# Formal multiple Eisenstein series and their derivations

Henrik Bachmann  Graduate School of Mathematics, Nagoya University, Nagoya,
Japan<EMAIL_ADDRESS>and  Jan-Willem van Ittersum  Max-
Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. Department
of Mathematics and Computer Science, University of Cologne, Weyertal 86-90
50931 Cologne, Germany<EMAIL_ADDRESS>Department of Mathematical
Sciences, University of Copenhagen, Universitetsparken 5
2100 Copenhagen Ø, Denmark<EMAIL_ADDRESS>
###### Abstract.

We introduce the algebra of formal multiple Eisenstein series and study its
derivations. This algebra is motivated by the classical multiple Eisenstein
series, introduced by Gangl–Kaneko–Zagier as a hybrid of classical Eisenstein
series and multiple zeta values. In depth one, we obtain formal versions of
the Eisenstein series satisfying the same algebraic relations as the classical
Eisenstein series. In particular, they generate an algebra whose elements we
call formal quasimodular forms. We show that the algebra of formal multiple
Eisenstein series is an $\mathfrak{sl}_{2}$-algebra by formalizing the usual
derivations for quasimodular forms and extending them naturally to the whole
algebra. Additionally, we introduce some families of derivations for general
quasi-shuffle algebras, providing a broader context for these derivations.
Further, we prove that a quotient of this algebra is isomorphic to the algebra
of formal multiple zeta values. This gives a novel and purely formal approach
to classical (quasi)modular forms and builds a new link between (formal)
multiple zeta values and modular forms.

###### Key words and phrases:

multiple zeta values, multiple Eisenstein series, derivations, quasi-shuffle
algebras, (quasi)modular forms

###### 2020 Mathematics Subject Classification:

Primary 11F11,11M32. Secondary 13N15,16T30

(_With an appendix by_ NILS MATTHES)

##  1\. Introduction

The purpose of this note is to propose a notion of formal multiple Eisenstein
series, and explain their intimate connection to formal multiple zeta values
and the $\mathfrak{sl}_{2}$-algebra of formal (quasi)modular forms. Before
outlining the surprisingly simple steps to obtain the formal space in Section
1.3, we first recall the ‘concrete’ objects we aim to ‘formalize’, i.e., the
notions of multiple Eisenstein series, multiple zeta values and (quasi)modular
forms.

###  1.1. Multiple zeta values

Multiple zeta values, which are defined for integers $r\geq 1$ and $k_{1}\geq
2,k_{2},\dots,k_{r}\geq 1$ by

$\displaystyle\zeta(k_{1},\ldots,k_{r})\coloneqq\sum_{m_{1}>\cdots>m_{r}>0}\frac{1}{m_{1}^{k_{1}}\cdots
m_{r}^{k_{r}}}$ (1.1)

are subject to many relations. Denote the $\mathbb{Q}$-algebra of all multiple
zeta values by $\mathcal{Z}$. Conjecturally, the _extended double shuffle
relations_ of multiple zeta values provide all algebraic relations among
multiple zeta values [IKZ]. These relations are obtained (after possible
regularization) from the two ways of expressing the product of multiple zeta
values—the ‘usual’ (stuffle) product of real numbers, and a (shuffle) product
from the iterated integral representation of multiple zeta values—which both
can be interpreted as quasi-shuffle products [H].

First, we recall the usual algebraic setup for multiple zeta values, following
[IKZ] and [H], and introduce formal multiple zeta values. Consider the
alphabet $\mathcal{L}_{z}=\\{z_{k}\mid k\geq 1\\}$ and let
$\mathfrak{H}^{1}=\mathbb{Q}\langle\mathcal{L}_{z}\rangle$ be the free algebra
over $\mathcal{L}_{z}\mspace{1.0mu}$. Define a product $\diamond$ on
$\mathbb{Q}\mathcal{L}_{z}$ by $z_{i}\diamond z_{j}=z_{i+j}$ for all $i,j\geq
1$. The corresponding quasi-shuffle product (3.1) $\ast=\ast_{\diamond}$ is
usually called the _stuffle product_ and $(\mathfrak{H}^{1},\ast)$ is a
commutative $\mathbb{Q}$-algebra. Let $(\mathfrak{H}^{0},\ast)$ be the
subalgebra of $(\mathfrak{H}^{1},\ast)$ generated by all words not starting in
$z_{1}\mspace{1.0mu}$. The linear map defined on generators by

$\displaystyle\begin{split}\zeta:\mathfrak{H}^{0}&\longrightarrow\mathcal{Z}\\\
z_{k_{1}}\cdots z_{k_{r}}&\longmapsto\zeta(k_{1},\ldots,k_{r})\end{split}$
(1.2)

is an algebra homomorphism from $(\mathfrak{H}^{0},\ast)$ to $\mathcal{Z}$.
This homomorphism can be extended to a homomorphism
$\zeta^{\ast}:\mathfrak{H}^{1}\to\mathcal{Z}[T]$, that is, we obtain elements
$\zeta^{\ast}(k_{1},\dots,k_{r})\in\mathbb{R}[T]$ for all
$k_{1},\dots,k_{r}\geq 1$ called the _stuffle regularized multiple zeta
values_ (see [IKZ]). In case $k_{1}\geq 2$, these coincide with the multiple
zeta values (1.1).

The second product is given by the shuffle product for which we consider the
alphabet given by the two letters $\mathcal{L}_{xy}=\\{x,y\\}$ and write
$\mathfrak{H}=\mathbb{Q}\langle\mathcal{L}_{xy}\rangle$. Define the product
$\diamond$ by $a\diamond b=0$ for all
$a,b\in\mathbb{Q}\mathcal{L}_{xy}\mspace{1.0mu}$. Denote the corresponding
quasi-shuffle product $\ast_{\diamond}$ by $\shuffle$. Through the
identification $z_{k}=x^{k-1}y$ we can view $\mathfrak{H}^{1}$ and
$\mathfrak{H}^{0}$ as subalgebras of $(\mathfrak{H},\shuffle)$. Due to the
iterated integral expression of multiple zeta values, one obtains that the map
(1.2) gives an algebra homomorphism from $(\mathfrak{H}^{0},\shuffle)$ to
$\mathcal{Z}$. There is also a unique extension of the map $\zeta$ to an
algebra homomorphism
$\zeta^{\shuffle}:(\mathfrak{H}^{1},\shuffle)\rightarrow\mathcal{Z}[T]$.

The two regularizations $\zeta^{\ast}$ and $\zeta^{\shuffle}$ differ and their
difference can be described explicitly (see [IKZ, Theorem 1]). The comparison
of shuffle- and stuffle-regularized multiple zeta values, leads to the
_extended double shuffle relations_. The space of _formal multiple zeta
values_ is then defined as $\mathfrak{H}^{1}$ modulo the extended double
shuffle relations (cf., Definition A.1). The class of a word $z_{k_{1}}\cdots
z_{k_{r}}$ is denoted by $\zeta^{f}\\!(k_{1},\dots,k_{r})$. We will give a new
definition of formal multiple zeta values (Definition 2.15) and then show that
they coincide with the one mentioned here (Theorem 2.17).

###  1.2. Multiple Eisenstein series

Multiple zeta values and (quasi)modular forms are connected in various ways.
For example, in the case $r=1$, a multiple zeta value is a Riemann zeta value,
which is the constant term of the Eisenstein series. The Eisenstein series of
weight $k\geq 2$ is given for
$\tau\in\mathbb{H}=\\{{\tau\in\mathbb{C}\mid}\operatorname{Im}(\tau)>0\\}$ by

$\displaystyle\mathbb{G}_{k}(\tau)\coloneqq\zeta(k)+\frac{(-2\pi
i)^{k}}{(k-1)!}\sum_{m,n\geq 1}n^{k-1}q^{mn}\qquad\qquad(q=e^{2\pi i\tau}).$
(1.3)

For even $k\geq 2$ these series are (quasi)modular forms for the full modular
group. In [GKZ] the authors defined double Eisenstein series, which have
double zeta values ((1.1) in the case $r=2$) as their constant terms, and
which can be seen as a natural depth two version of Eisenstein series. This
construction was generalized by the first author in [Ba1]. Given a _depth_
$r\geq 1$ and integers $k_{1},\dots,k_{r}\geq 2$ the _multiple Eisenstein
series_ are defined111In the case $k_{1}=2$ one needs to use Eisenstein
summation. See [Ba5] for details. for $\tau\in\mathbb{H}$ by

$\displaystyle\mathbb{G}_{k_{1},\dots,k_{r}}(\tau)\coloneqq\sum_{\begin{subarray}{c}\lambda_{1}\succ\dots\succ\lambda_{r}\succ
0\\\
\lambda_{i}\in\mathbb{Z}\tau+\mathbb{Z}\end{subarray}}\frac{1}{\lambda_{1}^{k_{1}}\cdots\lambda_{r}^{k_{r}}}\,,$
(1.4)

where the order $\succ$ on the lattice $\mathbb{Z}\tau+\mathbb{Z}$ is defined
by $m_{1}\tau+n_{1}\succ m_{2}\tau+n_{2}$ iff $m_{1}>m_{2}$ or
$m_{1}=m_{2}\wedge n_{1}>n_{2}$. The multiple Eisenstein series are
holomorphic functions, i.e. elements in $\mathcal{O}(\mathbb{H})$. Since
$\mathbb{G}_{k_{1},\dots,k_{r}}(\tau+1)=\mathbb{G}_{k_{1},\dots,k_{r}}(\tau)$,
which can be obtained directly by the above definition, the multiple
Eisenstein series possess a Fourier expansion

$\displaystyle\mathbb{G}_{k_{1},\dots,k_{r}}(\tau)=\zeta(k_{1},\dots,k_{r})\,+\,\sum_{n\geq
1}a_{n}\,q^{n}\qquad(a_{n}\in\mathcal{Z}[2\pi i]),$ (1.5)

which was calculated in depth $r=2$ in [GKZ] and for arbitrary depth by the
first author in [Ba1]. More precisely, it was shown that the multiple
Eisenstein series can be written as an explicit $\mathcal{Z}[2\pi i]$-linear
combination of the $q$-series

$\displaystyle
g(k_{1},\dots,k_{r})\coloneqq\sum_{\begin{subarray}{c}m_{1}>\dots>m_{r}>0\\\
n_{1},\dots,n_{r}>0\end{subarray}}\frac{n_{1}^{k_{1}-1}}{(k_{1}-1)!}\dots\frac{n_{r}^{k_{r}-1}}{(k_{r}-1)!}q^{m_{1}n_{1}+\dots+m_{r}n_{r}}.$
(1.6)

We denote the $\mathbb{Q}$-vector space spanned by all $\eqref{eq:defmes}$ by

$\displaystyle\mathcal{E}\coloneqq\langle\mathbb{G}_{k_{1},\dots,k_{r}}\mid
r\geq 0,k_{1},\dots,k_{r}\geq 2\rangle_{\mathbb{Q}}\,,$ (1.7)

where we set $\mathbb{G}_{k_{1},\dots,k_{r}}(\tau)=1$ for $r=0$. With a
similar argument as for multiple zeta values, one can see that $\mathcal{E}$
is an algebra. More precisely, the subspace
$\mathfrak{H}^{2}=\mathbb{Q}\langle z_{k}\mid k\geq 2\rangle$ of
$\mathfrak{H}^{1}$ is closed under the stuffle product $\ast$, i.e., we get a
$\mathbb{Q}$-subalgebra
$(\mathfrak{H}^{2},\ast)\subset(\mathfrak{H}^{1},\ast)$. The space
$\mathcal{E}$ can then be seen as the image of the $\mathbb{Q}$-linear map
$\mathbb{G}:\mathfrak{H}^{2}\rightarrow\mathcal{O}(\mathbb{H})$ defined on the
generators by $z_{k_{1}}\cdots
z_{k_{r}}\mapsto\mathbb{G}_{k_{1},\dots,k_{r}}\mspace{1.0mu}$. This map is an
algebra homomorphism ([Ba5]) with respect to the stuffle product $\ast$.

The Fourier expansion (1.5) and the discussion above raise the natural
question of whether there exist algebra homomorphisms
$\mathbb{G}^{\bullet}:(\mathfrak{H}^{1},\bullet)\rightarrow\mathcal{O}(\mathbb{H})$,
for $\bullet\in\\{\ast,\shuffle\\}$, such that
$\mathbb{G}^{\bullet}_{\mid\mathfrak{H}^{2}}=\mathbb{G}$ and such that
$\mathbb{G}^{\bullet}(k_{1},\dots,k_{r})$ has a Fourier expansion with
constant term $\zeta^{\bullet}(k_{1},\dots,k_{r})$ for any
$k_{1},\dots,k_{r}\geq 1$. The case $\bullet=\shuffle$ was positively answered
in [BT], where the authors constructed _shuffle regularized multiple
Eisenstein series_ $\mathbb{G}^{\shuffle}$ by relating the Goncharov coproduct
to the Fourier expansion of multiple Eisenstein series. Similarly, the case
$\bullet=\ast$ was solved in [Ba2] (see also [Ba5]), where _stuffle
regularized multiple Eisenstein series_ $\mathbb{G}^{\ast}$ were constructed,
using the deconcatination coproduct for the stuffle algebra. As described at
the beginning, for multiple zeta values it is conjectured that all relations
can be described by the extended double shuffle relations, which then lead to
the notion of formal multiple zeta values, given by symbols modulo these
relations. Similarly, one can ask the question of what relations are satisfied
by, for example, the stuffle regularized multiple Eisenstein series and how
their formal version should be defined. In the following, we will explain the
origin and motivation of formal multiple Eisenstein series, based on various
numerical calculations in recent years.

In both the construction of $\mathbb{G}^{\shuffle}$ and of
$\mathbb{G}^{\ast}$, generalizations of the $q$-series $g$ in (1.6) were used.
More precisely, for $k_{1},\dots,k_{r}\geq 1,d_{1},\dots,d_{r}\geq 0$ define
the $q$-series

$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}\coloneqq\sum_{\begin{subarray}{c}m_{1}>\dots>m_{r}>0\\\
n_{1},\dots,n_{r}>0\end{subarray}}\frac{n_{1}^{k_{1}-1}m_{1}^{d_{1}}}{(k_{1}-1)!}\cdots\frac{n_{r}^{k_{r}-1}m_{r}^{d_{r}}}{(k_{r}-1)!}q^{m_{1}n_{1}+\dots+m_{r}n_{r}}\,.$
(1.8)

The stuffle and shuffle regularized multiple Eisenstein series are
$\mathcal{Z}[2\pi i]$-linear combinations of these $q$-series. The $q$-series
(1.8) were first defined in [Ba2], where their algebraic structure was
described in detail. In particular, it was shown that their product can be
expressed by a quasi-shuffle product quite similar to the stuffle product. For
example, we have

$\displaystyle
g\genfrac{(}{)}{0.0pt}{}{2}{1}g\genfrac{(}{)}{0.0pt}{}{3}{2}=g\genfrac{(}{)}{0.0pt}{}{2,3}{1,2}+g\genfrac{(}{)}{0.0pt}{}{3,2}{2,1}+g\genfrac{(}{)}{0.0pt}{}{5}{3}-\frac{1}{12}g\genfrac{(}{)}{0.0pt}{}{3}{3}.$
(1.9)

If we call $k_{1}+\dots+k_{r}+d_{1}+\dots+d_{r}$ the _weight_ of (1.8), then
this product can be seen as a double indexed version of the usual stuffle
product (where we add componentwise to obtain the ‘stuffle’-terms), modulo the
lower weight term $g\genfrac{(}{)}{0.0pt}{}{3}{3}$. The coefficient of $q^{N}$
in these $q$-series can be related to sums over certain partitions of $N$ (see
[Ba4, BI]). Using the conjugation of partitions then naturally leads to a
family of relations among these $q$-series, which we call _swap invariance_.
For example, in the case $r=1$ this just means
$g\genfrac{(}{)}{0.0pt}{}{k}{d}=\frac{d!}{(k-1)!}g\genfrac{(}{)}{0.0pt}{}{d+1}{k-1}$,
which can also be obtained directly from the definition. Based on numerical
calculations ([Ba3]), the first author conjectured that all relations among
the $q$-series (1.8) are a consequence of this swap invariance and the quasi-
shuffle product. Moreover, it was observed that, conjecturally, every series
(1.8) can be written as a $\mathbb{Q}$-linear combination of the single
indexed version $g(k_{1},\dots,k_{r})$ in (1.6). Finally, numerical
calculations further suggest that multiple Eisenstein series
$\mathbb{G}_{k_{1},\dots,k_{r}}$ satisfy the same $\mathbb{Q}$-linear
relations as $g(k_{1},\dots,k_{r})$ modulo lower weight terms. The quasi-
shuffle product for the double indexed $g$ modulo lower weight terms can be
seen as a double indexed version of the stuffle product as we can see in
(1.9).

All these observations then lead to the notion of formal multiple Eisenstein
series, which should be understood as formal objects satisfying the same
relations as multiple Eisenstein series.

###  1.3. Formal multiple Eisenstein series

Motivated by the observations mentioned above, the algebra of formal multiple
Eisenstein series $\mathcal{G}^{\\!f}$ is defined (Definition 2.5) to be the
$\mathbb{Q}$-vector space spanned by symbols $G^{\\!f}\\!$, which are swap
invariant and whose product is given by (a double indexed version of) the
stuffle product. More precisely, for $r\geq 1,k_{1},\dots,k_{r}\geq
1,d_{1},\dots,d_{r}\geq 0$ we introduce formal variables
$G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$ and
impose the relations (swap invariance)

$\displaystyle\mathfrak{G}_{r}\genfrac{(}{)}{0.0pt}{}{X_{1},\dots,X_{r}}{Y_{1},\dots,Y_{r}}=\mathfrak{G}_{r}\genfrac{(}{)}{0.0pt}{}{Y_{1}+\dots+Y_{r},\dots,Y_{1}+Y_{2},Y_{1}}{X_{r},X_{r-1}-X_{r},\dots,X_{1}-X_{2}}\,\qquad(\text{for
all }r\geq 1),$ (1.10)

where $\mathfrak{G}_{r}$ denotes the generating series

$\displaystyle\mathfrak{G}_{r}\genfrac{(}{)}{0.0pt}{}{X_{1},\dots,X_{r}}{Y_{1},\dots,Y_{r}}:=\sum_{\begin{subarray}{c}k_{1},\dots,k_{r}\geq
1\\\ d_{1},\dots,d_{r}\geq
0\end{subarray}}G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}X_{1}^{k_{1}-1}\cdots
X_{r}^{k_{r}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\cdots\frac{Y_{r}^{d_{r}}}{d_{r}!}\,.$

We then define the _algebra of formal multiple Eisenstein series_
$\mathcal{G}^{\\!f}$ as the $\mathbb{Q}$-vector space spanned by these formal
symbols $G^{\\!f}\\!$ equipped with the _stuffle product_ (Definition 2.1),
e.g.,

$\displaystyle
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1}}{d_{1}}G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{2}}{d_{2}}$
$\displaystyle=G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}+G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{2},k_{1}}{d_{2},d_{1}}+G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1}+k_{2}}{d_{1}+d_{2}}\,,$
$\displaystyle
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1}}{d_{1}}G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{2},k_{3}}{d_{2},d_{3}}$
$\displaystyle=G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2},k_{3}}{d_{1},d_{2},d_{3}}+G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{2},k_{1},k_{3}}{d_{2},d_{1},d_{3}}+G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{2},k_{3},k_{1}}{d_{2},d_{3},d_{1}}+G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1}+k_{2},k_{3}}{d_{1}+d_{2},d_{3}}+G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}+k_{3}}{d_{1},d_{2}+d_{3}}\,.$

For example, in the case $r=1$ they satisfy, similarly to $g$, the relation
$G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k}{d}=\frac{d!}{(k-1)!}G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{d+1}{k-1}$.
The use of double indices is necessary to make sense of the swap invariance
(1.10). But often we will be interested in the case when $d_{1}=\dots=d_{r}=0$
and set

$\displaystyle G^{\\!f}\\!(k_{1},\dots,k_{r})\coloneqq
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{0,\dots,0}\,.$ (1.11)

Then the discussion above suggests that the symbols
$G^{\\!f}\\!(k_{1},\dots,k_{r})$ should satisfy the same relations as the
stuffle regularized multiple Eisenstein series222It should be noted the
stuffle regularized multiple Eisenstein series are not defined uniquely. So,
more precisely, the claim is that there is a choice in the stuffle
regularization of the multiple Eisenstein series so that they satisfy exactly
the same relations as their formal counterparts.. In fact, we will see that
the $G^{\\!f}\\!(k)$ satisfy exactly the algebraic relations as the classical
Eisenstein series (Theorem 2.10), which is quite surprising considering the
simple family of relations that we impose on $G^{\\!f}\\!$. Even though it
seems like the single indexed $G^{\\!f}\\!(k_{1},\dots,k_{r})$ in (1.11) just
give a small portion of the whole space $\mathcal{G}^{\\!f}$, we conjecture
(Conjecture 2.6) it is already spanned by them. This justifies calling
$\mathcal{G}^{\\!f}$ the algebra of formal multiple Eisenstein series.

