$\mathcal{M}^{\\!f}_{k}$ to denote the space of all formal modular forms of
weight $k\geq 0$.

######  Proposition 5.7.

We have
$\mathcal{M}^{\\!f}=\mathbb{Q}[G^{\\!f}\\!(4),G^{\\!f}\\!(6)]\cong\mathcal{M}$.

###### Proof.

The first isomorphism follows directly from Corollary 5.5 after computing
$\delta(G^{\\!f}\\!(4))=0$ and $\delta G^{\\!f}\\!(6)=0$. The second
isomorphism then follows from Theorem 5.4. ∎

######  Remark 5.8.

The definition of $\mathcal{M}^{\\!f}$ raises the question what the kernel of
$\delta$ is when considered on the whole space $\mathcal{G}^{\\!f}$. By the
direct formula in the proof of Proposition 4.11, we see that it contains all
$G^{\\!f}\\!(k_{1},\dots,k_{r})$, with $k_{1}\geq 3,k_{2},\dots,k_{r}\geq 2$;
these are the indices for which the classical multiple Eisenstein series (1.7)
converge absolutely. In fact, restricted to $\mathcal{E}^{\\!f}$, the elements
$G^{\\!f}\\!(k_{1},\dots,k_{r})$ with $k_{1}\geq 3,k_{2},\dots,k_{r}\geq 2$
capture the entire kernel of $\delta$, and every formal multiple Eisenstein
series in $\mathcal{E}^{\\!f}$ can be written as a polynomial in
$G^{\\!f}\\!(2)$ with coefficients given by these elements. This is similar to
a property of quasimodular forms, which can be written uniquely as a
polynomial in $\mathbb{G}_{2}$ with modular coefficients. However, note that
the kernel of $\delta$ on $\mathcal{G}^{\\!f}$ also contains elements with
non-admissible indices, for example, $G^{\\!f}\\!(1,2,1)\in\ker(\delta)$.

The following corollary follows from Proposition 5.7 and the fact that the
corresponding statement is true for modular forms. For modular forms, this
result follows from [KZ] and heavily depends on the analytic theory of modular
forms, in particular on Rankin’s method. It is an interesting open question to
obtain a _combinatorial_ proof of the following result, that is, by only using
the stuffle product and the swap.

######  Corollary 5.9.

For $k\geq 4$ we have

$\displaystyle\mathcal{M}^{\\!f}_{k}=\mathbb{Q}G^{\\!f}\\!(k)+\langle
G^{\\!f}\\!(a)\,G^{\\!f}\\!(b)\mid a+b=k,a,b\geq 4\text{
even}\rangle_{\mathbb{Q}}\,.$ (5.2)

In particular,
$\mathcal{M}^{\\!f}\subset\widetilde{\mathcal{M}}^{\\!f}\subset\operatorname{Fil}^{\mathrm{\operatorname{dep}}}_{2}\mathcal{G}^{\\!f}$.

Recall the projection $\pi:\mathcal{G}^{\\!f}\rightarrow\mathcal{Z}^{f}$ given
by (2.15).

######  Definition 5.10.

We define the algebra of formal cusp forms by
$\mathcal{S}^{\\!f}=\ker\pi_{|\mathcal{M}^{\\!f}}$. Write
$\mathcal{S}^{\\!f}_{k}$ to denote the space of all formal cusp forms of
weight $k\geq 0$.

The first example of a non-zero formal cusp form appears in weight
$12=\operatorname{lcm}(4,6)$.

$\displaystyle\Delta^{\mathfrak{f}}$
$\displaystyle=\frac{e(4)^{3}-e(6)^{2}}{1728}=2400\cdot 6!\cdot
G^{\\!f}\\!(4)^{3}-420\cdot 7!\cdot G^{\\!f}\\!(6)^{2}\,,$

where we write $e(k)=-\frac{2k!}{B_{k}}G^{\\!f}\\!(k)$ for even $k\geq 2$.
These elements correspond to the Eisenstein series $E_{k}\mspace{1.0mu}$,
which all have constant term $1$.

######  Proposition 5.11.

We have $\Delta^{\mathfrak{f}}\in\mathcal{S}^{\\!f}_{12}$ and
$D\Delta^{\mathfrak{f}}=e(2)\Delta^{\mathfrak{f}}$.

