We build a general framework to compute the envelope of certain deformations of PostLie algebras and apply it to operator-valued non-commutative probability by proposing new transforms allowing computations of conditionally free and conditionally monotone multiplicative convolutions of operator-valued non-commutative distributions.

§ INTRODUCTION

This article highlights a connection between operator-valued (conditionally) free multiplicative convolution of non-commutative distributions and the theory of operated sets via, most notably, a certain factorization holding for Matching $\mathcal{O}$ operators (a definition original to the present work to the best of our knowledge) presented here as a generalization of the celebrated Semenov-Tian-Shanskii factorization.

This work gives a continuation of [12] and reveals how post-lie structures give some conceptual explanation to the factorization of the $T$-transform.

In [35], Voiculescu introduced the non-commutative analogue to the classical Fourier–Melin transform, dubbed by him the $T$-transform, to provide an algorithmic answer to the problem of computing the spectral distribution in a tracial probability space of the self-adjoint $\sqrt{a}b\sqrt{a}$ where $a \geq 0,b$ are two free operators.
A few years later, K. Dykema [10] extended the work of Voiculescu and defined the operator-valued $T$ transform. He observed a curious factorization of this transform over the multiplication of two elements which are free with amalgamation, to which we refer as the twisted factorization of the $T$-transform [10].

In the first part of the present work, we revisit the $T$-transform of Dykema in the abstract and explain how it can be used to perform computations in the theory of deformations of post-Lie algebras.
In the second part, we exploit the abstract formalism developed in the first part to the study of operator-valued conditionally free [23, 7] and operator-valued conditionally monotone multiplicative convolutions [27] non-commutative distributions.

To be more precise, we utilize this abstract $T$-transform to prove the factorization (in a sense that will be made precise) for certain deformations of structures adjacent to a $\mathcal{O}$ Lie operator, the associated Grossman-Larson product namely.

The present work deals with Rota–Baxter operators and certain generalizations thereof introduced by Kuperschmidt and named $\mathcal{O}$-operator, as an operator form of the Yang-Baxter equations[20]. These operators have initially been introduced by Baxter [6] in the realm of probability theory, following the work by Spitzer [34]. Later on, Rota [30] formalized the notion of what is nowadays known as Rota–Baxter operator on an associative algebra.

Remarkably, the Rota–Baxter identity is consistent with passing to the associated commutator Lie algebra, which yields the notion of a Rota–Baxter operator on a Lie algebra.
Rota–Baxter operators on a Lie algebra were already considered in [32] in the context of the – operator form of the – classical Yang–Baxter equation. Kuperschmidt [20] studied what may be considered a class of generalized Rota–Baxter operators defined over Lie modules. They will be called $\mathcal{O}$-Lie operators (defined over a given Lie module) and are of prime interest to this work.

Recently, interest has shifted towards the development of a theory of operated groups. Starting with the work [17], where the notion of Rota–Baxter operator on a group (and more generally of a $\mathcal{O}$-group operator in the terminology of the present work) is introduced, the theory was further developed in [4]. Motivations come from describing a global version of the (integrated) Semenov-Tian-Shanskii factorization Theorem [31]. In [5], connections with the quantum Yang–Baxter equation are drawn using the theory of (skew-left) braces.
A variety of algebraic structures accompanies a Rota–Baxter(–Lie) operator and, more generally, a $\mathcal{O}$-Lie operator. We will call them adjacent structures to a given $\mathcal{O}$-Lie operator. For instance, to any such operator corresponds a postLie algebra [8]. PostLie algebras are central to the theory of numerical integration of differential equations on homogeneous spaces with vanishing curvature (see [8, 14, 24, 25] and references therein).
A postLie algebra yields a second Lie algebra, referred to as the adjacent Lie algebra (whereas the base Lie algebra constitutive of a postLie algebra is referred to as the controlling Lie algebra).
One notable feature of this adjacent Lie algebra is the possibility of realizing its universal envelope over the universal envelope of the controlling Lie algebra (in the pre-Lie case, the latter is trivial) [26]. For free pre- and postLie algebras, the duals of the resulting Hopf algebras are known as  Butcher–Connes–Kreimer (BCK) respectively Munthe-Kaas–Wright (MKW) Hopf algebras.

More precisely, in the postLie case building the Hopf algebra implies extending the postLie product to polynomials on Lie elements; this is done by unfolding the Guin–Oudom recursion [26]. It results in an operation defined on the universal envelope of the controlling Lie algebra constitutive of a $D$-algebra [3] – or postHopf algebra in [22]. Finally, the author in [15] introduced and studied the notion of Rota–Baxter operator on a cocommutative Hopf algebra; we call them Rota–Baxter–Hopf operators. It extends both the notions of Rota–Baxter group operator and Rota–Baxter–Lie operator. Following Kuperschmidt, we propose the notion of a $\mathcal{O}$-Hopf operator. In a sense, it is the operator form of a postHopf algebra.

In this operator form, and this is the main observation of the present work, unfolding the Guin–Oudom recursion is equivalent to exponentiating with respect to an associative product (denoted $\#$ in this work; this is not to be confused with the compositional product of [2])). Given two $\{{\bf s, t}\}$ $\mathcal{O}-$ Lie operators over a Lie module $(\g,\vartriangleleft,\mathfrak{a})$ with $[-,-]_{\g}=0$ (for the sake of simplicity, we will state a more general result) satisfying some compatibility conditions, ${\bf s} + u{\bf t}$ is a $\mathcal{O}$-Lie operator for any value of the parameter $u$. The product $\#$ yields a closed formula for the operator form (a $\mathcal{O}$-Hopf operator $[{\bf s} + u{\bf t}]$) of the associative product integrating the deformed Lie bracket $[-,-]_{{\bf s}+u{\bf t}}$ adjacent to the $\mathcal{O}$-Lie operator ${\bf s} + u{\bf t}$, see Theorem <ref>.

We address applying this abstract results toward non-commutative probability, focusing on conditionally free and monotone probability. In particular, we pinpoint a Matching $\mathcal{O}$ group operator governing relations between the various transforms introduced to this day to compute non-commutative convolutions. With our new formalism and the notion of the crossed product of groups, we can give definitions for the conditional operator-valued $T$-transform, proving that it factorizes over the product of conditionally free with amalgamation random variables.

In a publication to appear soon, we study applications of postLie structures to represent the $T$-transform of multiplicative infinitely divisible operator-valued distributions.

§.§ Contributions

Let $(\a, \vartriangleleft, \g)$ be a Lie module.
We list our contributions, first for the abstract part,

* We define a product $\#$ (see Section <ref>) and prove that its restriction to co-algebraic maps (this yields a group which we call the Guin–Oudom group) between the envelopes of $\a$ and $\g$ yields an algebraic Lie group, which turns out to be itself a $\mathcal{O}$-group. We give properties of the subset of $\mathcal{O}$ Hopf operators with respect to this product.

* Secondly, we show how to recover the Semenov–Tian–Shanskii (see Theorem <ref>) factorization from the commutativity of two Rota–Baxter–Hopf operators (it is a $\mathcal{O}$ Hopf operator on the Lie module $(\g, {\rm ad}, \g)$) with respect to the product $\#$.

* We define the notion of Matching $\mathcal{O}$-Hopf operators (see Section <ref>) and provide several examples

and secondly, for applications to operator-valued free probability theory

* We define the conditional $T$ transform and prove the transform factorizes over the product of conditional free random variables in Theorem <ref>

* We define the conditional H transform and prove the transform factorizes over the product of conditional monotone random variables in Theorem <ref>

* We state subordination equations in Theorem <ref>

§ BACKGROUND

§.§ Notations

In this work, the letters $H$ and $K$ always denote co-commutative Hopf algebras. We use subscripts to distinguish between structural morphisms in the case where there is more than one Hopf algebra at stake, for example $\eta_H$ (unit), $\Delta_H$ (coproduct), $\varepsilon_H$ (counit),…We use ${\rm Prim}(H)$ to denote the Lie algebra of the  primitive elements of a Hopf algebra $H$,

{\rm Prim} (H) = \{ X \in H\colon \Delta_H(x)=x\otimes \eta_{H}+\eta_H \otimes x \}.

In general, we reserve small-case letters to denote "infinitesimal" objects, such as Lie–Rota–Baxter operators (defined further down) or Lie algebras. Uppercase letters are used to denote "integrated" objects, such as universal envelopes, Hopf algebras, etc.

§.§ Basic settings

For the sake of completeness, several basic definitions are recalled in the following.

* A right Lie module is a triple $(\mathfrak{g},\vartriangleleft, \mathfrak{a})$ where

* $\mathfrak{g}$ and $\mathfrak{a}$ are Lie algebras,

* $\vartriangleleft$ is an action of the Lie algebra $\mathfrak{a}$ on the Lie algebra $\mathfrak{g}$ by derivations:

\begin{aligned}
[x,y]_{\g} \vartriangleleft a
&= [x \vartriangleleft a,y]_{\g} + [x,y \vartriangleleft a]_{\g}\\
x \vartriangleleft [a,b]_{\mathfrak{a}}
&= [(\cdot)\vartriangleleft a, (\cdot)\vartriangleleft b]_{\textrm{End}(\g)}(x)
\end{aligned}

for any $ x,y \in \g$ and $a,b \in \mathfrak{a}$.

A morphism between two right Lie modules $(\g,\vartriangleleft,\a)$ and $(\g^{\prime}, \vartriangleleft^{\prime},\a^{\prime})$ is a pair $(\varphi, \theta)$ of morphisms of Lie algebras $\varphi:\g\to\g^{\prime}$ and $\theta:\a\to\a^{\prime}$ such that for any $x\in \g$ and $a\in\a$:

\varphi(x) \vartriangleleft' \theta(a) = \varphi(x\vartriangleleft a).

* A right group module is a triple $(G,\cdot,\Gamma)$ where

* $G$ and $\Gamma$ are groups,

* $\cdot$ is a right action of $\Gamma$ on $G$ by automorphisms:

\begin{align*}
&(xy) \cdot a
= (x\cdot a)(y\cdot a), \\
&(x\cdot a) \cdot b = x \cdot (ab),
\end{align*}

for any $x,y \in G,~ a,b \in \Gamma$.

A morphism between two right group modules $(G,\cdot,\Gamma)$ and $(G^{\prime}, \cdot^{\prime},\Gamma^{\prime})$ is a pair $(\Phi, \Theta)$ of group morphisms $\Phi:G\to G^{\prime}$ and $\Theta:\Gamma\to\Gamma^{\prime}$ such that for any $x \in G$ and $k \in \Gamma$:

\Phi(h) \cdot^{\prime} \Theta(k) = \Phi(h\cdot k).

* A right Hopf module is a triple $(H,\preliediff,K)$ where

* $(H,\Delta_{H},\eta_{H},\mathcal{S}_H)$ and $(K,\Delta_{K},\eta_{K},\mathcal{S}_K)$ are cocommutative Hopf algebras.

* Using Sweedler's notation:

\begin{align*}
\begin{aligned}
(h_1h_2) \preliediff k
&= (h_1 \preliediff k_{(1)})(h_2\preliediff k_{(2)})\\
(h_{(1)} \preliediff k_{(1)})\otimes(h_{(2)} \preliediff k_{(2)})
&=(h\preliediff k)_{(1)} \otimes (h \preliediff k)_{(2)}\\
\mathcal{S}_H(h\preliediff k)
&= \mathcal{S}_H(h) \preliediff k
\end{aligned}
\qquad h,h_1,h_2 \in H,~ k \in K.
\end{align*}

A morphism between right Hopf modules, $(H,\preliediff,K)$ and $(H^{\prime}, \preliediff^{\prime},K^{\prime})$, is a pair $(\Phi, \Theta)$ of Hopf algebra morphisms, $\Phi:H\to H^{\prime}$ and $\Theta:K\to K^{\prime}$, such that for any $h\in H$ and $k\in K$,

\Phi(h) \preliediff^{\prime} \Theta(k)
= \Phi(h \preliediff k).

We explain the notion of universal envelope of a right Lie module $(\g,\vartriangleleft, \a)$. This is an object $(H,\preliediff, K)$ in the category of right Hopf modules together with an injection $(\iota_\g,\iota_\a)$ of right Lie modules:

\iota_\g:\g \to {\rm Prim}(H),
\qquad
~\iota_\a :\a \to {\rm Prim}(K),

which is the solution to the following universal problem: let $(H^{\prime},\preliediff,K^{\prime})$ be a right Hopf module and $(\varphi,\theta)$ a pair a morphism of right Lie modules, $\varphi : \g \to {\rm Prim}(H^{\prime})$ and $\theta : \a \to {\rm Prim}(K^{\prime})$. Then there exists a morphism $(\Phi,\Theta)$ of right Hopf modules, $\Phi:H\to H^{\prime}$ and $\Theta:K\to K^{\prime}$, such that

\Phi \circ \iota_{H}
= \varphi,
\qquad
\Theta \circ \iota_{K}
= \theta.

See Fig. <ref> for a diagrammatic formulation.

\[\begin{tikzcd}
H && {H^{\prime}} & K && {K^{\prime}} \\
{{\rm Prim}(H)} && {{\rm Prim}(H^{\prime})} & {{\rm Prim}(K)} && {{\rm Prim}(K^{\prime})} \\
& {\mathfrak{g}} &&& {\mathfrak{a}}
\arrow["\Phi", from=1-1, to=1-3]
\arrow["{\iota_{\mathfrak{g}}}", from=3-2, to=2-1]
\arrow[from=2-1, to=1-1]
\arrow[from=2-3, to=1-3]
\arrow["\varphi"', from=3-2, to=2-3]
\arrow["\Theta", from=1-4, to=1-6]
\arrow[from=2-4, to=1-4]
\arrow[from=2-6, to=1-6]
\arrow["{\iota_{a}}", from=3-5, to=2-4]
\arrow["\theta"', from=3-5, to=2-6]
\end{tikzcd}\]

Envelope of a right Lie module.

It is not difficult to build a universal envelope of a right Lie module $(\g,\vartriangleleft,\a)$. We call $\mathcal{U}(\g)$ and $\mathcal{U}(\a)$ the universal enveloping algebras of $\g$ and $\a$, respectively. We "extend/co-extend" the action of $\a$ on $\g$ to an action of $\mathcal{U}(\a)$ on $\mathcal{U}(\g)$. First, pick $a\in \a$ and set for any $x_1 \cdots x_n \in  \mathcal{U}(\g)$:

x_1\cdots x_n \preliediff a
:= \sum_{i=1}^n x_1\cdots x_{i-1}(x_i \vartriangleleft a)x_{i+1}\cdots x_n.

We need to check that this is well-defined, which amounts to verifying the equality

[x_1,x_2]_{\g} \preliediff a = x_1 x_2 \preliediff a - x_{2}x_1 \preliediff a.

It is implied by the fact that $\a$ acts by derivations on $\g$. Notice that $\preliediff$ is a morphism from the Lie algebra $\a$ to the Lie algebra ${\rm End}_{\g}(\mathcal{U}(\g))$. Therefore, we extend $\preliediff$ to an algebra morphism from $\mathcal{U}(\a)$ to ${\rm End}(\mathcal{U}(\g))$. A simple induction shows that $X_1X_2 \preliediff A = (X_1 \preliediff A_{(1)})(X_2 \preliediff A_{(2)})$, for any $A \in \mathcal{U}(\a)$ and $X_1,X_2 \in \mathcal{U}(\g)$.

We let $(\varphi,\theta)$ be a morphism of right Lie modules, $\iota_{\mathcal{U}(\g)}$, $\iota_{H^{\prime}}$ and $\iota_{\mathcal{U}(\mathfrak{a})}$, $\iota_{K^{\prime}}$ are the canonical injections of the Lie algebras of primitive elements of $\mathcal{U}(\g)$, $H^{\prime}$, $\mathcal{U}(\a)$, and $K^{\prime}$, respectively. Since $\iota_{H^{\prime}} \circ \varphi$ and $\iota_{K^{\prime}} \circ \theta$  are Lie algebra morphisms, we extend them to algebra morphisms $\Phi: \mathcal{U}(\g)\to H^{\prime}$ and $\Theta: \mathcal{U}(\a)\to K'$. With these definitions, the diagrams in Fig. <ref> are commutative.

We denote by LieM, GroupM, and HopfM the categories of Lie modules, Group modules, and Hopf modules, respectively. The constructions exposed above produce a functor

\mathcal{U}: {\rm LieM} \to {\rm HopfM},

which is the left adjoint of a functor ${\rm{HLM}}$ that associates with any Hopf module a Lie module via a restriction to primitive elements:

{\rm HLM} : {\rm HopfM} \to {\rm LieM}.

§.§ On the Semenov–Tian-Shanskii factorization

We refer the reader to [31].
We state in this section the Semenov–Tian–Shanskii factorization for factorizable Lie bialgebras.
We recall that a factorizable Lie bialgebra is the data of a Lie algebra $\g$ together with a tensor $\bm{r} \in g \otimes g$ such that:

* the symmetric tensor $I := r_{12} + r_{21}$ is invertible and ad-invariant,

* $\bm{r}$ satisfies the classical Yang–Baxter equation:

\begin{equation}
\label{eqn:cyb}
[r_{12},r_{13}] + [r_{12},r_{23}]+[r_{13},r_{23}]=0.
\end{equation}

Recall that (<ref>) is equivalent to:

\begin{equation*}
\end{equation*}

for any $x^{\star},y^{\star} \in \g^{\star}$. The above equation implies that $[-,-]_{\star}\colon \g^{\star}\otimes\g^{\star}\to\g^{\star}$ defined by

[-,-]_{\star}(x^{\star},y^{\star})= ad^{\star}(\bm{r}(x^{\star}))(y^{\star})-ad^{\star}(\bm{r}^{\star}(y))(x^{\star}),~x^{\star},y^{\star} \in \g^{\star}

is a Lie bracket, and therefore that $\bm{r}$ is a morphism of Lie algebras.
In the following, $\bm{r}_+ = \bm{r}$ and $\bm{r}_-=-r_{21}$. Therefore, $\bm{r}_+ = I + \bm{r}_-$ and $\bm{r}_-$ satisfy the classical Yang–Baxter equation.
We use the same symbols $\bm{r}_+$ (resp. $\bm{r}_-$) for the extension and coextension of $\bm{r}_+$ (resp. $\bm{r}_-$) to the envelope $\mathcal{U}(\g_+, [-,-]_+)$ (resp. $\mathcal{U}(\g),[-,-]_-$) with values in $\mathcal{U}(\g)$.
We denote by $G$ the group-like elements of $\mathcal{U}(\g)$ and $G_+$ (resp. $G_-$) the group-like elements of $\mathcal{U}(\g_+, [-,-]_+)$ (resp. $\mathcal{U}(\g_-, [-,-]_-)$)).

Let $(\mathfrak{g},\bm{r})$ be a factorizable Lie bialgebra.

* Then, for any $x \in \g$, there exists a unique couple $(x_-,x_+)$ with $x_- \in {\rm Im}(\bm{r}_-)$ and $x_+ \in {\rm Im}(\bm{r}_+)$ such that:

x = x_+ - x_-.

* Then any $X \in \mathcal{U}(\g)$ admits an unique representation:

X = \sum_{i} X^{(i)}_+ \otimes \mathcal{S}_{\mathcal{U}(\g)}(X^{(i)}_-),

such that:

\sum_i X^{(i)}_+ \otimes X^{(i)}_- =  (\bm{r}_+ \otimes \bm{r}_-) \circ \Delta \circ (I^{-1}(X)).

* Let $g \in G$ be a group-like element in $\mathcal{U}(\g)$. Then $g$ admits a unique representation:

g = g_+g_-^{-1},

where $g_+ \in \tilde{G}_+={\rm Im}(\bm{r}_+ : G_+\to G)$ and $g_- \in \tilde{G}_- = {\rm Im}(\bm{r}_-: G_- \to G)$.

§ KUPERSCHMIDT'S $\MATHCAL{O}$ OPERATORS AND BEYOND

We recall the definitions of an $\mathcal{O}$ -Lie operator on a Lie module $(\g, \vartriangleleft, \a)$ [21] and that of a $\mathcal{O}$ group operator [17] on a right group module $(G,\cdot,\Gamma)$ (see the previous section). We also formalize the notion of $\mathcal{O}$-Hopf operators  in the context of a right Hopf module $(H,\preliediff, K)$, as suggested in [15]. Recall first the definition of a post-Lie algebra; see, for example, [8].

A postLie algebra $(\g,\vartriangleleft)$ consists of a Lie algebra $(\g,[-,-]_{\g})$ and a binary operation $\vartriangleleft: \g \otimes \g \to \g$ satisfying for any $x,y,z \in \g$:

\begin{align}
[x,y]\vartriangleleft z
&= [x\vartriangleleft z,y]+[x,y\vartriangleleft z] \label{postlie1}\\
x \vartriangleleft [y,z]
&= (x \vartriangleleft y) \vartriangleleft z - x \vartriangleleft (y\vartriangleleft z)
- ((x \vartriangleleft z) \vartriangleleft y - x \vartriangleleft (z \vartriangleleft y)) \label{postlie2}
\end{align}

§.§ Types of  $\mathcal{O}$ operators

Let $(\mathfrak{g}, \vartriangleleft, \mathfrak{a})$ be a right Lie module. We call a $\mathcal{O}$-Lie operator a map ${\bf t} : \mathfrak{g} \to \mathfrak{a}$ satisfying for any $x,y \in \g$:

\begin{equation}
\label{eqn:olie}
[{\bf t}(x),{\bf t}(y)]_{\mathfrak{a}} = {\bf t}(x \vartriangleleft {\bf t}(y)) - {\bf t}(y \vartriangleleft {\bf t}(x)) + {\bf t}([x,y]_\mathfrak{g}).
\end{equation}

We call $\mathcal{O}$-Lie algebra the tuple $(\mathfrak{g}, \vartriangleleft, \mathfrak{a}, {\bf t})$. A morphism between two $\mathcal{O}$-Lie algebras, $(\mathfrak{g}, \vartriangleleft, \mathfrak{a}, {\bf t})$ and $(\mathfrak{g}^{\prime},\vartriangleleft^{\prime}, \mathfrak{g}^{\prime}, {\bf t}^{\prime})$, is a pair $(\varphi,\theta)$ of Lie algebra morphisms, $\varphi: \g \to \g^{\prime}$ and $\theta : \a \to a^{\prime}$, such that:

\theta \circ {\bf t}
= {\bf t^{\prime}} \circ \varphi.