By the Fourier expansion (1.5) of multiple Eisenstein series we see that
projecting onto the constant term gives a surjective algebra homomorphism from
the space of multiple Eisenstein series to multiple zeta values. This raises
the natural question of whether there is a formal analogue of this projection
onto the algebra of formal multiple zeta values $\mathcal{Z}^{f}$. We give a
positive answer to this question together with an explicit description of the
kernel of this ‘formal projection onto the constant term’ as follows.

######  Theorem 1.1.

There exists a surjective algebra homomorphism

$\displaystyle\pi:\mathcal{G}^{\\!f}\rightarrow\mathcal{Z}^{f},$

with $\pi(G^{\\!f}\\!(k_{1},\dots,k_{r}))=\zeta^{f}(k_{1},\dots,k_{r})$. The
kernel of $\pi$ is the ideal generated by all formal multiple Eisenstein
series which are _not_ of the form

$\displaystyle
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{1,\dots,1,k_{1},\dots,k_{r}}{d_{1},\dots,d_{s},0,\dots,0},$

for some $r,s\geq 0$ and $k_{1},\dots,k_{r}\geq 1,d_{1},\dots,d_{s}\geq 0$.

In fact, we will define the algebra of formal multiple zeta values as the
quotient by the above ideal (Definition 2.13) and then show that this algebra
is isomorphic to the classical definition of formal multiple zeta values
(Theorem 2.17). Theorem 1.1 offers a new perspective on the extended double
shuffle relations. Namely, it states that these relations are equivalent to
the combination of swap invariance, the stuffle product, and the relations
derived from dividing out the mentioned ideal.

###  1.4. Derivations and $\mathfrak{sl}_{2}$-algebras

The second and main part of this work focuses on the derivations of the
algebra $\mathcal{G}^{\\!f}$. This is motivated by the derivations for
classical (quasi)modular forms, which we recall now. For this, we first
introduce another normalization of the Eisenstein series and define

$G_{k}\coloneqq(-2\pi
i)^{-k}\mathbb{G}_{k}=-\frac{B_{k}}{2k!}+\frac{1}{(k-1)!}\sum_{m,n\geq
1}m^{k-1}\,q^{mn},\qquad(B_{k}=k\text{th Bernoulli number}).$ (1.12)

As is well-known, for even $k$ the relations

$\displaystyle\begin{split}\frac{k+1}{2}G_{k}&\>=\>\frac{1}{k-2}\,q\frac{d}{dq}G_{k-2}\,+\\!\\!\sum_{\begin{subarray}{c}k_{1}+k_{2}=k\\\
k_{1},k_{2}\geq 2\text{
even}\end{subarray}}\\!\\!G_{k_{1}}G_{k_{2}}\,,\qquad(k\geq 4)\\\
\frac{(k+1)(k-1)(k-6)}{12}\,G_{k}&\>=\>\\!\\!\sum_{\begin{subarray}{c}k_{1}+k_{2}=k\\\
k_{1},k_{2}\geq 4\text{
even}\end{subarray}}\\!\\!(k_{1}-1)(k_{2}-1)\,G_{k_{1}}G_{k_{2}}\,,\qquad(k\geq
6)\end{split}$ (1.13)

suffice to write every Eisenstein series $G_{k}$ with $k\geq 4$ as a
polynomial in $G_{4}$ and $G_{6}\mspace{1.0mu}$. Moreover, the relations
(1.13) imply that the space
$\widetilde{\mathcal{M}}=\mathbb{Q}[G_{2},G_{4},G_{6}]$ of quasimodular forms
(with rational coefficients) is closed under the operator $q\frac{d}{dq}$.
Even more, the algebra $\widetilde{\mathcal{M}}$ is an
$\mathfrak{sl}_{2}$-algebra (see [Z] or Section 4.3). In general, an
_$\mathfrak{sl}_{2}$ -algebra_ is an algebra together with three derivations
$W,D,\delta$ acting on this algebra and satisfying the following commutator
relations

$\displaystyle[W,D]=2D,\quad[W,\delta]=-2\delta,\quad[\delta,D]=W.$

In the case of quasimodular forms, these derivations are given by
$D=q\frac{d}{dq}$ and the other two derivations are determined by $\delta
G_{2}=-\frac{1}{2}$, $\delta G_{4}=\delta G_{6}=0$ and
$WG_{k}=kG_{k}\mspace{1.0mu}$. As one of the main results of this work
(Theorem 4.12), we will show that the algebra $\mathcal{G}^{\\!f}$ is an
$\mathfrak{sl}_{2}$-algebra which can be seen as a natural generalization of
the $\mathfrak{sl}_{2}$-algebra of quasimodular forms:

######  Theorem 1.2.

There exist derivations $W,D,\delta$ on $\mathcal{G}^{\\!f}$ such that

  1. (i)

$\mathcal{G}^{\\!f}$ is an $\mathfrak{sl}_{2}$-algebra _;_

  2. (ii)

the subalgebra
$\widetilde{\mathcal{M}}^{\\!f}=\mathbb{Q}[G^{\\!f}\\!(2),G^{\\!f}\\!(4),G^{\\!f}\\!(6)]\subset\mathcal{G}^{\\!f}$
is isomorphic to $\widetilde{\mathcal{M}}$ as an $\mathfrak{sl}_{2}$-algebra.

We refer to $\widetilde{\mathcal{M}}^{\\!f}$ as the algebra of _formal
quasimodular forms_. The derivations $W,D,\delta$ will be given explicitly.
For example, the derivations $W$ and $D$ on $\mathcal{G}^{\\!f}$ are given as
the $\mathbb{Q}$-linear maps defined on the generators by

$\displaystyle
DG^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$
$\displaystyle\coloneqq\sum_{j=1}^{r}k_{j}\,G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{j}+1,\dots,k_{r}}{d_{1},\dots,d_{j}+1,\dots,d_{r}},$
(1.14) $\displaystyle
WG^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$
$\displaystyle\coloneqq(k_{1}+\dots+k_{r}+d_{1}+\dots+d_{r})\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}.$
(1.15)

On the single indexed $G^{\\!f}\\!$ we will give an algebraic interpretation
of the action of $D$ in terms of double shuffle relations (see (4.15)), which,
for example, gives $DG^{\\!f}\\!(1)=G^{\\!f}\\!(3)-G^{\\!f}\\!(2,1)$. Applying
the projection $\pi$ to this, together with the fact that
$D\mathcal{G}^{\\!f}\in\ker\pi$, yields $\zeta^{f}\\!(3)=\zeta^{f}\\!(2,1)$.

The derivation $\delta$ will be much more involved (see its explicit formula
in the proof of Proposition 4.11) and its discovery is based on extensive
numerical experiments done by the authors. To describe it we will introduce
some general derivations for quasi-shuffle algebras in Section 3 and then
write $\delta$ as a sum of five of these derivations. As explained in Remark
4.13, it seems that all three derivations on $\mathcal{G}^{\\!f}$ are unique
with the property of the above theorem. The $\mathfrak{sl}_{2}$-algebra
structure of $\mathcal{G}^{\\!f}$ gives a natural definition of the space of
_formal modular forms_
$\mathcal{M}^{\\!f}\coloneqq\ker\delta_{\mid\widetilde{\mathcal{M}}^{\\!f}}$
and a notion of Rankin–Cohen brackets. The projection $\pi$ in Theorem 1.1
then also naturally leads to the space of _formal cusp forms_
$\mathcal{S}^{\\!f}\coloneqq\ker\pi_{\mid\mathcal{M}^{\\!f}}\mspace{1.0mu}$.
We show in Section 5 that these spaces are isomorphic to their classical
counterparts $\mathcal{M}$ and $\mathcal{S}$.

###  1.5. Overview

We end the Introduction by giving an overview of some of the spaces considered
in this work. Summarizing the discussion above, we have the following _‘formal
picture’_ :

$0$$\mathbb{Q}[\zeta^{f}\\!(2)]$$\mathcal{Z}^{f}$$\mathcal{S}^{\\!f}$$\mathcal{M}^{\\!f}$$\widetilde{\mathcal{M}}^{\\!f}$$\mathcal{E}^{\\!f}$?$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}{\mathcal{G}^{\\!f}}$$\mathcal{G}^{\\!f}$?$\cong$$\mathfrak{sl}_{2}$$\mathfrak{sl}_{2}$$\mathfrak{sl}_{2}$$\mathfrak{sl}_{2}$$\pi$

Here,
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}{\mathcal{G}^{\\!f}}$ is
the subalgebra of $\mathcal{G}^{\\!f}$ spanned by the single indexed
$G(k_{1},\dots,k_{r})$ in (1.11) and $\mathcal{E}^{\\!f}$ is the subalgebra of
$\mathcal{G}^{\\!f}$ spanned by all the symbols with $k_{1},\dots,k_{r}\geq
2$. The algebra $\mathcal{E}^{\\!f}$ can be seen as the formal version of the
algebra $\mathcal{E}$ spanned by the multiple Eisenstein series (1.4). We
conjecture that this space is also an $\mathfrak{sl}_{2}$-subalgebra of
$\mathcal{G}^{\\!f}$ isomorphic to $\mathcal{E}$ (Conjecture 4.18).

Multiple Eisenstein series were the original motivation for the project, and
they are the centerpiece of the _‘classical picture’_ :

$0$$\mathbb{Q}[\zeta(2)]$$\mathcal{Z}$$\mathbb{S}$$\mathbb{M}$$\widetilde{\mathbb{M}}$$\mathcal{E}$$\mathcal{E}^{\rm{reg}}$$?$$\subset\mathcal{O}(\mathbb{H})$??$\mathfrak{sl}_{2}$$\mathfrak{sl}_{2}$$\mathfrak{sl}_{2}$const.
term

Even though this classical picture motivated the formal one, not all
counterparts of the formal picture are clear classically. Here,
$\widetilde{\mathbb{M}}=\mathbb{Q}[\mathbb{G}_{2},\mathbb{G}_{4},\mathbb{G}_{6}]$
(with $\mathbb{M}$ and $\mathbb{S}$ the subspace of $\widetilde{\mathbb{M}}$
of modular and cusp forms respectively) and $\mathcal{E}^{\rm{reg}}$ denotes a
space of regularized multiple Eisenstein series (e.g., one of the
regularizations mentioned at the beginning). This algebra
$\mathcal{E}^{\rm{reg}}$ should give the correct counterpart for
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}{\mathcal{G}^{\\!f}}$,
but it is difficult to show that there exists a well-defined homomorphism as
no counterpart for $\mathcal{G}^{\\!f}$ is known.

In [BB] the authors were able to give a ‘rational analogue’ of the space
$\mathcal{G}^{\\!f}$. For this, denote by
$\mathcal{Z}_{q}\subset\mathbb{Q}\llbracket q\rrbracket$ the space spanned by
the $q$-series (1.8), which appeared in the regularization of multiple
Eisenstein series. Then it was shown in [BB] that there exists a surjective
algebra homomorphism $G:\mathcal{G}^{\\!f}\rightarrow\mathcal{Z}_{q}$ (see
Theorem 2.19), which conjecturally is also injective. We obtain the _‘rational
picture’_ :

$0$$\mathbb{Q}$$\mathbb{Q}$$\mathcal{S}$$\mathcal{M}$$\widetilde{\mathcal{M}}$$\mathcal{E}_{\mathbb{Q}}$?$\mathcal{Z}^{\circ}_{q}$?$\mathcal{Z}_{q}$?$\subset\mathbb{Q}\llbracket
q\rrbracket$?$\cong$$\mathfrak{sl}_{2}$$\mathfrak{sl}_{2}$$\mathfrak{sl}_{2}$$\mathfrak{sl}_{2}$const.
term

Here $\mathcal{Z}^{\circ}_{q}$ (resp., $\mathcal{E}_{\mathbb{Q}}$) denotes the
image of
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}{\mathcal{G}^{\\!f}}$
(resp. $\mathcal{E}^{\\!f}$) under $G$. Conjecturally, all these spaces are
isomorphic to their formal analogues and, therefore, are
$\mathfrak{sl}_{2}$-algebras, where the derivation $D$ is given by
$q\frac{d}{dq}$. The open problem is to show that the derivation $\delta$ in
$\mathcal{G}^{\\!f}$ also gives a well-defined map on these spaces. The
elements in the space $\mathcal{Z}_{q}$ can be viewed as _$q$ -analogues of
multiple zeta values_ (see [BK, BI]), since the limit $q\rightarrow 1$ yields
multiple zeta values.

Since the start of this project in late 2019, and their original definition at
that time, formal multiple Eisenstein series were studied in the case $r=2$ in
[BKM] and some of the results proven here were announced in [Ba4]. Later, in
[BB] the authors introduced a realization of the algebra $\mathcal{G}^{\\!f}$
in the space $\mathbb{Q}\llbracket q\rrbracket$ by constructing combinatorial
multiple Eisenstein series, which give the rational picture mentioned above.
In her thesis [Bu1] (see also [Bu2, Bu3]), Burmester then reinterpreted the
algebra $\mathcal{G}^{\\!f}$ by introducing a new algebraic setup, called the
balanced setup, which gives a different description of the stuffle product and
the swap invariance. She also gave a new independent proof of Theorem 1.1
using the balanced setup (see [Bu3, Theorem 7.4]). We recall her algebraic
setup in Section 6.

### Acknowledgements

The authors thank Annika Burmester, Masanobu Kaneko, Ulf Kühn and Don Zagier
for fruitful discussions. This project was partially supported by JSPS KAKENHI
Grants 19K14499, 21K13771 and 23K03030. The first author would like to thank
the University of Cologne for the support of a research stay in which the
authors worked on this project. The second author was supported by the SFB/TRR
191 “Symplectic Structure in Geometry, Algebra and Dynamics”, funded by the
DFG (Projektnr. 281071066 TRR 191). Part of the work was carried out during
two visits of the second author to Nagoya, and he thanks Nagoya University for
its hospitality and support. He also thanks Utrecht University, where he
worked when the authors initiated this work.

##  2\. Formal multiple Eisenstein series

Define the set $\mathcal{A}$, whose elements we call _letters_ , by

$\displaystyle\mathcal{A}=\left\\{\genfrac{[}{]}{0.0pt}{}{k}{d}\mid k\geq
1,\,d\geq 0\right\\}.$

We are interested in $\mathbb{Q}$-linear combinations of words in the letters
of $\mathcal{A}$, i.e., in elements of $\mathbb{Q}\langle\mathcal{A}\rangle$.
Here and in the following, we call the monic monomials in
$\mathbb{Q}\langle\mathcal{A}\rangle$ _words_. For $k_{1},\dots,k_{r}\geq 1$
and $d_{1},\dots,d_{r}\geq 0$, we will use the following notation to write
words in $\mathbb{Q}\langle\mathcal{A}\rangle$:

$\displaystyle\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}:=\genfrac{[}{]}{0.0pt}{}{k_{1}}{d_{1}}\dots\genfrac{[}{]}{0.0pt}{}{k_{1}}{d_{1}}\,,$

where the product on the right is the usual non-commutative product in
$\mathbb{Q}\langle\mathcal{A}\rangle$. The space $\mathfrak{H}^{1}$ defined in
the introduction can be viewed naturally as a subspace of
$\mathbb{Q}\langle\mathcal{A}\rangle$ via the inclusion

$\displaystyle\begin{split}\mathfrak{H}^{1}&\longrightarrow\mathbb{Q}\langle\mathcal{A}\rangle\\\
z_{k_{1}}\cdots
z_{k_{r}}&\longmapsto\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{0,\dots,0}.\end{split}$
(2.1)

We will extend the stuffle product defined on $\mathfrak{H}^{1}$ to
$\mathbb{Q}\langle\mathcal{A}\rangle$ in the following way.

######  Definition 2.1.

Define the _stuffle product_ $\ast$ on $\mathbb{Q}\langle\mathcal{A}\rangle$
as the $\mathbb{Q}$-bilinear product, which satisfies $1\ast w=w\ast 1=w$ for
any word $w\in\mathbb{Q}\langle\mathcal{A}\rangle$ and

$\displaystyle\genfrac{[}{]}{0.0pt}{}{k_{1}}{d_{1}}w\ast\genfrac{[}{]}{0.0pt}{}{k_{2}}{d_{2}}v=\genfrac{[}{]}{0.0pt}{}{k_{1}}{d_{1}}\left(w\ast\genfrac{[}{]}{0.0pt}{}{k_{2}}{d_{2}}v\right)+\genfrac{[}{]}{0.0pt}{}{k_{2}}{d_{2}}\left(\genfrac{[}{]}{0.0pt}{}{k_{1}}{d_{1}}w\ast
v\right)+\genfrac{[}{]}{0.0pt}{}{k_{1}+k_{2}}{d_{1}+d_{2}}(w\ast v)$

for any letters
$\genfrac{[}{]}{0.0pt}{}{k_{1}}{d_{1}},\genfrac{[}{]}{0.0pt}{}{k_{2}}{d_{2}}\in\mathcal{A}$
and words $w,v\in\mathbb{Q}\langle\mathcal{A}\rangle$.

This turns $\mathbb{Q}\langle\mathcal{A}\rangle$ into a commutative
$\mathbb{Q}$-algebra $(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$, as shown in
[H]. The algebra $(\mathfrak{H}^{1},\ast)$ can be viewed as a subalgebra of
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$ via (2.1). Most of the time, we
will work with the generating series of our objects. Let
$\mathcal{B}_{0}(\mathbb{Q}\langle\mathcal{A}\rangle)=\mathbb{Q}\langle\mathcal{A}\rangle$
and for $r\geq 1$ set

$\displaystyle\mathcal{B}_{r}=\mathcal{B}_{r}(\mathbb{Q}\langle\mathcal{A}\rangle):=\mathbb{Q}\langle\mathcal{A}\rangle\llbracket
X_{1},Y_{1},\dots,X_{r},Y_{r}\rrbracket\quad\text{and}\quad\mathcal{B}=\mathcal{B}(\mathbb{Q}\langle\mathcal{A}\rangle):=\bigoplus_{r\geq
0}\mathcal{B}_{r}(\mathbb{Q}\langle\mathcal{A}\rangle).\qquad$ (2.2)

For $r\geq 0$ consider the family of formal power series
$\mathfrak{A}=(\mathfrak{A}_{0},\mathfrak{A}_{1},\ldots)\in\mathcal{B}$, where
$\mathfrak{A}_{0}=1$ and for $r\geq 1$ one has

$\displaystyle\mathfrak{A}_{r}\genfrac{(}{)}{0.0pt}{}{X_{1},\dots,X_{r}}{Y_{1},\dots,Y_{r}}:=\sum_{\begin{subarray}{c}k_{1},\dots,k_{r}\geq
1\\\ d_{1},\dots,d_{r}\geq
0\end{subarray}}\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}X_{1}^{k_{1}-1}\dots
X_{r}^{k_{r}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\dots\frac{Y_{r}^{d_{r}}}{d_{r}!}\,.$

######  Definition 2.2.

We define the _swap_ as the linear map
$\sigma:\mathcal{B}\rightarrow\mathcal{B}$ given by $\sigma(f_{r})=(\sigma
f_{r})$ with

$\displaystyle\sigma\left(f_{r}\genfrac{(}{)}{0.0pt}{}{X_{1},\dots,X_{r}}{Y_{1},\dots,Y_{r}}\right)\coloneqq
f_{r}\genfrac{(}{)}{0.0pt}{}{Y_{1}+\dots+Y_{r},\dots,Y_{1}+Y_{2},Y_{1}}{X_{r},X_{r-1}-X_{r},\dots,X_{1}-X_{2}}\,.$
(2.3)

Note that the swap $\sigma$ is an involution.

######  Definition 2.3.

By applying a map $\rho:\mathcal{B}\to\mathcal{B}$ to the power series
$\mathfrak{A}$ one obtains a map
$f:\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}\langle\mathcal{A}\rangle$
by comparing coefficients, i.e.,
$\rho\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$ is defined
as the coefficient of $X_{1}^{k_{1}-1}\dots
X_{r}^{k_{r}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\dots\frac{Y_{r}^{d_{r}}}{d_{r}!}$
of $\rho\mathfrak{A}_{r}\mspace{1.0mu}$

In particular, we can think of the swap
$\sigma:\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}\langle\mathcal{A}\rangle$
as the linear map defined in the above sense.