###### Proof.

Using that

$G^{\\!f}\\!(4)=\frac{1}{5}DG^{\\!f}\\!(2)+\frac{2}{5}G^{\\!f}\\!(2)^{2},\qquad
G^{\\!f}\\!(6)=\frac{1}{70}D^{2}G^{\\!f}\\!(2)+\frac{6}{35}G^{\\!f}\\!(2)DG^{\\!f}\\!(2)+\frac{8}{35}G^{\\!f}\\!(2)^{3},$

we can write $\Delta^{\mathfrak{f}}$ explicitly as a polynomial in
$G^{\\!f}\\!(2),DG^{\\!f}\\!(2),D^{2}G^{\\!f}\\!(2)$ as follows

$\frac{1}{432}\Delta^{\mathfrak{f}}=48G^{\\!f}\\!(2)^{2}(DG^{\\!f}\\!(2))^{2}+32(DG^{\\!f}\\!(2))^{3}-32G^{\\!f}\\!(2)^{3}D^{2}G^{\\!f}\\!(2)\\\
-24G^{\\!f}\\!(2)(DG^{\\!f}\\!(2))(D^{2}G^{\\!f}\\!(2))-(D^{2}G^{\\!f}\\!(2))^{2}\,.$
(5.3)

In particular we see that no term with $G^{\\!f}\\!(2)^{6}$ appears, i.e., all
remaining terms are elements in the ideal $\mathfrak{N}$ (see (2.13)) and
therefore $\Delta^{\mathfrak{f}}\in\ker\pi$. The second statement follows by
taking the derivative of this equation and then using the Chazy equation
(5.1). ∎

######  Proposition 5.12.

We have
$\mathcal{M}^{\\!f}_{k}=\mathbb{Q}G^{\\!f}\\!(k)\oplus\mathcal{S}^{\\!f}_{k}\mspace{1.0mu}$.

###### Proof.

Every element $f\in\mathcal{M}^{\\!f}_{k}$ can be written as a polynomial in
$G^{\\!f}\\!(2),DG^{\\!f}\\!(2),D^{2}G^{\\!f}\\!(2)$. Suppose that
$f=aG^{\\!f}\\!(2)^{\frac{k}{2}}+f_{1}$ for some $a\in\mathbb{Q}$, where
$f_{1}$ contains monomials which contain $DG^{\\!f}\\!(2)$ or
$D^{2}G^{\\!f}\\!(2)$, i.e., $f_{1}\in\ker\pi$. By Corollary 2.12 we have
$G^{\\!f}\\!(k)=-\frac{B_{k}}{2k!}(-24G^{\\!f}\\!(2))^{\frac{k}{2}}+g_{1}$ for
some $g_{1}\in\ker\pi$, i.e.,
$G^{\\!f}\\!(k)\notin\mathcal{S}^{\\!f}_{k}\mspace{1.0mu}$. We conclude
$f+a\frac{2k!}{B_{k}}(-24)^{-\frac{k}{2}}G^{\\!f}\\!(k)\in\mathcal{S}^{\\!f}_{k}$
from which the statement follows. ∎

Analogously, the proof of many other statements known for (quasi)modular forms
can be translated to the formal setup. We invite the interested reader or
motivated students to do so.

##  6\. Balanced setup

In this section, we will introduce another setup that can be used to describe
the algebra $\mathcal{G}^{\\!f}$. This setup is motivated by the idea of
making the involution $\sigma$ simpler on the level of coefficients as it was
done in [Zu] for the $q$-series $g$ given in (1.8). It was introduced by
Burmester in her thesis [Bu1] and then studied in [Bu2] and [Bu3]. Let
$\mathbb{Q}\langle\mathcal{B}\rangle$ be the non-commutative polynomial ring
in the alphabet $\mathcal{B}=\\{b_{0},b_{1},\ldots\\}$, and let
$\mathcal{B}^{\ast}$ be the free monoid generated by the elements of
$\mathcal{B}$, to which we refer to as _words_. On
$\mathbb{Q}\langle\mathcal{B}\rangle$ we recursively define the _quasi-shuffle
product_ $\ast_{b}$ as the $\mathbb{Q}$-bilinear product satisfying

$\displaystyle b_{i}u\ast_{b}b_{j}v\>=\>b_{i}(u\ast b_{j}v)+b_{j}(b_{i}u\ast
v)+(b_{i}\diamond b_{j})(u\ast v)\qquad(i,j\geq 0\text{ and
}u,v\in\mathbb{Q}\langle\mathcal{B}\rangle)$

and $1\ast w=w\ast 1=w$ for any $w\in\mathbb{Q}\langle\mathcal{B}\rangle$.
Here $b_{i}\diamond b_{j}=\delta_{ij>0}\,b_{i+j}$ is an associative and
commutative product on $\mathbb{Q}\mathcal{B}$. As before, the space
$(\mathbb{Q}\langle\mathcal{B}\rangle,\ast)$ is a commutative
$\mathbb{Q}$-algebra (see [H] and also [HI]). The subspace
$\mathbb{Q}\langle\mathcal{B}\rangle^{0}$ of words not starting in $b_{0}$ is
closed under $\ast$ which gives rise to a commutative $\mathbb{Q}$-algebra
$(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast)$.