Note that equation (<ref>) is not linear; the sum of two $\mathcal{O}$-Lie operators is, in general, not an $\mathcal{O}$-Lie operator.

If $\mathfrak{a} = \mathfrak{g}$ and $\vartriangleleft$ is the adjoint action of the Lie algebra $\g$ over itself:

\begin{equation*}
x \vartriangleleft y =[x,y],~x,y \in \g,
\end{equation*}

then a $\mathcal{O}$-Lie operator is a Rota–Baxter operator (of weight one) on the Lie algebra $\g$.

Let $(G,\cdot, \Gamma)$ be a right group module. We call an $\mathcal{O}$-group operator a map ${\bf T}:G \to \Gamma$ satisfying for any $h,g \in G$,

\begin{equation}
{\bf T}(g){\bf T}(h)
= {\bf T}((g \cdot {\bf T}(h))h).
\end{equation}

We call $\mathcal{O}$-group the tuple $(G,\cdot,\Gamma, {\bf T})$. A morphism between two $\mathcal{O}$-groups, $(G,\cdot,\Gamma, {\bf T})$ and $(G^{\prime},\cdot^{\prime},\Gamma^{\prime}, {\bf T}^{\prime})$, is a pair $(\Phi,\Theta)$ of group morphisms $\Phi : G \to G^{\prime}$ and $\Theta : \Gamma \to \Gamma^{\prime}$ such that

\Theta \circ {\bf T}^{\prime}
= {\bf T}\circ \Phi.

If $\Gamma = G$ and $\cdot$ is the right adjoint action $Ad$ of $G$ on itself, then an $\mathcal{O}$-group operator is a Rota–Baxter group operator and $(G,Ad,G,{\bf T})$ is a Rota–Baxter group (of weight one) [17].

Let $(H, \preliediff, K)$ be a right Hopf module. We call an $\mathcal{O}$-Hopf operator a coalgebra map ${\bf T}: H \to K$ satisfying for any $e,f \in H$,

\begin{equation}
\label{def:Ohopfoperator}
{\bf T}(e){\bf T}(f)
={\bf T}((e \preliediff {\bf T}(f_{(1)}))f_{(2)}).
\end{equation}

We call $\mathcal{O}$-Hopf algebra the tuple $(H,\preliediff, K, {\bf T})$. A morphism between two $\mathcal{O}$-Hopf algebras $(H,\preliediff,K, {\bf T})$ and $(H^{\prime},\preliediff^{\prime},K^{\prime}, {\bf T}^{\prime})$, is a pair $(\Phi,\Theta)$ of Hopf algebra morphisms, $\Phi : H \to H^{\prime}$ and $\Theta : K \to K^{\prime}$, such that

\Theta \circ {\bf T}^{\prime}
= {\bf T}\circ \Phi.

§.§ Adjacent structures

Recall first the definition of a post-Lie algebra; see, for example, [8].

A postLie algebra $(\g,\vartriangleleft)$ consists of a Lie algebra $(\g,[-,-])$ and a binary operation $\vartriangleleft: \g \otimes \g \to \g$ satisfying for any $x,y,z \in \g$:

\begin{align}
[x,y]\vartriangleleft z
&= [x\vartriangleleft z,y]+[x,y\vartriangleleft z] \label{postlie1}\\
x \vartriangleleft [y,z]
&= (x \vartriangleleft y) \vartriangleleft z - x \vartriangleleft (y\vartriangleleft z)
- ((x \vartriangleleft z) \vartriangleleft y - x \vartriangleleft (z \vartriangleleft y)) \label{postlie2}
\end{align}

We propose here to review algebraic structures adjacent to an $\mathcal{O}$-operator of any type, that is, Lie, Hopf, group.
The following proposition is well known.

Let ${\bf t} : \g \to \a$ be an $\mathcal{O}$-Lie operator over $(\g,\vartriangleleft,\a)$. The binary operation $\vartriangleleft_{\bf t}\colon \g\otimes\g \to \g$ defined for any $x,y \in \g$ by:

\begin{equation*}
x \vartriangleleft_{\bf t} y = x \vartriangleleft {\bf t}(y).
\end{equation*}

yields a postLie algebra $(\g,\vartriangleleft_{\bf t})$.

The postLie axioms imply that the bracket defined, for $x,y \in \g$, by:

[x,y]_{\bf t}
:= x \vartriangleleft_{\bf t} y - y \vartriangleleft_{\bf t} x  + [x,y]_{\g}

is a Lie bracket. We denote by $\g_{\bf t}$ the Lie algebra with the same vector space as $\g$ but with bracket $[-,-]_{\bf t}$. Then, $(\g, \vartriangleleft_{\bf t})$ is postLie if and only if $(\g,\vartriangleleft_{\bf t}, \g_{\bf t})$ is a right Lie module. In that case,

({\rm id}_{\g},{\bf t}) : (\g,\vartriangleleft_{\bf t},\g_{\bf t}) \to (\g,\vartriangleleft,\a)

is a morphism of right Lie modules.
The converse statement of Proposition <ref> is also true; any postLie algebra is a right Lie module of the form $(\g, \vartriangleleft_{\bf t}, \g_{\mathfrak{t}})$ where ${\bf t} = \rm{id}_{\mathfrak{g}}$. It will be convenient to call such a triple $(\g, \vartriangleleft_{\bf t}, \g_{\bf t})$  a right post-Lie module.
Now we describe structures adjacent to a $\mathcal{O}$-Hopf operator $\bT$ on a right Hopf module $(H,\preliediff,K)$. See [22] for the proofs of the following propositions.

Consider the right Hopf module $(H,\preliediff,K)$ with $\mathcal{O}$-Hopf operator $\bT$.
Define the bilinear operation $\star_{\bT}$ between elements of $H$ by, for any $e,f \in H$:

\begin{equation}
\label{eqn:productohopfoperator}
e \star_{\bf T} f
:= (e \preliediff {\bf T} (f_{(1)}))f_{(2)}.
\end{equation}

Then $\star_{\bf T}$ is associative and $H_{\bf T} := (H, \star_{\bT}, \Delta)$ is a co-commutative Hopf algebra. We call $\mathcal{S}_{\bf T}$ its antipode. Additionally, the binary operation $\vartriangleleft_{\bT}\colon H \otimes H \to H$ defined for any $e,f \in H$ by:

\begin{equation*}
e \vartriangleleft_{\bf T} f = e \preliediff {\bf T}(f)
\end{equation*}

yields a right Hopf module $(H,\preliediff_{\bT},H_{\bT})$ and

({\rm id_H,\bT}) : (H,\preliediff_{\bT},H_{\bT}) \to (H, \preliediff, K)

is a morphism between right Hopf modules.

The Hopf algebra $H_{\bT}$ is called a postHopf algebra in [19]. It goes also by the name of $D$-algebra; see, for instance, [3]. We call such a triple  $(H,\preliediff_{\bT},H_{\bT})$ a postHopf module. We now describe structures adjacent to a $\mathcal{O}$-group operator on a group module $(G,\cdot,\Gamma)$.

Let ${\bf T} : G \to \Gamma$ be an $\mathcal{O}$-group operator over $(G,\cdot,\Gamma)$. Define for any $f,e \in G$ the product

f \star_{\bT} e
:= (f\cdot {\bf T}(e))e.

Then $\star_{\bT}$ is associative and gives a group $(G,\cdot_{\bT})$.
Additionally, the operation $\cdot_{\bf T}\colon G \times G \to G$ defined for any $f,e \in G$ by:

\begin{equation*}
f \cdot_{\bT} e
= f \cdot \bT(e)
\end{equation*}

yields a right group module $(G,\cdot_{\bT},G_{\bT})$ and

({\rm id_G}, {\bf T}) : (G,\cdot_{\bT},G_{\bT}) \to (G, \cdot, \Gamma).

is a morphism of right group-modules.

The inverse of an element $g\in G_{\bT}$ is $g^{-1}\cdot \bT(g)^{-1}$. In the remaining part of the paper, to lift ambiguities, we will refer to the product in the notations for inverses, for example

g^{-1^{\star_{\bT}}} = g^{-1^{\centerdot}}\cdot \bT(g)^{-1^{\circ}}

with $\centerdot$ the internal product on $G$ and $\circ$ the internal product on $\Gamma$. We will also allow ourselves to forget replace $\star_{\bT}$ by $\bT$ to keep the notations contained.

We call a right group module of the form $(G, \cdot_{\bT}, G_{\bT})$ a postGroup module.

We explain how a $\mathcal{O}$ -Lie operator ${\bf t}$ over a right Lie module $(\g,\vartriangleleft,\mathfrak{a})$ yields an $\mathcal{O}$-Hopf operator ${\bf T}$ on the universal envelope $(\mathcal{U}(\g),\preliediff, \mathcal{U}(\mathfrak{a}))$ of the Lie module $(\g,\vartriangleleft,\a)$. Given $E \in \mathcal{U}(\g)$ and $h \in \g$,
we define the map $[{\bf t}] : \mathcal{U}(\g) \to \mathcal{U}(\a)$ recursively by the system:

\begin{align}
\tag{GO}
\begin{aligned}
\label{eqn:RBop}
&[{\bf t}](Eh) := [{\bf t}](E) \mathbf{t}(h) - [{\bf t}](E\preliediff \mathbf{t}(h)), \\
&[{\bf t}](h)  := \mathbf{t}(h) \\
&\bf T(\mathbf{1})=\mathbf{1}.
\end{aligned}
\end{align}

Inspired  [26], we refer to (<ref>) as the  Guin–Oudom recursion with initial values ${\bf t}$.

Note that the above recursion is well-defined only for $\mathcal{O}$-Lie operators of weight $1$. In particular, it is not possible to extend an $\mathcal{O}$-Lie operator of weight $\lambda \in \mathbb{R}$ by using this recursion.

For any $\mathcal{O}$-Lie operator $\mathbf{t}$ on the right Lie module $(\mathfrak{g},\vartriangleleft, \mathfrak{a})$ the operator $\bf T$ obtained by the Guin–Oudom recursion (<ref>) with initial condition ${\bf t}$ is a coalgebra morphism.

We proceed by induction. For an integer $N \geq 1$, we denote by $\mathcal{U}(\g)^{(\leq N)}$ the image of $\bigoplus_{k \leq N} V^{\otimes k} \subset T(\g)$ by the natural projection. Since $\bf T$ equals $\mathbf{t}$ on primitive elements, we see that for any $h \in \g$, $[{\bf t}] \otimes [{\bf t}] \circ \Delta(h) = \Delta [{\bf t}](h)$. Let $n \geq 1$. We assume that $\bf T$ satisfies for any $E \in \mathcal{U}(\g)^{(\leq N)}:$

\begin{equation}
[{\bf t}] \otimes [{\bf t}] \circ \Delta (E)
= {\Delta} [{\bf t}](E).
\end{equation}

Now pick a – primitive – element $h \in \g$, then

\begin{align*}
\Delta [{\bf t}](Eh)
&=\Delta\big([{\bf t}](E) [{\bf t}](h)-[{\bf t}](E\preliediff [{\bf t}](h)) \big) \\
&=[{\bf t}](E_{(1)}) [{\bf t}](h)_{(1)} \otimes ([{\bf t}](E_{(2)}) [{\bf t}](h)_{(2)}) - [{\bf t}](E_{(1)}\preliediff [{\bf t}](h)_{(1)}) \otimes [{\bf t}](E_{(2)}\preliediff [{\bf t}](h)_{(2)}) \\
&=[{\bf t}](E_{(1)}) [{\bf t}](h)\otimes [{\bf t}](E_{(2)}) + [{\bf t}](E_{(1)})\otimes [{\bf t}](E_{(2)}) [{\bf t}](h) \\
&\hspace{2cm}-\big([{\bf t}](E_{(1)} \preliediff [{\bf t}](h)) \otimes [{\bf t}](E_{(2)}) + [{\bf t}](E_{(1)})\otimes [{\bf t}](E_{(2)}\preliediff [{\bf t}](h))\big)\\
&=\big([{\bf t}](E_{(1)}) {\bf t}(h)-[{\bf t}](E_{(1)} \preliediff {\bf t}(h))\big)\otimes [{\bf t}](E_{(2)})\\
&\hspace{2cm} + [{\bf t}](E_{(1)})\otimes \big([{\bf t}](E_{(2)}) {\bf t}(h)-[{\bf t}](E_{(2)}\preliediff {\bf t}(h))\big) \\
&= {[{\bf t}]} \otimes {[{\bf t}]} \circ \Delta (Eh).
\end{align*}

Let $\mathbf{t}$ be an $\mathcal{O}$-Lie operator on the Lie module $(\g,\vartriangleleft,\mathfrak{a})$. Then the operator $[{\bf t}]$ following from the Guin–Oudom recursion (<ref>) is a $\mathcal{O}$-Hopf operator on $(\mathcal{U}(\g), \preliediff, \mathcal{U}(\mathfrak{a}))$:

\begin{equation}
\label{eqn:secondcompat}
[{\bf t}](E)[{\bf t}](F) = [{\bf t}](E \star_{[{\bf t}]} F),
\quad E,F \in \univalgebra.
\end{equation}

We set for $E\otimes F \in \mathcal{U}(\g) \otimes \mathcal{U}(\g)$

\begin{align}
\alpha(E\otimes F) := [{\bf t}](E) [{\bf t}](F),
\quad
\beta(E\otimes F) := [{\bf t}](E\star_{[{\bf t}]}F)
\end{align}

We proceed by induction. The formula holds whenever $F$ is primitive. We assume $\alpha(E\otimes F) = \beta(E\otimes F)$ whenever $F \in \mathcal{U}(\g)^{\leq (N)}$ for a certain $N\geq 1$. Pick $h \in \g$ and $E,F \in Ker(\varepsilon)$. Then

\begin{align*}
[{\bf t}](E\star_{[{\bf t}]}(Fh))
& = [{\bf t}](E\star_{[{\bf t}]}(F\star_{[{\bf t}]}h)) - [{\bf t}](E\star_{[{\bf t}]}(F\preliediff \mathbf{t}(h))) \\
&=[{\bf t}](E\star_[{\bf t}] F) {\bf t}(h) - [{\bf t}](E)[{\bf t}](F\preliediff \mathbf{t}(h))\\
&=[{\bf t}](E) \big([{\bf t}](F)[{\bf t}](h) - [{\bf t}](F\preliediff {\bf t}(h)) \big)\\
&=[{\bf t}](E)\preliediff [{\bf t}](F\star_{[{\bf t}]}h) - [{\bf t}](F\preliediff \mathbf{t}(h))\\
&=[{\bf t}](E)[{\bf t}](Fh).
\end{align*}

Any $\mathcal{O}$-Hopf operator on $(\mathcal{U}(\g), \preliediff, \mathcal{U}(\mathfrak{a}))$ satisfies the recursion (<ref>) with $\bf t$ the restriction of ${\bf T}$ to $\g$; $[\bf t] = \bT$. Therefore, two $\mathcal{O}$ -Hopf operators coincide if their restrictions to $\g$  coincide.

Following the above corollary, we should forget about the brackets for the extension of a $\mathcal{O}$-Lie operator ${\bf t}$.

The envelope of the Lie algebra $\mathfrak{g}_{\bf t}$ is isomorphic to $H_{\bT}$

The proof follows from [13] and the observation that, with $\bT$ the $\mathcal{O}$-Hopf operator obtained by unfolding <ref> and $\star_{\bT}$ and $\vartriangleleft_{\bT}$ defined as above, for any $E,F \in \mathcal{U}(\g)$ and $f\in \g$

\begin{align*}
&E \star_{\bT}(F) = E \vartriangleleft_{\bT}(F_{(1)}) F_{(2)} \\
& E \vartriangleleft_{\bT} (Ff) = (E \vartriangleleft_{\bT} F) \vartriangleleft_{\bT}f)-E \vartriangleleft_{\bT} (F \vartriangleleft_{\bT}f)
\end{align*}

§.§ On the antipode of a postHopf algebra

For the entire section, we fix a Lie algebra module $(\g,\vartriangleleft,\a)$. Uppercase letters will be used for co-algebra morphisms, e.g $T\colon\mathcal{U}(\g)\to\mathcal{U}(\a)$. For the remaining part of the paper, bold letters ($\bm{t}, \bm{T}$ …) will be used to denote $\mathcal{O}$ operators. In this Section, we define a linear mapping between coalgebra morphisms from $\mathcal{U}(\g)$ to $\mathcal{U}(\a)$ and coalgebra diffeomorphisms of $\mathcal{U}(\g)$. We pinpoint a fixed-point equation satisfied by the antipode of $H_{\bf T}$ when $\bm{T}$ is a $\mathcal{O}$ Hopf operator.

Let $\bT\coloni \mathcal{U}(\g)\to \mathcal{U}(\a)$ be a coalgebra morphism. We define $e_{\bT} \colon \mathcal{U}(\g) \to \mathcal{U}(\g)$ as the unique linear  mapping satisfying:

\begin{equation*}
\varepsilon_K \circ e_{\bT}
= \varepsilon_H,
\end{equation*}

and for any element $f \in \mathcal{U}(\g)$:

\begin{align*}
&e_{\bT}(f) = f_{(1)} \preliediff \bT \circ e_{\bT}(f_{(2)}),
\end{align*}

using Sweedler's notation $\Delta_{ \mathcal{U}(\g)}(f)=f_{(1)} \otimes f_{(2)}$.

Let $x,y,z \in H$ are primitive elements of $H$ then

\begin{align*}
&= x\\
&=xy + x \preliediff \bT\circ e_T (y) + y \preliediff T\circ e_T(x) \\
&= xy
+ x \preliediff {T}(y)
+ y \preliediff { T}(x).\\
&= xyz
+ xy \preliediff {T}(z)
+ xz \preliediff {T}(y)
+ yz \preliediff {T}(x)\\
& \quad
+ x \preliediff  Te_T(yz)
+ y \preliediff {T}e_T(xz)
+ z \preliediff {T}e_T(xy).
\end{align*}

When $T$ is a $\mathcal{O}$-Hopf operator, implying that

\bT(xy)=\bT(x)\bT(y)-\bT(x\preliediff \bT(y)),

one gets:

\begin{align*}
&= xyz
+(x \preliediff {\bf T}(z))y
+x(y \preliediff {\bf T}(z)) \\
&\quad + (x \preliediff {\bf T}(y))z
+ x (z \preliediff {\bf T}(y)) \\
&\quad + (y \preliediff {\bf T}(x))z
+ y(z \preliediff {\bf T}(x)) \\
&\quad + (x \preliediff {\bf T}(y)) \preliediff {\bf T}(z)
+ x \preliediff {\bf T}(z \preliediff {\bf T}(y)) \\
&\quad + (y \preliediff {\bf T}(x)) \preliediff {\bf T}(z)
+ y \preliediff {\bf T}(z \preliediff {\bf T}(x)) \\
&\quad + (z \preliediff {\bf T}(x)) \preliediff {\bf T}(y)
+ z \preliediff {\bf T}(y \preliediff {\bf T}(x)).
\end{align*}

If $T$ is only assumed to be a coalgebra morphism, one gets instead:

\begin{align*}
&= xyz
+(x \preliediff {\bf T}(z))y
+x(Y \preliediff {\bf T}(z)) \\
&\quad + (x \preliediff {\bf T}(y))z
+ x (z \preliediff {\bf T}(y)) \\
&\quad + (y \preliediff {\bf T}(x))z
+ y(z \preliediff {\bf T}(x)) \\
&\quad +(x \preliediff {\bf T}(yz)+x\preliediff \bT(y \preliediff\bT(z))
+ x \preliediff \bT(z \preliediff \bT(y))  \\
&\quad + (y \preliediff {\bf T}(xz)+ y \preliediff \bT(x\preliediff\bT(z))
+ x \preliediff \bT(z \preliediff \bT(y)) \\
&\quad + (z \preliediff {\bf T}(xy)+z\preliediff \bT(x\preliediff\bT(y))
+ x \preliediff \bT(z \preliediff \bT(y)) .
\end{align*}

In the following propositions, we collect the basic properties of the transform $e_T$ of a coalgebra morphism.

The map $e_{T}$ is an isomorphism of coalgebras.

For any $f \in \mathcal{U}(\g)$, one has:

e_{T}(f) = f_{(1)} \preliediff T\circ e_{T}(f_{(2)})
=(\text{id} \preliediff T\circ e_{T}) \Delta(f).

The strategy of the proof consists in showing that
$e_{T}(f)_{(1)} \otimes e_{T}(f)_{(2)}$ and $e_{T}(f_{(1)})\otimes e_{T}(f_{(2)})$ satisfy the same fixed point equation (this will be sufficient since both equal $1\otimes 1$ if $f=1$).
To this end, we should extend $\preliediff $ to $\mathcal{U}(\g)\otimes \mathcal{U}(\g)$ by setting

f \otimes g \preliediff  c\otimes d = f \preliediff  c \otimes g \preliediff d, \quad  f,g,c,d \in H.

From $[f \preliediff T(g)]_{(1)} \otimes [f \preliediff T(g)]_{(2)}$ = $(f_{(1)} \preliediff T\circ g_{(1)}) \otimes (f_{(2)} \preliediff T(g_{(2)}))$,  for $f,g \in H$, it is not difficult to see that

\alpha \coloni f \mapsto e_{T}(f)_{(1)} \otimes e_{T}(f)_{(2)}

solves the fixed point equation:

\alpha (f)
= (f_{(1)} \otimes f_{(2)}) \preliediff T\circ 	\alpha (f_{(3)}).