######  Remark 2.4.

There are several other ways to interpret the _swap_ $\sigma$.

  1. (i)

Explicitly,

$\displaystyle\sigma\
\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}=\sum_{\begin{subarray}{c}a_{1},\ldots,a_{r}\geq
1\\\
|\underline{a}|=|\underline{d}|+r\end{subarray}}\sum_{\begin{subarray}{c}b_{1},\ldots,b_{r}\geq
0\\\
|\underline{b}|=|\underline{k}|-r\end{subarray}}C^{\underline{a},\underline{k}}_{\underline{b},\underline{d}}\,\genfrac{[}{]}{0.0pt}{}{a_{1},\ldots,a_{r}}{b_{1},\ldots,b_{r}},$
(2.4)

where for
$\underline{a},\underline{b},\underline{d},\underline{k}\in\mathbb{Z}^{r}$ the
constant $C^{\underline{a},\underline{k}}_{\underline{b},\underline{d}}$ is
given by

$C^{\underline{a},\underline{k}}_{\underline{b},\underline{d}}=(-1)^{|\underline{b}|}\prod_{j=1}^{r}\textstyle\binom{s_{j}(\underline{d})-s^{j}(\underline{a})+j-1}{a_{r-j+1}-1}\binom{k_{r-j+1}-1}{s_{j}(\underline{b})-s^{j}(\underline{k})+j-1}\frac{(a_{j}-1)!}{(k_{j}-1)!}(-1)^{s_{j}(\underline{k})+s^{j}(\underline{b})+j}$

and where for $\underline{\ell}\in\mathbb{Z}^{r}$ and $j=1,\ldots,r$, we write

$s_{j}(\underline{\ell})=\sum_{i=1}^{j}\ell_{i}\,,\qquad
s^{j}(\underline{\ell})=\sum_{i=r-j+2}^{r}\ell_{i}\,.$

  2. (ii)

The concept of ‘swap’ originates from the works of Écalle (see e.g., [E]),
where it acts as an important operator in his theory of moulds. In our
context, its relevance emerges when exploring the $q$-series (1.8), which was
a key motivator for introducing the space $\mathcal{G}^{\\!f}$. Their
coefficients can be described as sums over certain partitions, and this also
leads to a similar interpretation for the generating series: The change of
variables in the definition of the swap (2.3) can be easily described by
thinking of a Young diagram with rectangle widths $X_{j}$ and heights
$Y_{j}\mspace{1.0mu}$. The change of variables can then be read off by taking
the conjugate of this Young diagram as visualized in the following picture:

$X_{1}$$X_{2}$$X_{r-1}$$X_{r}$$Y_{1}$$Y_{2}$$Y_{r-1}$$Y_{r}$conjugate$Y_{1}+\dots+Y_{r}$$\,\,Y_{1}+\dots+Y_{r-1}$$\,Y_{1}+Y_{2}$$Y_{1}$$X_{r}$$X_{r-1}-X_{r}$$X_{2}-X_{3}$$X_{1}-X_{2}$

For example, $X_{1}$ gets replaced by $Y_{1}+\dots+Y_{r}\mspace{1.0mu}$, since
the new width of the top rectangle in the conjugated diagram comes from adding
up all the heights from the original diagram.

Formal multiple Eisenstein series are objects whose generating series are
invariant under the change of variables in (2.3).

######  Definition 2.5.

We define the space of _formal multiple Eisenstein series_ as

$\displaystyle\mathcal{G}^{\\!f}\coloneqq\faktor{(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)}{\mathcal{I}}\,$

where $\mathcal{I}$ is the ideal in
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$ generated by $\sigma(w)-w$ for
all $w\in\mathbb{Q}\langle\mathcal{A}\rangle$. For $k_{1},\dots,k_{r}\geq 1$
and $d_{1},\dots,d_{r}\geq 0$ we denote the class of the word
$w=\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$ in
$\mathcal{G}^{\\!f}$ by
$G^{\\!f}\\!(w)=G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$.
We define a weight grading and two filtrations on $\mathcal{G}^{\\!f}$, as
follows:

  1. (i)

Write

$\displaystyle\operatorname{wt}(w)$ $\displaystyle\coloneqq
k_{1}+\ldots+k_{r}+d_{1}+\ldots+d_{r},$ $\displaystyle\operatorname{lwt}(w)$
$\displaystyle\coloneqq d_{1}+\ldots+d_{r},$
$\displaystyle\operatorname{dep}(w)$ $\displaystyle\coloneqq r$

for the _weight_ , _lower weight_ and _depth_ of the word $w$ respectively.

  2. (ii)

Write $\mathcal{G}^{\\!f}_{k}$ for the subvectorspace of $\mathcal{G}^{\\!f}$
generated by $G^{\\!f}\\!(w)$ with $\operatorname{wt}(w)=k$.

  3. (iii)

Write
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{d}\mathcal{G}^{\\!f}$,
$\operatorname{Fil}^{\mathrm{\operatorname{dep}}}_{r}\mathcal{G}^{\\!f}$ for
the lower weight and depth filtration on $\mathcal{G}^{\\!f}$ respectively. We
shorten the notation when considering two filtrations at the same time, that
is

$\displaystyle\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{d}\mathcal{G}^{\\!f}$
$\displaystyle=\langle\,G^{\\!f}\\!(w)\mid
w\in\mathcal{A}^{\ast},\operatorname{lwt}(w)\leq d\,\rangle_{\mathbb{Q}}\,,$
(2.5)
$\displaystyle\operatorname{Fil}^{\mathrm{\operatorname{lwt},\operatorname{dep}}}_{d,r}\mathcal{G}^{\\!f}$
$\displaystyle=\langle\,G^{\\!f}\\!(w)\mid
w\in\mathcal{A}^{\ast},\operatorname{lwt}(w)\leq d\text{ and
}\operatorname{dep}(w)\leq r\,\rangle_{\mathbb{Q}}\,,\text{ etc.}$ (2.6)

In the special case $\operatorname{lwt}(w)=0$, we write

$\displaystyle G^{\\!f}\\!(k_{1},\dots,k_{r})\coloneqq
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{0,\dots,0}\,.$

The space of formal multiple Eisenstein series is a commutative
$\mathbb{Q}$-algebra $(\mathcal{G}^{\\!f},\ast)$ where each element is swap
invariant. Notice that the $\mathbb{Q}$-linear map

$\displaystyle\begin{split}G^{\\!f}\\!:(\mathfrak{H}^{1},\ast)&\longrightarrow\mathcal{G}^{\\!f}\\\
z_{k_{1}}\cdots z_{k_{r}}&\longmapsto
G^{\\!f}\\!(k_{1},\dots,k_{r})\end{split}$ (2.7)

is an algebra homomorphism. Even though the double indices are crucial for the
definition, a non-trivial conjecture is that the space $\mathcal{G}^{\\!f}$ is
already spanned by the singles indexed $G^{\\!f}\\!(k_{1},\dots,k_{r})$, i.e.,
that the mapping (2.7) is surjective. This conjecture is the formal version of
the conjectures in [Ba2, Conjecture 4.3], [BK, Conjecture 5 (B2)] and [BI,
Conjecture 3.15], of which only special cases are known (cf. [Ba2, Bu1, V]).

######  Conjecture 2.6.

The map (2.7) is surjective, i.e.

$\displaystyle\mathcal{G}^{\\!f}\simeq\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}{\mathcal{G}^{\\!f}}.$

A more refined version of the conjecture is that

$\operatorname{Fil}^{\mathrm{\operatorname{lwt},\operatorname{dep}}}_{d,r}{\mathcal{G}^{\\!f}}\subset\operatorname{Fil}^{\mathrm{\operatorname{lwt},\operatorname{dep}}}_{0,d+r}{\mathcal{G}^{\\!f}}\qquad\text{for
all }d,r\geq 0.$

######  Remark 2.7.

Conjecture 2.6 raises the natural question of whether we can give an explicit
description of the kernel of (2.7), that is, if there exists an explicit
description of an ideal $\mathfrak{R}\subset(\mathfrak{H}^{1},\ast)$, such
that $\mathcal{G}^{\\!f}\cong\faktor{(\mathfrak{H}^{1},\ast)}{\mathfrak{R}}$.
This would make it easier to compare the relations satisfied by (formal)
multiple Eisenstein series with those satisfied by (formal) multiple zeta
values. Conjecturally, the space of (stuffle regularized) multiple zeta values
is isomorphic to $\faktor{(\mathfrak{H}^{1},\ast)}{\mathfrak{E}}$, where
$\mathfrak{E}$ is the ideal generated by $w\ast v-w\shuffle v$ with
$w\in\mathfrak{H}^{1},v\in\mathfrak{H}^{0}$ (extended double shuffle
relations). Since multiple zeta values should satisfy at least those relations
of multiple Eisenstein series (as their constant terms), we would in
particular expect $\mathfrak{R}\subset\mathfrak{E}$.

######  Remark 2.8.

Let $\mathcal{G}^{\\!f}_{k}$ denote the space of formal multiple Eisenstein
series of weight $k$. The conjectured dimensions for them are as follows:

$\displaystyle\sum_{k\geq 0}\dim\mathcal{G}^{\\!f}_{k}\,X^{k}$
$\displaystyle\overset{?}{=}\frac{1}{1-X-X^{2}-X^{3}+X^{6}+X^{7}+X^{8}+X^{9}}$
$\displaystyle=1+X+2X^{2}+4X^{3}+7X^{4}+13X^{5}+23X^{6}+41X^{7}+73X^{8}+\dots\,.$

This conjecture emerges from combining the dimension conjectures for
$\mathcal{Z}_{q}\mspace{1.0mu}$ as discussed in [BK], with the hypothesis that
all relations in $\mathcal{Z}_{q}$ are derived from the swap invariance and
the stuffle product. It should be noted that the dimensions given in [BT,
Remark 4.14] corroborate these figures. Multiplying the above series by
$(1-X)$, we can extract the dimensions of the (admissible) multiple Eisenstein
series presented there.

###  2.1. Relations among formal multiple Eisenstein series

We will now show some relations among formal multiple Eisenstein series in
small depth. Recall the generating series of the formal multiple Eisenstein
series is denoted by

$\displaystyle\mathfrak{G}_{r}\genfrac{(}{)}{0.0pt}{}{X_{1},\dots,X_{r}}{Y_{1},\dots,Y_{r}}=\sum_{\begin{subarray}{c}k_{1},\dots,k_{r}\geq
1\\\ d_{1},\dots,d_{r}\geq
0\end{subarray}}G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}X_{1}^{k_{1}-1}\cdots
X_{r}^{k_{r}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\cdots\frac{Y_{r}^{d_{r}}}{d_{r}!}\,.$

Since the formal multiple Eisenstein series are swap invariant we have

$\displaystyle\mathfrak{G}_{r}\genfrac{(}{)}{0.0pt}{}{X_{1},\dots,X_{r}}{Y_{1},\dots,Y_{r}}=\mathfrak{G}_{r}\genfrac{(}{)}{0.0pt}{}{Y_{1}+\dots+Y_{r},\dots,Y_{1}+Y_{2},Y_{1}}{X_{r},X_{r-1}-X_{r},\dots,X_{1}-X_{2}}\,.$
(2.8)

On $\mathbb{Q}\langle\mathcal{A}\rangle$ we can define another product

$\shuffle$

by $w\raisebox{1.3pt}{\; \leavevmode\hbox to7.8pt{\vbox
to7.8pt{\pgfpicture\makeatletter\hbox{\hskip 3.89775pt\lower-3.89775pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69775pt}{0.0pt}\pgfsys@curveto{3.69775pt}{2.04224pt}{2.04224pt}{3.69775pt}{0.0pt}{3.69775pt}\pgfsys@curveto{-2.04224pt}{3.69775pt}{-3.69775pt}{2.04224pt}{-3.69775pt}{0.0pt}\pgfsys@curveto{-3.69775pt}{-2.04224pt}{-2.04224pt}{-3.69775pt}{0.0pt}{-3.69775pt}\pgfsys@curveto{2.04224pt}{-3.69775pt}{3.69775pt}{-2.04224pt}{3.69775pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.4034pt}{-1.81082pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scalebox{0.53}{$\shuffle$}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\;v=\sigma(\sigma(w)\ast\sigma(v))$
for $w,v\in\mathbb{Q}\langle\mathcal{A}\rangle$. One can easily check that
since $\sigma$ is an involution, this product is commutative and associative.
Due to the swap invariance of $\mathcal{G}^{\\!f}$, this product is the same
as the product $\ast$. This implies a large family of relations among elements
in $\mathcal{G}^{\\!f}$, i.e., $f\ast g-f\raisebox{1.3pt}{\; \leavevmode\hbox
to7.8pt{\vbox to7.8pt{\pgfpicture\makeatletter\hbox{\hskip
3.89775pt\lower-3.89775pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69775pt}{0.0pt}\pgfsys@curveto{3.69775pt}{2.04224pt}{2.04224pt}{3.69775pt}{0.0pt}{3.69775pt}\pgfsys@curveto{-2.04224pt}{3.69775pt}{-3.69775pt}{2.04224pt}{-3.69775pt}{0.0pt}\pgfsys@curveto{-3.69775pt}{-2.04224pt}{-2.04224pt}{-3.69775pt}{0.0pt}{-3.69775pt}\pgfsys@curveto{2.04224pt}{-3.69775pt}{3.69775pt}{-2.04224pt}{3.69775pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.4034pt}{-1.81082pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scalebox{0.53}{$\shuffle$}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\;g=0$ for all
$f,g\in\mathcal{G}^{\\!f}.$ These relations can be seen as an analogue of the
double shuffle relations for multiple zeta values. If $f$ and $g$ are of depth
$1$, these relations are given as follows.

######  Proposition 2.9.

For $k_{1},k_{2}\geq 1,d_{1},d_{2}\geq 0$ we have

$\displaystyle
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1}}{d_{1}}\,G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{2}}{d_{2}}=$
$\displaystyle\,\,G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},k_{2}}{d_{1},d_{2}}+G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{2},k_{1}}{d_{2},d_{1}}+G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1}+k_{2}}{d_{1}+d_{2}}$
$\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{\begin{subarray}{c}l_{1}+l_{2}=k_{1}+k_{2}\\\
e_{1}+e_{2}=d_{1}+d_{2}\end{subarray}}\\!\\!\left(\\!\binom{l_{1}-1}{k_{1}-1}\binom{d_{1}}{e_{1}}(-1)^{d_{1}-e_{1}}+\binom{l_{1}-1}{k_{2}-1}\binom{d_{2}}{e_{1}}(-1)^{d_{2}-e_{1}}\\!\right)G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{l_{1},l_{2}}{e_{1},e_{2}}$
$\displaystyle\quad+\frac{d_{1}!d_{2}!}{(d_{1}+d_{2}+1)!}\binom{k_{1}+k_{2}-2}{k_{1}-1}G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1}+k_{2}-1}{d_{1}+d_{2}+1}\,,$

where we sum over all $l_{1},l_{2}\geq 1$ and $e_{1},e_{2}\geq 0$, subject to
$l_{1}+l_{2}=k_{1}+k_{2}$ and $e_{1}+e_{2}=d_{1}+d_{2}\mspace{1.0mu}$, in the
second expression.

###### Proof.

The first expression is a direct consequence of the definition of the product
in $\mathcal{G}^{\\!f}$. For the second, one calculates the coefficient of
$X_{1}^{k_{1}-1}X_{2}^{k_{2}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\frac{Y_{1}^{d_{2}}}{d_{2}!}$
in $\mathfrak{G}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}\raisebox{1.3pt}{\;
\leavevmode\hbox to7.8pt{\vbox to7.8pt{\pgfpicture\makeatletter\hbox{\hskip
3.89775pt\lower-3.89775pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69775pt}{0.0pt}\pgfsys@curveto{3.69775pt}{2.04224pt}{2.04224pt}{3.69775pt}{0.0pt}{3.69775pt}\pgfsys@curveto{-2.04224pt}{3.69775pt}{-3.69775pt}{2.04224pt}{-3.69775pt}{0.0pt}\pgfsys@curveto{-3.69775pt}{-2.04224pt}{-2.04224pt}{-3.69775pt}{0.0pt}{-3.69775pt}\pgfsys@curveto{2.04224pt}{-3.69775pt}{3.69775pt}{-2.04224pt}{3.69775pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.4034pt}{-1.81082pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scalebox{0.53}{$\shuffle$}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\;\mathfrak{G}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{2}}$,
where

$\displaystyle\mathfrak{G}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}}{Y_{1}}\raisebox{1.3pt}{\;
\leavevmode\hbox to7.8pt{\vbox to7.8pt{\pgfpicture\makeatletter\hbox{\hskip
3.89775pt\lower-3.89775pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{3.69775pt}{0.0pt}\pgfsys@curveto{3.69775pt}{2.04224pt}{2.04224pt}{3.69775pt}{0.0pt}{3.69775pt}\pgfsys@curveto{-2.04224pt}{3.69775pt}{-3.69775pt}{2.04224pt}{-3.69775pt}{0.0pt}\pgfsys@curveto{-3.69775pt}{-2.04224pt}{-2.04224pt}{-3.69775pt}{0.0pt}{-3.69775pt}\pgfsys@curveto{2.04224pt}{-3.69775pt}{3.69775pt}{-2.04224pt}{3.69775pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.4034pt}{-1.81082pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\scalebox{0.53}{$\shuffle$}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\;$
$\displaystyle\mathfrak{G}_{1}\genfrac{(}{)}{0.0pt}{}{X_{2}}{Y_{2}}$
$\displaystyle\>=\>\mathfrak{G}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{2}}{Y_{1},Y_{2}-Y_{1}}+\mathfrak{G}_{2}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2},X_{1}}{Y_{2},Y_{1}-Y_{2}}+\frac{\mathfrak{G}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{1}}-\mathfrak{G}_{1}\genfrac{(}{)}{0.0pt}{}{X_{1}+X_{2}}{Y_{2}}}{Y_{1}-Y_{2}}\,.\qed$

Proposition 2.9 shows that the formal multiple Eisenstein series in depth two
give a realization of the formal double Eisenstein space introduced in [BKM,
Definition 2.1], since the latter are formal symbols satisfying the above
relations. It was then shown in [BKM, Theorem 4.4], that these relations can
be used to obtain the following relations.

######  Theorem 2.10.

For all $k_{1},k_{2}\geq 1$ with $k=k_{1}+k_{2}\geq 4$ even we have

$\displaystyle\begin{split}\frac{1}{2}\left(\binom{k_{1}+k_{2}}{k_{2}}-(-1)^{k_{1}}\right)G^{\\!f}\\!(k)=&\sum_{\begin{subarray}{c}j=2\\\
j\text{even}\end{subarray}}^{k-2}\left(\binom{k-j-1}{k_{1}-1}+\binom{k-j-1}{k_{2}-1}-\mathbf{1}_{j,k_{1}}\right)G^{\\!f}\\!(j)\,G^{\\!f}\\!(k-j)\\\
&\,+\frac{1}{2}\left(\binom{k-3}{k_{1}-1}+\binom{k-3}{k_{2}-1}+\mathbf{1}_{k_{1},1}+\mathbf{1}_{k_{2},1}\right)G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k-1}{1}\,.\end{split}$
(2.9)

###### Proof.

This follows by sending $P\genfrac{(}{)}{0.0pt}{}{j,k-j}{0,0}$ to
$G^{\\!f}\\!(j)G^{\\!f}\\!(k-j)$ in [BKM, Theorem 4.4] ∎

The following relations are special cases of Theorem 2.10, which will be used
later when dealing with formal (quasi)modular forms. They can be seen as the
formal version of the classical recursive formulas for Eisenstein series given
in (1.13).

######  Corollary 2.11.

  1. (i)

For even $k\geq 4$ we have

$\displaystyle\frac{k+1}{2}\,G^{\\!f}\\!(k)\>=\>G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k-1}{1}+\\!\\!\\!\sum_{\begin{subarray}{c}k_{1}+k_{2}=k\\\
k_{1},k_{2}\geq 2\text{
even}\end{subarray}}\\!\\!\\!G^{\\!f}\\!(k_{1})\,G^{\\!f}\\!(k_{2})\,.$

  2. (ii)

For all even $k\geq 6$ we have

$\displaystyle\frac{(k+1)(k-1)(k-6)}{12}\,G^{\\!f}\\!(k)\>=\>\\!\\!\\!\sum_{\begin{subarray}{c}k_{1}+k_{2}=k\\\
k_{1},k_{2}\geq 4\text{
even}\end{subarray}}\\!\\!\\!(k_{1}-1)(k_{2}-1)\,G^{\\!f}\\!(k_{1})\,G^{\\!f}\\!(k_{2})\,.$

Due to Euler we know that for $m\geq 1$ we have

$\displaystyle\zeta(2m)=-\frac{B_{2m}}{2(2m)!}(-2\pi
i)^{2m}=-\frac{B_{2m}}{2(2m)!}(-24\zeta(2))^{m}.$

As an analogue, in our formal setup we can show the following.