Define the $\mathbb{Q}$-linear involution
$\tau:\mathbb{Q}\langle\mathcal{B}\rangle^{0}\rightarrow\mathbb{Q}\langle\mathcal{B}\rangle^{0}$
by

$\tau(b_{k_{1}}b_{0}^{m_{1}}\cdots b_{k_{s}}b_{0}^{m_{s}})\coloneqq
b_{m_{s}+1}b_{0}^{k_{s}-1}\cdots
b_{m_{1}+1}b_{0}^{k_{1}-1}\qquad(k_{1},\ldots,k_{s}\geq 1\text{ and
}m_{1},\dots,m_{s}\geq 0).$

We now want to compare the algebra
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast_{\diamond})$ with
$(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast)$. For this define the
$\mathbb{Q}$-linear map
$\varphi:\mathbb{Q}\langle\mathcal{B}\rangle^{0}\rightarrow\mathbb{Q}\langle\mathcal{A}\rangle$
for all $r\geq 1$ on the generators by

$\displaystyle\sum_{\begin{subarray}{c}k_{1},\dots,k_{r}\geq 1\\\
m_{1},\dots,m_{r}\geq 0\end{subarray}}\varphi(b_{k_{1}}b_{0}^{m_{1}}\cdots
b_{k_{r}}b_{0}^{m_{r}})\,X_{1}^{k_{1}-1}\cdots
X_{r}^{k_{r}-1}Y_{1}^{m_{1}}\cdots
Y_{r}^{m_{r}}\coloneqq\mathfrak{A}\genfrac{(}{)}{0.0pt}{}{X_{1},\dots,X_{r}}{Y_{1},Y_{2}-Y_{1},\dots,Y_{r}-Y_{r-1}}\,.$

######  Proposition 6.1 ([Bu1, Theorem 6.4]).

The map
$\varphi:(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast_{b})\rightarrow(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$
is an isomorphism of graded $\mathbb{Q}$-algebras satisfying additionally
$\sigma\circ\varphi=\varphi\circ\tau$.

As a consequence of this proposition, we get

$\displaystyle\mathcal{G}^{\\!f}\cong\faktor{(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast)}{T}\,,$

where $T$ is the ideal in $(\mathbb{Q}\langle\mathcal{B}\rangle^{0},\ast)$
generated by $\tau(w)-w$ for all
$w\in\mathbb{Q}\langle\mathcal{B}\rangle^{0}$. In particular, the algebra
$\mathcal{G}^{\\!f}$ in [Bu3] agrees with the original definition of the
algebra of formal multiple Eisenstein series used here. The operators
$D,W,\delta$ can be easily translated into this setup. For example, for a word
$w=b_{k_{1}}b_{0}^{m_{1}}\cdots b_{k_{r}}b_{0}^{m_{r}}$ with
$k_{1},\dots,k_{r}\geq 1,m_{1},\dots,m_{r}\geq 0$ the operator $D$ is given by

$\displaystyle D(w)=\sum_{1\leq i\leq j\leq
r}k_{i}(m_{j}+1)\,b_{k_{1}}b_{0}^{m_{1}}\cdots b_{k_{i}+1}\cdots
b_{k_{j}}b_{0}^{m_{j}+1}\cdots b_{k_{r}}b_{0}^{m_{r}}.$

##  Appendix A Classical double shuffle and swap & stuffle

(by Nils Matthes)

To make this appendix self-contained, we will recall some facts on quasi-
shuffle algebras which were already mentioned in the Introduction or in
Section 3. All algebras are assumed to be commutative, associative, and unital
unless stated otherwise.

