \in N_{\sigma} } \uparrow^{i}_v (I_{a}(\sigma) \, \widehat{\curvearrowright} \, \tau) + \sum_{v \in N_{\tau}} \uparrow^{i}_v (I_{a}(\sigma) \, \widehat{\curvearrowright} \, \tau) \\ & = (\uparrow^{i} I_{a}( \sigma)) \, \widehat{\curvearrowright} \, \tau + \uparrow^{i}_{N_{\tau}} (I_{a}(\sigma) \, \widehat{\curvearrowright} \, \tau). \end{align*} $$

In the end, we get

$$ \begin{align*} a_{\widehat{\triangleright}}(x,y,z) - a_{\widehat{\triangleright}}(y,x,z) = I_{a}(\tau) \, \widehat{\curvearrowright} \, I_b( \uparrow^i \sigma) - I_b( \uparrow^i_{N_{\sigma}} ( I_{a}(\tau) \, \widehat{\curvearrowright} \, \sigma )). \end{align*} $$

The equality between the two expressions is given by Proposition 3.1.

Corollary 4.5. The bracket $ [[.,.]] $ defined by

$$ \begin{align*}[[x, y]] = [x, y]_{0} + x \, \widehat{\triangleright} \, y - y \, \widehat{\triangleright} \, x \end{align*} $$

for every $ x, y \in \mathcal {V} $ is a Lie bracket on $\mathcal {V}$ .

We denoted by $ U(\mathcal {V}_0) $ the enveloping algebra with the Lie bracket $ [.,.]_0 $ and by $ U(\mathcal {V}) $ the enveloping algebra with the Lie bracket $ [[.,.]] $ . We also set $ * $ to be the product obtained by the Guin-Oudom type procedure given in Section 2.

Following [Reference Linares, Otto and Tempelmayr34], one makes use of this isomorphism and the fact that the bracket $[.,.]_{0}$ vanishes on $\tilde {\mathcal {V}}$ to obtain the deformed coproduct given in [Reference Hairer32, Reference Bruned, Hairer and Zambotti9, Reference Bruned and Manchon11], by selecting a basis for $U(\mathcal {V})$ that is symmetric with respect to the elements of the basis of $\tilde {\mathcal {V}}$ . More precisely, along with the aid of the Poincare-Birkhoff-Witt theorem and after choosing to order the $X_{i}$ according to their indices, one obtains a basis of the form

$$ \begin{align*} B_{(\textbf{F}, \textbf{m})} = \ \prod_{i = 0, ..., d} X_{i}^{m_{i}} I_{a_{1}}(\sigma_{1}) \cdot \cdot \cdot I_{a_{n}}(\sigma_{n}), \end{align*} $$

where $\textbf {F} = I_{a_{1}}(\sigma _{1}) \cdot \cdot \cdot I_{a_{n}}(\sigma _n) $ ranges over all forests of planted trees and $\textbf {m} \in \mathbb {N}^{d+1}$ .

Remark 4.8. By virtue of Proposition 2.4 in Section 2, one obtains the following representation of $\mathfrak {\bar {g}}$ :

$$ \begin{align*} \rho : \mathfrak{\bar{g}} \rightarrow End_{\mathbb{R}}(U(\mathfrak{g})) \end{align*} $$

$$ \begin{align*} I_{a}(\sigma) \mapsto ( A \mapsto \sigma \widehat{\curvearrowright}^{a} A + I_{a}(\sigma)\cdot A ), \quad X_{i} \mapsto ( A \mapsto \uparrow^{i} A + X_{i} \cdot A ), \end{align*} $$

where $A = B_{(\textbf {F}, \textbf {m})} = \ \prod _{i = 0, ..., d} X_{i}^{m_{i}} I_{a_{1}}(\sigma _{1}) \cdot \cdot \cdot I_{a_{n}}(\sigma _{n}) $ is a basis element as above and $\cdot $ denotes the associative product in $U(\mathfrak {g})$ . We note that, on the second line, in order for us to be able to apply the grafting product, an element A can clearly be interpreted as a rooted tree by merging all the $I_{a_{i}}(\sigma _{i})$ on a single root and adjoining the appropriate label on that root.