######  Corollary 2.12 (Generalized Euler relation).

For $m\geq 1$ we have

$\displaystyle
G^{\\!f}\\!(2m)=-\frac{B_{2m}}{2(2m)!}(-24G^{\\!f}\\!(2))^{m}+Q_{2m},$

for $Q_{2m}\in D\widetilde{\mathcal{M}}^{\\!f}$, where
$\widetilde{\mathcal{M}}^{\\!f}=\mathbb{Q}[G^{\\!f}\\!(2),G^{\\!f}\\!(4),G^{\\!f}\\!(6)]$
is the space of formal quasimodular forms.

###### Proof.

First extend Corollary 2.11(i) to all $k=2m\geq 2$ by using the swap relation
$G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{1}{1}=G^{\\!f}\\!(2)$:

$\displaystyle
G^{\\!f}\\!(2m)=\frac{2}{2m+1}G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{2m-1}{1}+\frac{1}{3}{\bf
1}_{m=1}\,G^{\\!f}\\!(2)+\frac{2}{2m+1}\sum_{\begin{subarray}{c}a+b=m\\\
a,b\geq 1\end{subarray}}G^{\\!f}\\!(2a)\,G^{\\!f}\\!(2b)\,.$ (2.10)

Now, define the generating series.

$\displaystyle G(X)=\sum_{m\geq 1}G^{\\!f}\\!(2m)\,X^{2m},\qquad
F(X)=\sum_{m\geq
2}G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{2m-1}{1}\,X^{2m}=X^{2}\sum_{m\geq
1}\frac{DG^{\\!f}\\!(2m)}{2m}X^{2m}\,.$

Then (2.10) is equivalent to the following differential equation

$\displaystyle 2G(X)^{2}=G(X)+XG^{\prime}(X)-3G^{\\!f}\\!(2)X^{2}-2F(X)\,.$
(2.11)

Now assume we have a formal power series of the form $B(X)=\sum_{m\geq
1}b(2m)X^{2m}$ which satisfies the differential equation

$\displaystyle 2B(X)^{2}=B(X)+XB^{\prime}(X)-3b(2)X^{2}.$ (2.12)

Then the coefficients $b(2m)$ are all recursively determined by $b(2)$, i.e.,
for a fixed $b(2)$ the series $B(X)$ satisfying (2.12) is unique. Now one
checks by direct calculation that the formal power series

$\displaystyle B(X)=\sum_{m\geq 1}b(2m)X^{2m}\coloneqq$
$\displaystyle\frac{1}{2}-\frac{\sqrt{-24b(2)}X}{4}\coth\left(\frac{\sqrt{-24b(2)}X}{2}\right)$
$\displaystyle=$ $\displaystyle-\sum_{m\geq
1}\frac{B_{2m}}{2(2m)!}(-24b(2))^{m}X^{2m}$

is a solution of (2.12), i.e., $b(m)=-\frac{B_{2m}}{2(2m)!}(-24b(2))^{m}$ for
all $m\geq 1$. Since $G$ satisfies the differential equation (2.11), we see
that $G^{\\!f}\\!(2m)$ is also recursively determined by $G^{\\!f}\\!(2)$
and—due to the additional term $-2F(X)$—by $DG^{\\!f}\\!(2l)$ for $l<m$. The
statement follows inductively since any $G^{\\!f}\\!(2l)$ is contained in
$\widetilde{\mathcal{M}}^{\\!f}$ by Corollary 2.11. ∎

###  2.2. Formal multiple zeta values

In this section, we define the space of formal multiple zeta values.
Conjecturally, these satisfy exactly the same relations as multiple zeta
values. The definition in this work is equivalent to the usual definition of
formal multiple zeta values as formal symbols modulo the extended double
shuffle relations. The difference is that here we define the space of formal
multiple zeta values as a quotient of $\mathcal{G}^{\\!f}$. This approach has
the benefit of allowing connections to $q$-analogues of multiple zeta values
and modular form on a formal level, which is not directly possible with the
usual approach. This construction is motivated by the work [BI], where the
authors consider the behaviour of the $q$-series (1.8) as $q\rightarrow 1$. We
define the following two subsets of the alphabet $\mathcal{A}$.

$\displaystyle\mathcal{A}_{0}\coloneqq\left\\{\genfrac{[}{]}{0.0pt}{}{k}{0}\mid
k\geq
1\right\\}\,,\qquad\mathcal{A}^{1}\coloneqq\left\\{\genfrac{[}{]}{0.0pt}{}{1}{d}\mid
d\geq 0\right\\}\,.$

With this we define the following ideal in
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$ generated by the set
$\mathcal{A}^{*}\backslash((\mathcal{A}^{1})^{*}(\mathcal{A}_{0})^{*})$

$\displaystyle\mathfrak{N}\coloneqq\left(\mathcal{A}^{*}\backslash((\mathcal{A}^{1})^{*}(\mathcal{A}_{0})^{*})\right)_{\mathbb{Q}\langle\mathcal{A}\rangle}\,,$
(2.13)

where for an alphabet $\mathcal{L}$ by $\mathcal{L}^{*}$ we denote the set of
words in the letters $\mathcal{L}$, i.e., the free monoid generated by the
elements in $\mathcal{L}$. The generators of $\mathfrak{N}$ are exactly those
elements which are not of the form

$\displaystyle
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{1,\dots,1,k_{1},\dots,k_{r}}{d_{1},\dots,d_{s},0,\dots,0}\,,$
(2.14)

for some $k_{1},\dots,k_{r}\geq 1,d_{1},\dots,d_{s}\geq 0$.

######  Definition 2.13.

The algebra of _formal multiple zeta values_ is defined by

$\displaystyle\mathcal{Z}^{f}\coloneqq\faktor{\mathcal{G}^{\\!f}\,}{\mathfrak{N}}\,.$

The justification for the name formal multiple zeta values comes from the fact
that our notion is equivalent, up to the non-vanishing of $\zeta^{f}\\!(1)$ in
our case, to the one by Racinet [Rac] (see Theorem 2.17 below), which consists
of formal symbols satisfying the extended double shuffle relations. In
particular, we expect $\mathcal{Z}^{f}\cong\mathcal{Z}[T]$. Note that this
definition does not coincide with the definition of formal multiple zeta
values in the Introduction: it is the content of Theorem 2.17 that both
definitions are equivalent.

We denote the canonical projection of the space of formal multiple Eisenstein
series into the space of formal multiple zeta values by

$\displaystyle\pi:\mathcal{G}^{\\!f}\longrightarrow\mathcal{Z}^{f}\,.$ (2.15)

This projection can be seen as a formal version of the ‘projection onto the
constant term’.

######  Proposition 2.14.

The map
$\pi_{|\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}}:\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}\rightarrow\mathcal{Z}^{f}$
is surjective.

###### Proof.

All non-zero elements in $\mathcal{Z}^{f}$ are linear combinations of elements
of the form

$\displaystyle
f=G^{\\!f}\\!\Bigg{(}\genfrac{}{}{0.0pt}{}{1,\dots,1,}{d_{1},\dots,d_{s},}\underbrace{\genfrac{}{}{0.0pt}{}{1,\dots,1}{0,\dots,0}}_{j}\genfrac{}{}{0.0pt}{}{,k_{1},\dots,k_{r}}{,0,\dots,0}\Bigg{)}$
(2.16)

with $s,r\geq 0$, $j\geq 0$, $d_{s}\geq 1$ and $k_{1}\geq 2$. By induction on
$j$ one can apply the usual calculation used for the stuffle regularization
([BI, Proposition 4.18]) to show that $f$ can be written as

$\displaystyle
f\equiv\sum_{m=0}^{j}f_{m}\,G^{\\!f}\\!(1)^{m}\quad\mod\mathfrak{N}\,,$

where the $f_{m}$ are have the same shape as (2.16) with $j=0$. Such elements
can be expressed as products
$G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{1,\dots,1}{d_{1},\dots,d_{s}}G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{0,\dots,0}$
modulo $\mathfrak{N}$. By the definition of $\sigma$ it is easy to see that
$\sigma(\mathbb{Q}\langle\mathcal{A}^{1}\rangle)=\mathbb{Q}\langle\mathcal{A}_{0}\rangle$,
i.e., $\sigma
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{1,\dots,1}{d_{1},\dots,d_{s}}\in\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$.
Therefore, we can find a representative for the class of $f$ in
$\mathcal{Z}^{f}$ which is an element in
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$. ∎

######  Definition 2.15.

For $k_{1},\dots,k_{r}\geq 1$ we define the _formal multiple zeta value_
$\zeta^{f}\\!(k_{1},\dots,k_{r})$ by

$\displaystyle\zeta^{f}\\!(k_{1},\dots,k_{r})\coloneqq\pi(G^{\\!f}\\!(k_{1},\dots,k_{r}))\,.$

By Proposition 2.14 the $\zeta^{f}\\!$ span the space $\mathcal{Z}^{f}$ of
formal multiple zeta values. Recall that the derivation $D$ (see (1.14) in the
introduction) is given by

$\displaystyle
DG^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}\coloneqq\sum_{j=1}^{r}k_{j}\,G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{j}+1,\dots,k_{r}}{d_{1},\dots,d_{j}+1,\dots,d_{r}}\,.$
(2.17)

Since the formal multiple zeta values can be thought of as the ‘constant term’
of formal multiple Eisenstein series, they should vanish under taking the
derivative as we will see now.

######  Proposition 2.16.

We have $D\mathcal{G}^{\\!f}\subset\ker(\pi)$.

###### Proof.

By (2.17) we see that $D\mathcal{G}^{\\!f}\subset\mathfrak{N}$ since it is a
linear combination of words which all contain a letter
$\genfrac{[}{]}{0.0pt}{}{k}{d}$ with $k\geq 2$ and $d\geq 1$. ∎

This proposition implies that all the relations we proved so far for elements
in $\mathcal{G}^{\\!f}$ are also true in $\mathcal{Z}^{f}$ after setting all
terms involving derivatives, or, more generally, elements in $\mathfrak{N}$,
to zero. For example, as a direct consequence of Proposition 2.9 we obtain the
double shuffle relations in depth two: For $k_{1},k_{2}\geq 1$ we have

$\displaystyle\zeta^{f}\\!(k_{1})\,\zeta^{f}\\!(k_{2})$
$\displaystyle=\zeta^{f}\\!(k_{1},k_{2})+\zeta^{f}\\!(k_{2},k_{1})+\zeta^{f}\\!(k_{1}+k_{2})$
$\displaystyle=\sum_{l_{1}+l_{2}=k_{1}+k_{2}}\left(\binom{l_{1}-1}{k_{1}-1}+\binom{l_{1}-1}{k_{2}-1}\right)\zeta^{f}\\!(l_{1},l_{2})+\mathbf{1}_{k_{1}+k_{2},2}\,\zeta^{f}\\!(2)\,.$

In particular, we obtain the relation $\zeta^{f}\\!(3)=\zeta^{f}\\!(2,1)$ by
taking $k_{1}=1,k_{2}=2$. And as a consequence of Corollary 2.12 we get Eulers
relation, i.e., for $m\geq 1$ we have

$\displaystyle\zeta^{f}\\!(2m)=-\frac{B_{2m}}{2(2m)!}\left(-24\zeta^{f}\\!(2)\right)^{m}\,.$

The $\mathbb{Q}$-linear map defined on the generators by

$\displaystyle\begin{split}\zeta^{f}\\!:\mathfrak{H}^{1}&\longrightarrow\mathcal{Z}^{f}\\\
z_{k_{1}}\cdots
z_{k_{r}}&\longmapsto\zeta^{f}\\!(k_{1},\dots,k_{r})\end{split}$ (2.18)

is an algebra homomorphism with respect to the stuffle product $\ast$. This
follows from the fact that (2.7) is an algebra homomorphism and from the
definition of $\zeta^{f}\\!(k_{1},\dots,k_{r})$. The justification for calling
$\zeta^{f}\\!$ formal multiple zeta values comes from the following theorem,
stating that they conjecturally satisfy the same relations as (
$\ast$-regularized) multiple zeta values, namely the extended double shuffle
relations (cf. [IKZ, Rac]). Notice that, in contrast to [Rac], we have
$\zeta^{f}\\!(1)\neq 0$.

######  Theorem 2.17.

The formal multiple zeta values $\zeta^{f}\\!$ satisfy exactly the extended
double shuffle relations, i.e. the kernel of the map (2.18) is the ideal
generated by $w\ast v-w\shuffle v$ for $w\in\mathfrak{H}^{1}$ and
$v\in\mathfrak{H}^{0}$.

###### Proof.

In this proof, we make use of the notation and results in the appendix. First
observe that the element $F\in\mathcal{B}(\mathcal{Z}^{f})^{\times}$ given in
depth $r$ by

$\displaystyle\sum_{\begin{subarray}{c}k_{1},\dots,k_{r}\geq 1\\\
d_{1},\dots,d_{r}\geq
0\end{subarray}}\pi\left(G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}\right)X_{1}^{k_{1}-1}\cdots
X_{r}^{k_{r}-1}\frac{Y_{1}^{d_{1}}}{d_{1}!}\cdots\frac{Y_{r}^{d_{r}}}{d_{r}!}$

satisfies, by the definition of the ideal $\mathfrak{N}$, conditions (i) and
(ii) of Proposition A.4. Moreover, since the formal multiple Eisenstein series
satisfy the formula of the stuffle product, we also see
$F\in\mathcal{G}(\mathcal{Z}^{f})$. By Proposition A.5 we then find that the
coefficients of the corresponding $H\in\mathcal{G}_{X}(\mathcal{Z}^{f})$,
which is exactly the generating series of the formal multiple zeta values
$\mathcal{Z}^{f}(k_{1},\dots,k_{r})$, satisfy exactly the extended double
shuffle relations (as defined in Definition A.1). This implies the statement
in the theorem (cf. Remark A.2). ∎

######  Remark 2.18.

We define the _formal conjugated multiple zeta values_ for
$d_{1},\dots,d_{r}\geq 0$ by

$\displaystyle\xi^{f}(d_{1},\dots,d_{r})\coloneqq\pi\left(G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{1,\dots,1}{d_{1},\dots,d_{r}}\right).$

These span the space $\mathcal{Z}^{f}$ and satisfy the index shuffle product.
They can be seen as the formal analogues of the conjugated multiple zeta
values defined in [BI, Definition 1.3].

###  2.3. Realizations

Let $A$ be a $\mathbb{Q}$-algebra. For a subalgebra
$S\subseteq\mathcal{G}^{\\!f}$ we will call an algebra homomorphism
$\varphi:S\rightarrow A$ a _realization of $S$ in $A$_. Notice that finding a
realization of $\mathcal{G}^{\\!f}$ in $A$ is equivalent to finding an algebra
homomorphism

$\displaystyle\begin{split}\varphi:(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)&\longrightarrow
A\\\
w=\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}&\longmapsto\varphi(w)\end{split}$
(2.19)

which is $\sigma$-invariant, i.e. $\varphi(\sigma(w))=\varphi(w)$ for all
words $w\in\mathbb{Q}\langle\mathcal{A}\rangle$. Since the definition of
$\mathcal{G}^{\\!f}$ is motivated by multiple zeta values and multiple
Eisenstein series, a natural question is whether there exist corresponding
realizations of $\mathcal{G}^{\\!f}$. More precisely, since multiple zeta
values are just defined for indices $(k_{1},\dots,k_{r})\in\mathbb{Z}_{\geq
1}^{r}$ and multiple Eisenstein series are defined for indices
$(k_{1},\dots,k_{r})\in\mathbb{Z}_{\geq 2}^{r}$ a natural question is whether
there are $\sigma$-invariant algebra homomorphisms (2.19) for $A=\mathcal{Z}$
(resp. $A=\mathcal{O}(\mathbb{H})$) with
$\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{0,\dots,0}\mapsto\zeta(k_{1},\dots,k_{r})$
for $k_{1},\dots,k_{r}\geq 1$ (resp.
$\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{0,\dots,0}\mapsto\mathbb{G}_{k_{1},\dots,k_{r}}$
for $k_{1},\dots,k_{r}\geq 2$). In particular, one difficulty is that one has
to extend these maps to all
$\genfrac{[}{]}{0.0pt}{}{k_{1},\ldots,k_{r}}{d_{1},\ldots,d_{r}}$. The case
for $A=\mathcal{Z}$ was solved in [BI, Theorem 4.23]. The case for
$A=\mathcal{O}(\mathbb{H})$ remains open as mentioned in the Introduction.
Instead in [BB] the authors introduced a ‘rational version’ of multiple
Eisenstein series, which gives a realization in $A=\mathbb{Q}\llbracket
q\rrbracket$ such that $\varphi(\genfrac{[}{]}{0.0pt}{}{k}{0})=(-2\pi
i)^{-k}\mathbb{G}_{k}$ for $k\geq 2$ and $q=e^{2\pi i\tau}$. The construction
of the images of words in higher depth is inspired by the formula for the
Fourier expansion of the multiple Eisenstein series. Both constructions in
[BI] and [BB] start with the family of $q$-series $g(\cdots)$ defined by the
first author in [Ba2]; see (1.8). Notice that in the special case $r=1$ and
$k\geq 1$ one has
$g\genfrac{(}{)}{0.0pt}{}{k}{0}=\frac{1}{(k-1)!}\sum_{n>0}\sigma_{k-1}(n)q^{n}$.
The key observation is that the linear map
$\varphi_{g}:\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}\mapsto
g\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$ is
$\sigma$-invariant ([Ba2]). This map is not an algebra homomorphism with
respect to the product $\ast$, but one has $\varphi_{g}(w\ast
v)\equiv\varphi_{g}(w)\varphi_{g}(v)$ modulo terms of lower weight. In [BB]
the authors were able to correct these lower weight terms without losing the
$\sigma$-invariance to obtain the following:

######  Theorem 2.19 ([BB]).

There exists an algebra homomorphism
$\varphi_{G}:\mathcal{G}^{\\!f}\rightarrow\mathbb{Q}\llbracket q\rrbracket$
such that

$\displaystyle\varphi_{G}:G^{\\!f}\\!(k)\longmapsto
G(k)=-\frac{B_{k}}{2k!}+\frac{1}{(k-1)!}\sum_{n>0}\sigma_{k-1}(n)q^{n},\qquad(k\geq
2).$

Moreover, for any $f\in\mathcal{G}^{\\!f}$ we have
$\varphi_{G}(D(f))=q\frac{d}{dq}\varphi_{G}(f)$.

The image of $\varphi_{G}$ in Theorem 2.19 for an arbitrary
$G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$ is
given by the combinatorial bi-multiple Eisenstein series value defined
explicitly in [BB, Definition 6.4]. Conjecturally, the space of bi-multiple
Eisenstein series is isomorphic to $\mathcal{G}^{\\!f}$. In particular,
conjecturally, the realization $\varphi_{G}$ in Theorem 2.19 is injective.

To obtain a realization of $\mathcal{G}^{\\!f}$ in $\mathcal{Z}$ the authors
in [BI] consider the (regularized) limit of $q\rightarrow 1$ of (1.8) after
multiplication with $(1-q)^{k_{1}+\dots+k_{r}+d_{1}+\dots+d_{r}}$. Lower
weight terms also vanish under this limit and the resulting map is
$\sigma$-invariant. As a consequence, one obtains the following:

######  Theorem 2.20 ([BI]).

There exists an algebra homomorphism
$\varphi_{G}:\mathcal{G}^{\\!f}\rightarrow\mathcal{Z}$ such that

$\displaystyle\varphi_{\zeta}:G^{\\!f}\\!(k_{1},\dots,k_{r})\longmapsto\zeta^{\ast}(k_{1},\dots,k_{r}),\qquad(k_{1},\dots,k_{r}\geq
1).$

The image of $\varphi$ in Theorem 2.20 for an arbitrary
$G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$ is
given by the bi-multiple zeta value defined explicitly in [BI, Definition
4.22]. Note that the statement of Theorem 2.20 is also a consequence of
Theorem 2.17: If $\pi_{\mathcal{Z}}:\mathcal{Z}^{f}\rightarrow\mathcal{Z}$
denotes the projection of formal multiple zeta values to (stuffle regularized)
multiple zeta values, then we get a realization
$\pi_{\mathcal{Z}}\circ\pi:\mathcal{G}^{\\!f}\to\mathcal{Z}$, where $\pi$ is
given in (2.15). We should mention that the definition of the ideal
$\mathfrak{N}$ and our definition of $\mathcal{Z}^{f}$ are motivated by the
work [BI] for exactly this reason.