###  A.1. The extended double shuffle relations

####  A.1.1. Product structures on words

Let $\mathcal{L}_{z}=\\{z_{k}\mid k\geq 1\\}$ be the alphabet of letters
indexed by the positive integers and let
$\mathbb{Q}\langle\mathcal{L}_{z}\rangle$ be the free $\mathbb{Q}$-vector
space on the free monoid $\mathcal{L}_{z}^{\ast}$ of words on
$\mathcal{L}_{z}\mspace{1.0mu}$, including the empty word $1$. It can be
equipped with two product structures: the first is the index shuffle
product333The index shuffle product $\overline{\shuffle}$ is the shuffle
product on the alphabet $\mathcal{L}_{z}\mspace{1.0mu}$. One should not
confuse it with the shuffle product $\shuffle$ on the alphabet
$\mathcal{L}_{xy}\mspace{1.0mu}$. which is defined for words
$w,w_{1},w_{2}\in\mathcal{L}_{z}^{\ast}$ and letters
$z_{k_{1}},z_{k_{2}}\in\mathcal{L}_{z}$ by the recursive formula

$\displaystyle z_{k_{1}}w_{1}\overline{\shuffle}z_{k_{2}}w_{2}$
$\displaystyle=z_{k_{1}}(w_{1}\overline{\shuffle}z_{k_{2}}w_{2})+z_{k_{2}}(z_{k_{1}}w_{1}\overline{\shuffle}w_{2})$
$\displaystyle 1\overline{\shuffle}w$ $\displaystyle=w\overline{\shuffle}1=w.$

The second is the stuffle product, which is defined similarly to before, but
with one extra term:

$\displaystyle z_{k_{1}}w_{1}\ast z_{k_{2}}w_{2}$
$\displaystyle=z_{k_{1}}(w_{1}\ast
z_{k_{2}}w_{2})+z_{k_{2}}(z_{k_{1}}w_{1}\ast w_{2})+z_{k_{1}+k_{2}}(w_{1}\ast
w_{2})$ $\displaystyle 1\ast w$ $\displaystyle=w\ast 1=w.$

Each of these products turns $\mathbb{Q}\langle\mathcal{L}_{z}\rangle$ into a
$\mathbb{Q}$-algebra whose unit element is the empty word in both cases.
Together with the deconcatenation coproduct

$\displaystyle\Delta:\mathbb{Q}\langle\mathcal{L}_{z}\rangle$
$\displaystyle\rightarrow\mathbb{Q}\langle\mathcal{L}_{z}\rangle\otimes\mathbb{Q}\langle\mathcal{L}_{z}\rangle$
$\displaystyle z_{k_{1}}\ldots z_{k_{n}}$
$\displaystyle\mapsto\sum_{i=0}^{n}z_{k_{1}}\ldots z_{k_{i}}\otimes
z_{k_{i+1}}\ldots z_{k_{n}},$

both $(\mathbb{Q}\langle\mathcal{L}_{z}\rangle,\overline{\shuffle},\Delta)$ as
well as $(\mathbb{Q}\langle\mathcal{L}_{z}\rangle,\ast,\Delta)$ are
commutative Hopf algebras.

####  A.1.2. Commutative power series

Let $R$ be a commutative unital $\mathbb{Q}$-algebra. It is well known that
$\mathbb{Q}$-linear morphisms
$\varphi\in\operatorname{Hom}(\mathbb{Q}\langle\mathcal{L}_{z}\rangle,R)$ can
be identified with tuples of power series. More precisely, there is an
$R$-linear isomorphism

$\displaystyle\operatorname{Hom}(\mathbb{Q}\langle\mathcal{L}_{z}\rangle,R)$
$\displaystyle\rightarrow\prod_{n\geq 0}R\llbracket
X_{1},\ldots,X_{n}\rrbracket$ (A.1) $\displaystyle\varphi$
$\displaystyle\mapsto(\Phi_{n})_{n\geq 0},$

where $\Phi_{0}=\varphi(1)$ and
$\Phi_{n}(X_{1},\ldots,X_{n})=\sum_{k_{1},\ldots,k_{n}\geq
1}\varphi(z_{k_{1}}\ldots z_{k_{n}})\,X_{1}^{k_{1}-1}\ldots X_{n}^{k_{n}-1}$,
for $n\geq 1$. Under this correspondence, the convolution product
$\varphi\mspace{1.0mu}\star\mspace{1.0mu}\psi:=m_{R}\circ(\varphi\otimes\psi)\circ\Delta\in\operatorname{Hom}(\mathbb{Q}\langle\mathcal{L}_{z}\rangle,R)$,
where $m_{R}$ denotes the multiplication in $R$, corresponds to the element
$(\Xi_{n})_{n\geq 0}$ defined by

$\Xi_{n}(X_{1},\ldots,X_{n})=\sum_{i=0}^{n}\Phi(X_{1},\ldots,X_{i})\,\Psi_{n-i}(X_{i+1},\ldots,X_{n}).$