As one can play around using the commutation relations on the universal envelope $U(\mathfrak {g})$ , the linear representation given in the remark above gives us hints about the resulting $*$ product on the universal enveloping algebra. Indeed, we shall now proceed to prove that the product $*$ thus obtained is identical to the product $\star _{2}$ that governs the combinatorics of recentering in regularity structures. The product $\star _{2}$ is the dual of the $\Delta _{2}$ coproduct as was proven in [Reference Bruned and Manchon11]. Therefore, our Hopf algebra will be proven to be precisely the dual of the Hopf algebra ${\mathcal {H}}_{2}$ used in regularity structures for handling the combinatorics of recentering. For this, we shall use the following proposition that characterizes the $\star _{2}$ product and was proven in [Reference Bruned and Manchon11, Prop. 3.17]:

Proposition 4.9. Let $\sigma = X^{k} \prod _i I_{a_{i}}(\sigma _{i}) $ . Then one has

$$ \begin{align*} I_{b}(\sigma \star_{2} \tau) = \tilde{\uparrow}^{k}_{N_{\tau}} \left( \prod_i I_{a_{i}}(\sigma_{i}) \, \widehat{\curvearrowright} \, I_{b}(\tau) \right), \end{align*} $$

where $ \tilde {\uparrow }^{k}_{N_{ \tau }} $ is defined by

(15) $$ \begin{align} \tilde{\uparrow}^{k}_{N_{\tau}} = \sum_{k = \sum_{v \in N_{\tau}} k_v} \uparrow^{k_v}_{v}. \end{align} $$

Theorem 4.10. The product $\star _{2}$ coincides with the product $*$ obtained by the previous construction by unfolding the post-Lie operation onto the universal envelope $U(\mathcal {V}_0)$ . One has for every $\sigma = X^{k} \prod _i I_{a_{i}}(\sigma _{i}) $

$$ \begin{align*} I_{b}(\sigma \star_{2} \tau) = \sigma \, \widehat{\triangleright} \, I_{b}( \tau). \end{align*} $$

Proof. When $ \sigma = X_i $ or $ \sigma = I_a(\hat {\sigma }) $ , this is just a consequence of the definition of $ \widehat {\triangleright } $ . If $\sigma = X_i I_{a}(\hat {\sigma })$ ; we make use of Proposition 3.1 (here $ \tilde {\uparrow }^{e_i} $ = $ \uparrow ^{i} $ ):

$$ \begin{align*} I_{b}(\sigma \star_{2} \tau) & = \uparrow^{i}_{N_{\tau}} \left( I_{a}(\hat{\sigma} ) \, \widehat{\curvearrowright} \, I_{b}(\tau) \right) \\ & = I_{a}(\hat{\sigma} ) \, \widehat{\curvearrowright} \, I_{b}( \uparrow^{i} \tau) - I_{a-e_i}(\hat{\sigma} ) \, \widehat{\curvearrowright} \, I_{b}(\tau) \\ & = I_{a}(\hat{\sigma} ) \, \widehat{\triangleright} \, \left( X_i \, \widehat{\triangleright} \, I_{b}( \tau) \right) - I_{a-e_i}(\hat{\sigma} ) \, \widehat{\triangleright} \, I_{b}(\tau) \\ & = I_{a}(\hat{\sigma} ) X_i \, \widehat{\triangleright} \, I_{b}( \tau) - \left( I_{a}(\hat{\sigma} ) \, \widehat{\triangleright} \, X_i \right) \, \widehat{\triangleright} \, I_{b}( \tau) - I_{a-e_i}(\hat{\sigma} ) \, \widehat{\triangleright} \, I_{b}(\tau) \\ & = I_{a}(\hat{\sigma} ) X_i \, \widehat{\triangleright} \, I_{b}( \tau) - I_{a-e_i}(\hat{\sigma} ) \, \widehat{\triangleright} \, I_{b}(\tau) \\ & = X_i I_{a}(\hat{\sigma} ) \, \widehat{\triangleright} \, I_{b}( \tau), \end{align*} $$

where we have used the definition of $ \widehat {\triangleright } $ in the fourth (see the recursive definition in section $2 $ ) and fifth lines above. We can then proceed by induction in order to prove it for $ \sigma = X^k \prod _i I_{a_i}(\sigma _i) $ following the recursive definition of the product giving in Section 2.