In fact, for any $f \in H$:

\begin{align*}
\alpha (f)
=e_{T}(f)_{(1)} \otimes e_{ T}(f)_{(2)}
&= [ f_{(1)} \preliediff T\circ e_{ T}(f_{(2)})]_{(1)} \otimes  [f_{(1)} \preliediff T\circ e_{T}(f_{(2)})]_{(2)}\\
&=f_{(1)} \otimes f_{(2)} \preliediff T\circ e_{T}(f_{(3)})_{(1)} \otimes e_{T}(f_{(3)})_{(2)} \\
&= (\text{id} \preliediff T \circ \alpha)(\Delta \otimes \text{id})\Delta(f).
\end{align*}

On the other hand, from cocommutativity of $H$, we get:

\begin{align*}
e_{T}(f_{(1)}) \otimes e_{T}(f_{(2)})
&= [ f_{(1)} \preliediff T\circ e_{T}(f_{(2)})] \otimes  [f_{(3)} \preliediff T\circ e_{T}(f_{(4)})]\\
&=f_{(1)} \otimes f_{(3)} \preliediff T\circ e_{T}(f_{(2)}) \otimes e_{T}(f_{(4)}) \\
&=f_{(1)} \otimes f_{(2)} \preliediff T\circ e_{T}(f_{(3)}) \otimes e_{T}(f_{(4)}).
\end{align*}

This ends the proof.

In the following proposition, we denote by $T^{-1^*}$ the inverse the inverse of the coalgebra morphism $T$ with respect to the convolution product $*$ .

Let ${T} : \mathcal{U}(\g) \to \mathcal{U}(\a)$ be a coalgebra morphism. The inverse of $e_T$ for the compositional product $\circ$ on $Hom_{Vect}(\mathcal{U}(\g),\mathcal{U}(\g))$ admits the following closed formula:

\begin{align*}
= \mathrm{id} \preliediff {T}^{-1^{*}}.
\end{align*}

One has

\begin{align*}
(\mathrm{id}\preliediff T^{-1^{*}})\circ e_{{T}}
&= e_{{T}} \preliediff {T}^{-1^*} e_{{ T}} \\
&=\mathrm{id} \preliediff {T} e_{{T}} * {T}^{-1^{*}}e_{{T}} \\
&= \mathrm{id}\preliediff ({T} * {T}^{-1^*} )\circ e_{T} \\
&=\mathrm{id} \preliediff \eta_{\mathcal{U}(\a)}\circ\varepsilon_{\mathcal{U}(\g)} \\
\end{align*}

Let ${\bf T}: \mathcal{U}(\g) \to \mathcal{U}(\a)$ be an operator $\mathcal{O}$ -Hopf then
$$ \mathrm{id}\preliediff {\bf T}^{-1^{*}}
= e_{{\bf T}}^{-1} = \antipode_{\mathcal{U}(\a)} \circ e_{\bf T} \circ \antipode_{\mathcal{U}(\g)} = e_{{\bf T} \circ \mathcal{S}_{\mathcal{U}(\g)}}.

It is enough to prove $e_{\bT \circ \antipode_{\mathcal{U}(\g)}} = e_{\bT}^{-1}$. We compute $e_{\bT} \circ e_{\bT \circ \antipode_{\mathcal{U}(\g)}}$.

\begin{align*}
e_{\bT} \circ e_{\bT \circ \antipode_{\mathcal{U}(\g)}}
&= e_{\bT \circ \antipode_{\mathcal{U}(\g)}} \preliediff (\bT \circ e_{\bT}) * e_{\bT \circ \antipode_{\mathcal{U}(\g)}} \\
&= \mathrm{id} \preliediff(\bT \circ \antipode_{\mathcal{U}(\g)} \circ e_{\bT\circ \antipode_{\mathcal{U}(\g)}}) * (\bT \circ e_{\bT} \circ e_{\bT \circ \antipode_{\mathcal{U}(\g)}}) \\
&= \mathrm{id} \preliediff \bT \circ (\antipode_{\mathcal{U}(\g)} *_{\bT} e_{\bT}) \circ e_{\bT \circ \antipode_{\mathcal{U}(\g)}} \\
&= \mathrm{id} \preliediff \bT \circ ((\antipode_{\mathcal{U}(\g)} \preliediff e_{\bT})\cdot e_{\bT}) \circ e_{\bT \circ \antipode_{\mathcal{U}(\g)}} \\
&= \mathrm{id} \preliediff \bT \circ ((\antipode_{\mathcal{U}(\g)} (\mathrm{id} \preliediff e_{\bT}))\cdot e_{\bT}) \circ e_{\bT \circ \antipode_{\mathcal{U}(\g)}}, \\
&= \mathrm{id} \preliediff \bT \circ ((\antipode_{\mathcal{U}(\g)}\circ e_{\bT}) \cdot e_{\bT}) \circ e_{\bT \circ \antipode_{\mathcal{U}(\g)}  } \\
&=\mathrm{id} \preliediff \bT \circ \eta_{\mathcal{U}(\a))}\circ\varepsilon_{\mathcal{U}(\g)} = \mathrm{id} \preliediff \eta_{\mathcal{U}(\a)}\circ\varepsilon_{\mathcal{U}(\g)} = \mathrm{id}.
\end{align*}

Let ${\bf T}: \mathcal{U}(\g) \to \mathcal{U}(\a)$ be an $\mathcal{O}$-Hopf operator. The antipode $\mathcal{S}_{H_{{\bT}}}$ of the Hopf algebra $H_{{\bT}}$ satisfies

\begin{equation}
\label{eqn:antipodebt}
\mathcal{S}_{H_{{\bf T}}} = \mathcal{S}_{\mathcal{U}(\g)}\circ e_{{\bf T} \circ \antipode_{\mathcal{U}(\g)}} = e_{\bf T}\circ \antipode_{\mathcal{U}(\g)}
\end{equation}

\begin{align*}
{\rm id} \star_{\bf T} (e_{\bT} \circ \antipode_{\mathcal{U}(\g)}) &= ({\rm id} \preliediff \bT \circ e_{\bT} \circ \antipode_{\mathcal{U}(\g)})(e_{\bT} \circ \antipode_{\mathcal{U}(\g)}) \\
&=(\antipode_{\mathcal{U}(\g)}\circ e_{\bT} \circ \antipode_{\mathcal{U}(\g)})( e_{\bT} \circ \antipode_{\mathcal{U}(\g)}) \\
&= \eta_{\mathcal{U}(\a)} \circ \varepsilon_{\mathcal{U}(\g)}.
\end{align*}

\begin{align*}
(e_{\bT} \circ \antipode_{\mathcal{U}(\g)}) \star_{\bT} { \rm id } &= (\mathcal{S}_{\mathcal{U}(\g)}\circ e_{{\bf T}\circ \mathcal{S}_{\mathcal{U}(\g)}} \preliediff \bT)\cdot {\rm id} \\
&= (\mathcal{S}_{\mathcal{U}(\g)}\circ (e_{{\bf T}\circ \mathcal{S}_{\mathcal{U}(\g)}} \preliediff \bT))\cdot {\rm id} \\
&= (\mathcal{S}_{\mathcal{U}(\g)}\circ (e_{{\bf T}\circ \mathcal{S}_{\mathcal{U}(\g)}} \preliediff \bT\circ e_{\bT} \circ e_{{\bf T}\circ \mathcal{S}_{\mathcal{U}(\g)}}))\cdot {\rm id} \\
&= (\mathcal{S}_{\mathcal{U}(\g)}\circ (e_ {\bf T}\circ e_{{\bf T}\circ \mathcal{S}_{\mathcal{U}(\g)}}))\cdot {\rm id} \\
&= \antipode_{\mathcal{U}(\g)} \cdot {\rm id} \\
&= \eta_{\mathcal{U}(\a)} \circ \varepsilon_{\mathcal{U}(\g)}
\end{align*}

By inserting the defining fixed point equation for the $T$-transform, we obtain from equation (<ref>):

\begin{equation*}
S_{H_{\bf T}}
= e_{{\bf T}} \circ \mathcal{S}_H
= (\mathrm{id}\preliediff e_{{\bf T}}) \circ \mathcal{S}_H
= \mathcal{S}_H \preliediff (e_{{\bf T}} \circ \mathcal{S}_H)
= \mathcal{S}_H \preliediff \mathcal{S}_{H_{\bf T}}.
\end{equation*}

This formula for the antipode of the Hopf algebra $H_{{\bf T}}$ already appeared in [22], Remark 2.5.

Let $(G,\cdot,\Gamma)$ be a $\mathcal{O}$-group. Assume that both $G$ and $\Gamma$ are complete groups endowed with left invariant distances and that the action $\cdot$ is Lipschitz continuous. We denote by $d_G$ and $d_\Gamma$ the distances on $G$ and $\Gamma$ respectively. Suppose that we are given a strong contraction map $T\colon G \to \Gamma$   with $T(1_G)=T(1_\Gamma)$, for any $g,g' \in G$:

d_{\Gamma}(T(g),T(g^{\prime})) \leq \sigma_T d_{G}(g,g^{\prime})~(\sigma_T \in ]0,1[).

Under these hypotheses, the map $G \ni x \mapsto g \cdot T(x)\in G$ is a strong contraction and has an unique fixed point that we denote by $e_T(x)$. Furthermore, for any $x,y \in G$

d_G(e_T(x),e_T(y)) \leq (1-\sigma_T)^{-1}d_G(x,y).

All the above calculations can be readily adapted to prove that $e_T:G\to G$ has a continuous inverse given by:

e_T^{-1}(g)=g\cdot T(g)^{-1}

Notice that the right-hand side of the above equation is well defined for any continuous map from $G$ to $\Gamma$. Under the assumption that $T$ is a $\mathcal{O}$ group operator, we also have:

e_{\bT}(g)= (g^{-1})^{-1^{\cdot_\bT}}.

Here again, we can take the right-hand side of the above equation as a definition for $e_{\bT}(g)$ in the case where $\bT$ is a $\mathcal{O}$ group operator but not necessarily a strong contraction.

§.§ Matching $\mathcal{O}$ operators

We introduce matching $\mathcal{O}$ opeators of various types, see [9].

§.§.§ Matching $\mathcal{O}$-Lie operators

We begin with the notion of Matching Lie $\mathcal{O}$ operator.

Let $(\mathfrak{g}, \cdot, \mathfrak{a})$ be a Lie right module. A Matching Lie $\mathcal{O}$ operator is the data of a family of maps $({\bf t}_w)_{w\in \Omega}:G\to \Gamma$ such that for any $x,y \in \g$,

* and $w\in \Omega$, one has

[{\bf t}_w(x),{\bf t}_w(y)]_{\mathfrak{a}}
= {\bf t}_w(x \vartriangleleft {\bf t}_w(y))
- {\bf t}_{w}(y \vartriangleleft {\bf t}_w(x))
+ {\bf t}_w([x,y]_\mathfrak{g})

* and for any $w\neq w^{\prime} \in \Omega$:

\begin{equation}
\label{eqn:MLie}
\tag{MLie}
[{\bf t}_w(x),{\bf t}_{w^{\prime}}(y)]_{\mathfrak{a}} = {\bf t}_w(x \vartriangleleft {\bf t}_{w^{\prime}}(y)) - {\bf t}_{w^{\prime}}(y \vartriangleleft {\bf t}_w(x))
\end{equation}

Notice that given a matching $\mathcal{O}$-Lie operator ${\bf s, \bf t}$, then ${\bf s} + {\bf t}$ is a $\mathcal{O}$ Lie operator. This does not hold for any linear combinations of ${\bf s}$ and ${\bf t}$ except in the case $[-,-]_\g = 0$ (weight zero operators). In the next section, we provide an example of a matching Lie operator.

Let ${\bf s}: \mathfrak{g} \to \mathfrak{a}$ and ${\bf t}:\mathfrak{g}\to \mathfrak{a}$ be a matching $\mathcal{O}$-Lie operator on $(\mathfrak{g}, \vartriangleleft, \mathfrak{a})$ and call $\bS$ and $\bT$ their extension on $\univalgebra$ to an $\mathcal{O}$-Hopf operators on $(U(\g), \preliediff, U(\mathfrak{a}))$. Then for any $E,F \in \univalgebra$, the following relation holds

\begin{equation}
\label{eqn:firstMatching}
{\bT}(E \preliediff S_{\mathcal{U} (\mathfrak{a})}\bS(F_{(1)})) \star \bS(F_{(2)})
= {\bS}(F\preliediff S_{\mathcal{U}(\mathfrak{a})}\bT(E_{(1)})) \star \bT(E_{(2)})
\end{equation}

We prove the statement inductively on the total length $N=m+n$ with $E=E_1\ldots E_m$ and $F=F_1\cdots F_n$. The $N=2$ initialization is equivalent to the compatibility relation between ${\bf s}$ and ${\bf t}$. So let $N > 2$ and suppose that the result holds for elements $E,F$ with $m+n < N$ and pick two elements $E,F$ such that $m+n=N-1$. Pick a third element $x \in \g$. Two cases occur. We start by proving:

\begin{equation*}
{\bT}((Fx)\preliediff S_{\star}\bS(E_{(1)})) \star \bS(E_{(2)})
= {\bS}(E\preliediff S_{\star}\bT((Fx)_{(1)})) \star \bT((Fx)_{(2)})
\end{equation*}

These are lengthy but straightforward computations. We use the tag $(Rec. H.) $ to indicate that we use the recursive hypothesis.

\begin{align*}
\bS(&(Ex)\preliediff S_{\star}\bT(F_{(1)}))\star \bT(F_{(2)}) \\
&= \bS((E\preliediff S_{\star}\bT(F_{(1)}))(x \preliediff S_{\star}\bT(F_{(2)})))\star \bT(F_{(3)}) \\
&=\bS(E\preliediff (S_{\star}\bT(F_{(1)})\star S_{\star}\bS(x_{(1)}\preliediff S_{\star}\bT(F_{(2)})))\star\bS(x_{(2)} \preliediff S_{\star}\bT(F_{(3)}))\star \bT(F_{(4)}) \\
&=\bS(E\preliediff (S_{\star}(\bS(x_{(1)}\preliediff S_{\star}\bT(F_{(1)})) \star \bT(F_{(2)}))))\star\bS(x_{(2)} \preliediff S_{\star}\bT(F_{(3)}))\star \bT(F_{(4)}) & {\rm (Rec.H.)} \\
&=\bS(E\preliediff (S_{\star}(\bS(F_{(1)}\preliediff S_{\star}\bS(x_{(1)})) \star \bS(x_{(2)}))))\star\bT(F_{(2)} \preliediff S_{\star}\bS(x_{(3)}))\star \bS(x_{(4)}) &{\rm (Rec.H.)}\\
&=\bS([E \preliediff {S}_{\star}\bS(x_{(1)})]\preliediff (S_{\star}(\bS(F_{(1)}\preliediff S_{\star}\bS(x_{(2)})))))\star\bT(F_{(2)} \preliediff S_{\star}\bS(x_{(3)}))\star \bS(x_{(4)}) \\
&=\bS([E \preliediff S_{\star}\bS(x_{(1)})]\preliediff (S_{\star}(\bS(F_{(1)}\preliediff S_{\star}\bS(x_{(2)(1)})))))\star\bT(F_{(2)} \preliediff S_{\star}\bS(x_{(2)(2)}))\star \bS(x_{(4)}) \\
&=\bT((F_{(1)}\preliediff S_{\star}\bS(x_{(2)}))\preliediff S_{\star}\bS(E_{(1)} \preliediff S_{\star}\bS(x_{(1)(1)})))\star\bS(E_{(2)} \preliediff S_{\star}\bS(x_{(1)(2)}))\star \bS(x_{(4)}) & {\rm (Rec.H.)} \\
&=\bT((F_{(1)}\preliediff S_{\star}\bS(x_{(1)}))\preliediff S_{\star}\bS(E_{(1)} \preliediff S_{\star}\bS(x_{(2)})))\star\bS(E_{(2)} \preliediff S_{\star}\bS(x_{(3)}))\star \bS(x_{(4)}) \\
&= \bT(F \preliediff S_{\star}\bS((E\preliediff S_{\star}\bS(x_{(1)}))\star_{\bS} x_{(2)})) \star \bS(({E}\preliediff S_{\star}\bS(x_{(3)})) \star_{S} x_{(4)})\\
&=\bT(F \preliediff S_{\star}\bS(Ex_{(1)}))\star \bS(Ex_{(2)}).
\end{align*}

We now treat the second case

\begin{align*}
\bS(E
&\preliediff S_{\star}\bT(Fx_{(1)}))\star \bT(Fx_{(2)})\\
&= \bS(E \preliediff \big(S_{\star}\bT(x_{(1)}) \star S_{\star}\bT(F_{(1)} \preliediff S_{\star}\bT(x_{(2)}))\big))\star \bT(F_{(2)}\preliediff S_{\star}\bT(x_{(3)}))\star \bT(x_{(4)})\\
&=\bS((E \preliediff S_{\star}\bT(x_{(1)})) \preliediff S_{\star}\bT(F_{(1)} \preliediff S_{\star}\bT(x_{(2)}))))\star \bT(F_{(2)}\preliediff S_{\star}\bT(x_{(3)}))\star \bT(x_{(4)}) \\
&=\bT((F \preliediff S_{\star}\bT(x_{(1)})) \preliediff S_{\star}\bS(E_{(1)} \preliediff S_{\star}\bT(x_{(2)})))))\star \bS(E_{(2)} \preliediff S_{\star}\bT(x_{(2)}))\star \bT(x_{(2)}) & {\rm (Rec.H.)}\\
&=\bT((F \preliediff S_{\star}\bT(x_{(1)})) \preliediff S_{\star}\bS(E_{(1)} \preliediff S_{\star}\bT(x_{(2)})))))\star\bT(x_{(2)}\preliediff S_{\star} \bS(E_{(2)}))\star \bS(E_{(3)}) & {\rm (Rec.H.)} \\
&=\bT(((F \preliediff S_{\star}\bT(x_{(1)})) \preliediff S_{\star}\bS(E_{(1)} \preliediff S_{\star}\bT(x_{(2)}))))\star_{\bT}(x_{(3)}\preliediff S_{\star} \bS(E_{(2)})))\star \bS(E_{(3)}) \\
&= \bT(((F \preliediff [S_{\star}(\bS(E_{(1)} \preliediff S_{\star}\bT(x_{(1)})) \star \bT(x_{(2)})))\star {\bT}(x_{(3)}\preliediff S_{\star} \bS(E_{(2)})))]\star \bS(E_{(3)})\\
%&= \bT(((F \preliediff [S_{\star}(\bS({E \preliediff S_{\star}\bT(x_{(1)})}) \star \bT(x_{(2)})))\star {\bT}(x_{(3)}\preliediff S_{\star} \bS(E_{(1)})))]\star \bS(E_{(2)})\\
&=\bT(((F \preliediff [S_{\star}(\bT({x_{(1)} \preliediff S_{\star}\bS(E_{(1)})}) \star \bS(E_{(2)})))\star {\bT}(x_{(2)}\preliediff S_{\star} \bS(E_{(3)})))]\star \bS(E_{(4)}) &{\rm (Rec.H.)}\\
&=\bT(Fx \preliediff S_{\star}\bS(E_{(1)}))\star \bS(E_{(2)}).
\end{align*}

§.§.§ Matching $\mathcal{O}$-Hopf operators

Let $(H,\preliediff, K)$ be a Hopf right module. A Matching Hopf $\mathcal{O}$ operator is the data of a family of co-algebra morphisms $(\bT_w)_{w\in \Omega}:H\to K$ over such that for any $E,F \in H$,

* and $w\in \Omega$, one has

\bT_w(E\star_{\bT_w} F) = \bT_w(E) \star \bT_w(F),

* and for any $w,w^{\prime} \in \Omega$:

\begin{align}
\label{eqn:matchingHopf}
\tag{MHopf}
{\bT}_w(E_{(1)}\preliediff \cS_{K}\bT_{w^{\prime}}(F_{(1)})) \star \bT_{w^{\prime}}(F_{(2)}) = {\bT}_{w^{\prime}}(F_{(1)}\preliediff \cS_{K}\bT_{w}(E_{(1)})) \star \bT_{w}(E_{(2)})
\end{align}

We compare our notion of Matching $\mathcal{O}$ Hopf operators to distributivity (of one operation over the other). With the notations introduced so far, $\preliediff_{\bf T}$ distributes over $\preliediff_{\bf S}$ if, for $X,Y,Z$

(X \preliediff_{\bf S} Y) \preliediff_{\bf T} Z = (X \preliediff_{\bf T} Z) \preliediff_{\bf S} ( Y \preliediff_{\bf T} Z)

Inserting definitions, this will be implied by the following:

{\bf S}(Y)\star {\bf T}(Z)
= {\bf T}(Z)\star{\bf S}(Y \preliediff {\bf T}(Z))

or else:

{\bf T}(Z)\star{\bf S}(Y)
= {\bf S}(Y \preliediff \mathcal{S}_K{\bf T}(Z)) \star {\bf T}(Z)

and we see that $\{\bS, \bT\}$ is a Matching $\mathcal{O}$ Hopf operator if the ranges of $\bS$ and $\bT$ commute.

We move on to discussing the notion of Matching group $\mathcal{O}$ operators.

§.§.§ Matching $\mathcal{O}$-group operators

Let $(G, \cdot, \Gamma)$ be a right group module. A Matching group $\mathcal{O}$ operator is the data of a family of maps $(\bT_w)_{w\in \Omega}:G\to \Gamma$ such that for any $g,h \in G$,

* and $w\in \Omega$, one has

\bT_w(g\star_{\bT_w}h) = \bT_w(g) \star \bT_w(h),

* and for any $w,w^{\prime} \in \Omega$:

\begin{align}
\label{eqn:matchinggroup}
\tag{MGroup}
{\bT}_w(g\cdot \bT_{w^{\prime}}(h)^{-1})\bT_{w^{\prime}}(h)
= {\bT}_{w^{\prime}}(h\cdot \bT_{w}(g)^{-1}) \bT_{w}(g)
\end{align}

The following proposition is a simple a consequence of the definitions.

