††Corresponding author email [A2]<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Poisson dialgebras

,  Apurba Das  Department of Mathematics, Indian Institute of Technology
Kharagpur, Kharagpur-721302, West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Satyendra Kumar Mishra  and  Goutam
Mukherjee  TCG Centres for Research and Education in Science and Technology,
Institute for Advancing Intelligence, Salt Lake, Kolkata-700091, West Bengal,
India.  Academy of Scientific and Innovative Research (AcSIR), Ghaziabad-
201002, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The notion of Poisson dialgebras was introduced by Loday. In this article, we
propose a new definition with some modifications that is supported by several
canonical examples coming from Poisson algebra modules, averaging operators on
Poisson algebras, and differential Poisson algebras. We show that a Poisson
object in the category of linear maps has an associated Poisson dialgebra
structure. Conversely, starting from a Poisson dialgebra we describe a Poisson
object in the category of linear maps. These constructions yield a pair of
adjoint functors between the category of Poisson objects in the category of
linear maps and the category of Poisson dialgebras. There is a Lie $2$-algebra
associated with any Leibniz algebra. Here, we first obtain an associative
$2$-algebra starting from a dialgebra. Then, for a Poisson dialgebra, we
construct a graded space that inherits both a Lie $2$-algebra and an
associative $2$-algebra structure. In a particular case of Poisson dialgebras,
which we call ‘reduced Poisson dialgebra’, we obtain an associated $2$-term
homotopy Poisson algebra (of degree $0$).

###### Key words and phrases:

Poisson algebra, dialgebras, Leibniz algebras, Homotopy algebras.

2020 Mathematics Subject Classification: 17B63, 17A32, 17A30, 18N40.

##  1\. Introduction

The notion of Leibniz algebras was first introduced by Bloh [3]. A right
Leibniz algebra is a vector space $L$ equipped with a bracket $[-,-]$ such
that the map $[-,x]$ is a derivation of the bracket $[-,-]$ for each $x\in L$.
Leibniz algebras are non-commutative analog of Lie algebras, which appear in
different areas such as Poisson geometry, representation theory of Lie
algebras, and Nambu-Mechanics.

A dialgebra is a vector space equipped with two associative algebra products
satisfying three compatibility relations. Dialgebras were also introduced by
Loday [10]. Dialgebras play a dominant role in the theory of Leibniz algebras
generalizing the role of associative algebra in the theory of Lie algebras.
The notion of dialgebras is a non-commutative analog of associative algebras,
in particular, we have the following diagram.

Lie algebrasAssociative algebrasLeibniz algebrasDialgebrasnon-commutative
analognon-commutative analogenveloping algebra functor$\scriptstyle[x,y]=xy-
yx$eneveloping algebra functor$\scriptstyle[x,y]=x\dashv y-y\vdash x$

Loday and Pirashvili considered a tensor category of linear maps
$\mathcal{LM}$ in order to prove a Milnor-Moore type theorem for Leibniz
algebras in [11]. They showed that any Leibniz algebra is a Lie object in the
category of linear maps. The universal enveloping algebra functor from the
category of Leibniz algebras to the category of associative algebras factors
through the category of cocommutative Hopf algebras in $\mathcal{LM}$. Here,
we first show that there is an adjunction between the category of associative
algebra objects in the category $\mathcal{LM}$ and the category of dialgebras.
Thus, the notion of dialgebras is a non-commutative analogue of associative
algebras in the sense of [11].

Poisson algebras are central objects in Poisson geometry. Throughout the
article, by Poisson algebra we mean a vector space equipped with a Lie bracket
and an associative algebra product such that the Lie bracket is a derivation
of the algebra product, we refer to [1, 6, 7] for more details. The main goal
of this article is to study a non-commutative analog of Poisson algebras (in
the sense of [11]) known as Poisson dialgebra first introduced by Loday in
[10]. We redefine Poisson dialgebras consisting of a Leibniz algebra and a
dialgebra structure satisfying some compatibility relations (see Definition
4.1). We add two conditions on mixed skew-symmetry to the definition of Loday
[10]. In a sequel, we justify the modifications in the definition with several
canonical examples coming from dialgebras, Poisson bimodules, averaging
operators, and filtered dialgebras. Later on, we establish an adjunction
between the category of poisson dialgebras and Poisson algebra objects in
$\mathcal{LM}$.