One may wonder how the post-Lie algebra introduced here relates to the post-Lie algebra on planar rooted trees. In fact, our post-Lie algebra can be interpreted as a quotient of the free object defined in Section 2. Indeed, we can consider the $ X_i $ as terminal edges. Then, we look at the set recursively defined by

$$ \begin{align*} \mathfrak{T}_p = \Big\{ \Xi \prod_{i} A_i, \, A_i \in \{ I_a(\tau), \, \tau \in \mathfrak{T}_p, \, a \in \mathbb{N}^{d+1} \} \cup \{ X_{i} \}_{i = 0,..., d} \Big\}. \end{align*} $$

Here the product $ \Pi _i $ is not commutative and therefore $ \mathfrak {T}_p $ contains planar decorated trees. Below, we provide an example of such decorated trees:

Notice that now there is no decorations on the nodes. Edges decorated by $ X_i $ represent the symbol $ X_i $ . We define new spaces $ \mathcal {V}_p $ and $ \tilde {\mathcal {V}}_p $

$$ \begin{align*} \mathcal{V}_p & = \Big \langle \{ I_a(\tau), \, a \in \mathbb{N}^{d+1}, \, \tau \in \mathfrak{T}_p \} \cup \{ X_{i} \}_{i = 0,..., d} \Big \rangle_{\mathbb{R}}, \\ \tilde{\mathcal{V}}_p & = \Big \langle \{ I_a(\tau), \, a \in \mathbb{N}^{d+1}, \, \tau \in \mathfrak{T}_p \} \Big \rangle_{\mathbb{R}}. \end{align*} $$

One can define the leftmost grafing product on these trees denoted by $ \widehat {\triangleright }_l $ . One grafts on the leftmost spot at the right of the noise $\Xi $ . We impose one restriction which is that one cannot graft on top of the $X_i$ . This gives

(16) $$ \begin{align} I_{a}(\tau) \, \widehat{\triangleright}_l \, X_{i} = 0, \quad X_i \, \widehat{\triangleright}_l \, X_{j} = 0. \end{align} $$

Then, for recovering our post-Lie product, we need to quotient by certain relations that are given by the Lie bracket (12)

(17) $$ \begin{align} \begin{aligned} X_i X_j & = X_j X_i, \quad I_a(\tau) I_b(\sigma) =I_b(\sigma) I_a(\tau), \\ I_a(\tau)X_i & = X_i I_a(\tau) + I_{a-e_i}(\tau). \end{aligned} \end{align} $$

Using these relations, we work in the basis described by $ \mathfrak {T} $ where now the $ X_{i} $ appear before the planted trees. We shall slightly abuse notation by still using the term leftmost grafting for the resulting operation after taking the quotient and will still denote it by $ \widehat {\triangleright }_l $ . Then, one has that the leftmost grafting product coincides with the post-Lie product $\widehat {\triangleright }$ .

Proof. We proceed by induction, using the basis of $\mathcal {V}$ and making use of the recursive construction of $\mathfrak {T}$ afforded to us by the sets $\mathfrak {T}_{p}$ defined above. We want to show that for every $\tau $ and $\sigma $ in $\mathcal {V}$ , one has

$$ \begin{align*} \sigma \, \widehat{\triangleright}_l \, \tau = \sigma \, \widehat{\triangleright} \, \tau. \end{align*} $$

When $ \tau = X_i $ , this is just a consequence of (16) and (11). Now, we suppose that $ \tau = I_b(\hat {\tau }) $ where

$$ \begin{align*} \hat{\tau} = \Xi X^{\ell} \prod_{j=1}^r I_{a_j}(\tau_j). \end{align*} $$

Then, one has

(18) $$ \begin{align} \sigma \, \widehat{\triangleright}_l \, \tau = I_b\left( \Xi \sigma X^{\ell} \prod_{j=1}^r I_{a_j}(\tau_j) + \Xi X^{\ell} \sum_{n=1}^r \left( \sigma \, \widehat{\triangleright}_l \, I_{a_n}(\tau_n) \right) \prod_{j \neq n} I_{a_j}(\tau_j) \right), \end{align} $$

which is the recursive defintion of $ \widehat {\triangleright }_l $ . We apply the induction hypothesis to conclude on the fact that for every $ n \in \{ 1,...,r \} $ ,

$$ \begin{align*} \sigma \, \widehat{\triangleright}_l \, I_{a_n}(\tau_n) = \sigma \, \widehat{\triangleright} \, I_{a_n}(\tau_n). \end{align*} $$

For the first term on the right-hand side of (18), one has if $ \sigma = X_i $ ,

$$ \begin{align*} \Xi \sigma X^{\ell} \prod_{j=1}^r I_{a_j}(\tau_j) & = \Xi X_i X^{\ell} \prod_{j=1}^r I_{a_j}(\tau_j) \\ & = \Xi X^{\ell+e_i} \prod_{j=1}^r I_{a_j}(\tau_j) = \Xi \left( X_i \, \widehat{\triangleright} X^{\ell} \right) \prod_{j=1}^r I_{a_j}(\tau_j). \end{align*} $$