Let $({\bf T}_{w})_{w\in\Omega}$ be a Matching Hopf $\mathcal{O}$ operator. Then it restricts to a Matching Lie $\mathcal{O}$ operator on the primitive element of $H$ and to a Matching $\mathcal{O}$ group operator on the group-like elements of $H$.

§.§ Examples

§.§.§ Translations by the unit in a non-symmetric operad with multiplication

In this section, we exhibit a Matching $\mathcal{O}$ Lie operator for any given operad with multiplication $(\mathcal{P},\gamma,m,\unit)$:

* $\mathcal{P}=(\mathcal{P}(n))_{n\geq 1})$ is a collection of complex vector spaces with $\mathcal{P}(1)=\mathbb{C}$,

* $\circ$ denotes composition in the category of collections, given two collections $\mathcal{P}$ and $\mathcal{Q}$ of vector spaces:

(\mathcal{P}\circ \mathcal{Q})(n)
=\bigoplus_{\substack{p\geq 1,\\n_1+\cdots+n_p=n}}
\mathcal{P}(p)\otimes \mathcal{Q}(n_1)\otimes\cdots\otimes\mathcal{Q}(n_p),

* the letter $\gamma$ denotes an operadic composition, in particular:

\gamma : \mathcal{P} \circ \mathcal{P} \to \mathcal{P},

and $I\in\mathcal{P}(1)$ is the unit for $\gamma$, which means $\gamma(I \otimes x)=x$ and $\gamma(x,I^{\otimes |x|})=x$,

* $m\in \mathcal{P}(2)$ is a multiplication: it is an operator with arity $2$ in $\mathcal{P}$ satisfying:

m\circ ({\rm id}\otimes m)=m \circ m\otimes {\rm id}.

In the sequel, we denote by $\mathbb{C}[\mathcal{P}]_+$ the linear span of $\mathcal{P}(n),~n\geq 2$ and $\mathbb{C}[\mathcal{P}]$ the linear span of the $\mathcal{P}(n)$, $n\geq 1$. Multiplication $m$ induces an associative algebra product on $\mathbb{C}[\mathcal{P}]$:

p \centerdot q = \gamma(m\circ (p\otimes q)),~p,q \in \mathcal{P}.

We use the symbol $\lieinv{-}{-}$ for the Lie bracket on $\mathbb{C}[\mathcal{P}]$ associated with the multiplication $\centerdot$. We introduce the left and right translations with respect to $\centerdot$ by the unit $\unit$:

\begin{equation}
\begin{array}{cccc}
\lambda :   & \algop      & \to     & \algop \\
&    x 	    & \mapsto & I\centerdot x
\end{array},
\quad
\begin{array}{cccc}
\rho :  & \algop    & \to     & \algop \\
&    x 	  & \mapsto & x\centerdot I
\end{array}
\end{equation}

and set for brevity $\g = (\algop,\lieinv{-}{-})$. The operadic composition $\gamma$ produces a preLie product $\vartriangleleft$ on $\mathbb{C}[\mathcal{P}]_+$, the Gerstenhaber preLie product:

p \vartriangleleft q
= \sum_{i=1}^{|p|}\gamma(p\circ(\unit\otimes\ldots\otimes\unit\otimes q\otimes \unit \otimes \ldots \otimes \unit)),~p,q \in \mathbb{C}[\mathcal{P}]_+.

We denote by $\liediff{-}{-}$ the adjacent Lie bracket on $\liediff{-}{-}$:

\liediff{p}{q}
:= p \vartriangleleft q - q \vartriangleleft p,~p,q \in \mathbb{C}[\mathcal{P}]_+.

We denote by $\a$ the Lie algebra $(\mathbb{C}[\mathcal{P}]_+, \liediff{-}{-})$. Proofs of the following propositions follow from induction.

$(-\lambda)$ and $\rho$ are $\mathcal{O}$ Lie operators over the Lie module $(\g,\vartriangleleft,\a)$ satisfying:

\begin{align*}
&\lambda(p) \vartriangleleft \lambda(q) = \lambda(p \vartriangleleft \lambda(q) + q \centerdot p),~
&\rho(p) \vartriangleleft \rho(q) = \rho(p \vartriangleleft \rho(q) + p\centerdot q),
\end{align*}

Besides $(-\lambda,\rho)$ is a Matching $\mathcal{O}$-Lie operator;

\liediff{\lambda(p)}{\rho(q)}
=\lambda(p\vartriangleleft q) - \rho(q \vartriangleleft p),~p,q \in \mathbb{C}[\mathcal{P}].

We next provide formulae for the extensions of $\lambda$ and $\rho$ and $\lambda + \rho$ as $\mathcal{O}$-Hopf operators on the universal envelope of the Lie module $(\g,\vartriangleleft,\a)$. Pick a sequence $P$ of $n$ elements $p_1,\ldots,p_n$ in the Lie algebra $(\mathbb{C}[\mathcal{P}],[-,-]_{\centerdot})$ and a partition $\pi = \{\pi_1,\ldots,\pi_j\} \in {\rm Part}(n)$ with blocks $\pi_i$ of the interval $[n]$ and set for any $1 \leq j \leq k$:

P_{\pi_j} = (p_{j_1}\centerdot \cdots \centerdot p_{j_{p_k}}),~ \pi_j = {j_1 < \cdots < j_{p_k}}.

In the following, we denote by $\bar{\pi_j}$ the block $\pi_j$ endowed with the opposite of the natural order.
Let $P_1,\ldots,P_n \in (\mathbb{C}[\mathcal{P}],[-,-]_{\centerdot})$, then:

\begin{equation*}
[\boldsymbol{\rho}](p_1\cdots p_n) = \sum_{\{\pi_1,\ldots,\pi_k\} \in {\rm Part}(n)} \boldsymbol{\rho}(P_{\pi_1})\cdots \boldsymbol{\rho}(P_{\pi_k})
\end{equation*}

\begin{equation*}
[\boldsymbol{\lambda}](p_1\cdots p_n) = \sum_{\pi = \{\pi_1,\ldots,\pi_k\} \in {\rm Part}(n)} (-1)^n\boldsymbol{\lambda}(P_{\bar{\pi}_1}) \cdots \boldsymbol{\lambda}(P_{\bar{\pi}_k})
\end{equation*}

Recall that $\mathrm{sol}_1:\mathcal{U}(\mathbb{C}[\mathcal{P}]_+, \llbracket -,- \rrbracket) \to \mathbb{C}[\mathcal{P}]$ is the projector onto $\mathbb{C}[\mathcal{P}]$ parallel to

$$\pi_{nat}(\sum_{n\geq 2} \mathcal{S}^{(n)}(\mathbb{C}[\mathcal{P}])) \oplus \mathbb{C}\cdot 1.$$

(The first Eulerian idempotent). We set:

\rho^{(1)} := {\rm sol}_1 \circ \boldsymbol{\rho}

We define $\hat{m}\colon \mathcal{U}(\mathbb{C}\llbracket P \rrbracket, [-,-]_{\centerdot}) \rightarrow \mathbb{C}\llbracket P \rrbracket$ be defined by taking the $\centerdot$ product of all entries of a polynomial in $\mathcal{U}(\mathbb{C}\llbracket P \rrbracket, [-,-]_{\centerdot})$.
We prove recursively that $\rho^{(1)}(E_1 \cdots E_n) = \rho (\hat{m}(E_1\cdots E_n))$. If $n=1$, the assertion is trivial.
Pick $p_1 \cdots p_{n} \in \univalgebra$ and $h \in \mathbb{C}[\mathcal{P}]$. Then

\begin{align*}
\rho^{(1)}(p_1\cdots p_n h)&=\rho^{(1)}(p_1\cdots p_n)\preliediff \rho(h) - \rho^{(1)}((p_1\cdots p_n) \preliediff \rho(h))\\
&=\rho({\hat{m}(p_1\cdots p_n)}) \preliediff \rho(h) - \rho(\hat{m}(p_1\cdots (p_i \preliediff \rho(h)) \cdots p_n)) \\
&=\rho(\hat{m}(p_1 \cdots p_n h)) + \rho(p_1\cdots (p_i \preliediff \rho(h))\cdots p_n) - \rho(\hat{m}(p_1\cdots (p_i \preliediff \rho(h)) \cdots p_n)) \\
&=\rho(\hat{m}(p_1 \cdots p_n h))
\end{align*}

We should denote by $\mathbb{C}\llbracket \mathcal{P}\rrbracket$ the vector space of formal series on operators in $\mathcal{P}$. All of the above extends verbatim to $\mathbb{C}\llbracket \mathcal{P}\rrbracket$.
The following proposition is obtained by writing any group like element of $\mathbb{C}\llbracket \mathcal{P}\rrbracket$ as the exponential of a primitive element.

Whenever $P$ is a group like elements in $\mathbb{C}\llbracket \mathcal{P}\rrbracket$,

\begin{align}
[-\bm{\lambda}](P)= I \centerdot P^{-1},\quad [\bm{\rho}](P)= P \centerdot I,\quad  [\rho-\lambda] = [\rho] \# [\lambda]^{-1^{\#}} = P \centerdot I \centerdot P^{-1}
\end{align}

We might take here the opportunity to highlight connections with free probability theory. A certain transform, called the $T$-transform, plays a major role in the computations of the distribution of the product of two freely independent non-commutative random variables. Given a random variable $a$, $T_a$ is a series in $\mathbb{C}[[{\rm Hom} (B)]]$ where:

* $B$ is a complex unital algebra,

* ${\rm Hom}(B)$ stands for the endomorphism operad of $B$; ${\rm Hom}(B)(n)={\rm Hom}_{Vect_{\mathbb{C}}}(B^{\otimes n},B),~n\geq 1$.

The $T$-transform $T_a \in \mathbb{C}1\oplus \mathbb{C}[[{\rm Hom}(B)]]$ of a random variable $a$ is a series on multilinear maps on $B$ with values in $B$ which starts by $1$:

T_a = 1+\sum_{n\geq 1}T_a^{(n)},\quad T_a^{(n)}:B^{\otimes n} \rightarrow B

Given two freely independent random variables, the $T$-transform $T_{ab}$ of the product $ab$ factorizes as follows:

T_{ab}=T_a\circ(T_bIT_B^{-1})\centerdot T_b,

where $\centerdot$ is the pointwise product of series and $\circ$ is the composition product $\times$:

(A \circ B)^{(n)} (x_1,\ldots,x_n) = \sum_{\substack{k\geq 1,\\ q_1+\cdots+q_k=n}}A_k(B_{q_1}(x_1,\ldots,x_{q_1}),\ldots,B_{q_k}(x_{q_1+\cdots+q_{k-1}+1},\ldots,x_{n})).

We elaborate on this in the second part of this work.

§.§.§ Translations in the dendriform operad

In this section, we show that in the dendriform operad Dend, which is an operad with multiplication, we can define a second Matching $\mathcal{O}$ Lie operator, besides the one described in the previous section, also given by translations by the unit of Dend, but with respect to the two half-dendriform products. The dendriform operad is generated by two operators $\prec$ and $\succ$ with arity $2$ subject to the relations:

\begin{align*}
\prec \circ (\prec \otimes {\rm I})
&=\prec \circ (I \otimes (\prec + \succ)),\\
\prec \circ(\succ \otimes I)
&=\succ \circ (I \otimes \prec),\\
\succ \circ (\succ \otimes I)
&= ((\succ + \prec) \otimes I) \circ \succ .
\end{align*}

In the following we will use the notation:

m = \prec + \succ.

The above relations imply that $m$ is a product in Dend. We will use $\centerdot$ for the induced product on $\mathbb{C}[\mathcal{P}]$. Note that, for any $p,q \in \mathcal{C}[\mathcal{P}]$:

[p,q]_{\centerdot}=p \blacktriangleleft q - q \blacktriangleleft p

with $\blacktriangleleft$ the preLie product over $\mathbb{C}[\mathcal{P}]$ defined by:

p \blacktriangleleft q = p \prec q - q \succ p.

Next, we define:

\begin{equation}
\begin{array}{cccc}
\lambda_{\prec} : & \mathbb{C}[{\rm Dend}] & \to     & \mathbb{C}[{\rm Dend}] \\
&    x 	                & \mapsto & I\prec x
\end{array}, \quad
\begin{array}{cccc}
\rho_{\succ} :  &\mathbb{C}[{\rm Dend}] & \to     &\mathbb{C}[{\rm Dend}] \\
&    x 	                & \mapsto & x\succ I
\end{array}
\end{equation}

Recall the notations of the previous Section.

Both $\lambda_{\prec}$ and $\lambda_{\succ}$ are $\mathcal{O}$ Lie operators. Furthermore, $(\lambda_{\prec},\rho_{\succ})$ is a Matching $\mathcal{O}$ Lie operator on the Lie module $(\g,\vartriangleleft,\a)$.

Let us prove that $\lambda_{\prec}$ is a $\mathcal{O}$ Lie operator. The result for $\rho_{\succ}$ will follow from the same computations.

\begin{align*}
\lambda_{\prec}(x) \vartriangleleft \lambda_{\prec}(y)
&=\lambda(x) \prec \lambda(y)\\
&=((I \prec y) \prec x)+I \prec (x \vartriangleleft(I\prec y))\\
&=I \prec (y \prec x + x \succ y) + I \prec (x \vartriangleleft (I \prec y))\\
&=\lambda_{\prec}(x \vartriangleleft \lambda_{\prec}(y))+\lambda_{\prec}(y \prec x + x \succ y)\\
&=\lambda(x \vartriangleleft \lambda_{\prec}(y))+\lambda_{\prec}(x\centerdot y).
\end{align*}

This yields

\begin{align*}
\llbracket\lambda_{\prec}(x),\lambda_{\prec}(y)\rrbracket
=\lambda_{\prec}(x\vartriangleleft\lambda_{\prec}(y))-\lambda_{\prec}(y \vartriangleleft \lambda_{\prec}(x))+ \lambda ([x,y]_{\centerdot})
\end{align*}

Next, we show that $\lambda_{\prec}, \rho_{\succ}$ is a Matching $\mathcal{O}$ Lie operator.

\begin{align*}
\lambda_{\prec}(x) \vartriangleleft \rho_{\succ}(y)
&=\rho_{\succ}(y)\prec x + I \prec (x \vartriangleleft \rho_{\succ}(y)) \\
&=(y \succ I) \prec x + I \prec (x \vartriangleleft \rho_{\succ}(y)) \\
&= y \succ (I \prec x) + \lambda_{\prec}(x \vartriangleleft\rho_{\succ}(y)) \\
\end{align*}

and, by the same computations:

\begin{align*}
\rho_{\succ}(y) \vartriangleleft \lambda_{\prec}(x)
= y \succ (I \prec x) + \rho_{\succ}(y \vartriangleleft (I\prec x))
\end{align*}

The above two equations imply $\llbracket \lambda_{\prec}(x),\rho_{\succ}(y) \rrbracket=\lambda_{\prec}(x \vartriangleleft \rho_{\succ}(y))-\rho_{\succ}(y \vartriangleleft \lambda_{\prec}(y))$.

§.§.§ Chen Fliess series and multiplicative feedback

We refer the reader to [16] for details.
In the following, $\mathbb{R}^q\langle\langle X \rangle\rangle$ is the linear space of $q$-tuples of formal series on words with entries in an alphabet $X$. We use the symbol $\shuffle$ for the shuffle product between series in $\mathbb{R}^q\langle\langle X\rangle\rangle$.
Pick $k\geq 1$ and $\ell \geq 1$ two integers. Let $X$ and $X^{\prime}$ be two alphabets with $|X^{\prime}|=\ell+1$. Let  $c=(c_1,\ldots,c_k) \in \mathbb{R}^{k}\langle\langle X^{\prime} \rangle\rangle$ and $d=(d_1,\ldots,d_\ell) \in \mathbb{R}^\ell\langle\langle X\rangle\rangle$ be formal series.
Define the composition product $c \circ d$  as the formal series in $\mathbb{R}^k\langle\langle X \rangle\rangle$ defined by:

c \circ d = \sum_{\eta \in (X^{\prime})^{\star}} (c,\eta) \psi_d(\eta)(\emptyset)

where $\psi_d(\eta) \in {\rm End}(\mathbb{R}^k\langle\langle X \rangle\rangle)$ is defined by:

\begin{align*}
&\psi_d(x_i^{\prime} \eta) = \psi_d(x_i^{\prime}) \circ \psi_d(\eta), \\
&\psi_d(x^{\prime}_i)(e)=x_0(d_i\shuffle e)
\end{align*}

The multiplicative fixed composition product $e \curvearrowleft F$ between two formal series $e \in \mathbb{R}^{\ell}\langle\langle X \rangle\rangle$ and $F \in \mathbb{R}^m \langle\langle X \rangle\rangle$ on the same alphabet $X=\{x_0,\ldots,x_m\}$ is defined as

e \curvearrowleft F = \sum_{\eta\in X^{\star}} (e,\eta) \eta \curvearrowleft F = \sum_{\eta \in X^{\star}}(e,\eta)\bar{\varphi}_d(\eta)(\emptyset)

where $\varphi_d\coloni \mathbb{R}\langle\langle X \rangle\rangle \rightarrow {\rm End}(\mathbb{R}\langle\langle X \rangle\rangle)$ is defined by:

\begin{align*}
&\bar{\varphi}_d (x_0)(t)=x_0t, \\
&\bar{\varphi}_d (x_i)(t)=x_i(e_i\shuffle t).
\end{align*}

From Lemma 5.2 in <cit.>, for any $c \in \mathbb{R}^k\langle\langle X^{\prime} \rangle\rangle$, $e \in \mathbb{R}^{\ell}\langle\langle X \rangle\rangle$ and $F \in \mathbb{R}^m\langle \langle X \rangle \rangle$:

\begin{equation}
\label{eqn:dist}
c \circ (e \curvearrowleft F)=(c\circ d) \curvearrowleft F.
\end{equation}

With the associative group $\star$ product between two series $F,G \in \mathbb{R}^m\langle\langle X \rangle\rangle$ defined by:

F \star G = (F \curvearrowleft G) \shuffle G

one obtains that $((\mathbb{R}^k\langle\langle X \rangle\rangle,\shuffle),\curvearrowleft, (\mathbb{R}^m\langle\langle X\rangle\rangle,\star) )$ is a right group module. To any formal series $\mathbb{R}^m \langle\langle X \rangle\rangle$, one considers the left translation:

\lambda_c \colon \mathbb{R}^k\langle \langle X \rangle \rangle \to \mathbb{R}^m\langle\langle X \rangle\rangle, d \mapsto c \circ d

Then $\{\lambda_c,~ c \in \mathbb{R}^m\langle\langle X \rangle\rangle\}$ is a Matching $\mathcal{O}$-group operator. As it can be easily checked from (<ref>):

\begin{align*}
c \circ (d \curvearrowleft (c^{\prime} \circ d^{\prime})^{-1^{\star}})\star (c^{\prime} \circ d^{\prime}) &= ((c \circ (d \curvearrowleft (c^{\prime} \circ d^{\prime})^{-1^{\star}})) \curvearrowleft (c^{\prime}\circ d^{\prime}))\shuffle (c^{\prime}\circ d^{\prime})\\
&=(c \circ (d \curvearrowleft ((c^{\prime} \circ d^{\prime})^{-1^{\star}}\star(c^{\prime}\circ d^{\prime}))))\shuffle (c^{\prime}\circ d^{\prime})\\
&=(c \circ d) \shuffle (c^{\prime} \circ d)\\
&=c^{\prime} \circ (d^{\prime} \curvearrowleft (c \circ d)^{-1^{\star}}) \star (c\circ d)
\end{align*}

The computations above also prove that :
\lambda_c \# \lambda_{c^{\prime}} = \lambda_{c \shuffle c^{\prime}}.
We obtain  the Theorem 6.1 in <cit.> as an immediate corollary of the computations above, with the notations of  <cit.>: $(c$ $ d) $  $d^{\prime} = e_{\lambda_{d^{\prime}}}((e_{\lambda_{d}})(c))=e_{\lambda_d \# \lambda_{d^{\prime}}}(c) = e_{\lambda_{d\shuffle d^{\prime}}}(c) = c$ $ (d \shuffle d^{\prime})$.

§ SEMENOV-TIAN-SHANSKII FACTORIZATION AND $\MATHCAL{O}$ OPERATORS

In this section, we develop the deformation theoretic perspective on the STS factorization.
Hereafter, the subset of all $\mathcal{O}$ -Hopf operators on a Hopf module $(H,\preliediff,K)$ will be denoted $G^{\mathcal{O}}_{\Delta}$ (or $G^{\mathcal{O}}_{\Delta}(H,K)$ to eliminate ambiguities).

§.§ Guin-Oudom group

The objective of this section is to give a Lie algebraic interpretation to the recursion (<ref>) by defining the structure of a postLie algebra on $\varepsilon$-co-cocycles from $H$ to $K$. This adds a novel perspective on the Guin–Oudom construction and will be useful in the next section regarding computing the quantization of linear combinations of matching $\mathcal{O}$-operators (a notion introduced subsequently).
The association of a Lie algebra and corresponding group to each right Hopf module is functorial, but we need a slight variation of the notion of morphism between two right Hopf modules. In this section, given two right Hopf modules $(H,\preliediff,K)$ and $(H^{\prime}, \preliediff^{\prime},K^{\prime})$, a morphism with source $(H,\preliediff, K)$ and target $(H^{\prime}, \preliediff^{\prime}, K^{\prime})$ will be a pair of $(\Phi^{\prime}, \Theta)$, $\Phi^{\prime}:H^{\prime}\to H$ and $\Theta:K\to K^{\prime}$ such that for $h^{\prime} \in H^{\prime}$ and $k \in K$

\Phi^{\prime}(h^{\prime} \preliediff^{\prime} \Theta(k))
= \Phi^{\prime}(h^{\prime}) \preliediff k.