The notion of Lie $2$-algebras was introduced by Baez and Crans as the
categorification of Lie algebras [2]. They showed that Lie $2$-algebras are
equivalent to $2$-term homotopy Lie algebras ($2$-term $L_{\infty}$-algebras
[8]). It has been observed in [15] that Leibniz algebras are closely related
to Lie 2-algebras. Homotopy algebras such as homotopy associative algebras,
homotopy Lie algebras, and homotopy Poisson algebras have been extensively
studied, we refer to [2, 8, 9, 12, 5, 4, 17, 16] to mention some of these
studies. A $2$-term homotopy Poisson algebra is a Lie $2$-algebra endowed with
an associative multiplication satisfying some compatibilities, which in turn
gives a homotopy Poisson algebra of finite type [9, 12]. We first show that a
dialgebra naturally gives rise to an associative $2$-algebra structure. This
can be thought of as the dialgebra analogue of the construction given in [15].
Next, we will further generalize the construction for Poisson dialgebras.
Given a Poisson dialgebra, we construct a graded space that inherits both a
Lie $2$-algebra and an associative $2$-algebra structure. Additionally, if the
given Poisson dialgebra is reduced (i.e., a Poisson dialgebra with $x\dashv
y=x\vdash y,$ for all $x,y\in P$), then the induced associative $2$-algebra
structure becomes associative and satisfies all the compatibilities of a
$2$-term homotopy Poisson algebra.

The organization of the paper is as follows: In Section 2, we recall the
definitions of Poisson algebras, dialgebras, Leibniz algebras, tensor category
of linear maps and related results. In Section 3, we construct an associative
$2$-algebra from a dialgebra. We propose a new definition of Poisson
dialgebras in Section 4 followed by different natural examples. We show that
dialgebras, Poisson algebra bimodules, differential Poisson algebras, and
averaging operators on Poisson algebras give a Poisson dialgebra structure. We
also define a graded Poisson dialgebra associated to a filtered dialgebra. We
describe two adjoint functors between the categories of Poisson dialgebras and
Poisson algebra objects in the category $\mathcal{LM}$. The last section is
devoted to homotopy structures related to Poisson dialgebras. For a Poisson
dialgebra $(P,\dashv,\vdash,[-,-])$, we define a graded space $P\oplus J$ that
simultaneously carries a Lie $2$-algebra and an associative $2$-algebra
structure. In the particular case of a reduced Poisson dialgebra, we obtain a
$2$-term homotopy Poisson algebra (degree $0$) on the graded space $P\oplus
J$.

##  2\. Preliminaries

In this section, we recall some definitions and fix notations. Let
$\mathbb{K}$ be a field of characteristic zero. All the linear maps and tensor
products are over the field $\mathbb{K}$ unless otherwise stated. Let
$(A,\mu)$ is an associative algebra and $M$ be an $A$-bimodule. Then, we
denote $ab:=\mu(a,b)$ for all $a,b\in A$. We denote the left and right module
actions of $A$ on $M$ by $a\cdot m$ and $m\cdot a$, respectively for all $a\in
A$ and $m\in M$.

###  2.1. Poisson algebras

A Poisson algebra is a $\mathbb{K}$-vector space $P$ equipped with an
associative algebra product $\mu$ and a Lie bracket $[-,-]$ such that the
following compatibility relation holds.

(1) $[x,yz]=y[x,z]+[x,y]z,\quad\forall x,y,z\in P.$

A map $\varphi:P\rightarrow P^{\prime}$ is called a homomorphism of Poisson
algebras if for all $x,y\in P$,

$\varphi(xy)=\varphi(x)\varphi(y)\quad\mbox{and}\quad\varphi[x,y]_{P}=[\varphi(x),\varphi(y)]_{P^{\prime}}.$

We denote the category of Poisson algebras by $\mathsf{Pois}$.

######  Definition 2.1.

Let $(P,\mu,[-,-])$ be a Poisson algebra. Then a $\mathbb{K}$-vector space is
called a Poisson-bimodule over $P$ if $M$ is simultaneously an associative
algebra bimodule over $(P,\mu)$ and a Lie algebra module over $(P,[-,-])$ such
that the compatibility relation (1) holds whenever two of the entries are in
$P$ and one of the entries is in $M$.

###  2.2. Associative dialgebras

An associative dialgebra $(D,\dashv,\vdash)$ is a vector space $D$ equipped
with two linear maps $\dashv,\vdash:D\otimes D\rightarrow D$ (called the left
product and the right product, respectively) such that the following
identities hold

(2) $(x\dashv y)\dashv z=x\dashv(y\vdash z),$ (3) $(x\dashv y)\dashv
z=x\dashv(y\dashv z),$ (4) $(x\vdash y)\dashv z=x\vdash(y\dashv z),$ (5)
$(x\dashv y)\vdash z=x\vdash(y\vdash z),$ (6) $(x\vdash y)\vdash
z=x\vdash(y\vdash z),$

for all $x,y,z\in D$. Note that identities (3) and (6) imply that the left and
right products are associative. Thus, a dialgebra is a vector space $D$ with
two associative products $\dashv$ and $\vdash$ satisfying