If $ \sigma = I_a(\hat {\sigma }) $ , then

$$ \begin{align*} \Xi \sigma X^{\ell} \prod_{j=1}^r I_{a_j}(\tau_j) & = \Xi I_a(\hat{\sigma}) X^{\ell} \prod_{j=1}^r I_{a_j}(\tau_j) \\ & = \Xi \sum_{m} {\ell \choose m} X^{\ell - m}I_{a-m}(\hat{\sigma}) \prod_{j=1}^r I_{a_j}(\tau_j), \end{align*} $$

where we have used (17) repeatedly. In the end, one obtains

$$ \begin{align*} \sigma \, \widehat{\triangleright}_l \, \tau = I_b\left( \Xi \left(\sigma \, \widehat{\triangleright} \, X^{\ell} \right) \prod_{j=1}^r I_{a_j}(\tau_j) + \Xi X \sum_{n=1}^r \left( \sigma \, \widehat{\triangleright} \, I_{a_n}(\tau_n) \right) \prod_{j \neq n} I_{a_j}(\tau_j) \right), \end{align*} $$

which is exactly the recursive definition of $ \widehat {\triangleright } $ .

Below, we illustrate Proposition 4.11. We first compute the leftmost grafting and we use (17)

However, one has

We conclude by using the identification of consecutive edges of the form $ X^k $ at the right location of the noise $ \Xi $ to node decoration $ k $ which gives

5 A post-Lie algebra for multi-indices

Recall the Lie algebras of derivations on $\mathbb {R}[[z_{k}, z_{n}]]$ defined in [Reference Linares, Otto and Tempelmayr34], which are

$$ \begin{align*} \tilde{L} = \Big \langle \{ z^{\gamma}D^{(n)} \}_{[\gamma] \geq 0} \Big \rangle_{\mathbb{R}}, \end{align*} $$

which, equipped with the pre-Lie product $ \blacktriangleright $ defined in [Reference Linares, Otto and Tempelmayr34] by

$$ \begin{align*} z^{\gamma}D^{(n)} \blacktriangleright z^{\gamma'}D^{(n')} = \sum_{\beta'} (z^{\gamma}D^{(n)})^{\gamma'}_{\beta'} z^{\beta'} D^{(n')} \end{align*} $$

is also a pre-Lie algebra. We also consider

$$ \begin{align*} L = \Big \langle \{ z^{\gamma}D^{(n)} \}_{[\gamma] \geq 0} \cup \{ \partial_{i} \}_{i=0,...,d} \Big \rangle_{\mathbb{R}}. \end{align*} $$

Here $\gamma $ is a multi-index and $n \in \mathbb {N}_{0}^{d+1}$ . Note that we do not impose the condition $ |\gamma |_{{\mathfrak {s}}} - |n|_{{\mathfrak {s}}}> 0$ . The Lie bracket $ [\cdot ,\cdot ] $ on $ L $ is defined by

$$ \begin{align*} \begin{aligned} & [ z^{\gamma}D^{(n)}, z^{\gamma'}D^{(n')} ] = \sum_{\beta'} (z^{\gamma}D^{(n)})^{\gamma'}_{\beta'} z^{\beta'} D^{(n')} - \sum_{\beta} (z^{\gamma'}D^{(n')})^{\gamma}_{\beta} z^{\beta} D^{(n)} \\ & [\partial_i,\partial_j] = 0, \quad [z^{\gamma}D^{(n)}, \partial_i] = n_i z^{\gamma}D^{(n-e_i)} - \sum_{\beta} (\partial_i)^{\gamma}_{\beta} z^{\beta} D^{(n)}, \end{aligned} \end{align*} $$

where the $ \beta $ and $ \beta ' $ are such that $ [\beta ] \geq 0, [\beta '] \geq 0 $ .