##  3\. Quasi-shuffle algebras and their derivations

###  3.1. Quasi-shuffle products

Suppose we have a set $\mathcal{L}$, called the set of _letters_ , and a
commutative and associative product $\diamond$ on $\mathcal{L}$. Extending
this product bi-linearly to $\mathbb{Q}\mathcal{L}$ gives a commutative non-
unital $\mathbb{Q}$-algebra $(\mathbb{Q}\mathcal{L},\diamond)$. We will be
interested in $\mathbb{Q}$-linear combinations of multiple letters of
$\mathcal{L}$, i.e., in elements of $\mathbb{Q}\langle\mathcal{L}\rangle$.
Here and in the following we call monic monomials in
$\mathbb{Q}\langle\mathcal{L}\rangle$ _words_. Moreover, we call the degree of
this monomial the _length_ of the word. In the special case
$\mathcal{L}=\mathcal{A}$, the following definition recovers Definition 2.1.

######  Definition 3.1.

Define the _quasi-shuffle product_ $\ast_{\diamond}$ on
$\mathbb{Q}\langle\mathcal{L}\rangle$ as the $\mathbb{Q}$-bilinear product
which satisfies $1\ast_{\diamond}w=w\ast_{\diamond}1=w$ for any word
$w\in\mathbb{Q}\langle\mathcal{L}\rangle$ and

$\displaystyle
aw\ast_{\diamond}bv=a(w\ast_{\diamond}bv)+b(aw\ast_{\diamond}v)+(a\diamond
b)(w\ast_{\diamond}v)$ (3.1)

for any letters $a,b\in\mathcal{L}$ and words
$w,v\in\mathbb{Q}\langle\mathcal{L}\rangle$.

This gives a commutative $\mathbb{Q}$-algebra
$(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$ as shown in [H]. In
the special case that $a\diamond b=0$ for all $a,b\in A$, we obtain the usual
shuffle product $\ast_{\diamond}=\shuffle$.

All quasi-shuffle algebras over the same alphabet are isomorphic due to the
results in [H]. In particular, any quasi-shuffle algebra
$(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$ is isomorphic to the
shuffle algebra $(\mathbb{Q}\langle\mathcal{L}\rangle,\shuffle)$ with the
isomorphism given by

$\displaystyle\log_{\diamond}:(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$
$\displaystyle\longrightarrow(\mathbb{Q}\langle\mathcal{L}\rangle,\shuffle)$
(3.2) $\displaystyle a_{1}\cdots a_{r}$ $\displaystyle\longmapsto$ (3.3)
$\displaystyle\\!\\!\\!\sum_{\begin{subarray}{c}1\leq l\leq r\\\
i_{1}+\dots+i_{l}=r\\\ i_{1},\dots,i_{l}\geq
1\end{subarray}}\frac{(-1)^{r-l}}{i_{1}\cdots
i_{l}}(a_{1}\diamond\dots\diamond a_{i_{1}})(a_{i_{1}+1}\diamond\dots\diamond
a_{i_{1}+i_{2}})\cdots(a_{i_{1}+\dots+i_{l-1}+1}\diamond\cdots\diamond
a_{r})\,\,.$ (3.4)

Its inverse is given by the algebra homomorphism

$\displaystyle\exp_{\diamond}:(\mathbb{Q}\langle\mathcal{L}\rangle,\shuffle)$
$\displaystyle\longrightarrow(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$
$\displaystyle a_{1}\cdots a_{r}$ $\displaystyle\longmapsto$
$\displaystyle\\!\\!\\!\sum_{\begin{subarray}{c}1\leq l\leq r\\\
i_{1}+\dots+i_{l}=r\\\ i_{1},\dots,i_{l}\geq
1\end{subarray}}\frac{1}{i_{1}!\cdots i_{l}!}(a_{1}\diamond\dots\diamond
a_{i_{1}})(a_{i_{1}+1}\diamond\dots\diamond
a_{i_{1}+i_{2}})\cdots(a_{i_{1}+\dots+i_{l-1}+1}\diamond\cdots\diamond
a_{r})\,.$

In particular, for any derivation $\Theta$ on
$(\mathbb{Q}\langle\mathcal{L}\rangle,\shuffle)$, the map

$\Theta_{\diamond}\coloneqq\exp_{\diamond}\circ\,\Theta\circ\log_{\diamond}$

is a derivation on $(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$.

###  3.2. Derivations

In this section, we consider examples of derivations on
$(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$. Some of these
derivations were studied previously in [KK] for the shuffle algebra, but it
seems that not much research is done on derivations for quasi-shuffle algebras
in general. We start with a simple construction, which gives a derivation on
the level of words coming from derivations on the level of letters. Let
$\varphi:\mathbb{Q}\mathcal{L}\rightarrow\mathbb{Q}\mathcal{L}$ be a
$\mathbb{Q}$-linear map. We extend $\varphi$ to a $\mathbb{Q}$-linear map
$\Theta^{\varphi}:\mathbb{Q}\langle\mathcal{L}\rangle\rightarrow\mathbb{Q}\langle\mathcal{L}\rangle$
by $\Theta^{\varphi}(1)=0$ and for a word $w=a_{1}\cdots a_{r}$ with
$a_{1},\dots,a_{r}\in\mathcal{L}$ by

$\displaystyle\Theta^{\varphi}(w)=\Theta^{\varphi}(a_{1}\cdots
a_{r})=\sum_{j=1}^{r}a_{1}\cdots\varphi(a_{j})\cdots a_{r}\,.$ (3.5)

######  Proposition 3.2.

If $\varphi$ is a derivation on $(\mathbb{Q}\mathcal{L},\diamond)$, then
$\Theta^{\varphi}$ is a derivation on
$(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$.

###### Proof.

We need to show
$\Theta^{\varphi}(w\ast_{\diamond}v)=\Theta^{\varphi}(w)\ast_{\diamond}v+w\ast_{\diamond}\Theta^{\varphi}(v)$
for all words $v,w\in\mathbb{Q}\langle\mathcal{L}\rangle$. From now on, we
write $\Theta$ for $\Theta^{\varphi}$. We proceed by induction on the sum of
the length of $w$ and $v$. First of all, note the result clearly holds if $w$
or $v$ is the empty word. In particular, the smallest length case is clear.
For the general case we get by definition that for $a,b\in\mathcal{L}$ and
words $v,w\in\mathbb{Q}\langle\mathcal{L}\rangle$

$\displaystyle\Theta(aw$ $\displaystyle\ast_{\diamond}bv)$
$\displaystyle\>=\>$
$\displaystyle\Theta(a(w\ast_{\diamond}bv)+b(aw\ast_{\diamond}v)+(a\diamond
b)(w\ast_{\diamond}v))$ $\displaystyle\>=\>$
$\displaystyle\varphi(a)(w\ast_{\diamond}bv)+a\,\Theta(w\ast_{\diamond}bv)+\varphi(b)(aw\ast_{\diamond}v)+b\,\Theta(aw\ast_{\diamond}v)\,+$
$\displaystyle\qquad\Theta(a\diamond b)(w\ast_{\diamond}v)+(a\diamond
b)\,\Theta(w\ast_{\diamond}v)$ $\displaystyle\>=\>$
$\displaystyle\varphi(a)(w\ast_{\diamond}bv)+a\,(\Theta(w)\ast_{\diamond}bv)+a\,(w\ast_{\diamond}\Theta(bv))+\varphi(b)(aw\ast_{\diamond}v)+b\,(\Theta(aw)\ast_{\diamond}v)\,+$
$\displaystyle\qquad b\,(aw\ast_{\diamond}\Theta(v))+\varphi(a\diamond
b)(w\ast_{\diamond}v)+(a\diamond
b)\,(\Theta(w)\ast_{\diamond}v+w\ast_{\diamond}\Theta(v)),$

where we used the induction hypothesis. On the other hand, we have

$\displaystyle\Theta(aw)$
$\displaystyle\ast_{\diamond}bv+aw\ast_{\diamond}\Theta(bv)$
$\displaystyle\>=\>$
$\displaystyle\varphi(a)w\ast_{\diamond}bv+a\Theta(w)\ast_{\diamond}bv+aw\ast_{\diamond}\varphi(b)v+aw\ast_{\diamond}b\,\Theta(v)$
$\displaystyle\>=\>$
$\displaystyle\varphi(a)(w\ast_{\diamond}bv)+b(\varphi(a)w\ast_{\diamond}v)+(\varphi(a)\diamond
b)(w\ast_{\diamond}v)+a(\Theta(w)\ast_{\diamond}bv)+b(a\Theta(w)\ast_{\diamond}v)\,+$
$\displaystyle+(a\diamond
b)(\Theta(w)\ast_{\diamond}v)+a(w\ast_{\diamond}\Theta(b)v)+\varphi(b)(aw\ast_{\diamond}v)+(a\diamond\varphi(b))(w\ast_{\diamond}v)\,+$
$\displaystyle+a(w\ast_{\diamond}b\,\Theta(v))+b(aw\ast_{\diamond}\Theta(v))+(a\diamond
b)(w\ast_{\diamond}\Theta(v))\,.$

Hence,

$\Theta(aw\ast_{\diamond}bv)-\Theta(aw)\ast_{\diamond}bv+aw\ast_{\diamond}\Theta(bv)=\bigl{(}\varphi(a\diamond
b)-(\varphi(a)\diamond b)-(a\diamond\varphi(b))\bigr{)}(w\ast_{\diamond}v),$

which vanishes by using $\varphi$ is a derivation. ∎

Later we will need the following generalization of Proposition 3.2 for linear
maps $\varphi$ which are not (necessarily) derivations with respect to
$\diamond$, but satisfy the following identity

$\displaystyle 2\varphi(a\diamond b\diamond c)=\varphi(a\diamond b)\diamond
c+\varphi(b\diamond c)\diamond a+\varphi(c\diamond a)\diamond b,$ (3.6)

which is equivalent to the condition (3.7) on $\gamma$ below. Clearly, if
$\varphi$ is a derivation on $(\mathbb{Q}\mathcal{L},\diamond)$ it satisfies
(3.6), but, since $(\mathbb{Q}\mathcal{L},\diamond)$ is a non-unital algebra,
this identity does not imply that $\varphi$ is a derivation. Note that the map
$\Theta^{\varphi}$ below agrees with the map in the previous proposition in
case $\varphi$ is a derivation.

######  Lemma 3.3.

Let $\varphi:\mathbb{Q}\mathcal{L}\rightarrow\mathbb{Q}\mathcal{L}$ be a
$\mathbb{Q}$-linear map. Let
$\gamma:\mathbb{Q}\mathcal{L}\times\mathbb{Q}\mathcal{L}\to\mathbb{Q}\mathcal{L}$
be the ‘failure of $\varphi$ being a derivation’, i.e.,

$\displaystyle\gamma(a,b)\coloneqq\varphi(a\diamond b)-\varphi(a)\diamond
b-a\diamond\varphi(b)\qquad(a,b\in\mathbb{Q}\mathcal{L}).$

Assume $\gamma(a,b)=0$ for $a,b\in\operatorname{im}\diamond$ and

$\displaystyle\gamma(a\diamond c,b)+\gamma(a,b\diamond c)=\gamma(a,b)\diamond
c\qquad(a,b,c\in\mathbb{Q}\mathcal{L}).$ (3.7)

Then the $\mathbb{Q}$-linear map
$\Theta^{\varphi}:\mathbb{Q}\langle\mathcal{L}\rangle\rightarrow\mathbb{Q}\langle\mathcal{L}\rangle$
defined by

$\displaystyle\Theta^{\varphi}(a_{1}\cdots a_{r})$
$\displaystyle\coloneqq\sum_{j=1}^{r}a_{1}\cdots\varphi(a_{j})\cdots
a_{r}-\frac{1}{2}\sum_{j=1}^{r-1}a_{1}\cdots
a_{j-1}\,\gamma(a_{j},a_{j+1})\,a_{j+2}\cdots a_{r}$

is a derivation on $(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$.

###### Proof.

Again, we want to show
$\Theta^{\varphi}(w\ast_{\diamond}v)=\Theta^{\varphi}(w)\ast_{\diamond}v+w\ast_{\diamond}\Theta^{\varphi}(v)$
for all words $w=w_{1}\cdots w_{r}$ and $v=v_{1}\cdots v_{s}\mspace{1.0mu}$.
Setting $W=\\{w_{1},\dots,w_{r}\\}$ and $V=\\{v_{1},\dots,v_{s}\\}$ we have

$\displaystyle w\ast_{\diamond}v=\sum_{\begin{subarray}{c}u=u_{1}\cdots
u_{t}\\\ \max(r,s)\leq t\leq
r+s\end{subarray}}\binom{u}{w,v}_{\\!\\!\diamond}\,u\,$

for some non-negative integers $\binom{u}{w,v}_{\\!\\!\diamond}$ and $u_{j}\in
W\cup V\cup(W\diamond V)$.

First, consider some $u_{j}\in W\cup V$ in the sum above, and write
$u=u^{\prime}u_{j}u^{\prime\prime}$. If $\varphi$ acts on
$u_{j}\mspace{1.0mu}$, then the same contribution
$u^{\prime}\,\varphi(u_{j})\,u^{\prime\prime}$ appears in both
$\Theta^{\varphi}(w\ast_{\diamond}v)$ and
$\Theta^{\varphi}(w)\ast_{\diamond}v+w\ast_{\diamond}\Theta^{\varphi}(v)$.
Moreover, if both $u_{j},u_{j+1}\in W$ (resp. $V$), write
$u=u^{\prime}u_{j}u_{j+1}u^{\prime\prime}$. Then, if $\gamma$ acts on this
pair, the same contribution

$-\frac{1}{2}u^{\prime}\,\gamma(u_{j},u_{j+1})\,u^{\prime\prime}$

appears in both $\Theta^{\varphi}(w\ast_{\diamond}v)$ and
$\Theta^{\varphi}(w)\ast_{\diamond}v$ (resp.
$w\ast_{\diamond}\Theta^{\varphi}(v)$).

Next, consider letters $w_{m}\in W$ and $v_{n}\in V$. This gives rise to words
with letter $w_{m}\diamond v_{n}$ as well as to words with $w_{m}$ and $v_{n}$
as consecutive letters, i.e., $u=u^{\prime}(w_{m}\diamond
v_{n})u^{\prime\prime}$, $u=u^{\prime}w_{m}v_{n}u^{\prime\prime}$ or
$u=u^{\prime}v_{n}w_{m}u^{\prime\prime}$ for some words
$u^{\prime},u^{\prime\prime}$. Then, the term
$\Theta^{\varphi}(w\ast_{\diamond}v)$ contains

$u^{\prime}(\varphi(w_{m}\diamond
v_{n}))u^{\prime\prime}-\frac{1}{2}u^{\prime}\gamma(w_{m},v_{n})u^{\prime\prime}-\frac{1}{2}u^{\prime}\gamma(v_{n},w_{m})u^{\prime\prime}.$

Moreover, the terms
$\Theta^{\varphi}(w)\ast_{\diamond}v+w\ast_{\diamond}\Theta^{\varphi}(v)$
contain

$u^{\prime}(\varphi(w_{m})\diamond
v_{n})u^{\prime\prime}+u^{\prime}(w_{m}\diamond\varphi(v_{n}))u^{\prime\prime}$

with the same multiplicity. Note that by definition of $\gamma$ these together
vanish.

The last case that we have to consider is $u=u^{\prime}(w_{m}\diamond
v_{n})u_{j+1}u^{\prime\prime}$. First, assume $u_{j+1}\in V\cup W$. W.l.o.g.,
say $u_{j+1}=w_{m+1}\in W$. Then, the term
$\Theta^{\varphi}(w\ast_{\diamond}v)$ contains

$-\frac{1}{2}u^{\prime}(\gamma(w_{m}\diamond
v_{n},w_{m+1}))u^{\prime\prime}-\frac{1}{2}u^{\prime}(\gamma(w_{m},w_{m+1}\diamond
v_{n}))u^{\prime\prime}$

Moreover, the term $\Theta^{\varphi}(w)\ast_{\diamond}v$ contains

$-\frac{1}{2}u^{\prime}(\gamma(w_{m},w_{m+1})\diamond v_{n})u^{\prime\prime}.$

Now, these contributions vanish by assumption (3.7) on $\gamma$. Finally,
assume $u_{j+1}\in V\diamond W$ with $u=u^{\prime}(w_{m}\diamond
v_{n})u_{j+1}u^{\prime\prime}$. Then, the term
$\Theta^{\varphi}(w\ast_{\diamond}v)$ contains

$-\frac{1}{2}u^{\prime}(\gamma(w_{m}\diamond v_{n},u_{j+1})u^{\prime\prime}$

Note that, as both $w_{m}\diamond v_{n}$ and $u_{j+1}$ are in the image of
$\diamond$, this contribution is zero.

Observe that we have covered all words appearing in
$\Theta^{\varphi}(w\ast_{\diamond}v),\Theta^{\varphi}(w)\ast_{\diamond}v$ and
$w\ast_{\diamond}\Theta^{\varphi}(v)$ and that each word appear with zero
coefficient in

$\Theta^{\varphi}(w\ast_{\diamond}v)-\Theta^{\varphi}(w)\ast_{\diamond}v-w\ast_{\diamond}\Theta^{\varphi}(v),$

i.e., $\Theta^{\varphi}$ is a derivation. ∎

######  Proposition 3.4.

For any $a\in\mathcal{L}$ the linear maps

$\displaystyle\Theta^{[a}:\mathbb{Q}\langle\mathcal{L}\rangle$
$\displaystyle\longrightarrow\mathbb{Q}\langle\mathcal{L}\rangle,$
$\displaystyle\quad\Theta^{a]}:\mathbb{Q}\langle\mathcal{L}\rangle$
$\displaystyle\longrightarrow\mathbb{Q}\langle\mathcal{L}\rangle$
$\displaystyle a_{1}\cdots a_{r}$
$\displaystyle\longmapsto\mathbf{1}_{a_{1}=a}a_{2}\cdots a_{r},$
$\displaystyle\quad a_{1}\cdots a_{r}$
$\displaystyle\longmapsto\mathbf{1}_{a_{r}=a}a_{1}\cdots a_{r-1},$

are derivations on $(\mathbb{Q}\langle\mathcal{L}\rangle,\shuffle)$.

###### Proof.

Can be shown easily by induction on the length of a word. ∎

######  Corollary 3.5.

For any $a\in\mathcal{L}$ the linear maps

$\displaystyle\Theta^{[a}_{\diamond}:\mathbb{Q}\langle\mathcal{L}\rangle,$
$\displaystyle\longrightarrow\mathbb{Q}\langle\mathcal{L}\rangle,$
$\displaystyle a_{1}\cdots a_{r}$ $\displaystyle\longmapsto\sum_{1\leq l\leq
r}\mathbf{1}_{a_{1}\diamond\cdots\diamond
a_{l}=a}\frac{(-1)^{l+1}}{l}a_{l+1}\cdots a_{r},$
$\displaystyle\Theta^{a]}_{\diamond}:\mathbb{Q}\langle\mathcal{L}\rangle,$
$\displaystyle\longrightarrow\mathbb{Q}\langle\mathcal{L}\rangle,$
$\displaystyle a_{1}\cdots a_{r}$ $\displaystyle\longmapsto\sum_{1\leq l\leq
r}\mathbf{1}_{a_{r-l+1}\diamond\cdots\diamond
a_{r}=a}\frac{(-1)^{l+1}}{l}a_{1}\cdots a_{r-l},$

are derivations with respect to the quasi-shuffle product $\ast_{\diamond}$.

###### Proof.

This is a direct consequence of Proposition 3.4 by applying the isomorphisms
$\log_{\diamond}:(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})\to(\mathbb{Q}\langle\mathcal{L}\rangle,\shuffle)$
and its inverse $\exp_{\diamond}$. ∎

######  Proposition 3.6.

Let $a\in\mathcal{L}$ and let
$\varphi:\mathbb{Q}\mathcal{L}\rightarrow\mathbb{Q}\mathcal{L}$ be a linear
map. Then the linear map

$\displaystyle\Theta^{\varphi,a}:\mathbb{Q}\langle\mathcal{L}\rangle$
$\displaystyle\longrightarrow\mathbb{Q}\langle\mathcal{L}\rangle,$
$\displaystyle a_{1}\cdots a_{r}$
$\displaystyle\longmapsto\sum_{j=1}^{r-1}\mathbf{1}_{a_{j}=a}\,a_{1}\cdots
a_{j-1}\varphi(a_{j+1})\cdots
a_{r}-\sum_{j=2}^{r}\mathbf{1}_{a_{j}=a}\,a_{1}\cdots\varphi(a_{j-1})a_{j+1}\cdots
a_{r}$

is a derivation with respect to the shuffle product $\shuffle$.