Now define an automorphism

$\displaystyle\sigma:\prod_{n\geq 0}R\llbracket X_{1},\ldots,X_{n}\rrbracket$
$\displaystyle\rightarrow\prod_{n\geq 0}R\llbracket
X_{1},\ldots,X_{n}\rrbracket$ $\displaystyle(\Phi_{n})_{n\geq 0}$
$\displaystyle\mapsto(\sigma(\Phi_{n}))_{n\geq 0},$

where
$\sigma(\Phi_{n})(X_{1},\ldots,X_{n}):=\Phi_{n}(X_{1}+\ldots+X_{n},X_{1}+\ldots+X_{n-1},\ldots,X_{1})$.
Via the isomorphism (A.1), it induces an automorphism
$\sigma^{*}:\operatorname{Hom}(\mathbb{Q}\langle\mathcal{L}_{z}\rangle,R)\rightarrow\operatorname{Hom}(\mathbb{Q}\langle\mathcal{L}_{z}\rangle,R)$.

####  A.1.3. The extended double shuffle relations

Given a $\mathbb{Q}$-linear morphism
$\varphi\in\operatorname{Hom}(\mathbb{Q}\langle\mathcal{L}_{z}\rangle,R)$, we
define a new morphism $\varphi_{\rm
corr}\in\operatorname{Hom}(\mathbb{Q}\langle\mathcal{L}_{z}\rangle,R)$ by
sending a word $z_{k_{1}}\ldots z_{k_{n}}$ to its coefficient in the power
series

$\exp\left(\sum_{n=2}^{\infty}\frac{(-1)^{n}}{n}\varphi(z_{n})\cdot
z_{1}\right)\in R\langle\\!\langle z_{1}\rangle\\!\rangle.$

######  Definition A.1.

We say that $\varphi$ satisfies the _extended double shuffle relations_ , if,
for all words $w_{1},w_{2}\in\mathcal{L}_{z}^{*}$, we have that:

  * (i)

$\varphi(w_{1}\ast w_{2})=\varphi(w_{1})\,\varphi(w_{2})$;

  * (ii)

$\sigma^{*}(\varphi_{\rm
corr}\star\varphi)(w_{1}\overline{\shuffle}w_{2})=\sigma^{*}(\varphi_{\rm
corr}\star\varphi)(w_{1})\,\sigma^{*}(\varphi_{\rm corr}\star\varphi)(w_{2})$.

######  Remark A.2.

This definition is equivalent to Racinet’s definition ([Rac]) in the case
where $\varphi(z_{1})=0$, a condition which is, in fact, part of Racinet’s
definition. The key point is that, by [I, Proposition 8],
$\sigma^{*}(\varphi_{\rm corr}\star\varphi)$ is a homomorphism for the shuffle
product if and only if the map

$(\varphi_{\rm
corr}\star\varphi)\circ\pi_{\mathcal{L}_{z}}:\mathfrak{H}^{1}\rightarrow R$

is a homomorphism for the shuffle product on
$\mathfrak{H}^{1}=\mathbb{Q}+\mathbb{Q}\langle x,y\rangle y\mspace{1.0mu}$.
Here
$\pi_{\mathcal{L}_{z}}:\mathfrak{H}^{1}\rightarrow\mathbb{Q}\langle\mathcal{L}_{z}\rangle$
is the $\mathbb{Q}$-linear isomorphism defined by mapping $x^{k_{1}-1}y\cdots
x^{k_{n}-1}y$ to $z_{k_{1}}\ldots z_{k_{n}}\mspace{1.0mu}$. Then one checks,
by the usual duality of the underlying Hopf algebras, that this is also
equivalent to $\varphi$ having the EDS property as defined in [IKZ, Definition
1].

###  A.2. Commutative power series

In this section, we fix a $\mathbb{Q}$-algebra $R$.

####  A.2.1. Preliminaries

Let

$\mathcal{B}(R):=\prod_{n=0}^{\infty}R\llbracket
X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n}\rrbracket$

be the $R$-module of tuples of formal power series in an increasing number of
variables, where $R^{\times}$ denotes the group of units of $R$ (cf., Equation
(2.2)). It is a noncommutative, associative, unital $R$-algebra with the
concatenation product