Definition 5.1. The Lie algebra $L_{0}$ is the Lie algebra with underlying vector space L and the Lie bracket $[ x, y]_{0}$ which is defined as $[ x, y]_{0} = 0$ for $x, y \in \tilde {L}$ and $x, y \in \langle \partial _{i} \rangle _{\mathbb {R}}$ . We then define

$$ \begin{align*} [z^{\gamma}D^{(n)} , \partial_{i}]_{0} = n_{i}z^{\gamma}D^{(n-e_{i})}. \end{align*} $$

Proof. One carefully checks that the axioms of a post-Lie algebra hold. The proof is similar to the one for decorated trees. More specifically the axioms for a post-Lie algebra are

$$ \begin{align*} x \, \widehat{\blacktriangleright} \, [y,z]_0 = [x \, \widehat{\blacktriangleright} \, y,z]_0 + [y, x \, \widehat{\blacktriangleright} \, z]_0 \end{align*} $$

and

$$ \begin{align*} [x, y]_{0} \, \widehat{\blacktriangleright} \, z = a_{\widehat{\blacktriangleright}}(x,y,z) - a_{\widehat{\blacktriangleright}}(y,x,z). \end{align*} $$

As in the previous section, one easily checks that the first property is true. We prove the second property by distinguishing cases.

Case 1: If $x, y \in \langle \partial _{i} \rangle _{\mathbb {R}}$ , then either $z \in \langle \partial _{i} \rangle _{\mathbb {R}}$ and the property holds trivially, or $z \in \tilde {L}$ and we have $0$ on the left-hand side and only two non-zero terms on the right-hand side that cancel each other out. Hence, the property holds in this case.

Case 2: $x, y \in \tilde {L}$ and $z \in \langle \partial _{i} \rangle _{\mathbb {R}}$ , then all terms are $0$ by definition of the product $\widehat {\blacktriangleright }$ and the property holds trivially.

Case 3: If $x, y, z \in \tilde {L}$ , we notice that, in this case, the second property is equivalent to

$$ \begin{align*} [x, y]_{0} \, \widehat{\blacktriangleright} \, z = a_{{\blacktriangleright}}(x,y,z) - a_{\blacktriangleright}(y,x,z), \end{align*} $$

which is simply the pre-Lie property for the product $\blacktriangleright $ .

Case 4: If $y, z \in \tilde {L}$ and $x \in \langle \partial _{i} \rangle _{\mathbb {R}}$ , then from the left-hand side, we have

$$ \begin{align*} [x,y]_0 \ \widehat{\blacktriangleright} \ z = n_{i}z^{\gamma}D^{(n-e_i)} \blacktriangleright z^{\gamma '}D^{(n')}. \end{align*} $$

From the right-hand side, the following terms remain:

$$ \begin{align*} \begin{aligned} a_{\widehat{\blacktriangleright}}(x,y,z) & = z^{\gamma}D^{(n)} \, \widehat{\blacktriangleright} \, ( X_i \, \widehat{\blacktriangleright} \, z^{\gamma '}D^{(n')}) - (z^{\gamma}D^{(n)} \, \widehat{\blacktriangleright} \, X_i) \, \widehat{\blacktriangleright} \, z^{\gamma '}D^{(n')} \\ & = z^{\gamma}D^{(n)} \, \widehat{\blacktriangleright} \, (\partial_{i} z^{\gamma '}D^{n'}) \end{aligned} \end{align*} $$

because $z^{\gamma }D^{(n)} \, \widehat {\blacktriangleright } \, \partial _{i} = 0 $ . Then

$$ \begin{align*} \begin{aligned} a_{\widehat{\blacktriangleright}}(y,x,z) & = \partial_{i} \, \widehat{\blacktriangleright} \, (z^{\gamma}D^{(n)}) \, \widehat{\blacktriangleright} \, z^{\gamma '}D^{(n')}) - (\partial_{i} \, \widehat{\blacktriangleright} \, z^{\gamma}D^{(n)}) \, \widehat{\blacktriangleright} \, z^{\gamma '}D^{(n')} \\ & = \ \partial_{i}(z^{\gamma}D^{(n)} \blacktriangleright z^{\gamma '}D^{(n')} ) - (\partial_{i}z^{\gamma}D^{(n)})\blacktriangleright z^{\gamma '}D^{(n')}. \end{aligned} \end{align*} $$

Finally, one obtains

$$ \begin{align*} a_{\widehat{\blacktriangleright}}(x,y,z) - a_{\widehat{\blacktriangleright}}(y,x,z) & = z^{\gamma}D^{(n)} \, \blacktriangleright \, (\partial_{i} z^{\gamma '}D^{(n'}) - \partial_{i}(z^{\gamma}D^{(n)} \blacktriangleright z^{\gamma '}D^{(n')} ) \\ & + (\partial_{i}z^{\gamma}D^{(n)})\blacktriangleright z^{\gamma '}D^{(n')}, \end{align*} $$

which is equal to $n_{i}z^{\gamma }D^{(n-e_i)} \blacktriangleright z^{\gamma '}D^{(n')}$ after unwinding definitions, as desired.