###### Proof.

Similar as in the proof of Lemma 3.3, for words
$w,v\in\mathbb{Q}\langle\mathcal{L}\rangle$ write

$\displaystyle w\shuffle v=w_{1}\cdots w_{r}\shuffle v_{1}\cdots
v_{s}=\sum_{u=u_{1}\cdots u_{r+s}}\binom{u}{w,v}\,u\,$

for some non-negative integers $\binom{u}{w,v}$. Setting
$W=\\{w_{1},\dots,w_{r}\\}$ and $V=\\{v_{1},\dots,v_{s}\\}$ we have $u_{j}\in
W\cup V$. Assume we have a word $u$ in the sum on the right with $u_{j}=a$ and
$u_{j}\in W$. If $u_{j+1}\in V$, then there also exists a word $u^{\prime}$ in
the shuffle product which equals $u$ but with $u_{j+1}$ and $u_{j}$
interchanged and $\binom{u}{w,v}=\binom{u^{\prime}}{w,v}$. In this case the
terms in $\Theta^{\varphi,a}$ cancel out. Therefore, the only contribution to
$\Theta^{\varphi,a}(w\shuffle v)$ with $u_{j}=a$ comes from those $u$, where
$u_{j},u_{j+1}\in W$ or $u_{j},u_{j+1}\in V$ (similarly $u_{j},u_{j-1}\in W$
or $u_{j},u_{j-1}\in V$). But these terms come exactly from the terms in
$\Theta^{\varphi,a}(w)\shuffle v$ and $w\shuffle\Theta^{\varphi,a}(v)$
respectively. ∎

Later we will need the following generalization of Proposition 3.6.

######  Lemma 3.7.

Let $(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$ be a quasi-shuffle
algebra and $S\subset\mathcal{L}$ such that
$\operatorname{im}(\diamond)\subset\mathbb{Q}\mathcal{L}\backslash S$. For
each $a\in S$ let $\varphi_{a}:\mathbb{Q}\mathcal{L}\to\mathbb{Q}\mathcal{L}$
be a linear map such that $\varphi_{a}(b\diamond c)=\varphi_{a}(b)\diamond c$
for any $b,c\in\mathcal{L}$ with $b\not\in S$ and
$\varphi_{a}(a^{\prime}\diamond c)=\varphi_{a^{\prime}}(a\diamond c)$ for all
$a,a^{\prime}\in S$ and $c\in\mathcal{L}$. Then the map $\Theta_{S}=\sum_{a\in
S}\Theta^{\varphi_{a},a}$ is a derivation on
$(\mathbb{Q}\langle\mathcal{L}\rangle,\ast_{\diamond})$.

###### Proof.

Notice that if $S=\\{a\\}$ and $\varphi(b\diamond c)=\varphi(b)\diamond c$ for
all $b,c\in\mathcal{L}$ then the statement follows immediately from
Proposition 3.6 since
$\Theta^{\varphi,a}=\exp_{\diamond}\circ\,\Theta^{\varphi,a}\circ\log_{\diamond}$.
For general $S$, we proceed as in the proof of Lemma 3.3: We have

$\displaystyle w\ast_{\diamond}v=w_{1}\cdots w_{r}\ast_{\diamond}v_{1}\cdots
v_{s}=\sum_{\begin{subarray}{c}u=u_{1}\cdots u_{t}\\\ \max(r,s)\leq t\leq
r+s\end{subarray}}\binom{u}{w,v}_{\\!\\!\diamond}\,u\,$

for some non-negative integers $\binom{u}{w,v}_{\\!\\!\diamond}$ and $u_{j}\in
W\cup V\cup(W\diamond V)$. Suppose $u_{j}=a$ for some $a\in S$. Moreover,
without loss of generality, assume $u_{j}=a\in W$. We distinguish two
possibilities for $u_{j+1}\mspace{1.0mu}$. First, notice that we can treat the
case $u_{j+1}\in V\cup W$ in a similar way as in the proof of Proposition 3.6.
Next, we suppose $u_{j+1}=w_{m}\diamond v_{n}\mspace{1.0mu}$ and again
distinguish two cases. First, if $w_{m}\not\in S$, the corresponding term in
$\Theta^{\varphi,a}(w\ast_{\diamond}v)$ also appears in
$\Theta^{\varphi,a}(w)\ast_{\diamond}v$ with the same multiplicity as
$\varphi_{a}(w_{m}\diamond v_{n})=\varphi_{a}(w_{m})\diamond
v_{n}\mspace{1.0mu}$. Next, if $w_{m}=a^{\prime}$ for some $a^{\prime}\in S$,
then $w$ contains two elements of $S$ consecutively, i.e., $u_{j}=w_{m-1}=a$
and $w_{m}=a^{\prime}$ for some $a,a^{\prime}\in S$. Therefore, the same term
with opposite sign (and the same multiplicity) occurs for $u_{j+1}=a^{\prime}$
and $u_{j}=a\diamond v_{n}\mspace{1.0mu}$, since
$\varphi_{a}(a^{\prime}\diamond v_{n})=\varphi_{a^{\prime}}(a\diamond v_{n})$.
Similar contributions in $\Theta_{S}(w)\ast_{\diamond}v$ cancel by the same
argument. ∎

##  4\. Derivations on $\mathcal{G}^{\\!f}$

In this section, we consider derivations on the space $\mathcal{G}^{\\!f}$.
All the derivations we consider come from derivations on
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$. Recall that we defined the
alphabet $\mathcal{A}$ by

$\displaystyle\mathcal{A}=\left\\{\genfrac{[}{]}{0.0pt}{}{k}{d}\mid k\geq
1,\,d\geq 0\right\\}\,.$

The stuffle product $\ast$ defined in Definition 2.1 is then the quasi-shuffle
product $\ast_{\diamond}$ associated to the product $\diamond$ on
$\mathbb{Q}\mathcal{A}$ defined for $k_{1},k_{2}\geq 1$ and $d_{1},d_{2}\geq
0$ by

$\displaystyle\genfrac{[}{]}{0.0pt}{}{k_{1}}{d_{1}}\diamond\genfrac{[}{]}{0.0pt}{}{k_{2}}{d_{2}}=\genfrac{[}{]}{0.0pt}{}{k_{1}+k_{2}}{d_{1}+d_{2}}\,.$

We often use the results for derivations on quasi-shuffle algebras from
Section 3.

Note that a derivation $\Theta$ on
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$ restricts to $\mathcal{G}^{\\!f}$
if and only if
$[\Theta,\sigma]\mathbb{Q}\langle\mathcal{A}\rangle\subset\mathcal{I}$.
Namely, a derivation $\Theta$ restricts to $\mathcal{G}^{\\!f}$ if and only
$\Theta\mathcal{I}\subset\mathcal{I}$. An element of $\mathcal{I}$ can be
written as $(\sigma(w)-w)u$ for some words $w,u$. Now, we have

$\Theta((\sigma(w)-w)u)=([\Theta,\sigma]w+\sigma(dw)-dw)u+(\sigma(w)-w)du\equiv([\Theta,\sigma]w)u\mod\mathcal{I},$

from which we conclude that $\Theta\mathcal{I}\subset\mathcal{I}$ if and only
if $[\Theta,\sigma]\mathbb{Q}\langle\mathcal{A}\rangle\subset\mathcal{I}$. In
what follows we consider _$\sigma$ -equivariant derivations_, i.e.,
$[\Theta,\sigma]=0$. In particular, every $\sigma$-equivariant derivation
restricts to a derivation on $\mathcal{G}^{\\!f}$.

###  4.1. A derivation of weight $-1$

Recall that any endomorphism of $\mathcal{B}$ defines an endomorphism on
$\mathbb{Q}\langle\mathcal{A}\rangle$ (see Definition 2.3). Therefore, for all
$j=1,\ldots,r$ we define the following maps on $r$-tuples

$\displaystyle p_{j}(X_{1},\ldots,X_{r})$
$\displaystyle=(X_{1},\ldots,X_{j-1},X_{j+1},\ldots,X_{r})$ (4.1)
$\displaystyle c_{j,j+1}(Y_{1},\ldots,Y_{r})$
$\displaystyle=(Y_{1},\ldots,Y_{j-1},Y_{j}+Y_{j+1},Y_{j+2},\ldots,Y_{r}),$
(4.2)

where we set $Y_{r+1}=0$. Note that $p_{j}$ removes the $j$th element and
$c_{j,j+1}$ contracts the $j$th and $j+1$th element. As a special case,
$c_{r,r+1}(Y_{1},\ldots,Y_{r})=(Y_{1},\ldots,Y_{r-1})$. We abbreviate
$\underline{X}=(X_{1},\ldots,X_{r})$ and similarly we define $\underline{Y}$.
Then, define $\varphi_{j}^{+},\varphi_{j}^{-}:\mathcal{B}\to\mathcal{B}$ by

$\displaystyle\varphi_{j}^{+}f_{r}(\underline{X},\underline{Y})\coloneqq\begin{cases}f_{r-1}(p_{j}(\underline{X}),c_{j,j+1}(\underline{Y}))&1\leq
j\leq r\\\ 0&\text{else.}\end{cases}$ (4.3)
$\displaystyle\varphi_{j}^{-}f_{r}(\underline{X},\underline{Y})\coloneqq\begin{cases}f_{r-1}(p_{j}(\underline{X}),c_{j-1,j}(\underline{Y}))&2\leq
j\leq r\\\ 0&\text{else.}\end{cases}$ (4.4)

and define $\omega:\mathcal{B}\to\mathcal{B}$ by

$\omega\coloneqq\sum_{j\geq 1}(\varphi_{j}^{+}-\varphi_{j}^{-}).$

This induces an endomorphism
$\omega:\mathbb{Q}\langle\mathcal{A}\rangle\rightarrow\mathbb{Q}\langle\mathcal{A}\rangle$.

######  Proposition 4.1.

The map $\omega$ is a $\sigma$-equivariant derivation on
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$.

###### Proof.

For the equivariance, we compute

$\displaystyle\varphi_{j}^{+}\sigma f_{r}(\underline{X},\underline{Y})$
$\displaystyle=\varphi_{j}^{+}f_{r}(Y_{1}+\dots+Y_{r},\dots,Y_{1}+Y_{2},Y_{1},X_{r},X_{r-1}-X_{r},\dots,X_{1}-X_{2})$
$\displaystyle=\sigma
f_{r-1}(X_{1},\ldots,X_{r-j},X_{r-j+2},\ldots,X_{r},Y_{1},\ldots,Y_{r-j},Y_{r-j+1}+Y_{r-j+2},\ldots,Y_{r})$
$\displaystyle=\sigma\varphi_{r-j+1}^{+}f_{r}(\underline{X},\underline{Y}).$

Hence,

$\displaystyle\varphi_{j}^{+}\,\sigma=\sigma\,\varphi_{r-j+1}^{+}\qquad\text{and
similarly}\qquad\varphi_{j}^{-}\,\sigma=\sigma\,\varphi_{r-j+2}^{-}\,.$ (4.5)

These identities directly imply the equivariance.

For the second part, note that $\omega$ acts on words by

$\displaystyle\omega\genfrac{[}{]}{0.0pt}{}{k_{1},\ldots,k_{r}}{d_{1},\ldots,d_{r}}$
$\displaystyle=\mathbf{1}_{(k_{r},d_{r})=(1,0)}\genfrac{[}{]}{0.0pt}{}{k_{1},\ldots,k_{r-1}}{d_{1},\ldots,d_{r-1}}+\sum_{j=1}^{r-1}\mathbf{1}_{k_{j}=1}\genfrac{[}{]}{0.0pt}{}{k_{1},\ldots,k_{j-1},k_{j+1},\ldots,k_{r}}{d_{1},\ldots,d_{j-1},d_{j}+d_{j+1},\ldots,d_{r}}$
$\displaystyle\qquad-\sum_{j=2}^{r}\mathbf{1}_{k_{j}=1}\genfrac{[}{]}{0.0pt}{}{k_{1},\ldots,k_{j-1},k_{j+1},\ldots,k_{r}}{d_{1},\ldots,d_{j-1}+d_{j},d_{j+1},\ldots,d_{r}}.$

Hence, $\omega$ is a sum of the derivation $\Theta^{a]}_{\diamond}$ in
Corollary 3.5 with $a=\genfrac{[}{]}{0.0pt}{}{1}{0}$ and the derivation
$\Theta^{\varphi,a}$ in Lemma 3.7 with $a=\genfrac{[}{]}{0.0pt}{}{1}{d}$ and
$\varphi\genfrac{[}{]}{0.0pt}{}{k^{\prime}}{d^{\prime}}=\genfrac{[}{]}{0.0pt}{}{k^{\prime}}{d+d^{\prime}}$
for all $d\geq 0$. ∎

Note that in the special case $d_{1}=\dots=d_{r}$ the derivation $\omega$ is
given by

$\omega\genfrac{[}{]}{0.0pt}{}{k_{1},\ldots,k_{r}}{0,\ldots,0}=\mathbf{1}_{k_{1}=1}\genfrac{[}{]}{0.0pt}{}{k_{2},\ldots,k_{r}}{0,\ldots,0}.$

In particular, $\omega$ is the zero map if the sequence $(k_{1},\ldots,k_{r})$
is admissible. In the next section, we will see that the derivation $\omega$
allows to define the concept of regularization to all elements in
$\mathcal{G}^{\\!f}$.

###  4.2. Derivations and polynomial representations

######  Lemma 4.2.

Let $A=\bigoplus_{n\geq 0}A_{n}$ a graded $\mathbb{Q}$-algebra and
$d\in\mathrm{End}(A)$ such that $d(A_{n})\subset A_{n-k}$ for some $k>0$ (with
$A_{-n}\coloneqq\\{0\\}$ for $n>0$). Assume $a\in A$ is such that $d(a)=1$.
Then, $d$ is a derivation if and only if for any $f\in A$ one has

$\displaystyle f=\sum_{i=1}^{n}f_{i}\,a^{i}\qquad\text{for
$f_{1},\ldots,f_{n}\in A$ satisfying $d(f_{i})=0$}.$ (4.6)

###### Proof.

Let $f\in A$ be given and assume that $d$ is a derivation. We prove the
existence of an expansion of the form (4.6) by induction on the _depth_
$p(f)=p=\min\\{i\geq 0\mid d^{i+1}(f)=0\\}$. Observe that since $d$ a
derivation of weight $-k$ such a $p$ always exists. Also, observe that if
$p=0$ we can write $f=f_{0}\mspace{1.0mu}$.

Now, suppose the result is proven for elements of depth at most $p-1$ and $f$
is of depth $p$. Then, since $d$ is a derivation,

$d^{p}\Bigl{(}f-\frac{d(f)}{p}\,a\Bigr{)}=d^{p}(f)-\frac{d^{p}(f)}{p}\,p=0.$

Hence, both $f-\frac{d(f)}{p}\,a$ and $d(f)$ are of depth at most $p-1$, and
the expansion of the form (4.6) follows inductively.

Next, let $f,g\in A$ be given. Assume that both can be written as a polynomial
in $a$ with coefficient in the kernel of $d$, explicitly,

$f=\sum_{i}f_{i}\,a^{i},\qquad g=\sum_{j}g_{j}\,a^{j}$

with $d(f_{i})=0$ and $d(g_{i})=0$ for all $i$. Then, it follows from the
following explict computation that $d$ is a derivation:

$\displaystyle d(f\,g)$
$\displaystyle=d\Bigl{(}\sum_{i,j}f_{i}\,g_{j}\,a^{i+j}\Bigr{)}$
$\displaystyle=\sum_{i,j}(i+j)\,f_{i}\,g_{j}\,a^{i+j-1}$
$\displaystyle=\sum_{i,j}i\,f_{i}\,g_{j}\,a^{i-1}\,a^{j}+\sum_{i,j}j\,f_{i}\,g_{j}\,a^{i}\,a^{j-1}$
$\displaystyle=d(f)\,g+f\,d(g).$

######  Example 4.3.

Recall $\mathfrak{H}=\mathbb{Q}\langle x,y\rangle$ and
$\mathfrak{H}^{1}=\mathbb{Q}\oplus\mathfrak{H}y$. Recall that
$\mathfrak{H}^{1}$ is an algebra with respect to both the shuffle and stuffle
product. Now, the map $\delta_{y}:\mathfrak{H}^{1}\to\mathfrak{H}^{1}$ given
by

$\delta_{y}(w)=\begin{cases}v&\text{if }w=yv\text{ for some word }v\\\
0&\text{else.}\end{cases}$

is a derivation with respect to both products. Note that $\delta_{y}(w)=0$
precisely if $w$ is admissible. Hence, by Lemma 4.2, it follows that every
$w\in\mathfrak{H}^{1}$ can be written as a polynomial in $y$ with admissible
coefficients.

Similarly, on $\mathcal{G}^{\\!f}$ we obtain:

######  Corollary 4.4.

Every element $f\in\mathcal{G}^{\\!f}$ can be written as

$f=\sum_{i}f_{i}\,G^{\\!f}\\!(1)^{i}$

for $f_{i}\in\mathcal{G}^{\\!f}$ with $\omega(f_{i})=0$.

######  Remark 4.5.

It is natural to say that an index
$\genfrac{[}{]}{0.0pt}{}{k_{1},\ldots,k_{r}}{d_{1},\ldots,d_{r}}$ is
_admissible_ precisely if $\omega
G^{\\!f}\\!\genfrac{[}{]}{0.0pt}{}{k_{1},\ldots,k_{r}}{d_{1},\ldots,d_{r}}=0$.
In the case $d_{1}=\ldots=d_{r}=0$ this notion of admissibility coincides with
the usual one, i.e. $k_{1}\geq 2$.

###  4.3. $\mathfrak{sl}_{2}$-algebras

An algebra $A$ is called an _$\mathfrak{sl}_{2}$ -algebra_ if it forms a
module over the Lie algebra $\mathfrak{sl}_{2}$ for which the elements of
$\mathfrak{sl}_{2}$ act by derivations. More explicitly, there exist three
derivations $D,W,\delta\in\operatorname{Der}(A)$ such that $(D,W,\delta)$
forms an _$\mathfrak{sl}_{2}$ -triple_. This means that the satisfy the
commutator relations

$\displaystyle[W,D]=2D,\quad[W,\delta]=-2\delta,\quad[\delta,D]=W\,.$ (4.7)

As the main example of an $\mathfrak{sl}_{2}$-algebra, we now discuss the
algebra of quasimodular forms. Let $\mathbb{H}$ be the _upper half plane_ ,
i.e., the set of complex numbers $\tau$ for which $\mathrm{Im}\,(\tau)>0$. The
group $\Gamma:=\mathrm{SL}_{2}(\mathbb{Z})$ acts on $\mathbb{H}$ by _Möbius
transformations_. Given $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$,
$\phi:\mathbb{H}\to\mathbb{C}$, $\tau\in\mathbb{H}$ and $k\in\mathbb{Z}$, we
let $\gamma\tau=\frac{a\tau+b}{c\tau+d}$ and define the _slash operator_ in
weight $k$ by

$(\phi|_{k}\gamma)(\tau):=(c\tau+d)^{-k}\phi(\gamma\tau).$

A modular form of weight $k$ is a holomorphic function
$\phi:\mathbb{H}\to\mathbb{C}$ such that $\phi|_{k}\gamma=\phi$ for all
$\gamma\in\Gamma$ and such that $\phi$ admits a holomorphic $q$-expansion
around infinity $(q=e^{2\pi\mathrm{i}\tau})$, i.e.,

$\phi=\sum_{n\geq 0}a_{n}\,q^{n}\qquad\text{for some }a_{n}\in\mathbb{C}.$

In order to obtain _modular forms_ Eisenstein considered the series

$\frac{1}{2}\sum_{\begin{subarray}{c}\omega\in\mathbb{Z}\tau+\mathbb{Z}\\\
\omega\neq 0\end{subarray}}\frac{1}{\omega^{k}}\qquad(k\geq 3).$

These _Eisenstein series_ are non-zero modular forms of weight $k$ whenever
$k$ is even—equal to $\mathbb{G}_{k}$ in (1.3). In case $k=2$, we write
$\omega=m\tau+n$ and replace the sum over $\omega$ by
$\sum_{m}\sum_{n}\mspace{1.0mu}$, where $(m,n)\neq(0,0)$. Then, the resulting
series equal $\mathbb{G}_{2}$ and

$(\mathbb{G}_{2}|_{2}\gamma)(\tau)=\mathbb{G}_{2}-\pi\mathrm{i}\frac{c}{c\tau+d}\,.$

Hence, $\mathbb{G}_{2}$ is not a modular form.