$\displaystyle\odot:\mathcal{B}(R)\times\mathcal{B}(R)$
$\displaystyle\rightarrow\mathcal{B}(R)$ $\displaystyle((F_{n}),(G_{n}))$
$\displaystyle\mapsto(H_{n}),$

where

$H_{n}\genfrac{(}{)}{0.0pt}{}{X_{1},\ldots,X_{n}}{Y_{1},\ldots,Y_{n}}\coloneqq\sum_{i=0}^{n}F_{i}\genfrac{(}{)}{0.0pt}{}{X_{1},\ldots,X_{i}}{Y_{1},\ldots,Y_{i}}\,G_{n-i}\genfrac{(}{)}{0.0pt}{}{X_{i+1},\ldots,X_{n}}{Y_{i+1},\ldots,Y_{n}}\qquad(n\geq
0).$

The unit element is the tuple $\mathbbm{1}:=(1,0,0,\ldots)$ and
$F\in\mathcal{B}(R)$ has an inverse $F^{\odot-1}$ for $\odot$ if and only if
$F_{0}\in R^{\times}$, in which case $F^{\odot-1}$ can be defined recursively
by

$F^{\odot-1}_{0}=F_{0}^{-1},\qquad
F^{\odot-1}_{n}=\sum_{i=0}^{n-1}(-1)^{i+1}\frac{F_{n-i}}{F_{0}}F^{\odot-1}_{i}.$

Note that there is an embedding

$\displaystyle\lambda:R\llbracket t\rrbracket$
$\displaystyle\rightarrow\mathcal{B}(R)$ $\displaystyle P(t)=\sum_{n\geq
0}a_{n}t^{n}$ $\displaystyle\mapsto\lambda(P):=(a_{n})_{n\geq 0}$ (A.2)

of $R$-algebras. In other words, every power series in one variable can be
viewed as an element of $\mathcal{B}(R)$.

The algebra $\mathcal{B}(R)$ is equipped with an involution (the _swap_ , cf.,
Definition 2.2)

$\displaystyle\sigma:\mathcal{B}(R)$ $\displaystyle\rightarrow\mathcal{B}(R)$
$\displaystyle(F_{n})$ $\displaystyle\mapsto(\sigma(F_{n})),$

where

$\sigma(F_{n})\genfrac{(}{)}{0.0pt}{}{X_{1},\ldots,X_{n}}{Y_{1},\ldots,Y_{n}}:=F_{n}\genfrac{(}{)}{0.0pt}{}{Y_{1}+\ldots+Y_{n},Y_{1}+\ldots+Y_{n-1},\ldots,Y_{1}}{X_{n},X_{n-1}-X_{n},\ldots,X_{1}-X_{2}}.$

The space $\mathcal{B}(R)$ contains two distinguished $R$-submodules

$\mathcal{M}_{X}(R):=\prod_{n=0}^{\infty}R\llbracket
X_{1},\ldots,X_{n}\rrbracket,\qquad\mathcal{M}_{Y}(R):=\prod_{n=0}^{\infty}R\llbracket
Y_{1},\ldots,Y_{n}\rrbracket,$

which clearly are both subalgebras.

######  Proposition A.3.

For all $F\in\mathcal{M}_{Y}(R)$ and $G\in\mathcal{M}_{X}(R)$, we have

$\sigma(F\odot G)=\sigma(G)\odot\sigma(F).$

###### Proof.

This is a straightforward computation. For every $n\geq 0$, we have on one
hand

$\sigma((F\odot
G)_{n})=\sum_{i=0}^{n}F_{i}(X_{n},\ldots,X_{n-i+1}-X_{n-i+2})\,G_{n-i}(Y_{1}+\ldots+Y_{n-i},\ldots,Y_{1})$

and on the other hand

$\displaystyle(\sigma(G)\odot\sigma(F))_{n}$
$\displaystyle=\sum_{i=0}^{n}G_{i}(Y_{1}+\ldots+Y_{i},\ldots,Y_{1})\,F_{n-i}(X_{n},\ldots,X_{i+1}-X_{i+2})$
$\displaystyle=\sum_{i=0}^{n}G_{n-i}(Y_{1}+\ldots+Y_{n-i},\ldots,Y_{1})\,F_{i}(X_{n},\ldots,X_{n-i+1}-X_{n-i+2}),$

which are clearly equal, since $R$ is commutative. ∎

####  A.2.2. A bijection

Now define a map

$\displaystyle\Phi:\mathcal{M}_{X}(R)^{\times}$
$\displaystyle\rightarrow\mathcal{B}(R)^{\times}$ $\displaystyle F$
$\displaystyle\mapsto\sigma(F)\odot c(F)^{\odot-1}\odot F,$

where $c:\mathcal{M}_{X}(R)\rightarrow\prod_{n=0}^{\infty}R$ is the map which
in each component sends a power series to its constant term. In the other
direction, define

$\displaystyle\Psi:\mathcal{B}(R)^{\times}$
$\displaystyle\rightarrow\mathcal{M}_{X}(R)^{\times}$ $\displaystyle F$
$\displaystyle\mapsto c_{X}(F),$

where $c_{X}:\mathcal{B}(R)\rightarrow\mathcal{M}_{X}(R)$ is the map which in
each component sets $Y_{1}=\ldots=Y_{n}=0$.