For the entire section, we fix a right Hopf module $(H,\preliediff,K)$ (recall that both $H$ and $K$ are supposed to be graded connected Hopf algebras). If $H$ is cocommutative, we denote by $G_{\Delta}$ the group of coalgebra morphisms from $H$ to $K$.
We denote by $\delta$ the vector space of $\varepsilon$-co-cocycles from $H$ to $K$:

\begin{equation*}
\delta = \{ \alpha : H \to K, ~\Delta \circ \alpha = (\eta_K \circ \varepsilon_{H} \otimes \alpha + \alpha \otimes \eta_K\circ\varepsilon_{H}) \circ \Delta  \}.
\end{equation*}

It is well known that the space $\delta$ is a Lie algebra for the convolution Lie bracket:

\begin{equation}
[\alpha, \beta]_{*} = \alpha * \beta - \beta * \alpha, \qquad \alpha,\beta \in \delta.
\end{equation}

Furthermore, under the assumption that $H$ and $K$ are graded conilpotent, the exponential convolution $\exp_{*}$ is a bijection from $\delta$ to the convolution group $G_{\Delta}$ of all coalgebra morphisms from $H$ to $K$.

Let $\alpha, \beta \in \delta$ be two $\varepsilon$-co-cocyles and define for any $f \in H$:

\begin{equation}
(\alpha \preliediff^{*} \beta)(f)
:= -\alpha \circ (\mathrm{id}\preliediff \beta)(f)
= -\alpha(f_{(1)} \preliediff \beta(f_{(2)})).
\end{equation}

Then $\alpha \preliediff^{*} \beta$ is an $\varepsilon$-co-cocycle and $(\delta, \preliediff^{*}, [-,-]_*)$ is a postLie algebra.

Let $\alpha,\beta,\gamma \in \delta$. First, one has to prove that $\alpha \preliediff^{*} \beta$ is an $\varepsilon$-co-cocycle.

\begin{align*}
\Delta (\alpha \preliediff^{*} \beta)
&= \Delta (\alpha) \preliediff^{*} \Delta(\beta) \\
&=(\eta_{\mathcal{U}(\g)}\circ\varepsilon_{\mathcal{U}(\a)} \otimes \alpha + \alpha \otimes \eta_{\mathcal{U}(\g)}\circ\varepsilon_{\mathcal{U}(\a)}) \preliediff^{*} (\eta_{\mathcal{U}(\g)}\circ\varepsilon_{\mathcal{U}(\a)} \otimes \beta + \beta \otimes \eta_{\mathcal{U}(\g)}\circ\varepsilon_{\mathcal{U}(\a)})
\end{align*}

Since $\varepsilon_{\mathcal{U}(\g)} (\mathrm id \preliediff^{*} \beta)
= \varepsilon_{\mathcal{U}(\a)} \circ \beta
= \varepsilon_{\mathcal{U}(\g)} (\mathrm id \preliediff^{*} \alpha)
= \varepsilon_{\mathcal{U}(\a)} \circ \alpha
= 0$, we get

\Delta (\alpha \preliediff^{*} \beta) = ((\alpha \preliediff \beta) \otimes \varepsilon + \varepsilon \otimes (\alpha \preliediff^{*} \beta))\circ \Delta.

We now prove the postLie axioms. First, one sees that $\preliediff^{*}$ distributes over the convolution $*$. Second,

\begin{align*}
(\alpha \preliediff^{*} \beta) \preliediff^{*} \gamma
&= -(\alpha \preliediff^{*} \beta)(\mathrm{id}\preliediff \gamma) = \alpha((\mathrm{id} \preliediff \gamma) \preliediff \beta) + \alpha (\mathrm{id}\preliediff \beta(\mathrm{id}\preliediff \gamma)) \\
&=\alpha(\mathrm{id} \preliediff \gamma * \beta) + \alpha \preliediff^{*}(\beta\preliediff^{*} \gamma) \\
&=-\alpha\preliediff^{*} (\gamma * \beta) + \alpha \preliediff^{*} (\beta \preliediff^{*} \gamma)
\end{align*}

Anti-symmetrizing in $\beta$ and $\gamma$ yields

(\alpha \preliediff^{*} \beta) \preliediff^{*} \gamma - \alpha \preliediff^{*}(\beta \preliediff^{*} \gamma) = \alpha \preliediff^{*}[\beta,\gamma]_{*}.

The following proposition is a simple consequence if the definitions.

Let $(\Phi^{\prime},\Theta)\coloni (H,\preliediff,K) \to (H^{\prime}, \preliediff^{\prime},K^{\prime}) $ be a morphism between two right Hopf modules $(H,\preliediff,K)$ and $(H^{\prime}, \preliediff^{\prime},K^{\prime})$ and denote by $\delta$ and $\delta^{\prime}$ their respective postLie algebras introduced in Definition <ref>. Then

\Theta^*\Phi^{\prime}_*:
\delta \to \delta^{\prime},~\alpha\mapsto \Theta\circ\alpha\circ\Phi^{\prime}

is a morphism between postLie modules:

\Theta^*\Phi^{\prime}_*(\alpha\preliediff^{*} \beta) = \Theta^*\Phi^{\prime}_*(\alpha)\preliediff^{\prime,*}\Theta^*\Phi^{\prime}_*(\alpha)(\beta).

For the rest of the paper, we set for two $\varepsilon$-co-cocycles $\alpha,\beta \in \delta$:

\begin{equation}
\label{eqn:postliebrck}
[\alpha,\beta]_{\#} = [\alpha,\beta]_{*} + \alpha \preliediff^{*} \beta - \beta \preliediff^{*} \alpha
\end{equation}

Our objective is now to define an algebra structure on linear homomorphisms from $H$ to $K$ whose commutator Lie algebra contains $(\delta,[\alpha,\beta]_{\#})$.

Assume that $H$ is co-commutative. We define $\#$ to be the bilinear associative product defined over linear maps from $H$ to $K$

\begin{equation}
(A \# B)(f)
= A \Big(f_{(1)} \preliediff \antipode_{K}\big(B(f_{(2)})_{(1)}\big)\Big) B(f_{(2)})_{(2)},
\end{equation}

where $A,B\coloni H \to K$ and $f \in H$. The unit of the product $\#$ is the convolutional unit $\varepsilon_H \circ \eta_K$.

We prove that $\#$ is an associative product. This follows from the following computation. Pick $A,B,C$ linear maps from $H$ to $K$ and $f \in H$, then

\begin{align*}
((A\# B)\# C) (f)
&= (A\# B)(f_{(1)} \preliediff \antipode_{K}(C(f_{(2)})_{(1)})) C(f_{(2)})_{(2)} \\
&=A\Big[[f_{(1)} \preliediff \antipode_{K}(C(f_{(2)})_{(1)})]_{(1)} \preliediff \antipode_{K}\big(B\big([f_{(1)} \preliediff \antipode_{K}(C(f_{(2)})_{(1)})]_{(2)}\big)_{(1)}\big)\Big]\\
&\hspace{1cm}  B\big([f_{(1)} \preliediff \antipode_{K}(C(f_{(2)})_{(1)})]_{(2)}\big)_{(2)} C(f_{(2)})_{(2)} \\
&=A\Big[[f_{(1)} \preliediff \antipode_{K}(C(f_{(3)})_{(1)(1)})] \preliediff \antipode_{K}\big(B\big([f_{(2)} \preliediff \antipode_{K}(C(f_{(3)})_{(1)(2)})]\big)_{(1)}\big)\Big]\\
&\hspace{1cm}  B\big([f_{(2)} \preliediff \antipode_{K}(C(f_{(3)})_{(1)(2)})]\big)_{(2)}   C(f_{(3)})_{(2)} \\
&=A\Big[[f_{(1)} \preliediff \big[\antipode_{K}(C(f_{(3)})_{(1)(1)})  \antipode_{K}\big(B\big(f_{(2)} \preliediff \antipode_{K}(C(f_{(3)})_{(1)(2)})\big)_{(1)}\big)\big]\Big]\\
&\hspace{1cm}  B\big([f_{(2)} \preliediff \antipode_{K}(C(f_{(3)})_{(1)(2)})]\big)_{(2)}  C(f_{(3)})_{(2)}\\
&=A\Big[[E_{(1)} \preliediff \big[\antipode_{K}\big(B\big(E_{(2)} \preliediff \antipode_{K}(C(E_{(3)})_{(1)(2)})\big)_{(1)}  C(E_{(3)})_{(1)(1)}\big)\big]\Big]\\
&\hspace{1cm} B\big([E_{(2)} \preliediff \antipode_{K}(C(E_{(3)})_{(1)(2)})]\big)_{(2)} C(E_{(3)})_{(2)} \\
&=A\Big[[E_{(1)} \preliediff \antipode_{K}\big(B\big([E_{(2)} \preliediff \antipode_{K}(C(E_{(3)})_{(1)})]\big)_{(1)}  C(E_{(3)})_{(2)}\big)\big)\Big]\\
&\hspace{1cm}  B\big([E_{(2)} \preliediff \antipode_{K}(C(E_{(3)})_{(1)})]\big)_{(2)}   C(E_{(3)})_{(3)} \\
&= A \# (B \# C)(E),
\end{align*}

where in the ante-penultimate equality we have used co-commutativity of $H$.

It is straightforward that for the above-defined product $\#$, one has $\delta \subset {\rm Com}({\rm End}(H,K), \#)$ (the commutator Lie algebra of ${\rm End}(H,K)$  for the product $\#$).

We denote by $G^{inv}_{\#}$ the set of all invertible elements in $({\rm End}(H,K), \#)$.
For the following proposition, we specialize the general setting to the one where $H$ is a reduced graded conilpotent coalgebra (and thus the envelope of its primitive elements by the Milnore–Moore–Cartier theorem).

Let $(\g,\a,\vartriangleleft)$ be a Lie module.
The set $G_{\Delta}$ of all coalgebra morphisms from $\mathcal{U}(\g)$ to $\mathcal{U}(\a)$ is a group for the product $\#$ with the unit $\varepsilon_{\mathcal{U}(\a)} \circ \eta_{\mathcal{U}(\g)}$; $G_{\Delta} \subset G^{inv}_{\#}$.

The fact that the set $G_{\Delta}$ of all coalgebra morphisms is stable by $\#$ is a simple consequence of the fact that $\preliediff$ and $\mathcal{S}_{K}$ are compatible with the coproducts. It remains to show that any coalgebra morphism $A: \mathcal{U}(\g) \to \mathcal{U}(\a)$ is invertible. This follows from the following:
$$ A = \varepsilon_{\mathcal{U}(\a)} \circ \eta_{\mathcal{U}(\mathfrak{g})} + \bar{A}

where $\bar{A}$ is pointwise nilpotent for $\#$: for any $E \in \mathcal{U}(\g)$ there exists $n(E)$ such that $A^{\# n(E)}(E)=0.$

We use $G^{GO}$ for the group $({G_{\Delta},\#})$.

Let ${\bT}:\mathcal{U}(\g) \to \mathcal{U}(\a)$ be a $\mathcal{O}$-Hopf operator. Then $\bf T$ is invertible for the product $\#$ and ${\bf T}^{\# -1} = {\bf T} \circ \mathcal{S}_{\mathcal{U}(\g)}$.

The previous proposition has the following simple corollary.

Let ${\bf T} \in G_{\#}$ be a $\mathcal{O}$-Hopf operator, then

({\bf T}^{-1^*})^{-1^{\#}} = ({\bf T}^{-1^{\#}})^{-1^{*}}.

We have to prove that ${\bf T} \circ \mathcal{S}_H$ is a right- and left-inverse to ${\bf T}$ for the product $\# $. First, we compute:

\begin{align*}
{\bf T} \# ({\bf T} \circ \antipode_{\mathcal{U}(\g)})
&= {\bf T}({\rm id} \preliediff \antipode_{\mathcal{U}(\a)} \circ {\bf T } \circ \antipode_{\mathcal{U}(\g)})  *  ({\bf T} \circ \antipode_{\mathcal{U}(\g)}) \\
&={\bf T}({\rm id} \preliediff ({\bf T} \circ \antipode_{H_{\bf T}} \circ \antipode_{\mathcal{U}(\g)}))  *  ({\bf T} \circ \antipode_{\mathcal{U}(\g)}) \\
&={\bf T}({\rm id} \preliediff {\bf T} \circ e_{\bf T})  *  ({\bf T} \circ \antipode_{\mathcal{U}(\g)}) \\
&=({\bf T} \circ e_{\bf T})  *  ({\bf T} \circ \antipode_{\mathcal{U}(\g)}) \\
&={\bf T} \circ ((e_{\bf T} \preliediff {\bf T} \circ \antipode_{\mathcal{U}(\g)})\cdot \antipode_{\mathcal{U}(\g)}) \\
&= {\bf T}({\rm id} \cdot \antipode_{\mathcal{U}(\g)}) \\
&= {\bf T} \circ (\eta_{\mathcal{U}(\a)}\circ\varepsilon_{\mathcal{U}(\g)}) = \eta_{\mathcal{U}(\a)} \circ \varepsilon_{\mathcal{U}(\g)}\end{align*}

And second:

\begin{align*}
({\bf T}\circ \antipode_{\mathcal{U}(\g)}) \# {\bf T} &= {\bf T}(\antipode_{\mathcal{U}(\g)} \preliediff \antipode_{\mathcal{U}(\a)} \circ {\bf T})  *  {\bf T} \\
&={\bf T}((\antipode_{\mathcal{U}(\g)} \preliediff {\bf T} \circ \antipode_{H_{\bf T}}) *_{\bf T} {\rm id}) \\
&={\bf T}((\antipode_{\mathcal{U}(\g)} \preliediff ({\bf T} \circ \antipode_{\mathcal{U}(\g)} \circ e_{{\bf T} \circ \antipode_{\mathcal{U}(\g)}}))  *_{\bf T} {\rm id} ) \\
&= {\bf T}((\antipode_{\mathcal{U}(\g)} \circ e_{\bf T \circ \antipode_{\mathcal{U}(\g)}})  *_{\bf T} {\rm id}) \\
&={\bf T}(\antipode_{H_{\bf T}}  *_{\bf T}{\rm id}) = {\bf T} \circ (\eta_{\mathcal{U}(\a)} \circ \varepsilon_{\mathcal{U}(\g)}) = {\eta_{\mathcal{U}(\a)}} \circ \varepsilon_{\mathcal{U}(\g)}.
\end{align*}

This is a continuation of Remark <ref>. It is obvious that the product # makes sense in the setting of Remark <ref> as a product between the continuous map from $G$ to $\Gamma$. However, when it comes to evaluating the Lipschitz modulus of the product

\begin{align*}
d_\Gamma(S\#T(g), S\#T(g')) &= d_{\Gamma}(S(g\cdot T(g)^{-1})T(g),S(g'\cdot T(g')^{-1})T(g')) \\
&\leq d_{\Gamma}(S(g\cdot T(g)^{-1}), S(g'\cdot T(g')^{-1}))+d_{\Gamma}(T(g),T(g^{\prime})) \\
&= \sigma_{S}(d_{G}(g\cdot T(g)^{-1}, g'\cdot T(g')) + \sigma_T d_G(g,g') \\
&=\sigma_S (d_G (g,g')+d_G(T(g)^{-1},T(g')^{-1}))+\sigma_T d_G(g,g')
\end{align*}

When $G$ and $\Gamma$ are matrix subgroups of a group of unitary matrices (equipped with the Hilbert Schmidt distance), $d_G(T(g)^{-1},T(g')^{-1})) = d_G(T(g),T(g'))$ and

d_{\Gamma}(S \# T(g),S\# T(g'))\leq (\sigma_S + \sigma_T + \sigma_S\sigma_T)d_{G}(g,g').

Thus, $S\#T$ is a strong contraction map in the case where:

\sigma_S \leq \frac{1-\sigma_T}{1+\sigma_T}\quad(\Leftrightarrow \sigma_T \leq \frac{1-\sigma_S}{1+\sigma_S})

$$e_{\bT}= g \cdot (\bT^{-1^{\#}}(g))^{-1}$$

§.§ Exponential mapping

Define the exponential mapping
$\exp_{\#} : \delta \to {\rm Hom}_{\rm Vect}(H,K)$ by:

\begin{equation}
\label{hushexp}
\exp_{\#}(\alpha)
= \varepsilon \circ \eta_{H} + \sum_{n\geq 1} \frac{1}{n!} \alpha^{\# n}.
\end{equation}

The exponential mapping $\exp_{\#}$ defined in (<ref>) is a bijection from $\delta$ to the group $G_{\#}$.

It is enough to prove that $\exp_{\#}$ is surjective. We give an abstract argument. Let $\mathcal{U}(\delta, [-,-]_{*})$ be the envelope of the convolution Lie algebra $(\delta, [-,-]_{*})$. The envelope of $(\delta, [-,-]_{\#})$ is isomorphic to $\mathcal{U}(\delta, [-,-]_{*})$ equipped with the Guin–Oudom product

A \#_{\rm fo.} B
:= (A \preliediff^{*}_{\rm fo.} B_{(1)}) \cdot B_{(2)}.

From the universal property of $\mathcal{U}(\delta, [-,-]_{\#})$, we get the existence of an algebra morphism $I:\mathcal{U}(\delta, [-,-]_{\#}) \to ({\rm End}(H,K),\#)$. A simple recursion shows that $I$ is also an algebra morphism over $\mathcal{U}(\delta,[-,-]_{*})$.
We denote by $\hat{\mathcal{U}}(\delta, [-,-]_{*})$ and $\hat{\mathcal{U}}(\delta, [-,-]_{\#})$ the completions of the respective universal enveloping algebras. Also, the two formal exponentials, $\exp_{*_{\rm fo.}}$ and $\exp_{\#_{\rm fo.}}$, are bijections from the completion of $\delta$, denoted $\hat{\delta}$ to the group-like elements of $\hat{\mathcal{U}}(\delta, [-,-]_{*})$ and $\hat{\mathcal{U}}(\delta, [-,-]_{\#})$, which are isomorphic as sets.

Pick ${\bf T} \in G_{\#}$. The convolution exponential is a bijection from $\delta$ to the convolution group $G_{\Delta}$: there exists $t \in \delta$ such that $\exp_{*}(t)={\bf T}$. Thanks to the postLie Magnus expansion, we can find an element $\Omega(t)\in \hat{\delta}$ such that $\exp_{*_{fo}.}(t)=\exp_{{\#_{\rm fo.}}}(\Omega(t))$. Applying $I$ yields the result.

We compare the exponential mapping $\exp_\#$ and the Guin–Oudom recursion (<ref>).

Let $(\g,\vartriangleleft,\a)$ be a Lie module.
Let ${\bf T}\coloni \univalgebra\to \mathcal{U}(\a)$ be a $\mathcal{O}$-Hopf operator then, with $\Gamma^{\bT} = (\Gamma^{\bT}_n:\mathcal{U}(\g)\to\mathcal{U}(\a))_{n \geq 1}$ the sequence defined by

\Gamma^{\bT}_n := {\bf T} \circ {\rm sol}_n : \univalgebra \to \mathcal{U}(\a):

the following formula holds, for any $f \in H$:

\begin{equation}
\Gamma^{\bT}_{n+1}(f) = \Gamma^{\bT}_n * \Gamma^{\bT}_1 (f)- \Gamma^{\bT}_n(f_{(1)} \preliediff \Gamma^{\bT}_{1}(f_{(2)})) = \frac{1}{n!}(\Gamma^{\bT}_1)^{\# n} (f)
\end{equation}

\bT = \exp_{\#}(\bT \circ {\rm sol}_1).

Let $n\geq 1$ be an integer greater than $1$. First, recall that $e_{n+1}=e_{n}\cdot {\rm sol}_1$ where $\cdot$ denotes the convolution product with respect to the concatenation product over $\univalgebra$. Second, recall that

\Delta \circ {\rm sol}_1
= ({\rm sol}_1 \otimes \varepsilon + \varepsilon \otimes {\rm sol}_1) \circ \Delta

Then, for any $E \in \univalgebra$, one gets

\begin{align*}
e_n *_{\bT} {\rm sol}_1 (E)
&= e_n(E_{(1)}) \preliediff \bT((sol_{1}(E_{2}))_{(1)}) \cdot {\rm sol}_1(E_{(2)})_{(2)} \\
&=e_{n}(E_{(1)})\preliediff \bT({\rm sol}_1(E_{(2)})) + e_{n}(E_{(1)}){\rm sol}_1(E_{(2)})
\end{align*}

e_{n} *_{\bf T} {\rm sol}_1
= e_{n+1} + e_{n} \preliediff \bT \circ {\rm sol}_1.

We apply the above formula to get

\begin{align*}
{\bf T}(e_{n+1})
&= {\bf T}(e_n *_{{\bf T}} {\rm sol}_1 - e_n \preliediff {\bf T}\circ {\rm sol}_1) \\
&=\gamma_n * \gamma_1 - {\bf T}(e_n \preliediff {\bf T} \circ {\rm sol}_1) \\
&=\gamma_n * \gamma_1 - \gamma_n (\mathrm{id} \preliediff
\gamma_1)
& \textrm{Proposition } \eqref{prop:solande}
\end{align*}

This ends the proof.

It is possible to give a more general result than Theorem <ref>. As an application, we show that the subset of elements of $G_{\#}$ whose inverse with respect to $\#$ can be computed by precomposition with the antipode of $H$ is much larger than the subset of $\mathcal{O}$ - Hopf operators.

Let ${t} \in \delta$ and set, for any integer $n\geq 1$,

{t}_n = {t} \circ sol_{n}~ \textrm{ and }
{T} = \exp_{\#}(t).

Then, for any integer $p\geq 1$,

\begin{equation*}
T \circ {\rm sol}_p=\sum_{\lambda \vdash p} \frac{1}{|\lambda|!}{\bf t}_{\lambda_1} \#\cdots\#\, {\bf t}_{\lambda_{|\lambda|}},
\end{equation*}

where $\lambda \vdash p$ denotes a composition $(\lambda_1,\ldots,\lambda_{|\lambda|})$ of $p$.