Since any remaining cases are covered by symmetry, we have a post-Lie algebra. To finish the proof, we notice that the second claim of the lemma is immediate from the definitions of the brackets and the product $\widehat {\blacktriangleright }$ .

Remark 5.6. The map $ \Psi $ defined (9) can be extended to a pre-Lie morphism between the vector space $ \tilde {\mathcal {V}}_0 $ given by

$$ \begin{align*} \tilde{\mathcal{V}}_0 & = \Big \langle \{ I_a(\tau), \, a \in \mathbb{N}^{d+1}, \, \tau \in \mathfrak{T}_0 \} \Big \rangle_{\mathbb{R}} \end{align*} $$

and $ \tilde {L} $ . We denote this extension by $ \hat {\Psi } $ and it is given for every $ I_a(\sigma ) \in \tilde {\mathcal {V}}_0 $ by

$$ \begin{align*} \hat{\Psi}(I_a(\sigma)) =\Psi(\sigma) \frac{D^{(a)}}{a!}. \end{align*} $$

One also has

$$ \begin{align*} \hat{\Psi}( I_a(\sigma) \, \widehat{\curvearrowright} \, I_b(\tau) ) = \hat{\Psi}( I_a(\sigma) )\blacktriangleright \hat{\Psi}(I_b(\tau)) \end{align*} $$

where the pre-Lie product on the decorated trees is the one given by (10). To obtain a post-Lie morphism, one has to extend the map $ \hat {\Psi } $ on the $ X_i $ by setting

$$ \begin{align*} \hat{\Psi}(X_i) = \partial_i. \end{align*} $$

For example, one easily checks that

$$ \begin{align*} \hat{\Psi}(X_i \, \widehat{\triangleright} \, I_{a}(\sigma)) = \hat{\Psi}(\uparrow^{i} I_{a}(\sigma)) = \partial_{i} \, \widehat{\blacktriangleright} \, \Psi(\sigma)\frac{D^{(a)}}{a!} = \hat{\Psi}(X_i) \, \widehat{\triangleright} \, \hat{\Psi}(I_{a}(\sigma)), \end{align*} $$

where here $ \uparrow ^{i} $ applied to a decoration $ \prod _{j=1}^{k} X^{\ell _j} $ gives

$$ \begin{align*} \uparrow^{i} \prod_{j=1}^{k} X^{\ell_j} = X_i \prod_{j=1}^{k} X^{\ell_j} + \sum_{j=1}^k X^{\ell_{j} + e_i} \prod_{j' \neq j} X^{\ell_{j'}}. \end{align*} $$

Similarly, for the bracket, one gets

$$ \begin{align*} \hat{\Psi}([X_i, I_{a}(\sigma)]) = \hat{\Psi}(I_{a-e_{i}}(\sigma)) = \Psi(\sigma)\frac{D^{(a-e_{i})}}{(a-e_i)!} = [\hat{\Psi}(X_i), \hat{\Psi}(I_{a}(\sigma))]. \end{align*} $$

Due to the condition $ [\gamma ] \geq 0 $ , for any element of $ \tilde {L} $ , there exists a pre-image in $ \mathcal {V}_0 $ via $ \hat {\Psi } $ .

We are now able to use the main theorem in [Reference Ebrahimi-Fard, Lundervold and Munthe-Kaas24] which generalizes the Guin-Oudom theorem on the Lie enveloping algebra of a pre-Lie algebra to the case of a post-Lie algebra in order to obtain the following result:

Remark 5.8. Using the post-Lie morphism $ \hat {\Psi } $ , one has a direct connection between the product on multi-indices and the one on decorated trees. Indeed, the map $ \hat {\Psi } $ is in particular a Lie morphism for $[.,.]$ and so has a natural extension to $U(\mathfrak {g})$ . Hence, it commutes with the construction in Section 2. By virtue of this, it extends to a morphism of Hopf algebras and so one has for $ A, B \in U(L_0)$ ,

$$ \begin{align*} A \star B = \hat{\Psi}(x) \star \hat{\Psi}(y) = \hat{\Psi} ( x * y ), \end{align*} $$

where $ x, y \in U(\mathcal {V}_0) $ are such that $ A = \hat {\Psi }(x) $ and $ B = \hat {\Psi }(y) $ .