(7) $x\dashv(y\dashv z)=x\dashv(y\vdash z),$ (8) $(x\vdash y)\dashv
z=x\vdash(y\dashv z),$ (9) $(x\dashv y)\vdash z=(x\vdash y)\vdash z),$

for all $x,y,z\in D$. A linear map $f:D\rightarrow D^{\prime}$ is called a
homomorphism of dialgebras if $f(x\dashv y)=f(x)\dashv f(y)$ and $f(x\vdash
y)=f(x)\vdash f(y),$ for all $x,y\in D$. We denote the category of associative
dialgebras by $\mathsf{Dias}$.

We recall some interesting examples of dialgebras.

  * •

Let $A$ be an associative algebra, then $(A,\dashv,\vdash)$ is a dialgebra,
where $a\dashv b=ab=a\vdash b,~{}\forall a,b\in A$.

  * •

Let $A$ be an associative algebra and $M$ be an $A$-bimodule. If
$f:M\rightarrow A$ is an $A$-bimodule map, then $(M,\dashv,\vdash)$ is a
dialgebra, where the left and right products are given by

$m\dashv n=m\cdot f(n),\quad\mbox{and}\quad m\vdash n=f(m)\cdot
n,~{}~{}~{}\forall m,n\in M.$

  * •

Let $A$ be an associative algebra and $d:A\rightarrow A$ be a linear map. If
$(A,d)$ is a differential associative algebra, i.e., $d(ab)=(da)b+a(db),$ for
all $a,b\in A$. Then $(A,\dashv,\vdash)$ is a dialgebra, where the left and
right products are given by

$a\dashv b=ad(b),\quad\mbox{and}\quad a\vdash b=d(a)b,~{}~{}~{}\forall a,b\in
A.$

######  Remark 2.2.

Let $\mathsf{As}$ denotes the category of associative algebras. Any
associative algebra is a dialgebra. Thus, we have a functor
$\mathsf{inc}:\mathsf{As}\rightarrow\mathsf{Dias}$. For a dialgebra
$(D,\dashv,\vdash)$, let us consider the quotient space $D_{As}$ of $D$ by the
subspace spanned by the elements $x\dashv y-x\vdash y,~{}\forall x,y\in D$.
Then $D_{As}$ is an associative algebra with the product induced by the
products $\dashv,\vdash:D\otimes D\rightarrow D$. It yields a functor
$\mathsf{(-)_{As}}:\mathsf{Dias}\rightarrow\mathsf{As}$ called
associativization functor. The functor $\mathsf{(-)_{As}}$ is left adjoint to
the functor $\mathsf{inc}:\mathsf{As}\rightarrow\mathsf{Dias}$.

A $\mathbb{K}$-vector space $M$ is called a left module over a dialgebra
$(D,\dashv,\vdash)$ if there are linear maps $\dashv,\vdash:D\otimes
M\rightarrow M$ satisfying the equations (2)-(6) with two of the entries in
$D$ and one of the entries in $M$. Similarly, we can define the notion of
right modules over $(D,\dashv,\vdash)$.

######  Definition 2.3.

A $\mathbb{K}$-vector space $M$ is a bimodule over a dialgebra
$(D,\dashv,\vdash)$ if there are linear maps $\dashv,\vdash:M\otimes
D\rightarrow M$ and $\dashv,\vdash:D\otimes M\rightarrow M$ such that the
equations (2)-(6) hold true with two of the entries in $D$ and one of the
entries in $M$.

###  2.3. Leibniz algebras

A (right) Leibniz algebra $(\mathfrak{g},[-,-])$ is a $\mathbb{K}$-vector
space equipped with a linear map
$[-,-]:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$ (bracket)
satisfying the Leibniz identity

$[[x,y],z]=[[x,z],y]+[x,[y,z]],\quad\forall x,y,z\in\mathfrak{g}.$

If we consider the quotient of $\mathfrak{g}$ by the ideal generated by the
elements of the form $[x,x],~{}x\in\mathfrak{g}$, then it gives a Lie algebra
$\mathfrak{g}_{Lie}$ associated to the Leibniz algebra $(\mathfrak{g},[-,-])$.

Let us also recall the following result from [10].

######  Proposition 2.4 ([10]).

For a dialgebra $(D,\dashv,\vdash)$, the pair $(D,[-,-])$ is a Leibniz algebra
with the bracket

$[x,y]:=x\dashv y-y\vdash x,\quad\forall x,y\in D.$

###### Proof.

It is straightforward to check that $(D,[-,-])$ is a Leibniz algebra. ∎

Furthermore, it easily follows that the above result gives a functor from the
category $\mathsf{Dias}$ to the category $\mathsf{Leib}$ of Leibniz algebras.