Let $t:\g \to \a$ be a linear map and set ${\bf T}=\exp_{\#}(t\circ sol_1)$. Then

{\bf T}^{\# -1} = {\bf T} \circ \mathcal{S}_H.

In particular,

({\bf T}^{\# -1})^{* -1} = ({\bf T}^{* -1})^{\# -1}.

We only treat the case where ${t}_n = 0$ if $n > 1$. The general case follows by the same lines of arguments. First, we set

\begin{align*}
\gamma_{n,p}
= ({t} \circ {\rm sol}_1)^{\# n}\circ {\rm sol}_p
\end{align*}

The objective is to find a recursion on $(\gamma_{n,p})_{n,p}$. Let $n\geq 1$ and $p\geq 1$. By using the definition of $\sharp$, we get

\begin{align*}
\gamma_{n+1,p} =  (({t} \circ {\rm sol}_1)^{\# n} * ({t}\circ {\rm sol}_1))\circ {\rm sol}_p-({t} \circ {\rm sol}_1)^{\# n}({\rm id} \preliediff ({t} \circ {\rm sol}_1))\circ {\rm sol}_p
\end{align*}

Second, since the PBW isomorphism is a bijection of coalgebras, one has, in fact:

\begin{align*}
\Delta {\rm sol}_p
= \sum_{\substack{a,b \geq 0\\ a+b=p}} {\rm sol}_a \otimes {\rm sol}_b
\end{align*}

Because $sol_1 \circ sol_p=0$ if $p > 1$, one gets:

\begin{align}
\label{eqn:rec}
\gamma_{n+1,p}
= \gamma_{n,p-1} * ({t} \circ {\rm sol}_1)- \gamma_{n,p-1}({\rm id} \preliediff ({t} \circ {\rm sol}_1)) = \gamma_{n,p-1} \# (t \circ sol_1).
\end{align}

For $n > 0$, one has

\begin{align*}
\gamma_{n+1,1}
& = (({t} \circ {\rm sol}_1)^{\# n} * ({t}\circ{\rm sol}_1)) \circ {\rm sol}_1 - ({t} \circ {\rm sol}_1)^{\# n}(\mathrm{id}\preliediff ({t} \circ {\rm sol}_1))\circ {\rm sol}_1 \\
&= 0
\end{align*}

The result follows by unfolding the recursion (<ref>).

§.§ Factorization

Let $\{\bm{s},\bm{t}\}$ be a matching $\mathcal{O}$ operator on a Lie module $(\g,\a,\vartriangleleft)$. With the notation introduced so far:

[\bm{s} + \bm{t}] = [\bm{s}] \# [\bm{t}] = [\bm{t}] \# [\bm{s}].

Let $\{\bf{s},\bf{t}\}$ be a Matching $\mathcal{O}$ Lie operator. It is easy to see that the matching relations (<ref>) are equivalent to the vanishing of the Lie bracket of ${\bf s}\circ sol_1$ with ${\bf t}\circ sol_1$:

[{\bf s}\circ sol_1,{\bf t}\circ sol_1]_{\#}=0.

Recovering the classical Semenov-Tian-Shanskii factorization.

\begin{align*}
\end{align*}

Recall that ${\bm{R}}$ and ${\hat{\bm{R}}}$ are the Rota–Baxter–Hopf extending and coextending ${\bm{r}}$ and $\hat{\bm{r}}$ to the Hopf algebra $\univalgebra$.
Since ${\bm{r}}+\hat{{\bm{r}}}=-{\rm id}_{\mathfrak{g}}$ and ${\hat{\bm{R}}}\#{\bm{R}}$ is a Rota–Baxter–Hopf operator, we have $\hat{\bm{R}}\# {\bm{R}}=\mathcal{S}_{\mathcal{U}(\mathfrak{g})}$. We obtain the following factorization for any $X \in \univalgebra$:

\begin{equation}
X = (\mathcal{S}_{\univalgebra}\circ \hat{\bm{R}})(e_{\bm{R}}^{-1}(X_{(1)})) \mathcal{S}_{\univalgebra}({\bm{R}}(X_{(2)})).
\end{equation}

This last formula must be compared with Corollary 23 of [13] wherein $R_+=\mathcal{S}_{\univalgebra}\circ \hat{\bm{R}}$ and $R_-={\bm{R}}$.
Let $x \in \mathfrak{g}$, then we get
$$ \mathcal{S}_{\univalgebra}(\exp_{\star_{\bm{R}}}(x))
={\hat{\bm{R}}}\# {\bm{R}}(\exp_{\star_{\bm{R}}}(x))

Recall that we have the following formula for the right action $\preliediff$ (the Adjoint action) of $\univalgebra$ over itself, for any element $X,Y\in\univalgebra$:

X \preliediff Y
= \mathcal{S}_{\univalgebra}(Y_{(1)})XY_{(2)}.

It is not difficult to see that:

\begin{equation}
\label{eqn:expreflection}
\exp_{\bf \hat{R}}(x)
\end{equation}

The formula $e_{\bm{R}}^{-1}=\mathrm{id}\preliediff {\bm{R}}^{\star -1}=\mathrm{id}\preliediff \mathcal{S}_{\univalgebra}\circ {\bm{R}}$ yields, for any $x \in \mathfrak{g}$:

=\exp_{\star_{\bm{R}}}({\bm{R}}(x))\preliediff \mathcal{S}_{\univalgebra}\circ{\bm{R}}(\exp_{\star_{\bm{R}}}({\bm{R}}(x)))

The last formula together with (<ref>) yield:

\begin{equation}
\label{eqn:ttransformreflection}
\end{equation}

Finally, we obtain:

\mathcal{S}_{\univalgebra}(\exp_{\star_{\bm{R}}}(x))

This last equation implies the equality

\exp_{\bm{R}}(x)

Assume that $H$ is cocommutative. The compositional inverse of the $T$ transform yields a group morphism from the group $G_{\#}$ to the group $G_{\circ}$ of coalgebra morphisms on $H$ equipped with the compositional product;

\begin{equation}
e^{-1^{\circ}} : G_{\#}\to \big(\mathrm{End}_{\Delta}(H,H),\circ\big).
\end{equation}

Theorem <ref> admits the following reformulation. The group $G_{\circ}$ acts via pre-composition on the convolution group $G_{*}$ via group morphisms,

(A * B) \cdot X
= (A * B) \circ X = (A \circ X) * (B \circ X).

Then Theorem <ref> together with Proposition <ref> implies that $e^{-1}$ is a $\mathcal{O}$ operator on the group module $(G_{*},\cdot,G_{\circ})$. And, with the notations put in place in the previous Section, for any $A,B \in G_{*}$

A \# B
= A *_{e^{-1^{\circ}}} B,

since $S_K \circ A = A^{-1^{*}}$.

From [22], a set-solution to the Quantum Yang Baxter equation is given by

R_{\#}:G_{\#}\times G_{\#}\to G_{\#} \times G_{\#},~R_{\#}(\bS,\bT) = (\bS \# \bT \# (\bS \circ e_{\bT}^{-1^{\circ}})^{-1^{\#}}, \bS \circ e_{\bT}^{-1^{\circ}} )

§ OPERATOR-VALUED FREE PROBABILITY THEORY

The objectives of this Section are, first, to reformulate the relations between transforms in non-commutative probability (such as the $T$ transform, the cumulants transform $K$) and their properties with respect to the multiplication of independent (free or monotone independent, even conditionally free or conditionally independent) random variables by using $\mathcal{O}$-groups. Next, we explain how the cross product of Hopf algebras allows for a neat and clean analogy on the relations between transforms and their conditional counterparts (which so far only were defined in the setting of scalar non-commutative probability theory). Finally, we continue exploring the geometry of $\mathcal{O}$ operators by providing some abstract constructions motivated by non-commutative probability theory, so we have chosen to include them here.

We refer to [33] for basic definitions about operator-values free probability theory and to [10, 11] for definition of the various transform used in this section (we recall relevant definitions below though).
Through the section, we consider an operator-valued algebraic probability space $(\mathcal{A}, E,\mathcal{B})$.

Let $a,b$ be two random variables and suppose that $\mathbb{E}(a)=\mathbb{E}(b)=1$.

Consider their cumulants series:

K_a = 1+R_a = 1+\sum_{n\geq 1} R^{(n)}_a\in \mathbb{C}[[End\,B]],~K_b=1+\sum_{n\geq 1} R^{(n)}_b \in \mathbb{C}[[End\,B]],

with for $x\in \{a,b\}$, $b_1,\ldots,b_n \in B$

R^{(n)}_x \in {\rm Hom}(B^{\otimes n},B),\quad R^{(n)}_x(b_1,\ldots,b_n)=\kappa_{n+1}(ab_1,ab_2,\ldots,b_na)

and $\kappa_{n}$ denotes the $n^{\rm th}$ free cumulant.

Recall that we denote by $\mathbb{C}[[End\,B]]$ the formal series with entries in the endomorphisms operad of $B$:

\begin{align}
\alpha \in \mathbb{C}[[End \,B]], \quad \alpha = \sum_{n\geq 1} \alpha^{(n)},\, \alpha^{(n)}\colon B^{\otimes n} \to B.
\end{align}

We denote by $\mathbb{C}_0\llbracket End B\rrbracket$ the group supported by formal series in $\mathbb{C}\oplus \mathbb{C}[[End \,B]]$ starting with $1$ and equipped with the Cauchy product $\centerdot$. We let $\Gamma$ the group supported by formal series in $\mathbb{C}[[End \,B]]$ starting with the unit and equipped with the composition product $\circ$:

(\alpha \circ \beta)^{(n)}=\sum_{\substack{q_1+\cdots q_k=n, \\ k,q\geq 1}} \alpha^{(k)}\circ (\beta^{(q_1)}\otimes\cdots\otimes\beta^{(q_k)}).

The $T$ transform $T_x \in \mathbb{C}_{0}\llbracket End B\rrbracket$of a random variable $x\in\mathcal{A}$ is defined recursively via the fixed point equation:

K_x = T_x \circ (I\centerdot K_x)

It has been proved that [10]

T_{ab} = T_a \circ ({T_b\centerdot I\centerdot T^{-1^{\centerdot}}_b}) \centerdot T_b

Recall that $(\mathbb{C}_0[End B],\cdot,\Gamma)$ is a right group-module with the right action $\cdot$ defined by, for any $\alpha,\beta \in \CC[[End\,B]]$ :

(1 + \alpha) \cdot \beta = 1 + \alpha\circ\beta.

Generic elements of $\mathbb{C}_0[End B]$ will be denoted $A,B\ldots$ while elements of $\Gamma$ will be denoted $\alpha,\beta$...
The right translation $\bm{\rho}$ by the unit is a $\mathcal{O}$ group operator, as well as $\bm{\lambda}\circ \mathcal{S}$ $(=\bm{\lambda}^{-1^{\#}})$ where $\mathcal{S}$ denotes the inversion map with respect to the group law of $G$ and $\lambda$ is the left translation by the unit $I = {\rm id}_B$ (the operator $\bm{\lambda}$ alone is $\mathcal{O}$ - group operator but with respect to opposite group $G^{op}$). The pair $(\bm{\lambda}^{-1^{\#}}, \bm{\rho})$ is a Matching $\mathcal{O}$-group operator, as proved in Example <ref>:

\bm{\rho} \colon A \mapsto A \centerdot I,\quad \bm{\lambda} \colon A \mapsto I\centerdot A.

This implies, see Section <ref>, that $\mathbb{C}_0\llbracket End B\rrbracket$ can be endowed with three different group products, $\star_{\bm{\lambda}^{-1^{\#}}}$, $\star_{\bm{\rho}},$  and $ \star_{\bm{\lambda}^{-1^\#}\# \bm{\rho}}$, for any $A,B \in \mathbb{C}_0\llbracket End B\rrbracket$:

A \star_{\bm{\lambda}^{-1^{\#}}} B = (A\cdot (I\centerdot B^{-1^{\centerdot}} ))\centerdot B,\quad
A \star_{\bm{\rho}} B = (A\cdot (B\centerdot I) ))\centerdot B

A \star_{\bm{\lambda}^{-1^{\#}}\#\bm{\rho}} B = (A \cdot (B\centerdot I\centerdot B^{-1^{\centerdot}})) \centerdot B.

The fact that $(\bm{\lambda}^{-1^{\#}}, \bm{\rho})$ is a Matching $\mathcal{O}$-operator implies the following important relations, see Theorem <ref>:

\begin{align}
\label{eqn:subordination}
A \star_{\bm{\lambda}^{-1^{\#}}\#\bm{\rho}} B
% &= (A \cdot (I\centerdot B^{-1^{{\star_{\rho}}}}))\star_{\bm{\rho}} B = (A \cdot ((B^{-1^{\bm{\star_{\lambda^{-1^{\#}}}}}} \centerdot I)))\star_{{\bm{\lambda^{-1^{\#}}}}} B \\
&= (A \cdot_{\bm{\lambda}}( B^{-1^{{\star_{\rho}}}}))\star_{\bm{\rho}} B = (A \cdot_{\bm{\rho}^{-1^{\#}}} (B^{-1^{\bm{\star_{\lambda^{-1^{\#}}}}}})) \star_{{\bm{\lambda^{-1^{\#}}}}} B
% &= (A \cdot (I\centerdot B^{-1^{\star_{\bm{\lambda}}}}))\star_{\bm{\rho}} B \,\,= \quad (A \cdot ((B^{-1^{\star_{\bm{\rho}}}} \centerdot I)))\star_{{\bm{\lambda^{-1^{\#}}}}} B
\end{align}

We shall now indicate how those products connect to the other non-commutative independences and related convolutions (both additive and multiplicative).
To begin with, for operator-valued monotone multiplicative convolution, similar relations hold but for their $H$-transform $H_a$ (and $H_b$) defined by (again in our notations):

\begin{align}
\label{eqn:htransform}
H_a = (T^{-1^{\centerdot}}_a \cdot (\frac{I}{1-I}))^{-1^{\star_{\bm{\lambda}}}}= (\Sigma_a)^{-1^{\star_{\bm{\lambda}}}},\quad H_{ba} = H_a \star_{\bm{\lambda}}H_b .
\end{align}

whenever $a-1$ and $b-1$ are monotone independent $(a\prec b)$ and where we have denoted

\Sigma_a = T^{-1^{\centerdot}}_a \cdot \frac{I}{1-I}.

Let us see how $H_a$ can be expressed in the free cumulants series $K_a$ of $x$. First, by definition, the formal series $H_a$ satisfies the fixed point equation:

\begin{align}
H_a = (T_a\cdot \frac{I}{1-I}) \cdot_{\lambda} H_a
\end{align}

This fixed point equation implies that $H_a$ is also the unique solution $A_a$ to

A_a = T_a \cdot ((1-I\centerdot H_a)^{-1}\centerdot(I\centerdot A_a)).

On the other hand,

\begin{align*}
K_a\cdot ((1-I\centerdot H_a)^{-1} \centerdot I) &= T_a \cdot ((I\centerdot K_a) \circ ((1-I\centerdot H_a)^{-1}\centerdot I)) \\
&=  T_a \cdot ((1-I\centerdot H_a)^{-1}\centerdot I \centerdot K_a\circ ((1-I\centerdot H_a)^{-1} \centerdot I))
\end{align*}

And thus we derive the following formula for $H_a$

\begin{align}
\label{eqn:formulahx}
H_a = A_a = K_a\cdot ((1-I\centerdot H_a)^{-1} \centerdot I) = K_a \cdot_{\bm{\rho}}(1-I\centerdot H_a)^{-1}.
\end{align}

Let us stop a moment on the relations (<ref>). We observe that the relations between $\Sigma_a$ and $H_a$ on the one hand is the same as the relation between $T_a$ and $K_a$ on the other hand, a double inversion, namely.
Additionally pre-composition by the map $\frac{I}{1-I}$ (that is the action of $\frac{I}{1-I}$ through $\cdot$) is equivariant with respect the action $\cdot_{\bm{\lambda^{-1^{\#}}}\#\bm{\rho}}$:

(A \cdot_{\bm{\lambda^{-1^{\#}}}\#\bm{\rho}}B)\cdot (\frac{I}{1-I} )= (A \cdot (\frac{I}{1-I}))\cdot_{\bm{\lambda^{-1^{\#}}}\#\bm{\rho}}(B\cdot (\frac{I}{1-I})).

The above equivariance relation can be proved by using the explicit formulae for the product $\star_{\bm{\lambda}\#\bm{\rho}}$, see Proposition <ref>.
Finally, the relations (<ref>) can be described using the concept of subordination in non-commutative probability.
In fact, whenever $a,b$ are free independent then:

\begin{align*}
H_{ab} = \Sigma_{ab}^{-1^{\star_{\bm{\lambda}}}}= ((\Sigma_a \cdot_{\bm{\rho}} (\Sigma_y^{-1^{\bm{\star_{\lambda}}}}))\star_{\bm{\lambda}} \Sigma_b )^{-1^{\star_{\bm{\lambda}}}} = H_b \star_{\lambda} (\Sigma_a \cdot_{\bm{\rho}}(\Sigma_y^{-1^{\star_{\bm{\lambda}}}}))^{-1^{\star_{\bm{\lambda}}}}).
\end{align*}

Setting $H_{a \dashv b} :=  (\Sigma_a \cdot_{{\bm{\rho}}} (\Sigma_b^{-1^{\star_{\bm{\lambda}}}}))^{-1^{\star_{\bm{\lambda}}}} $, we see that $H_{a \dashv b}$ satisfies the fixed point equation :

\begin{align*}
H_{a \dashv b} = (\Sigma^{-1}_x \cdot_{\bm{\rho}}(\Sigma_b^{-1^{\star_{\bm{\lambda}}}})) \cdot_{\bm{\lambda}} H_{a \dashv b} &= ((\Sigma_a^{-1}\cdot_{\bm{\lambda}} \Sigma^{-1^{\star_{\bm{\lambda}}}}_a)\cdot ({\bm{\lambda}}(\Sigma_a)\circ{\bm{\rho}}(\Sigma_b^{-1^{\star_{\bm{\lambda}}}})) \cdot_{{\bm{\lambda}}} H_{a \vdash b} \\
&=\Sigma_a^{-1^{\star_{\bm{\lambda}}}}\cdot (\bm{\lambda}(\Sigma_a) \circ {\bm{\rho}}(\Sigma^{-1^{\star_{\bm{\lambda}}}}_b)\circ \bm{\lambda}(H_{a\vdash b})).
\end{align*}

We use the relations (<ref>) to write :

\begin{align*}
(\bm{\lambda}(\Sigma_a) \circ {\bm{\rho}}(\Sigma^{-1^{\star_{\bm{\lambda}}}}_y)\circ \bm{\lambda}(H_{a\vdash b})) &= {\bm{\rho}}(\Sigma^{-1^{\star_{\bm{\lambda}}}}_b \cdot_{\bm{\lambda}} ( (\Sigma_a \cdot_{\bm{\rho}} \Sigma^{-1^{\star_{\bm{\lambda}}}}_b)^{-1^{\star_{\bm{\lambda}}}} ))\circ \bm{\lambda}(\Sigma_a \cdot_{\bm{\rho}} \Sigma^{-1^{\star_{\bm{\lambda}}}}_b) \circ \bm{\lambda}(H_{a\vdash b}) \\
&={\bm{\rho}}(\Sigma^{-1^{\star_{\bm{\lambda}}}}_b \cdot_{\bm{\lambda}} H_{a\vdash b})
\end{align*}

We obtain finally the following fixed point equation :

\begin{align}
\label{eqn:subordinationfixedpointeqn}
H_{a\vdash b} = H_a \cdot_{\bm{\rho}} (H_b \cdot_{\bm{\lambda}} H_{a\vdash b})
\end{align}

We develop further the use of our formalism toward conditional independence in the next sections.

§.§ Conditional independences

We first give the definitions of conditional free and monotone independence. We refer to [1, 29, 28] and [27] for the definitions in the context of operator-valued probability spaces and to [18, 7] for their definitions in the scalar case.

Let $(\mathcal{A}, \varphi, \psi, B)$ a conditional operator-valued probability space; $\mathcal{A}$ is an unital complex algebras, $\mathcal{B}$ is acting on $\mathcal{A}$ from the right and from the left, $\varphi$ and $\psi$ are two $B$-bimodule morphisms.

We say that two variables $a_1$ and $a_2$ are conditionally free if they are free with respect to $\psi$ and for any $p_1,\ldots p_n \in\langle a_1\rangle \cup \langle a_2 \rangle$:

\begin{equation*}
\varphi(p_1\cdots p_n) = \varphi(p_1)\cdots \varphi(p_n)
\end{equation*}

whenever $p_j\in\mathcal{A}_{i_j},\psi(p_j)=0$ and $i_j\neq i_{j+1}$.