In fact, the transformation of $\mathbb{G}_{2}$ leads us to the definition of
a _quasi_ modular form, which are functions satisfying a more general modular
transformation property than the Eisenstein series and of which
$\mathbb{G}_{2}$ is the first non-modular example.

######  Definition 4.6.

A quasimodular form of weight $k$ and depth at most $p$ for $\Gamma$ is a
function $\phi:\mathbb{H}\to\mathbb{C}$ such that

  1. (i)

there exist $\phi_{0},\ldots,\phi_{p}$ so that for all $\tau\in\mathbb{H}$ and
all $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma$ one has

$\displaystyle(\phi|_{k}\gamma)(\tau)=\phi_{0}(\tau)+\phi_{1}(\tau)\frac{c}{c\tau+d}+\ldots+\phi_{p}(\tau)\Bigl{(}\frac{c}{c\tau+d}\Bigr{)}^{\\!p}.$
(4.8)

  2. (ii)

$\phi$ admits a holomorphic $q$-expansion around infinity
$(q=e^{2\pi\mathrm{i}\tau})$, i.e., $\phi=\sum_{n\geq 0}a_{n}q^{n}$ for some
$a_{n}\in\mathbb{C}.$ Similarly, $\phi_{0},\ldots,\phi_{p}$ are required to
admit such an expansion.

Note that if $\phi$ is a quasimodular form, the functions
$\phi_{0},\ldots,\phi_{p}$ are quasimodular forms uniquely determined by
$\phi$ (the function $\phi_{r}$ has weight $k-2r$ and depth $\leq p-r$) [Z].
Quasimodular forms of depth $0$ are modular forms. The Eisenstein series
$\mathbb{G}_{2}$ is an example of a quasimodular form of weight $2$ and depth
$1$.

The differential operator

$D:=\frac{1}{2\pi\mathrm{i}}\frac{\mathrm{d}}{\mathrm{d}\tau}=q\frac{\mathrm{d}}{\mathrm{d}q}$

preserves the space of quasimodular forms (cf., (1.13). Recall the $G_{k}$ and
$\mathbb{G}_{k}$ differ by a constant, as defined by (1.12).). Besides $D$, an
important differential operator on quasimodular forms is the operator $\delta$
defined by $\phi\mapsto 2\pi\mathrm{i}\,\phi_{1}$ (with $\phi_{1}$ defined by
the quasimodular transformation property (4.8)). Then, ${\delta
G_{2}=-\frac{1}{2}}$ and, in fact, this property together with the fact that
$\delta$ annihilates modular forms defines $\delta$ completely. Lastly, one
quasimodular forms one defines $W$ as the diagonal operator multiplying a form
$\phi$ by its weight. The algebra of quasimodular forms is an
$\mathfrak{sl}_{2}$-algebra with respect to these three operators.

A polynomial in the derivatives of two modular forms may be modular [Ran].
This is the case for the _Rankin–Cohen brackets_ of modular forms $f$ and $g$
of weight $k$ and $\ell$ respectively, defined by [C]

$\displaystyle[f,g]_{n}:=\sum_{\begin{subarray}{c}r,s\geq 0\\\
r+s=n\end{subarray}}(-1)^{r}\,\binom{k+n-1}{s}\,\binom{l+n-1}{r}\,D^{r}\\!f\,D^{s}\\!g\quad\quad(n\geq
0).$ (4.9)

In fact, one can show that for every $\mathfrak{sl}_{2}$-algebra $A$, the
Rankin–Cohen bracket of $f,g\in\ker\delta$ satisfies

$\delta[f,g]_{n}=0.$

Here, the Rankin–Cohen bracket $[f,g]_{n}$ is defined by the same formula
(4.9), where, now, $D$ is the derivation of weight $2$ on $A$.

######  Example 4.7.

We now give another example of two $\mathfrak{sl}_{2}$-algebras closely
related to the $\mathfrak{sl}_{2}$-action defined on $\mathcal{G}^{\\!f}$ in
the next section. For this, we extend our alphabet $\mathcal{A}$ to the set

$\displaystyle\widetilde{\mathcal{A}}$
$\displaystyle\coloneqq\left\\{\genfrac{[}{]}{0.0pt}{}{k}{d}\mid
k,d\in\mathbb{Z}\right\\}\,.$

On $\widetilde{\mathcal{A}}$, we define $\diamond$ as before, i.e., for
$k_{1},k_{2},d_{1},d_{2}\in\mathbb{Z}$ set

$\displaystyle\genfrac{[}{]}{0.0pt}{}{k_{1}}{d_{1}}\diamond\genfrac{[}{]}{0.0pt}{}{k_{2}}{d_{2}}=\genfrac{[}{]}{0.0pt}{}{k_{1}+k_{2}}{d_{1}+d_{2}}\,.$

We obtain an algebra $(\mathbb{Q}\mathcal{\widetilde{\mathcal{A}}},\diamond)$
which is, in contrast to $(\mathbb{Q}\mathcal{A},\diamond)$, unital with
neutral element $\genfrac{[}{]}{0.0pt}{}{0}{0}$. On
$\mathbb{Q}\mathcal{\widetilde{\mathcal{A}}}$ we define for $k,d\geq 0$ the
following three linear maps

$\displaystyle\widetilde{D}_{\diamond}:\genfrac{[}{]}{0.0pt}{}{k}{d}\mapsto
k\genfrac{[}{]}{0.0pt}{}{k+1}{d+1},\qquad\widetilde{W}_{\diamond}:\genfrac{[}{]}{0.0pt}{}{k}{d}\mapsto(k+d)\genfrac{[}{]}{0.0pt}{}{k}{d},\qquad\widetilde{\delta}_{\diamond}:\genfrac{[}{]}{0.0pt}{}{k}{d}\mapsto
d\genfrac{[}{]}{0.0pt}{}{k-1}{d-1}.$

By a direct calculation, one can check that these maps are derivations on
$(\mathbb{Q}\mathcal{\widetilde{\mathcal{A}}},\diamond)$ and satisfy the
commutator relations (4.7), i.e.
$(\mathbb{Q}\mathcal{\widetilde{\mathcal{A}}},\diamond)$ is an
$\mathfrak{sl}_{2}$-algebra. Denote by $\ast$ the quasi-shuffle product on
$\mathbb{Q}\mathcal{\widetilde{\mathcal{A}}}$ obtained by $\diamond$. By
Proposition 3.2, we obtain derivations
$\widetilde{D},\widetilde{W},\widetilde{\delta}$ on the quasi-shuffle algebra
$(\mathbb{Q}\langle\widetilde{\mathcal{A}}\rangle,\ast)$. These are explicitly
given by

$\displaystyle\widetilde{D}:\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$
$\displaystyle\mapsto\sum_{j=1}^{r}k_{j}\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j}+1,\dots,k_{r}}{d_{1},\dots,d_{j}+1,\dots,d_{r}},$
$\displaystyle\widetilde{\delta}:\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$
$\displaystyle\mapsto\sum_{j=1}^{r}d_{j}\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j}-1,\dots,k_{r}}{d_{1},\dots,d_{j}-1,\dots,d_{r}},$

and $\widetilde{W}$ is the multiplication by
$k_{1}+\dots+k_{r}+d_{1}+\dots+d_{r}\mspace{1.0mu}$. Again, these derivations
satisfy the commutator relations (4.7) and
$(\mathbb{Q}\langle\widetilde{\mathcal{A}}\rangle,\ast)$ becomes an
$\mathfrak{sl}_{2}$-algebra. In the next section, we will define similar
derivations on the subalgebra $(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$ and
show that these are $\sigma$-equivariant. This will give derivations on the
quotient $\mathcal{G}^{\\!f}$. In fact, the derivations $D$ and $W$ will just
be given by the restrictions of $\widetilde{D}$ and $\widetilde{W}$ to the
subspace $\mathbb{Q}\langle\mathcal{A}\rangle$. But notice that the derivation
$\widetilde{\delta}$ is not defined on
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$ and we will need to correct it in
a certain sense. Since
$(\mathbb{Q}\langle\widetilde{\mathcal{A}}\rangle,\ast_{\diamond})$ seems to
be a natural $\mathfrak{sl}_{2}$-algebra, one can ask if there is a natural
extension of $\sigma$ to $\mathbb{Q}\langle\widetilde{\mathcal{A}}\rangle$,
such that $\widetilde{D},\widetilde{W}$ and $\widetilde{\delta}$ become
$\sigma$-equivariant.

###  4.4. $\mathfrak{sl}_{2}$-action on $\mathcal{G}^{\\!f}$

In this section, we introduce three derivations $D,W$ and $\delta$ on
$\mathcal{G}^{\\!f}$ and show that these give $\mathcal{G}^{\\!f}$ the
structure of an $\mathfrak{sl}_{2}$-algebra. First, we will define the maps
$D,W$ and $\delta$ on $\mathcal{B}$. By Definition 2.3 they restrict to maps
$\mathbb{Q}\langle\mathcal{A}\rangle\to\mathbb{Q}\langle\mathcal{A}\rangle$
and we show they are $\sigma$-equivariant derivations.

######  Definition 4.8.

The $\mathbb{Q}$-linear maps $D$ and $W$ on $\mathcal{B}$ are defined by
$(f_{r})\mapsto(Df_{r})$ and $(f_{r})\mapsto(Wf_{r})$ respectively, where

$\displaystyle
Df_{r}(\underline{X},\underline{Y})\coloneqq\sum_{j=1}^{r}\frac{\partial}{\partial
X_{j}}\frac{\partial}{\partial Y_{j}}f_{r}(\underline{X},\underline{Y})$
(4.10)

and

$\displaystyle
Wf_{r}(\underline{X},\underline{Y})\coloneqq\sum_{j=1}^{r}\left(\frac{\partial}{\partial
X_{j}}X_{j}+Y_{j}\frac{\partial}{\partial
Y_{j}}\right)f_{r}(\underline{X},\underline{Y})$ (4.11)

The maps $D$ and $W$ restrict to $(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$
(see Definition 2.3).

######  Proposition 4.9.

The maps $D$ and $W$ are $\sigma$-equivariant derivations on
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$.

###### Proof.

First notice that on the level of coefficients the maps $D,W$ are given by

$\displaystyle
D\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}=\sum_{j=1}^{r}k_{j}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j}+1,\dots,k_{r}}{d_{1},\dots,d_{j}+1,\dots,d_{r}}$
(4.12)

and

$\displaystyle
W\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}=(k_{1}+\dots+k_{r}+d_{1}+\dots+d_{r})\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}.$
(4.13)

That $D$ and $W$ are derivations therefore follows from Proposition 3.2 by
noticing that $\genfrac{[}{]}{0.0pt}{}{k}{d}\mapsto
k\genfrac{[}{]}{0.0pt}{}{k+1}{d+1}$ and
$\genfrac{[}{]}{0.0pt}{}{k}{d}\mapsto(k+d)\genfrac{[}{]}{0.0pt}{}{k}{d}$,
similar as in Example 4.7, are both derivations on
$(\mathbb{Q}\mathcal{A},\diamond)$. Now, as

$\sum_{j=1}^{r}\frac{\partial}{\partial X_{j}}\frac{\partial}{\partial
Y_{j}}\qquad\text{and}\qquad\sum_{j=1}^{r}\frac{\partial}{\partial
X_{j}}X_{j}+Y_{j}\frac{\partial}{\partial Y_{j}}$

are invariant under the change of coordinates in $\sigma$, i.e., under

$\genfrac{(}{)}{0.0pt}{}{X_{1},\dots,X_{r}}{Y_{1},\dots,Y_{r}}\mapsto\genfrac{(}{)}{0.0pt}{}{Y_{1}+\dots+Y_{r},\dots,Y_{1}+Y_{2},Y_{1}}{X_{r},X_{r-1}-X_{r},\dots,X_{1}-X_{2}}\,,$

it follows that $D$ and $W$ commute with $\sigma$. ∎

At first sight, the following definition of $\delta$ might seem unnatural.
Note, however, that our numerical computations in Sage suggest that $\delta$
is the unique $\sigma$-equivariant derivation on
$\mathbb{Q}\langle\mathcal{A}\rangle$ of weight $-2$ (up to multiplication by
a constant). Moreover, as we prove in the next results, this derivation forms
an $\mathfrak{sl}_{2}$-triple together with the derivations $D$ and $W$, which
(as we explain in the next section) give rise to the usual
$\mathfrak{sl}_{2}$-action on the subspace of $\mathcal{G}^{\\!f}$ of formal
quasimodular forms.

######  Definition 4.10.

We define the linear map $\delta:\mathcal{B}\rightarrow\mathcal{B}$ by
$\delta(f_{r})=(\delta f_{r})$ with $\delta
f_{r}(\underline{X},\underline{Y})$ given by

$\displaystyle(X_{1}Y_{1}+\ldots+X_{r}Y_{r})f_{r}(\underline{X},\underline{Y})$
$\displaystyle\qquad+\sum_{j\geq
1}\biggl{(}-(X_{j}-X_{j+1}+Y_{j})\frac{\varphi_{j}^{+}}{2}-(X_{j-1}-X_{j}+Y_{j})\frac{\varphi_{j}^{-}}{2}+\Bigl{(}\frac{\varphi_{j}^{+}}{2}\Bigr{)}^{2}-\Bigl{(}\frac{\varphi_{j}^{-}}{2}\Bigr{)}^{2}\biggr{)}f_{r}(\underline{X},\underline{Y}).$

By Definition 2.3 we obtain a $\mathbb{Q}$-linear map
$\delta:\mathbb{Q}\langle\mathcal{A}\rangle\rightarrow\mathbb{Q}\langle\mathcal{A}\rangle$.

######  Proposition 4.11.

The map $\delta$ is a $\sigma$-equivariant derivation on
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$.

###### Proof.

The fact that $\delta$ commutes with $\sigma$ follows directly by observing
$X_{1}Y_{1}+\ldots+X_{r}Y_{r}$ is invariant under the change of coordinates
dictated by $\sigma$ and by making use of the relations (4.5).

We now show that $\delta$ is a derivation. For this notice that on the level
of coefficients, $\delta$ is explicitly given by

$\displaystyle\delta\coloneqq\delta^{1}-\frac{1}{2}\bigl{(}\delta^{2}+\delta^{3}+\delta^{4}+\delta^{5}\bigr{)}\mspace{1.0mu},$

where
$\delta^{j}:\mathbb{Q}\langle\mathcal{A}\rangle\rightarrow\mathbb{Q}\langle\mathcal{A}\rangle$
($1\leq j\leq 5$) are the linear maps defined on the generators by

$\displaystyle\delta^{1}\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$
$\displaystyle\coloneqq\sum_{j=1}^{r}\mathbf{1}_{k_{j}>1}\,d_{j}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j}-1,\dots,k_{r}}{d_{1},\dots,d_{j}-1,\dots,d_{r}}$
$\displaystyle\qquad-\frac{1}{2}\sum_{j=1}^{r-1}\mathbf{1}_{\begin{subarray}{c}k_{j+1}=1\end{subarray}}\,d_{j+1}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j},k_{j+2},\dots,k_{r}}{d_{1},\dots,d_{j}+d_{j+1}-1,d_{j+2},\dots,d_{r}}$
$\displaystyle\qquad-\frac{1}{2}\sum_{j=1}^{r-1}\mathbf{1}_{k_{j}=1}\,d_{j}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j-1},k_{j+1},\dots,k_{r}}{d_{1},\dots,d_{j-1},d_{j}+d_{j+1}-1,\dots,d_{r}},$
$\displaystyle\delta^{2}\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$
$\displaystyle\coloneqq\mathbf{1}_{\begin{subarray}{c}k_{r}=2\\\
d_{r}=0\end{subarray}}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r-1}}{d_{1},\dots,d_{r-1}}\,-\,\frac{1}{2}\,\mathbf{1}_{\begin{subarray}{c}k_{r-1}=k_{r}=1\\\
d_{r-1}=d_{r}=0\end{subarray}}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r-2}}{d_{1},\dots,d_{r-2}},$
$\displaystyle\delta^{3}\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$
$\displaystyle\coloneqq\mathbf{1}_{\begin{subarray}{c}k_{r}=1\\\
d_{r}=1\end{subarray}}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r-1}}{d_{1},\dots,d_{r-1}},$
$\displaystyle\delta^{4}\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$
$\displaystyle\coloneqq\sum_{j=2}^{r}\mathbf{1}_{\begin{subarray}{c}\\!\\!\\!k_{j}=1\\\
\\!\\!\\!k_{j-1}>1\end{subarray}}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j-1}-1,k_{j+1},\dots,k_{r}}{d_{1},\dots,d_{j-1}+d_{j},d_{j+1},\dots,d_{r}}\,$
$\displaystyle\qquad-\sum_{j=1}^{r-1}\mathbf{1}_{\begin{subarray}{c}\\!\\!\\!k_{j}=1\\\
\\!\\!\\!k_{j+1}>1\end{subarray}}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j-1},k_{j+1}-1,\dots,k_{r}}{d_{1},\dots,d_{j-1},d_{j}+d_{j+1},\dots,d_{r}},$
$\displaystyle\delta^{5}\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{r}}{d_{1},\dots,d_{r}}$
$\displaystyle\coloneqq\sum_{j=1}^{r-1}\mathbf{1}_{k_{j}=2}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j-1},k_{j+1},\dots,k_{r}}{d_{1},\dots,d_{j-1},d_{j}+d_{j+1},\dots,d_{r}}\,$
$\displaystyle\qquad-\sum_{j=1}^{r-1}\mathbf{1}_{k_{j+1}=2}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j},k_{j+2},\dots,k_{r}}{d_{1},\dots,d_{j}+d_{j+1},d_{j+2},\dots,d_{r}}$
$\displaystyle\qquad+\frac{1}{2}\sum_{j=1}^{r-2}\mathbf{1}_{k_{j+1}=k_{j+2}=1}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j},k_{j+3},\dots,k_{r}}{d_{1},\dots,d_{j}+d_{j+1}+d_{j+2},d_{j+3},\dots,d_{r}}$
$\displaystyle\qquad-\frac{1}{2}\sum_{j=1}^{r-2}\mathbf{1}_{k_{j}=k_{j+1}=1}\,\genfrac{[}{]}{0.0pt}{}{k_{1},\dots,k_{j-1},k_{j+2},\dots,k_{r}}{d_{1},\dots,d_{j-1},d_{j}+d_{j+1}+d_{j+2},\dots,d_{r}}.$

Now, we show each $\delta^{i}$ is a derivation:

  1. (i)

For $\delta^{1}$ use the derivation $\Theta^{\varphi}$ of Lemma 3.3 with
$\varphi:\genfrac{[}{]}{0.0pt}{}{k}{d}\mapsto\mathbf{1}_{k>1}d\genfrac{[}{]}{0.0pt}{}{k-1}{d-1}$.
Note

$\gamma\biggl{(}\genfrac{[}{]}{0.0pt}{}{k_{1}}{d_{1}},\genfrac{[}{]}{0.0pt}{}{k_{2}}{d_{2}}\biggr{)}=\bigl{(}\mathbf{1}_{k_{1}=1}d_{1}+\mathbf{1}_{k_{2}=1}d_{2}\bigr{)}\genfrac{[}{]}{0.0pt}{}{k_{1}+k_{2}-1}{d_{1}+d_{2}-1}.$

  2. (ii)

For $\delta^{2}$ use the derivation $\Theta^{a]}$ of Corollary 3.5 with
$a=\genfrac{[}{]}{0.0pt}{}{2}{0}$.

  3. (iii)

For $\delta^{3}$ also use the derivation $\Theta^{a]}$ of Corollary 3.5, now
with $a=\genfrac{[}{]}{0.0pt}{}{1}{1}$.

  4. (iv)

For $\delta^{4}$ consider the derivation $\Theta_{S}$ of Lemma 3.7, where
$S=\\{\genfrac{[}{]}{0.0pt}{}{1}{d^{\prime}}\mid d^{\prime}\geq 0\\}$ and for
$a=\genfrac{[}{]}{0.0pt}{}{1}{d^{\prime}}$, we let
$\varphi_{a}:\genfrac{[}{]}{0.0pt}{}{k}{d}\mapsto\mathbf{1}_{k>1}\genfrac{[}{]}{0.0pt}{}{k-1}{d+d^{\prime}}$.