######  Proposition A.4.

The maps $\Phi$ and $\Psi$ define mutually inverse bijections between
$\mathcal{M}_{X}(R)^{\times}$ and the subset of $\mathcal{B}(R)^{\times}$
consisting of those elements $F$ which satisfy the following two conditions:

  * (i)

$\sigma(F)=F$;

  * (ii)

There exist elements $G\in\mathcal{M}_{Y}(R)^{\times}$ and
$H\in\mathcal{M}_{X}(R)^{\times}$ such that $c(G)=\mathbbm{1}$ and such that
$F=G\odot H$.

###### Proof.

We begin by proving that $\Phi(F)$ satisfies conditions (i) and (ii) above,
for every $F\in\mathcal{M}_{X}(R)^{\times}$. For condition (i), this follows
directly from Proposition A.3 since $\sigma(F)\in\mathcal{M}_{Y}(R)$. For
condition (ii), it is clear that $\Phi(F)=G\odot H$ for
$H=F\in\mathcal{M}_{X}(R)^{\times}$ and $G=\sigma(F)\odot
c(F)^{\odot-1}\in\mathcal{M}_{Y}(R)^{\times}$. Moreover,
$c(G)=c(\sigma(F))\odot c(F)^{\odot-1}=\mathbbm{1}$, since $c$ is an algebra
homomorphism for $\odot$ and $c(\sigma(F))=c(F)$. Hence $\Phi(F)$ satisfies
conditions (i) and (ii) of the proposition.

We next prove that $\Phi$ and $\Psi$ are mutually inverse. Indeed, we have

$\Psi(\Phi(F))=c_{X}(\sigma(F)\odot c(F)^{\odot-1}\odot F)=c(\sigma(F))\odot
c(F)^{-1}\odot F=F,$

since $c_{X}$ agrees with $c$ on the subspace
$\mathcal{M}_{Y}(R)\subset\mathcal{B}(R)$ and is the identity on
$\mathcal{M}_{X}(R)$. On the other hand, since $c(G)=\mathbbm{1}$, we have

$\Phi(\Psi(G\odot H))=\Phi(H)=\sigma(H)\odot c(H)^{\odot-1}\odot H.$

In order to prove that this equals $G\odot H$, since $H$ is invertible, it
suffices to verify that $G=\sigma(H)\odot c(H)^{\odot-1}$. Indeed, since
$G\odot H$ is $\sigma$-invariant, we have

$G\odot H=\sigma(G\odot H)=\sigma(H)\odot\sigma(G),$

by Proposition A.3. Applying $c_{X}$ to both sides of this equality, we see
that $H=c(H)\odot\sigma(G)$, and multiplying both sides with $c(H)^{\odot-1}$,
then applying $\sigma$ yields the desired equality $G=\sigma(H)\odot
c(H)^{\odot-1}$. ∎

####  A.2.3. Hopf algebra structure

We next equip $\mathcal{B}(R)$ with the structure of a complete Hopf algebra
over $R$. To begin with, define a homomorphism of $R$-algebras by $(k\geq
1,d\geq 0\text{ and }(k,d)\neq(1,0))$

$\displaystyle\Delta_{\ast}:R[X,Y]$ $\displaystyle\rightarrow
R[X,Y]\otimes_{R}R[X,Y]$ $\displaystyle X^{k-1}Y^{d}$ $\displaystyle\mapsto
X^{k-1}Y^{d}\otimes 1+1\otimes X^{k-1}Y^{d}$
$\displaystyle\qquad+\sum_{\begin{subarray}{c}k_{1},k_{2}\geq 1\\\
k_{1}+k_{2}=k\end{subarray}}\sum_{\begin{subarray}{c}d_{1},d_{2}\geq 0\\\
d_{1}+d_{2}=d\end{subarray}}\binom{d_{1}+d_{2}}{d_{1}}\,X^{k_{1}-1}Y^{d_{1}}\otimes
X^{k_{2}-1}Y^{d_{2}}.$