We say that the two variables $a_1$ and $a_2$ are conditionally monotone independent if they are monotone independent with respect to $\psi$ and for any $p_1,\ldots,p_n \in \langle a_1 \rangle_0 \cup \langle a_2\rangle_0$, $p_i \in \mathcal{A}_{i_j}$, $i_j \neq i_{j+1}$

\varphi(p_1\cdots p_n)= \varphi(p_1)\varphi(p_2\cdots p_n),\quad \varphi(p_1\cdots p_n)= \varphi(p_1p_2\cdots )\varphi(p_n)

and for any $2 \leq i \leq n-1$:

\varphi(p_1\cdots p_n) = \varphi(p_1\cdots p_{i-1})(\varphi(p_i)-\psi(p_i))\varphi(p_{i+1}\cdots p_n) + \varphi(p_1\cdots p_{i-1}\psi(p_i)p_{i+1}\cdots p_n)

Given $\x = \{x_1,\ldots,x_n\} \in \mathcal{A}$, we denote by $\varphi_x$ and $\psi_x$ the joint distribution of $\x$ with respect to $\varphi$ and $\psi$, respectively:

\begin{align}
\varphi_{\x}(p(X_1,\ldots,X_n)) = \varphi(p(\x)),\quad \psi_{\x}(p(X_1,\ldots,X_n)) = \psi(p(\x))
\end{align}

for any non-commutative polynomial $p\in \mathbb{C}[X_1,\ldots,X_n]$. Then given $a_1$ conditionally monotone independent from $a_2$, there exists two conditionally free random variables $\tilde{a_1},\tilde{a_2}$ in some operator-valued conditional probability space $(\tilde{\mathcal{A}}, \tilde{\varphi}, \tilde{\psi}, \mathcal{B})$ whose distributions under $\varphi$ are equal to the distribution of $a_1$ and $a_2$ under $\varphi$ and

\psi_{\tilde{a}_1}(p) = p(1),\quad \psi_{a_2}(q)=\psi_{\tilde{a}_2}(q),\quad p \in \mathbb{C}[X_2],~q \in \mathbb{C}[X_1]

such that the following equality between linear functions on $\mathbb{C}[X_1,X_2]$ holds:

\begin{align}
\varphi_{a_1,a_2} = \varphi_{\tilde{a}_1, \tilde{a}_2}.
\end{align}

§.§ Cross products and conditional independences

We leverage the interpretation that we gave for the $T$-transform in Voiculescu's free probability theory, namely as the composition of two antipodes, see Proposition <ref>, to define the relevant $T$-transform in the setting of operator-valued conditional freeness. In particular, we explain that going to this conditional setting among to taking the crossed product of the Hopf algebra $H_{\bm{\lambda}^{-1^{\#}}\#\bm{\rho}}$ with the Hopf algebra $H_{\varepsilon}$, from the Hopf-algebraic perspective.

We recall the cross product construction[<https://ncatlab.org/nlab/show/crossed+product+algebra>] in the particular setting of cocommutative Hopf algebras and of groups. Let $(H,\preliediff,K)$ be a right Hopf module.
The crossed product of $K$ and $H$ over the action $\preliediff$ is the Hopf algebra denoted $K \ltimes_{\preliediff} H$, which is equal as a coalgebra to $(K \otimes H, \tau_{23}\circ\Delta_K \otimes \Delta_H)$ and is equipped with the following unital associative product:

\begin{equation}
(k^a \ltimes h^a) (k^b \ltimes h^b) =   k^ak^b_{(1)} \ltimes (h^a \preliediff k^b_{(2)}) h^b.
\end{equation}

On easily checks that $1_K \otimes 1_H$ is the unit for the above defined product.
The antipode of the crossed product $K\ltimes_{\theta} H$ Hopf algebra $S_{K\ltimes_{\preliediff} H}:K\ltimes_{\preliediff} H \to K\ltimes_{\preliediff} H $ satisfies the following formula:

\begin{align*}
\mathcal{S}_{K\ltimes_{\preliediff}H}(k\ltimes h) = \mathcal{S}_K(k_{(1)}) \otimes (\mathcal{S}_{H}(h) \preliediff \mathcal{S}_K(k_{(2)}) ).
\end{align*}

If $(G,\cdot, \Gamma)$ is a right group module, the crossed product of $G$ and $\Gamma$ over $\cdot$ is the group supported by the set $\Gamma \times H$ with the product defined by

(k^a \ltimes h^a)
(k^b \ltimes h^b) =   k^ak^b_{(1)} \ltimes (h^a \cdot k^b_{(2)}) h^b.

§.§.§ Conditional $T$ transform

Recall the definition of the Hopf algebras $H_{\bf T}$, given a $\mathcal{O}$ Hopf operator over a right Hopf module $(K,H,\preliediff)$ and the definition of $\preliediff_{\bf T}$, a right action of the Hopf algebra $H_{\bf T}$ on the Hopf algebra $H$.
Inspired by applications to non-commutative probability developed below, we define the $T$-transform of $\bf T$ conditionally to the convolution unit $\eta_K\circ \varepsilon_{H}$ of $H$.
Notice that in our notations, and as Hopf algebras, $H_{\eta_K\circ \varepsilon_H}=H$ ($\eta_K\circ \varepsilon_H$ is a $\mathcal{O}$-Hopf operator on the right Hopf-module $(K,H,\preliediff)$).

Define the morphism of coalgebra

$$e_{\bT\,|\,\varepsilon}\colon H\otimes H \to H \otimes H$$

by, for any $k\otimes h \in H\otimes H$

\begin{align*}
e_{\bT\,|\,\varepsilon}(k\otimes h) &:= \mathcal{S}_{H_{\bT} {\ltimes}_{\preliediff_{\bT}} H} \circ (\mathcal{S}_{H} \otimes \mathcal{S}_{H})(k \otimes h) \\
& = (S_{H_{\bT}} \circ \mathcal{S}_{H})(k_{(1)}) \,\otimes (h \preliediff_{\bT}  ((\mathcal{S}_{H_{\bT}} \circ \mathcal{S}_{H})(k_{(2)}))) \\
& =e_{\bT}(k_{(1)}) \,\otimes (h \preliediff_{\bT}  e_{\bT}(k_{(2)}))
\end{align*}

Remark that whenever $h$ and $k$ are group-like elements (in a adequate completion of $H$), the following formula is in hold:

\begin{align*}
e_{\bT \, | \, \varepsilon}(k\otimes h) = (k^{-1})^{-1^{\star_{\bT}}}\otimes h \cdot_{\bT} (k^{-1})^{-1^{\star_{\bT}}}.
\end{align*}

The same formula can be used to define the relative T transform of a $\mathcal{O}$-group operator in the setting of $\mathcal{O}$-group and crossed products of groups. More precisely, with $(G,\cdot,\Gamma)$ a right group-module, and $\bT\colon G \to \Gamma$ a $\mathcal{O}$- group operator, with the notations introduced in Definition, for any $(k,h)\in G \times G$ :

\begin{align}
e_{\bT\,|\,\varepsilon}(k,h):=\big((k^{-1})^{-1^{\star_{\bT}}}, h \cdot_{\bT} (k^{-1})^{-1^{\star_{\bT}}}\big) \in G_\bT \times G.
\end{align}

Recall the definition of the product in $G_\bT \ltimes_{\cdot_\bT} G$ :

\begin{align}
\label{eqn:producttwistedgp}
(k_1,h_1)(k_2,h_2) = (k_1\star_\bT k_2, (h_1\cdot_\bT k_2)h_2 ),\,\,k_1,k_2 \in G_{\bT},\,h_1,h_2 \in G.
\end{align}

Notice that $\mathcal{S}_H \otimes \mathcal{S}_H = \mathcal{S}_{H_\varepsilon \ltimes_{\preliediff_{\varepsilon}}} H$. Note that we can consider as well the composition between the antipodes of the Hopf algebras $H_{\bT} {\ltimes}_{\preliediff_{\bT}} H$ and $H_{\bS} {\ltimes}_{\preliediff_{\bS}} H$ since both are supported on tensor product $H\otimes H$ and define more generally:

\begin{align}
e_{\bT | \bS}(k\otimes h) &:= \mathcal{S}_{H_{\bT} {\ltimes}_{\preliediff_{\bT}} H} \circ \mathcal{S}_{H_{\bS} {\ltimes}_{\preliediff_{\bS}} H^{op}} (k\otimes h) \\
&= \mathcal{S}_{H_\bT} \circ \mathcal{S}_{H^{op}_\bS}(k_{(1)})\otimes h \preliediff ((\bS\circ \mathcal{S}_{H^{op}_\bS})\star (\bT \circ \mathcal{S}_{H_\bT} \circ \mathcal{S}_{H^{op}_\bS}))(k_{(2)})
\end{align}

where we recall that $\star$ is the convolution product between linear functional from $H$ to $K$.
The following proposition states that in cases of interest, we can express this more general relative, two arguments, $T$-transform by using the product $\#$ and the relative $T$-transform of Definition <ref>.

Assume that $\bT$ and $\bS^{-1^{\#}}$ are compatible $\mathcal{O}$ Hopf Operators then:

\begin{align*}
e_{\bT | \bS} = e_{\bS^{-1^{\#}} \# \bT \,|\,\varepsilon}.
\end{align*}

Recall that $\mathcal{S}_{H_\bT} = e_\bT \circ \mathcal{S}_H$ (see Theorem <ref>). Then

\begin{align*}
\mathcal{S}_{H_{\bT}} \circ \mathcal{S}_{H^{op}_{\bS}} = e_\bT \circ e_{\bS \circ \mathcal{S}}= e_{\bS\circ\mathcal{S} \# \bT}= e_{\bS^{-1^{\#}} \# \bT}  = \mathcal{S}_{\bS^{-1^\#} \# \bT} \circ \mathcal{S}_H = \mathcal{S}_{\bT \# \bS^{-1^\#}} \circ \mathcal{S}_H
\end{align*}

It follows that

\begin{align*}
(\bS\circ \mathcal{S}_{H^{op}_{\bS}})\star (\bT \circ \mathcal{S}_{H_\bT} \circ \mathcal{S}_{H_\bS})
&=(\mathcal{S}_H \circ \bS) \star (\mathcal{S}_H \circ \bT \circ \mathcal{S}_{H_\bS}) && {\rm eqn.\,\,}\eqref{def:Ohopfoperator}, \eqref{eqn:productohopfoperator}\\
&= \mathcal{S}_H \circ ((\bT \circ \mathcal{S}_{H_\bS}) \star \bS )\\
&=(\bT^{-1^\#} \# \bS )^{-1^\star}\\
&=(\bT^{-1^\#} \# \bS)  \circ \mathcal{S}_{\bT^{-1^\#} \# \bS} && {\rm Theorem\,\,} \ref{thm:semenovtianshanskii}\\
&=(\bT \# \bS^{-1^{\#}})  \circ \mathcal{S}_H \circ \mathcal{S}_{\bT^{-1^\#} \# \bS} && {\rm Proposition \,\,} \ref{prop:inverseOhopfoperator}\\
&=(\bT \# \bS^{-1^{\#}}) \circ e_{\bT \# \bS^{-1^{\#}}} && {\rm Theorem \,\, \ref{thm:tttransformtwoantipodes}}
\end{align*}

We specialize this construction to the setting exposed at the beginning of the section. Pick $T,T^c \in G$ two formal series of multilinear functionals on the algebra $B$ starting with one, then (with $I={\rm id}_B$)

\begin{equation}
\label{eqn:ttransformlambda}
e_{\bm{\lambda}\,|\,I\varepsilon}(T,T^c) = (e_{\bm{\lambda}}(T), T^c \cdot_{{\bm \lambda}} e_{\bm{\lambda}}(T)) = (e_{\bm{\lambda}}(T), T^c\cdot (I\centerdot e_{\lambda}(T)).
\end{equation}

Recall that we set $K_a$, respectively $K_a^c$, the formal series of the free cumulants of $x$, respectively, the formal series of the conditionally free cumulants of $x$

\begin{align}
\label{eqn:cumulantsmoments}
(C_x^{\psi})^{-1^{\circ}} = I \centerdot(1+ K_a \centerdot I)^{-1}, \quad (C_x^{\varphi})^{-1^{\circ}} = I \centerdot (1+ K^c_x\circ (C_x^{\psi} \circ  (C_x^{\varphi})^{-1^{\circ}} \centerdot I))^{-1}
\end{align}

where the multilinear series $C^{\psi}_x$ and $C^{\varphi}_x$ are the operator-valued moment transforms of $x$ [10]:

\begin{align}
C_x^{\varphi}=\sum_{n\geq 0} \varphi(b(ab)^n),\quad C_x^{\psi}=\sum_{n\geq 0} \psi(b(ab)^n).
\end{align}

(We have noted  $\varphi(b(ab)^n)$ for the multilinear function $(b_0,\ldots,b_n)\mapsto \varphi(b_0ab_1\cdots ab_n)$ ). In our notations and with the multilinear series $\tilde{C}^{\psi}_x$ (starting with $1$) defined through the relation ${C}^{\psi}_x = I\centerdot \tilde{C}^{\psi}_x$,

\begin{align}
\label{eqn:momentdoubleinversion}
\tilde{C}^{\psi}_x = ((1+ K_a \centerdot I)^{-1})^{-1^{\star_{\bm{\lambda}}}}
\end{align}

and $\tilde{C}_x^{\psi}$ is the solution to the fixed point equation (the moments-cumulants relations)

\begin{align}
\label{eqn:dependencehc}
\tilde{C}^{\psi}_x = (1+ K_a \centerdot I) \cdot_{\bm{\lambda}}\tilde{C}^{\psi}_x
\end{align}

We note that, with $C_x^{\psi}=I+I\centerdot \bar{C}_x^{\psi} \centerdot I$:

H_a = \bar{C}_x^{\psi} (1+I\centerdot \bar{C}_x^{\psi})^{-1}

In fact, the defining relation:

\begin{align*}
I = C_x^{\psi} \centerdot (1+(K_a \circ C_x^{\psi}) \centerdot C_x^{\psi})^{-1^{\centerdot}}
\end{align*}

implies, since the left translation by $I$ is injective :

\begin{align*}
\bar{C}^{\psi}_x \centerdot I=(K_a \circ C^{\psi}_x) \centerdot C_x^{\psi},
\end{align*}

and for the same reason, but applied to the right translation by $I$:

\begin{align*}
\bar{C}^{\psi}_x = (K_a \circ C^{\psi}_x)\centerdot (1+ I\centerdot \bar{C}^{\psi}_x)
\end{align*}

It is easily seen that this last fixed point equation is equivalent to the fixed point equation (<ref>) but for $(\bar{C}_x^{\psi} (1+I\centerdot \bar{C}_x^{\psi})^{-1}$ and the formula is proved.

The purpose of the next two theorems are to demonstrate utility of our formalism in defining, first, the operator-valued conditionally free $T$ - transform which may be used to compute the operator-valued conditionally free multiplicative convolution and also the conditional $H$ transform.
Given two formal series $(H,K) \in \CC_0\llbracket End B \rrbracket \times \CC_0\llbracket End B\rrbracket$, we will use $(H,K)^{-1^{\ltimes_{\bm{\lambda}}}}$ for the inverse of $(H,K)$ with respect to the product of $ \CC_0\llbracket End B \rrbracket _{\bm{\lambda}}\ltimes_{\cdot_{\bm{\lambda}}}\CC_0\llbracket End B \rrbracket$, see (<ref>).

Pick $a,b$ two conditionally free random variables in an algebraic conditional probability space $(\mathcal{A},\varphi,\psi)$. Additionally, we assume that $\psi(a)=\psi(b)=1=\varphi(a)=\varphi(b)$.
For $x\in \{a,b\}$, we define the multilinear series $T_x \in \CC_0\llbracket End B \rrbracket$ and $T_x^c\in \CC_0\llbracket End B \rrbracket$ through, see (<ref>)

\begin{align}
\label{eqn:conditionalttransform}
(K_a, K^c_x) := e_{\bm{\lambda}\,|\,\varepsilon}(T_x,T_x^c) = ((T_x,T^c_x)^{-1^{\centerdot}})^{-1^{\ltimes_{\bm{\lambda}}}},
\end{align}

then the following factorization is in hold:

\begin{equation}
(T_{ab},T_{ab}^c) = (T_{a},T_{a}^c)\ltimes_{\bm{\lambda}^{-1^\#} \# \bm{\rho} }(T_{b},T_{b}^c)=(T_a\star_{\bm{\lambda}^{-1^\#} \# \bm{\rho}}T_a, (T^c_a \cdot_{\bm{\lambda}^{-1^\#} \# \bm{\rho}} T_b)\centerdot T^c_b) .
\end{equation}

We call the pair $(T_x,T^c_x)$ the conditional operator-valued $T$-transform of $x$.  We made the restrictive assumptions $\psi(x)=1=\varphi(x)$. These can be relaxed to $\psi(x)=\alpha$ and $\varphi(x)=\beta$ invertible elements of $B$. Instead of considering multilinear functions series starting with 1, we allow these series to start (to have degree zero component) with any invertible element of $B$, we call the set of such series $\mathbb{C}_{\times}\llbracket \textrm{End} B\rrbracket$. Likewise, the set of series with no degree zero component but starting with a degree one component equal to $\alpha I$ for some invertible $\alpha$ in $B$ is called $\Gamma_\times$. The left and right translations by the identity $I$ from $\mathbb{C}_{\times}\llbracket \textrm{End} B\rrbracket$ to $\Gamma_\times$ are compatible $\mathcal{O}$ operators and the equation (<ref>) define the conditional operator-valued $T$-transform under these weaker assumptions.

The proof of Theorem (<ref>) is based on the following Lemma, the operator-valued version of the Lemma 2.2 in [29]

Let $a,b$ be two conditionally free random variables with $\varphi(a)=\varphi(b)=1=\psi(a)=\psi(b) $and define the multilinear series $K_{a \dashv b}$, $K_{a \vdash b}$

\begin{align}
\label{eqn:equationdeuxun}
&K_b \cdot_{\bm{\lambda}} (K_a \cdot_{\bm{\rho}} K_{a \vdash b}) = K_{a \vdash b}, \quad K_a  \cdot_{\bm{\rho}} (K_b \cdot_{\bm{\lambda}} K_{a \dashv b}) = K_{a \dashv b}
\end{align}

\begin{align}
\label{eqn:condcumulantsmult}
&K^c_{ab} =  (K^c_{a} \cdot_{\bm{\rho}} K_{a \vdash b}) \centerdot (K^c_{b} \cdot_{\bm{\lambda}}K_{a \dashv b}), \\
&K_{ab} = K_b \star_{\bm{\lambda}} K_{a \dashv b}= K_a \star_{\bm{\rho}}
K_{a \vdash b}	         \label{eqn:freecumulantsmult}
\end{align}

Notice that (<ref>) implies (<ref>) when $\varphi = \psi$. It is enough to prove (<ref>).
We use the notations introduced in the proof of Lemma 2.2 in [29].Additionally, we set $[2n]=\{1<\hat{1}<2<\hat{2}<\cdots n < \hat{n}\}=[n]\cup \widehat{[n]}$. First, we prove

\begin{align}
\label{eqn:krewerasun}
\kappa^c_n(a_1b_1,\ldots, a_nb_n)=\sum_{\pi \in {\rm NC}_0(2n)}\mathcal{K}^c(a_1,b_1,\ldots,a_n,b_n)
\end{align}

or equivalently

\begin{align}
\label{eqn:krewerasdeux}
\mathcal{K}^c_{\pi}(a_1b_1,\ldots,a_nb_n) = \sum_{\substack{\sigma \in {\rm NC}_S(2n)\\ \sigma \vee \hat{0}_n = \hat{\pi} }} \mathcal{K}^c_{\sigma}(a_1,b_1,\ldots,a_n,b_n)
\end{align}

with $\mathcal{K}^c_{\pi}$ the partitioned conditional free cumulants, with $\pi_1,\ldots,\pi_n$ the irreductible components of the partition $\pi$ and $a_1,\ldots,a_n$ random variables

\begin{align*}
&\mathcal{K}_{\pi}(a_1,\ldots,a_n) = K_{\pi_1}(a_{\pi_1})\cdots K_{\pi_n}(a_{\pi_1}) \\
&\mathcal{K}_{\pi_j}(a_1,\ldots,a_n) = \kappa_{|V_j|}(a_{v^j_1}\mathcal{K}_{\pi_j^1}(a_{\pi_j^1}),\ldots,\mathcal{K}_{\pi_j^{p-1}}(a_{\pi_j^{p-1}})a_{v^j_p})
\end{align*}

\begin{align*}
&\mathcal{K}^c_{\pi}(a_1,\ldots,a_n)   = K^c_{\pi_1}(a_{\pi_1})\cdots K^c_{\pi_n}(a_{\pi_1}) \\
&\mathcal{K}^c_{\pi_j}(a_1,\ldots,a_n) = \kappa^c_{|V_j|}(a_{v^j_1}\mathcal{K}_{\pi_j^1}(a_{\pi_j^1}),\ldots,\mathcal{K}_{\pi_j^{p-1}}(a_{\pi_j^{p-1}})a_{v^j_p})
\end{align*}

where $V_j$ is the outer block of $\pi_j$, $V_j=\{v^j_1,\ldots,v^j_p\}$ and the $\pi^k_j$ is the restriction of $\pi_j$ to $\{v^j_k+1, v^j_{k+1}-1\}$ and $(\kappa^c_n)_{n\geq 1}$ is the sequence of operator valued conditionally free cumulants,

\begin{align}
K^c_a = 1+ \sum_{n\geq 1} K^c_a {}^{(n)},\quad K^c_a{}^{(n)}(x_1,\ldots,x_n)=\kappa^c_{n+1}(ax_1,\ldots, x_na)
\end{align}

The same inductive arguments as in the proof of Lemma 2.2 [29] yields equation (<ref>).
Before continuing with the proof, we remark that a partition $\alpha=(\alpha_+,Kr(\alpha_+)) \in \mathrm{NC}_0(n)$ has the following restriction property, whenever $O=\{o_1 < \cdots < o_k\} \subset [n]$ (or $\widehat{O}=\{\widehat{o}_1 < \cdots < \widehat{o}_k\} \subset \widehat{[n]}$) is an outer block of $\alpha$, the restriction of $\alpha$ to an interval $I=\rrbracket o_i,o_{i+1}\llbracket$ is a non-crossing partition, that we will call $\alpha^{\prime}$ such that $\{\{ \delta \}\} \cup \alpha^{\prime} $ is a non-crossing partition in ${\rm NC}_0(\{\delta \}\cup I)$ with the convention that $i_1 = \hat{\delta}$, $\delta < i_1 < i_2 < \hat{i}_2 < \cdots i_k < \hat{i}+k$ ( if instead $O \subset \widehat{[n]}$, we add $\delta$ with the convention that $\delta$ is greater than the elements in $I$).
Now since $\pi = (\pi_+, Kr(\pi_+))$ has two outer block, the one containing $1$ and the other containing $n$, one gets

\begin{align}
\label{eqn:unun}
&\kappa^c_n((ab) x_1,\ldots, x_{n-1}(ab)) \\
&= \sum_{p=1}^n \sum_{\substack{I, J}} \sum_{\alpha,\beta} \kappa^c_{p}(a\mathcal{K}_{\alpha_1}((1,b,a,b,\ldots,a,b)(1,x_{I_1 \cap{\widehat{[n]}}})),\ldots,a\mathcal{K}_{\alpha_{p-1}}((1,b,a,b,\ldots,a,b)(1,x_{I_{p-1}\cap{\widehat{[n]}}})),a) \nonumber\\
&\hspace{1cm}\kappa^c_{n-p}(b,\mathcal{K}_{\beta_1}((x_{J_1\cap [n]},1)(a, b,\ldots,a, b,a,1)b,\ldots,\mathcal{K}_{\beta_{n-p-1}}((x_{J_{n-p-1}\cap [n]},1)(a,b,\ldots,a,1))b)\nonumber
\end{align}

where the second and third sums ranges overs non intersecting intervals of integers $I_1,\ldots,I_p$ and $J_1 < \cdots <J_{n-q}$ with $I_1 <\cdots I_{p-1} < J_1 < \cdots < J_{n-p-1}$ of $\{1,\ldots,\hat{n}\}$ such that $I_1,\ldots,I_p \cup J_1 \cdots < J_{n-q}$ covers $\{1,\ldots,2n\}$ but a subset of $n$ points.
The partitions $\alpha$ are non-crossing partitions in ${\rm NC}_0(\{\delta \}\cup\{ I \})$ and the partitions $\beta$ are irreducible non-crossing partitions in ${\rm NC}_0(J \cup \{\delta \})$.