  5. (v)

Using Proposition 3.6, we find

$\delta^{5}=\sum_{\begin{subarray}{c}d\geq 0\\\
a=\genfrac{[}{]}{0.0pt}{}{2}{d}\end{subarray}}\exp_{\diamond}\circ\,\Theta^{\varphi_{a},a}\circ\,\log_{\diamond},$

where
$\varphi_{a}:\genfrac{[}{]}{0.0pt}{}{k^{\prime}}{d^{\prime}}\mapsto\genfrac{[}{]}{0.0pt}{}{k^{\prime}}{d+d^{\prime}}$
for $a=\genfrac{[}{]}{0.0pt}{}{2}{d}$. Namely, the letter
$\genfrac{[}{]}{0.0pt}{}{2}{d}$ appears as a $\diamond$-product of two
letters:
$\genfrac{[}{]}{0.0pt}{}{2}{d}=\genfrac{[}{]}{0.0pt}{}{1}{d_{1}}\diamond\genfrac{[}{]}{0.0pt}{}{1}{d_{2}}$
for any $d_{1}+d_{2}=d$. Therefore, $\Theta^{\varphi_{a},a}$ does not commute
with $\log_{\diamond}\mspace{1.0mu}$, since in the definition of
$\log_{\diamond}$ in (3.2) we could have for some $1\leq j\leq l$ that
$i_{j}=2$ with $a=a_{i_{1}+\dots+i_{j}+1}\diamond
a_{i_{1}+\dots+i_{j}+2}\mspace{1.0mu}$. This gives rise to the additional
terms with factor $\frac{1}{2}$ in $\delta^{5}$. ∎

######  Theorem 4.12.

The space $\mathcal{G}^{\\!f}$ is an $\mathfrak{sl}_{2}$-algebra.

###### Proof.

By the previous propositions, it remains to show that the derivations
$D,W,\delta\in\operatorname{Der}(\mathcal{G}^{\\!f})$ form an
_$\mathfrak{sl}_{2}$ -triple_. Using the explicit formulas for $D,W$ in (4.12)
and for $\delta^{i}$ in the proof of Proposition 4.11, one directly find
$[D,W]=-2D$. Also, it is not hard to verify that

$[\delta^{i},W]=2\delta^{i}\qquad(i=1,2,3,4,5)$

and

$[\delta^{1},D]=W\qquad\text{and}\qquad[\delta^{i},D]=0\qquad(i=2,3,4,5).\qed$

######  Remark 4.13.

The derivations $D,W,\omega$ and $\delta$ are of weight $2,0,-1,-2$
respectively. Numerical experiments in Sage suggest that these are the only
$\sigma$-equivariant derivations on
$(\mathbb{Q}\langle\mathcal{A}\rangle,\ast)$ of weight between $-2$ and $2$
(up to multiplication by a constant). In particular, there seems not to exist
a non-trivial $\sigma$-equivariant derivation of weight $1$.

######  Remark 4.14.

The map $[\omega,\delta]=\omega\delta-\delta\omega$ is a non-trivial
$\sigma$-equivariant derivation of weight $-3$. Moreover, the map
$\omega_{3}:\mathcal{B}\to\mathcal{B}$ defined by
$\omega_{3}(f_{r})=(\omega_{3}f_{r})$ conjecturally yields a
$\sigma$-equivariant derivation of weight $-3$, where $\omega_{3}$ acts on
$f_{r}$ by

$\displaystyle\omega_{3}\coloneqq\,$
$\displaystyle\sum_{j=1}^{r}(X_{j}-X_{j+1}+Y_{j})^{2}\,\varphi_{j}^{+}-\sum_{j=2}^{r}(X_{j-1}-X_{j}-Y_{j})^{2}\,\varphi_{j}^{-}$
$\displaystyle+\sum_{j=1}^{r-1}(-X_{j}+X_{j+2}-Y_{j}-Y_{j+1})\,(\varphi_{j}^{+})^{2}$
$\displaystyle-\sum_{j=2}^{r-1}(X_{j-1}-4X_{j}+3X_{j+1}-3Y_{j}+Y_{j+1})(\varphi_{j}^{-})^{2}$
$\displaystyle+\frac{1}{3}\sum_{j=1}^{r-2}(\varphi_{j}^{+})^{3}-\frac{1}{3}\sum_{j=2}^{r-2}(\varphi_{j}^{-})^{3}.$

In particular, $[\omega,\delta]$ and $\omega_{3}$ seem to form a basis for the
space of derivations of weight $-3$. Explicitly, on elements in
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$ the
derivation $\omega_{3}$ is given by

$\displaystyle\omega_{3}\,$ $\displaystyle G^{\\!f}\\!(k_{1},\ldots,k_{r})=$
$\displaystyle\sum_{j=1}^{r}(-1)^{k_{j}+1}\,\binom{2}{k_{j}-1}\,G^{\\!f}\\!(k_{1},\ldots,k_{j-1},k_{j}+k_{j+1}-3,\ldots,k_{r})$
$\displaystyle-\sum_{j=2}^{r}(-1)^{k_{j}+1}\,\binom{2}{k_{j}-1}\,G^{\\!f}\\!(k_{1},\ldots,k_{j-1}+k_{j}-3,k_{j+1},\ldots,k_{r})$
$\displaystyle+\sum_{j=1}^{r-1}\mathbf{1}_{(k_{j},k_{j+1})\in\\{(1,1),(2,1)\\}}\,(-1)^{k_{j}+1}\,G^{\\!f}\\!(k_{1},\ldots,k_{j-1},k_{j}+k_{j+1}+k_{j+2}-3,\ldots,k_{r})$
$\displaystyle-\sum_{j=2}^{r-1}(\mathbf{1}_{(k_{j},k_{j+1})=(1,1)}-4\cdot\mathbf{1}_{(k_{j},k_{j+1})=(2,1)}+3\cdot\mathbf{1}_{(k_{j},k_{j+1})=(1,2)})$
$\displaystyle\hskip
180.0ptG^{\\!f}\\!(k_{1},\ldots,k_{j-1}+k_{j}+k_{j+1}-3,k_{j+2},\ldots,k_{r})$
$\displaystyle+\frac{1}{3}\mathbf{1}_{(k_{1},k_{2},k_{3})=(1,1,1)}\,G^{\\!f}\\!(k_{4},\ldots,k_{r}),$

where $G^{\\!f}\\!$ of a sequence containing non-positive entries is
considered to be $0$. It remains an interesting open question to find all
$\sigma$-equivariant derivations on $\mathbb{Q}\langle\mathcal{A}\rangle$.

###  4.5. Restriction to
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$ and
the space $\mathcal{E}^{\\!f}$

We provide explicit formulas for the derivations $D$ and $\delta$ on
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$ (the
subspace which conjecturally equals $\mathcal{G}^{\\!f}$). In particular,
these formulas imply the following.

######  Proposition 4.15.

The subspace
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$ is an
$\mathfrak{sl}_{2}$-subalgebra of $\mathcal{G}^{\\!f}$.

###### Proof.

From the explicit formulas in the proof of Proposition 4.11, we directly see
that $\delta$ keeps the space
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$
invariant:

$\displaystyle\begin{split}\delta\,G^{\\!f}\\!(k_{1},\dots,k_{r})&=-\frac{1}{2}\mathbf{1}_{k_{1}=2}\,G^{\\!f}\\!(k_{2},\dots,k_{r})+\frac{1}{4}\mathbf{1}_{k_{1}=k_{2}=1}G^{\\!f}\\!(k_{3},\dots,k_{r})\\\
&\qquad+\frac{1}{2}\sum_{j=1}^{r-1}\mathbf{1}_{\begin{subarray}{c}\\!\\!\\!k_{j}=1\\\
\\!\\!\\!k_{j+1}>1\end{subarray}}\,G^{\\!f}\\!(k_{1},\dots,k_{j-1},k_{j+1}-1,\dots,k_{r})\\\
&\qquad-\frac{1}{2}\sum_{j=2}^{r}\mathbf{1}_{\begin{subarray}{c}\\!\\!\\!k_{j}=1\\\
\\!\\!\\!k_{j-1}>1\end{subarray}}\,G^{\\!f}\\!(k_{1},\dots,k_{j-1}-1,k_{j+1},\dots,k_{r}).\end{split}$
(4.14)

Moreover, following the same proof as in [BB, Proposition 6.30 and Corollary
6.31] we have

$\displaystyle DG^{\\!f}\\!(k_{1},\dots,k_{r})=G^{\\!f}\\!\bigl{(}z_{2}\ast
z_{k_{1}}\cdots z_{k_{r}}-z_{2}\shuffle z_{k_{1}}\cdots z_{k_{r}}\bigr{)},$
(4.15)

where the right-hand side is given by the map $\eqref{eq:Gffromh1}$ and
$\shuffle$ denotes the shuffle product on $\mathfrak{H}^{1}$ defined in the
introduction. Hence, the action of the $\mathfrak{sl}_{2}$-triple restricts to
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$. ∎

######  Remark 4.16.

_Define_ $\delta$ and $D$ on $\mathfrak{H}^{1}$ by the formulas (4.14) and
(4.15). It might be interesting to consider the algebra
$(\mathfrak{H}^{1},\ast)$ modulo the relations which ensure that it becomes an
$\mathfrak{sl}_{2}$-algebra. For example, since $D:w\mapsto z_{2}\ast
w-z_{2}\shuffle w$ is not a derivation on $(\mathfrak{H}^{1},\ast)$, we would
impose the relation $D(w\ast v)-D(w)\ast v-w\ast D(v)=0$ for all
$w,v\in\mathfrak{H}^{1}$. In the special case $v=w=z_{1}\mspace{1.0mu}$, one
finds $z_{4}-2z_{2}z_{2}+2z_{3}z_{1}=0$. This corresponds to the first
relation among the elements in
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$,
i.e., $G^{\\!f}\\!(4)=2G^{\\!f}\\!(2,2)-2G^{\\!f}\\!(3,1)$. The relations
obtained this way might be a first step in answering the question raised in
Remark 2.7, namely, to find an ideal
$\mathfrak{R}\subset(\mathfrak{H}^{1},\ast)$, such that
$\mathcal{G}^{\\!f}\cong\faktor{(\mathfrak{H}^{1},\ast)}{\mathfrak{R}}$.

As a formal analogue of the space $\mathcal{E}$ defined in (1.7), we define
the following subspace of
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$

$\displaystyle\mathcal{E}^{\\!f}\coloneqq\langle
G^{\\!f}\\!(k_{1},\dots,k_{r})\mid r\geq 0,k_{1},\dots,k_{r}\geq
2\rangle_{\mathbb{Q}}\,.$ (4.16)

By the definition of the stuffle product, we see that $\mathcal{E}^{\\!f}$ is
a subalgebra. Clearly, the algebra $\mathcal{E}^{\\!f}$ is closed under $W$
and by (4.14) it is also closed under $\delta$. But from (4.15) we do not get
immediately that $\mathcal{E}^{\\!f}$ is closed under $D$, since, for example,
we have

$z_{2}\ast z_{3}-z_{2}\shuffle
z_{3}=(z_{2}z_{3}+z_{3}z_{2}+z_{5})-(z_{2}z_{3}+3z_{3}z_{2}+6z_{4}z_{1})=z_{5}-2z_{3}z_{2}-6z_{4}z_{1}$

and therefore

$\displaystyle
DG^{\\!f}\\!(3)=G^{\\!f}\\!(5)-2G^{\\!f}\\!(3,2)-6G^{\\!f}\\!(4,1).$ (4.17)

Here $G^{\\!f}\\!(4,1)$ is, a priori, not an element of $\mathcal{E}^{\\!f}$,
but we can give the following alternative expression for $DG^{\\!f}\\!(k)$,
which shows that $DG^{\\!f}\\!(k)\in\mathcal{E}^{\\!f}$ for all $k\geq 2$.

######  Proposition 4.17.

For $k\geq 2$ we have

$\displaystyle
DG^{\\!f}\\!(k)=(2k-1)G^{\\!f}\\!(k+2)-G^{\\!f}\\!(2,k)-\sum_{j=2}^{k}(k+j-1)\,G^{\\!f}\\!(k+2-j,j).$

###### Proof.

Using Proposition 2.9 for $d_{1}=d_{2}=0$ gives for $k_{1},k_{2}\geq 1$ with
$k_{1}+k_{2}\geq 3$

$\displaystyle\frac{(k_{1}+k_{2}-3)!}{(k_{1}-1)!(k_{2}-1)!}DG^{\\!f}\\!(k_{1}+k_{2}-2)$
$\displaystyle=G^{\\!f}\\!(k_{1},k_{2})+G^{\\!f}\\!(k_{2},k_{1})+G^{\\!f}\\!(k_{1}+k_{2})$
(4.18)
$\displaystyle\qquad-\sum_{j=1}^{k_{1}+k_{2}-1}\\!\left(\\!\binom{j-1}{k_{1}-1}+\binom{j-1}{k_{2}-1}\\!\right)G^{\\!f}\\!(j,k_{1}+k_{2}-j).$
(4.19)

The formula in the proposition follows by taking $2k$-times the case
$k_{1}=1,k_{2}=k+1$ and subtracting the case $k_{1}=2,k_{2}=k$. ∎

######  Conjecture 4.18.

  1. (i)

The algebra $\mathcal{E}^{\\!f}$ is an $\mathfrak{sl}_{2}$-subalgebra of
$\mathcal{G}^{\\!f}$.

  2. (ii)

The $\mathbb{Q}$-linear map defined by

$\displaystyle\mathcal{E}^{\\!f}$ $\displaystyle\longrightarrow\mathcal{E}$
$\displaystyle G^{\\!f}\\!(k_{1},\dots,k_{r})$
$\displaystyle\longmapsto\mathbb{G}_{k_{1},\dots,k_{r}}$

is an algebra isomorphism. In particular, $\mathcal{E}$ is also an
$\mathfrak{sl}_{2}$-algebra.

Note that by (4.14), for (i) it suffices to show that $\mathcal{E}^{\\!f}$ is
closed under the derivation $D$.

######  Remark 4.19.

Notice that the $\mathfrak{sl}_{2}$-structure on $\mathcal{G}^{\\!f}$ does not
restrict to the quotient, i.e., the space of formal multiple zeta values
$\mathcal{Z}^{f}=\faktor{\mathcal{G}^{\\!f}\,}{\mathfrak{N}}$ is not an
$\mathfrak{sl}_{2}$-algebra for the restricted operators. The derivation $D$
is well-defined on the quotient and becomes the zero map on this quotient
(c.f., Proposition 2.16), but the derivation $\delta$ is not well-defined. For
example, $\delta
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{2}{1}=G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{1}{0}$
and $\pi G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{2}{1}=0$, but $\zeta^{f}\\!(1)=\pi
G^{\\!f}\\!\genfrac{(}{)}{0.0pt}{}{1}{0}\neq 1$.

##  5\. Formal quasimodular forms

In this section, we will be interested in a subalgebra of $\mathcal{G}^{\\!f}$
isomorphic to the $\mathfrak{sl}_{2}$-algebra of quasimodular forms:

######  Definition 5.1.

We define the algebra of _formal quasimodular forms
$\widetilde{\mathcal{M}}^{\\!f}$_ as the smallest
$\mathfrak{sl}_{2}$-subalgebra of $\mathcal{G}^{\\!f}$ which contains
$G^{\\!f}\\!(2)$.

######  Remark 5.2.

By Corollary 5.5, $\widetilde{\mathcal{M}}^{\\!f}$ is a _non-trivial_ proper
finitely generated $\mathfrak{sl}_{2}$-subalgebra of $\mathcal{G}^{\\!f}$. Are
there more non-trivial subalgebras? The first subalgebra one could consider is
$\mathbb{Q}[G^{\\!f}\\!(k)\mid k\geq 1]$. But one can check that this algebra
is not closed under the operator $D$; for example, it does not contain
$DG^{\\!f}\\!(1)$. Secondly, note that by Proposition 4.15 the algebra
$\operatorname{Fil}^{\mathrm{\operatorname{lwt}}}_{0}\mathcal{G}^{\\!f}$ is
also an $\mathfrak{sl}_{2}$-subalgebra. Conjecturally, this subalgebra is not
a proper subalgebra (see Conjecture 2.6). In addition to the algebra
$\mathcal{E}^{\\!f}$ defined in (4.16), the authors were unable to find other
possible candidates for the $\mathfrak{sl}_{2}$-subalgebras. In particular, it
would be interesting to know if $\widetilde{\mathcal{M}}^{\\!f}$ might be the
only non-trivial finitely generated $\mathfrak{sl}_{2}$-subalgebra of
$\mathcal{G}^{\\!f}$.

As a consequence of the relations we proved in Section 2.1, we obtain the
following.

######  Corollary 5.3.

The Ramanujan differential equations are satisfied in $\mathcal{G}^{\\!f}$,
i.e.,

$\displaystyle DG^{\\!f}\\!(2)$
$\displaystyle=5G^{\\!f}\\!(4)-2G^{\\!f}\\!(2)^{2}\,,$ $\displaystyle
DG^{\\!f}\\!(4)$
$\displaystyle=14G^{\\!f}\\!(6)-8G^{\\!f}\\!(2)\,G^{\\!f}\\!(4)\,,$
$\displaystyle DG^{\\!f}\\!(6)$
$\displaystyle=\frac{120}{7}G^{\\!f}\\!(4)^{2}-12G^{\\!f}\\!(2)\,G^{\\!f}\\!(6)\,.$

###### Proof.

Recall Corollary 2.11(i), which is equivalent to

$\displaystyle
DG^{\\!f}\\!(k)=\frac{k(k+3)}{2}G^{\\!f}\\!(k+2)-k\sum_{\begin{subarray}{c}k_{1}+k_{2}=k+2\\\
k_{1},k_{2}\text{ even}\end{subarray}}G^{\\!f}\\!(k_{1})G^{\\!f}\\!(k_{2})\,.$

The formulas for $DG^{\\!f}\\!(2)$ and $DG^{\\!f}\\!(4)$ follow directly from
this and the formula for $DG^{\\!f}\\!(6)$ follows after using
$G^{\\!f}\\!(8)=\frac{6}{7}G^{\\!f}\\!(4)^{2}$, which is a consequence of
Corollary 2.11(ii). ∎

######  Theorem 5.4.

We have $\widetilde{\mathcal{M}}^{\\!f}\cong\widetilde{\mathcal{M}}$ as
$\mathfrak{sl}_{2}$-algebras.

###### Proof.

By the previous corollary, clearly
$G^{\\!f}\\!(2),G^{\\!f}\\!(4),G^{\\!f}\\!(6)$ are elements of
$\widetilde{\mathcal{M}}^{\\!f}$. Moreover, $G^{\\!f}\\!(2),G^{\\!f}\\!(4)$
and $G^{\\!f}\\!(6)$ are algebraically independent over $\mathbb{Q}$ as
follows from Theorem 2.19 and the well-known fact that this is true for
$G(2),G(4),G(6)\in\widetilde{\mathcal{M}}\subset\mathbb{Q}\llbracket
q\rrbracket$. Hence, the algebra
$\mathbb{Q}[G^{\\!f}\\!(2),G^{\\!f}\\!(4),G^{\\!f}\\!(6)]$ is free and
contained in $\widetilde{\mathcal{M}}^{\\!f}$. Since this is also an
$\mathfrak{sl}_{2}$-subalgebra of $\mathcal{G}^{\\!f}$, on which the action by
the $\mathfrak{sl}_{2}$-triple $D,W,\delta$ is the same, the result follows. ∎

The following results on formal (quasi)modular forms follow directly from the
previous result, as these statements are known to be true for (quasi)modular
forms. Nevertheless, we want to formulate these results in the language of
formal objects and give a self-contained proof. For example, the following
Corollary follows from Corollary 2.11.

######  Corollary 5.5.

The algebra of formal quasimodular forms $\widetilde{\mathcal{M}}^{\\!f}$
satisfies:

  1. (i)

$G^{\\!f}\\!(k)\in\widetilde{\mathcal{M}}^{\\!f}$ for all even $k\geq 2$.

  2. (ii)

$\widetilde{\mathcal{M}}^{\\!f}=\mathbb{Q}[G^{\\!f}\\!(2),G^{\\!f}\\!(4),G^{\\!f}\\!(6)]=\mathbb{Q}[G^{\\!f}\\!(2),DG^{\\!f}\\!(2),D^{2}G^{\\!f}\\!(2)]$.

  3. (iii)

The Chazy equation is satisfied, i.e.,

$\displaystyle
D^{3}G^{\\!f}\\!(2)+24\,G^{\\!f}\\!(2)\,D^{2}G^{\\!f}\\!(2)-36(DG^{\\!f}\\!(2))^{2}=0\,.$
(5.1)

######  Definition 5.6.

We define the algebra of _formal modular forms_ by
$\mathcal{M}^{\\!f}=\ker\delta_{\mid\widetilde{\mathcal{M}}^{\\!f}}$. Write