Since $\Delta_{\ast}$ respects the degree of monomials, it uniquely extends to
an $R$-algebra homomorphism $\widehat{\Delta}_{\ast}:R\llbracket
X,Y\rrbracket\rightarrow R\llbracket
X,Y\rrbracket\,\widehat{\otimes}_{R}\,R\llbracket X,Y\rrbracket$, where
$\widehat{\otimes}$ denotes the completed tensor product. Finally, it extends
to an $R$-linear map

$\widehat{\Delta}_{\ast}:\mathcal{B}(R)\rightarrow\mathcal{B}(R)\;\widehat{\otimes}\;\mathcal{B}(R)$

by the requirement that $\Delta_{\ast}$ is an algebra homomorphism for the
concatenation product $\odot$. The triple
$(\mathcal{B}(R),\odot,\widehat{\Delta}_{\ast})$ is then a complete bialgebra,
and one can show that it is in fact a Hopf algebra. We let
$\mathcal{G}(R)\subset\mathcal{B}(R)$ denote the subspace of group-like
elements, i.e., elements $F=(F_{n})_{n\geq 0}$ which satisfy
$\widehat{\Delta}_{*}(F)=F\otimes F$ and $F_{0}=1$. In fact,
$\mathcal{G}(R)\subset\mathcal{B}(R)$ is a subgroup for the concatenation
product.

Our next goal is to state and prove a variant of Proposition A.4 for group-
like elements. First of all, the map
$\Psi:\mathcal{B}(R)^{\times}\rightarrow\mathcal{M}_{X}(R)^{\times}$ which
sends all variables $Y_{i}=0$ respects the subspace of group-like elements,
hence induces a map

$\Psi:\mathcal{G}(R)\rightarrow\mathcal{G}_{X}(R),$

where $\mathcal{G}_{X}(R)$ denotes the group-like elements of the Hopf
subalgebra $\mathcal{M}_{X}(R)\subset\mathcal{B}(R)$. Hence, if in addition
$F=G\odot H$ with $G,H$ as in Proposition A.4, then $H\in\mathcal{G}_{X}(R)$.
We next state a refinement of this result.

######  Proposition A.5.

Let $F\in\mathcal{B}(R)^{\times}$ satisfy conditions (i) and (ii) of
Proposition A.4. Then the following are equivalent:

  1. (i)

$F\in\mathcal{G}(R)$;

  2. (ii)

$H\in\mathcal{G}_{X}(R)$, and $\sigma(\lambda(F_{\rm corr})\odot
c_{X}(F))\in\mathcal{G}_{Y}(R)$, where $\lambda$ is defined by (A.2.1),

$F_{\rm
corr}:=\exp\left(\sum_{n=2}^{\infty}\frac{(-1)^{n}}{n}f_{n,0}\,t^{n}\right),$

and $f_{n,0}$ is the coefficient of $X^{n-1}$ in $F_{1}\in R\llbracket
X,Y\rrbracket$.

###### Proof.

We first prove that $\text{{(i)}}\Rightarrow\text{{(ii)}}$. By the above
discussion, we already know that
$H\in\mathcal{G}_{X}(R)\subset\mathcal{G}(R)$. Therefore, since
$\mathcal{G}(R)$ is a group, we also know that $G\in\mathcal{G}_{Y}(R)$.
Moreover, by Proposition A.4, we have

$G=\sigma(F)\odot c(F)^{\odot-1}=\sigma(F)\odot\lambda(F_{\rm
corr})\odot\lambda(\exp(-f_{1,0}t)),$

where the last equality follows from [HI, Equation (32)]. Note that
$\lambda(\exp(-f_{1,0}t))\in\mathcal{G}(R)$. Hence, using Proposition A.3 and
again the fact that $\mathcal{G}(R)$ is a group, we see that
$\sigma(\lambda(F_{\rm corr})\odot F)\in\mathcal{G}_{Y}(R)$. Hence, also
$\sigma(\lambda(F_{\rm corr})\odot c_{X}(F))\in\mathcal{G}_{Y}(R)$ as was to
be shown.

Conversely, if both $H=c_{X}(F)$ and $\sigma(\lambda(F_{\rm corr})\odot
c_{X}(F))$ are group-like, then so is $G\odot H$, since

$G\odot H=\sigma(c_{X}(F))\odot c(F)^{\odot-1}\odot
c_{X}(F)=\sigma(\lambda(F_{\rm corr})\odot
c_{X}(F))\odot\lambda(\exp(-f_{1,0}t))\odot c_{X}(F)$

is a product of group-like elements, where the first equality follows from
Proposition A.4. ∎

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