Notice that $\mathcal{K}_{\alpha}((1,b,a,b,\ldots,a,b)(1,x_{I \cap \widehat{[n]}}))$ is non-zero only when $\{\delta\}\in \alpha$ ($\{ \delta\}\in\alpha_+$ which implies that $Kr(\alpha_+)$ is an irreducible partition $\{i_1,\hat{i}_2,\hat{i}_3,\ldots,\hat{i}_k\}$ with if $\delta \cup I=\{\delta < i_1 < i_2 < \hat{i}_2\cdots < i_k < \hat{i}_k \}$
). Moreover, for any $x=(x_1,\ldots,x_k)$

\begin{align} \label{eqn:undeux}
\kappa_{k}((b,ab,\ldots,ab)(1,x))&=\sum_{\alpha}\mathcal{K}_{\alpha}((1,b,a,b,\ldots,a,b)(1,x)) \\
&= \sum_{p,s}\kappa(b
\kappa_{s_1}((x_{I_1},1)(ab,\ldots,ab,a)), \ldots, \kappa_{s_p}((x_{I_p},1)(ab,\ldots,ab,a))b) \label{eqn:undeuxprime}
\end{align}

where $I_j=\llbracket s_1+\cdots s_j,s_1+\cdots s_{j+1}\rrbracket$ and $p\geq 2$.
The same reasoning yields

\begin{align}
\kappa_{k}((x,1)(ab,\ldots,ab,a))&=\sum_{\alpha}\mathcal{K}_{\alpha}((x,1)(ab,\ldots,ab,a)) \label{eqn:untrois}\\
&= \sum_{p,s}\kappa(a
\kappa_{s_1}((x_{J_1},1)(b,\ldots,ab)), \ldots, \kappa_{s_p}((x_{J_p},1)(b,ab,\ldots,ab))a) \label{eqn:untroisprime}
\end{align}

Together with the equation inferred from (<ref>) and (<ref>), (<ref>):

\begin{align}
&\kappa^c_n((ab) x_1,\ldots, x_{n-1}(ab)) \nonumber\\
&= \sum_{p}\sum_{s,t} \kappa^c_{p}(a\kappa_{s_1}((1,b,a,b,\ldots,a,b)(1,x_{I_1 \cap{\widehat{[n]}}})),\ldots,a\kappa_{s_p}((1,b,a,b,\ldots,a,b)(1,x_{I_{p-1}\cap{\widehat{[n]}}})),a) \\
&\hspace{1cm}\kappa^c_{n-p}(b,\kappa_{t_1}((x_{J_1},1)(a, b,\ldots,a, b,a,1)b,\ldots,\kappa_{t_{n-p-1}}((x_{J_{n-p-1}},1)(a,b,\ldots,a,1))b) \nonumber
\end{align}

The results now follow by noticing that (<ref>) and (<ref>) the generating series of the
$$(x_1,\ldots,x_k) \mapsto \kappa_{k}((ab,x_1\ldots,ab x_{k-1},a))$$
$$(x_1,\ldots,x_k) \mapsto\kappa_{k}((b, x_1ab,\ldots,x_k ab))$$
satisfy the fixed point equations in (<ref>) respectively.

Notice that $K_a \cdot_{\bm{\rho}} K_{a \vdash b} = K_{a \dashv b}$ and $K_b \cdot_{\bm{\lambda}} K_{a \dashv b}	 = K_{a \vdash b}	 $. Hence (<ref>) is equivalent to

\begin{align}
K^c_{ab} = (K^c_{a} \cdot_{\bm{\rho}} K_{a \vdash b}) \centerdot (K^c_{b} \cdot_{\bm{\lambda}}(K_a \cdot_{\bm{\rho}} K_{a \vdash b} )) =  (K^c_{a} \cdot_{\bm{\rho}} (K_b \cdot_{\bm{\lambda}} K_{a \dashv b})) \centerdot (K^c_{b} \cdot_{\bm{\lambda}}K_{a \dashv b}).
\end{align}

It is also possible to write the equations (<ref>) and (<ref>) by using the product $\ltimes_{\rho}$ and the diagonal group action $\cdot_{\bm{\lambda}}$ of $\mathbb{C}_{0}[End B]_{\bm{\lambda}}$ on  $\mathbb{C}_{0}[End B]\times \mathbb{C}_{0}[End B]$.
Notice first that $K_{ab} = K_a \star_{\bm{\rho}}(K_b \cdot_{\bm{\lambda}} K_{a \dashv b})$. Then, we claim that

\begin{align}
(K_{ab}, K^c_{ab}) = (K_a,K^c_a)\ltimes_{\bm{\rho}}((K_b,K_b^c)\cdot_{\bm{\lambda}}K_{a \dashv b})
\end{align}

Let us first notice the following relations

K_{a \dashv b} = K_a \cdot_{\bm{\rho}}(T_b \cdot_{\bm{\lambda}} K_{ab}) = T^{-1^{\centerdot}}_b \star_{\bm{\lambda}} K_{ab}

The proof of the factorization of the conditional $T$-transform is a straightforward computations based on Lemma <ref>

\begin{align*}
K^c_{ab} &= (K^c_{a} \cdot_{\bm{\rho}} K_{a \vdash b}) \centerdot (K^c_{b} \cdot_{\bm{\lambda}}(K_a \cdot_{\bm{\rho}} K_{a \vdash b} ))\\
&= ((T^c_a \cdot_{\bm{\lambda}} K_a)\cdot_{\bm{\rho}} K_{a \vdash b})\centerdot (T^c_b \cdot_{\bm{\lambda}} (K_b \star_{\bm{\lambda}} K_{a \dashv b})) \\
&= ((T^c_a \cdot_{\bm{\lambda}} K_a)\cdot_{\bm{\rho}} K_{a \vdash b})\centerdot (T^c_b \cdot_{\bm{\lambda}} K_{ab}) \\
&=((T^c_a \cdot_{\bm{\lambda}} K_a)\cdot_{\bm{\rho}} (K_b \cdot_{\bm{\lambda}} K_{a \dashv b}))\centerdot (T^c_b \cdot_{\bm{\lambda}} K_{ab})\\
&= (T^c_a \cdot (\bm{\lambda}(K_a)\circ \bm{\rho}(K_b \cdot_{\bm{\lambda}} K_{a \dashv b}) ))\centerdot  (T^c_b \cdot_{\bm{\lambda}} K_{ab}) \\
&= (T^c_a \cdot (\bm{\lambda}(K_a)\circ \bm{\rho}(T_b \cdot_{\bm{\lambda}} (K_b\star_{\bm{\lambda}}K_{a \dashv b}) ))\centerdot  (T^c_b\cdot_{\bm{\lambda}} K_{ab}) \\
&=(T^c_a \cdot (\bm{\lambda}(K_a)\circ \bm{\rho}(T_b \cdot_{\bm{\lambda}} K_{ab} ))\centerdot  (T^c_b\cdot_{\bm{\lambda}} K_{ab})
\end{align*}

We then use the fact that $(\bm{\rho},\bm{\lambda})$ is a matching $\mathcal{O}$-operator to write

\begin{align*}
\bm{\lambda}(K_a)\circ \bm{\rho}(T_b \cdot_{\bm{\lambda}} K_{ab} )& = \bm{\lambda}((K_a \cdot_{\rho} (T_b \cdot_{\bm{\lambda}} K_{ab}) ) \cdot_{\rho} (T_b \cdot_{\bm{\lambda}} K_{ab})^{-1^{\star_{\bm{\rho}}}} )\circ \bm{\rho}(T_b \cdot_{\bm{\lambda}} K_{ab} ) \\
& = \bm{\rho}((T_b \cdot_{\bm{\lambda}} K_{ab}) \cdot_{\bm{\lambda}}(K_a \cdot_{\rho} (T_b \cdot_{\bm{\lambda}} K_{ab}))^{-1^{\star_{\bm{\lambda}}}}) \circ \bm{\lambda}(K_a \cdot_{\rho} (T_b \cdot_{\bm{\lambda}} K_{ab})) && \textrm{see eqn.\,\,}\eqref{eqn:matchinggroup} \\
& = \bm{\rho}((T_b \cdot_{\bm{\lambda}} K_{ab}) \cdot_{\bm{\lambda}}(T_b^{-1^{\centerdot}}\star_{\bm{\lambda}}K_{ab})^{-1^{\star_{\bm{\lambda}}}}) \circ \bm{\lambda}(T_b^{-1^{\centerdot}}\star_{\bm{\lambda}}K_{ab}) \\
& = \bm{\rho}(T_b\cdot_{\bm{\lambda}} (T_b^{-1^{\centerdot}})^{-1^{\star_{\bm{\lambda}}}}) \circ \bm{\lambda}(T_b^{-1^{\centerdot}}\star_{\bm{\lambda}}
K_{ab}) \\
&=  \bm{\rho}^{-1^{\#}}(T_b^{-1^{\centerdot}} \cdot_{\bm{\lambda}^{-1^{\#}}} T_b^{-1^{\star_{\bm{\lambda}^{-1^{\#}}}}}) \circ \bm{\lambda}^{-1^{\#}}(T_b) \circ \bm{\lambda}(K_{ab})
\end{align*}

where the last equality follows the fact that $(T_b^{-1^{\centerdot}})^{-1^{\star_{\bm{\lambda}}}} = (T_b^{-1^{\star_{\bm{\lambda}^{-1^{\#}}}}})^{-1^{\centerdot}}$.

$$T_b^{-1^{\centerdot}} \cdot_{\bm{\lambda}^{-1^{\#}}} T_b^{-1^{\star_{\bm{\lambda}^{-1^{\#}}}}} =  T_b^{-1^{\star_{\bm{\lambda}^{-1^{\#}}}}}
we infer

\begin{align*}
K^c_{ab} &= (T^c_a \cdot (\bm{\rho}^{-1^{\#}}(T_b^{-1^{\star_{\bm{\lambda}^{-1^{\#}}}}})  \circ \bm{\lambda}^{-1^{\#}}(T_b)) \cdot_{\bm{\lambda}} K_{ab})\centerdot (T^c_b\cdot_{\bm{\lambda}} K_{ab})\\
&= ((T^c_a \cdot (\bm{\rho}\#\bm{\lambda}^{-1^{\#}})(T_b) \centerdot T^c_b ) \cdot_{\bm{\lambda}} K_{ab}
\end{align*}

and the result follows from Theorem <ref> and Proposition the formulae in Section <ref> recalled at the beginning of the Section.

§.§.§ Conditional $H$ transform

To state the next theorem, we set $E = \frac{I}{1-I} \in \Gamma$.

Define the two formal series $H_x$ and $H_x^c$ in $\mathbb{C}_0[End B]$ through :

$$(H_x, H^c_x) := e_{\bm{\lambda}\,|\,\varepsilon}(T_x\cdot E,T_x^c\cdot E) = ((T_x\cdot E,T^c_x\cdot E)^{-1^{\centerdot}})^{-1^{\ltimes_{\bm{\lambda}}}} = (H_x, (T^c_x \cdot E) \cdot_{\lambda} H_x ).

for $x$ a self-adjoint element of $\mathcal{A}$ with $\phi(x)=\psi(x)=1$.
Pick $a,b$ two conditionally monotone random variables with $\phi(a)=\psi(a)=1=\phi(b)=\psi(b)$, then:

\begin{align}
(H_{ba},H_{ba}^c) = (H_{a},H_{a}^c)\ltimes_{\bm{\lambda}}(H_{b},H_{b}^c) .
\end{align}

Notice that

\begin{align*}
K^c_x \cdot ((1-I\centerdot H_x)^{-1}\centerdot I) &= T^c_x \circ (I\centerdot K) \circ ((1-I\centerdot H_x)^{-1}\centerdot I)\\
&=T^c_x \circ ((1-I\centerdot H_x)^{-1})\centerdot I \centerdot (K_x\circ (I\centerdot (1-I\centerdot H_x)^{-1})) \\
&=T^c_x \circ ((1-I\centerdot H_x)^{-1}) \centerdot (I\centerdot H_x))\\
\end{align*}

Similar computations as done right before Theorem <ref> yield

\begin{align}
\label{eqn:momentshc}
H^c_x = \bar{C}_x^{\varphi}(1+I\centerdot \bar{C}_x^{\varphi})
\end{align}

We let $\tilde{a}$ and $\tilde{b}$ be two conditionally free random variables in a conditional probability space $(\mathcal{A},\psi,\varphi,B)$ as in Theorem <ref>. Since $\tilde{a}$ and $\tilde{b}$ are conditionally free,

T^{c}_{\tilde{b}\tilde{a}} = (T^c_{\tilde{b}}\cdot_{\bm{\lambda}^{-1^{\#}}\#{\bm \rho}} T_{\tilde{a}})\centerdot T^{c}_{\tilde{a}}

Thus we infer

\begin{align}
(T^c_{\tilde{b}\tilde{a}} \cdot E) \cdot_{\bm{\lambda}} H_{\tilde{b}} &=((T^c_{\tilde{b}} \cdot E)\cdot_{\bm{\lambda}} H_{\tilde{b}})\centerdot ((T^c_{\tilde{a}} \cdot E) \cdot_{\bm{\lambda}}H_{\tilde{b}}))
\end{align}

We note that $H_{\tilde{b}\tilde{a}}=H_{\tilde{b}}$ since $\tilde{a}$ is distributed as $1\in\mathcal{A}$ (see equation (<ref>) with respect to $\psi$ and thus from the previous equation we infer that

\begin{align}
H^c_{\tilde{b}\tilde{a}} = H^c_{\tilde{b}} \centerdot ((T^c_{\tilde{a}} \cdot E) \cdot_{\bm{\lambda}}H_{\tilde{b}})) = H^c_{b} \centerdot ((T^c_{\tilde{a}} \cdot E)\cdot_{\bm{\lambda}} H_{{b}})
\end{align}

where the second equality follows from the fact that $\tilde{b}$ is distributed as $b$ with respect to $\psi$ and $\varphi$. Following Remark <ref> and equation (<ref>), $H^c_{\tilde{b}\tilde{a}}$ does only depend upon the moments of $ \tilde{b}\tilde{a}$ under $\varphi$, which are equal to the moments of $ba$ and $H^c_{\tilde{b}\tilde{a}}=H^c_{{b}{a}}$. Also, $H_{\tilde{a}}=1$ implies

H^c_{{b}{a}} = H^c_{\tilde{b}\tilde{a}} = H^c_{b} \centerdot (H^c_{\tilde{a}}\cdot_{\bm{\lambda}} H_{{b}})

Now since $\tilde{a}$ is distributed as $a$ under $\varphi$, equation (<ref>) implies $H^c_{\tilde{a}}=H^c_{{a}}$ and the result follows.

We end this section with the computations of the subordination equation for the multiplicative convolution for operator-valued conditional freeness, that is the conditional counterpart of equation (<ref>)

\begin{align*}
H^c_{ab} &= T^c_{a} \cdot ((\bm{\lambda}^{-1^{\#}} \# \bm{\rho})(T_y)\circ \frac{I}{1-I}\circ \bm{\lambda}(H_{ab})) \centerdot (T^c_y\cdot (\frac{I}{1-I}\circ \bm{\lambda}(H_{ab}))) \\
&=T^c_{a} \cdot (E\circ (\bm{\lambda}^{-1^{\#}} \# \bm{\rho})(T_y \cdot E)\circ \bm{\lambda}(H_{ab})) \centerdot (T^c_y\cdot (\frac{I}{1-I}\circ \bm{\lambda}(H_{ab}))) \\
&=T^c_{a} \cdot (E\circ (\bm{\lambda}^{-1^{\#}} \# \bm{\rho})(T_y \cdot E)\circ \bm{\lambda}(H_{b}\star_{\bm{\lambda}}H_{a\vdash b})) \centerdot (T^c_y\cdot (\frac{I}{1-I}\circ \bm{\lambda}(H_{b}\star_{\bm{\lambda}}H_{a\vdash b}))) \\
&=T^c_{a} \cdot (E\circ (\bm{\lambda}^{-1^{\#}} \# \bm{\rho})(T_y \cdot E)\circ \bm{\lambda}(H_{b}\star_{\bm{\lambda}}H_{a\vdash b})) \centerdot (H^c_b\cdot_{\bm{\lambda}}H_{a\vdash b}) \\
&=T^c_{a} \cdot (E\circ \bm{\rho}((T^{-1^{\centerdot}}_y \cdot E)^{-1^{\star_{\bm{\lambda}}}})\bm{\lambda}((T^{-1^{\centerdot}}_y \cdot E))\circ \bm{\lambda}(H_{b}\star_{\bm{\lambda}}H_{a\vdash b})) \centerdot (H^c_b\cdot_{\bm{\lambda}}H_{a\vdash b}) \\
&=T^c_{a} \cdot (E\circ \bm{\rho}(H_y)\circ\bm{\lambda}(H_y^{-1^{\star_{\bm{\lambda}}}})\circ \bm{\lambda}(H_{b}\star_{\bm{\lambda}}H_{a\vdash b})) \centerdot (H^c_b\cdot_{\bm{\lambda}}H_{a\vdash b})\\
&=T^c_{a} \cdot (E\circ \bm{\rho}(H_y)\circ\bm{\lambda}(H_{a\vdash b})) \centerdot (H^c_b\cdot_{\bm{\lambda}}H_{a\vdash b}) \\
&=T^c_{a} \cdot (E\circ \bm{\lambda}(H_{a\vdash b}\cdot_{\bm{\rho}}(H_y\cdot_{\bm{\lambda}}H_{a\vdash b})^{-1^{\star_{\bm{\rho}}}})\circ\bm{\rho}(H_y\cdot_{\bm{\lambda}}H_{a\vdash b})) \centerdot (H^c_b\cdot_{\bm{\lambda}}H_{a\vdash b}) && {\rm see\,\,} \eqref{eqn:matchinggroup} \label{eqn:usematching}\\
&=T^c_{a} \cdot (E\circ \bm{\lambda}(H_a)\circ\bm{\rho}(H_y\cdot_{\bm{\lambda}}H_{a\vdash b})) \centerdot (H^c_b\cdot_{\bm{\lambda}}H_{a\vdash b}) \\
&=T^c_{a} \cdot (E\circ \bm{\lambda}(H_a)\circ\bm{\rho}(H_y\cdot_{\bm{\lambda}}H_{a\vdash b})) \centerdot (H^c_b\cdot_{\bm{\lambda}}H_{a\vdash b}) \\
&=H^c_a\circ\bm{\rho}(H_y\cdot_{\bm{\lambda}}H_{a\vdash b})) \centerdot (H^c_b\cdot_{\bm{\lambda}}H_{a\vdash b})
\end{align*}

Note that we have used once again the defining relation of Matching $\mathcal{O}$-operator.
We set $H_{a\dashv b}=H_y\cdot_{\bm{\lambda}}H_{a\vdash b}$. We have proved the following theorem

Let $a,b$ be two conditionally free random variables with $ \varphi(a)=\varphi(b)=1=\psi(a)=\psi(b)$, then

\begin{equation*}
H^c_{ab} = (H^c_a\cdot_{\bm{\rho}}H_{a\dashv b}) \centerdot (H^c_b\cdot_{\bm{\lambda}}H_{a\vdash b})
\end{equation*}

\begin{equation*}
H_{a\dashv b} = H_y \cdot_{\bm{\lambda}} ( H_a \cdot_{\bm{\rho}}H_{a\dashv b}),\quad H_{a\vdash b} = H_a \cdot_{\bm{\rho}} (H_y \cdot_{\bm{\lambda}} H_{a\vdash b})
\end{equation*}

[1]
Infinite divisibility and a non-commutative boolean-to-free bercovici–pata
Journal of Functional Analysis, 262(1):94–123, 2012.

[2]
Andrei Alexandrovich Agrachev and Revaz Valer'yanovich Gamkrelidze.
Chronological algebras and nonstationary vector fields.
Itogi Nauki: Problemy geometrii, 11:135–176, 1980.

[3]
Mahdi J Hasan Al-Kaabi, Kurusch Ebrahimi-Fard, Dominique Manchon, and Hans
Algebraic aspects of connections: from torsion, curvature, and
post-lie algebras to gavrilov's double exponential and special polynomials.
arXiv preprint arXiv:2205.04381, 2022.

[4]
Valeriy G Bardakov and Vsevolod Gubarev.
Rota–baxter operators on groups.
arXiv preprint arXiv:2103.01848, 2021.

[5]
Valeriy G Bardakov and Vsevolod Gubarev.
Rota—baxter groups, skew left braces, and the yang—baxter
Journal of Algebra, 596:328–351, 2022.

[6]