###  2.4. The category of linear maps

The category $\mathcal{LM}$ of linear maps is defined as follows:

  * •

The objects are the $\mathbb{K}$-linear maps $f:V\rightarrow W$, where $V$ and
$W$ are $\mathbb{K}$-vector spaces.

  * •

A morphism $(\alpha^{\prime},\alpha)$ from $V\xrightarrow{f}W$ and
$V^{\prime}\xrightarrow{f^{\prime}}W^{\prime}$ is a commutative diagram in the
category of vector spaces, given by

$\begin{CD}V@>{\alpha^{\prime}}>{}>V^{\prime}\\\
@V{f}V{}V@V{}V{f^{\prime}}V\\\ <EMAIL_ADDRESS>
For an object $V\xrightarrow{f}W$, by upstairs we mean the vector space $V$
and by downstairs we mean the vector space $W$. If $V\xrightarrow{f}W$ and
$V^{\prime}\xrightarrow{f^{\prime}}W^{\prime}$ be two objects in the category
$\mathcal{LM}$, then a tensor product of these two objects can be defined as
follows:

$f\otimes_{\mathcal{LM}}f^{\prime}:V\otimes_{\mathbb{K}}W^{\prime}\oplus
W\otimes_{\mathbb{K}}V^{\prime}\rightarrow W\otimes_{\mathbb{K}}W^{\prime},$

where

$f\otimes_{\mathcal{LM}}f^{\prime}:=f\otimes_{\mathbb{K}}\mathsf{1}+\mathsf{1}\otimes_{\mathbb{K}}f^{\prime}.$

The tensor product of two morphisms $(\alpha^{\prime},\alpha)$ and
$(\beta^{\prime},\beta)$ is given by

$(\alpha^{\prime},\alpha)\otimes_{\mathcal{LM}}(\beta^{\prime},\beta):=(\alpha^{\prime}\otimes_{\mathbb{K}}\beta+\alpha\otimes_{\mathbb{K}}\beta^{\prime},\alpha\otimes_{\mathbb{K}}\beta).$

######  Definition 2.5.

An associative algebra object in the category $\mathcal{LM}$ is an object
$M\xrightarrow{f}A$ with a morphism

$\bar{\mu}=(\mu^{\prime},\mu):(M\xrightarrow{f}A)\otimes_{\mathcal{LM}}(M\xrightarrow{f}A)\rightarrow(M\xrightarrow{f}A)$

such that

(10)
$\bar{\mu}(\mathsf{1}\otimes_{\mathcal{LM}}\bar{\mu})=\bar{\mu}(\bar{\mu}\otimes_{\mathcal{LM}}\mathsf{1}).$

The downstairs of the above condition (10) implies that $A$ is an associative
algebra with the product $\mu$. The upstairs implies that the map
$\mu^{\prime}$ yields an $A$-bimodule structure on $M$. Moreover, since
$\bar{\mu}=(\mu^{\prime},\mu)$ is a morphism in $\mathcal{LM}$, we have

$f(\mu^{\prime}(m,a))=\mu(f(m),a),~{}\quad\mbox{and
}f(\mu^{\prime}(a,m))=\mu(a,f(m)),\quad\forall m\in M,a\in A.$

Therefore, we have the following result.

######  Proposition 2.6.

An associative algebra object $M\xrightarrow{f}A$ in the category
$\mathcal{LM}$ is equivalent to an $A$-bimodule map $f:M\rightarrow A$, where
$A$ is an associative algebra and $M$ is an $A$-bimodule. ∎

Let $M\xrightarrow{f}A$ be an associative algebra in the category
$\mathcal{LM}$. Then we have a dialgebra structure $(M,\dashv,\vdash)$, where
the products $\dashv,\vdash$ are defined by

(11) $m\dashv n:=m\cdot f(n)\quad\mbox{and}\quad m\vdash n:=f(m)\cdot
n,\quad\forall m,n\in M.$

######  Proposition 2.7.

If $M\xrightarrow{f}A$ is an associative algebra object in the category
$\mathcal{LM}$. Then, $(M,\dashv,\vdash)$ is a dialgebra and $f:M\rightarrow
A$ is a dialgebra homomorphism.

###### Proof.

It is easy to check that the products $\dashv,\vdash$ defined by equation (11)
are associative and satisfy equations (7)-(9). Therefore, $(M,\dashv,\vdash)$
is a dialgebra.

Moreover, since $f:M\rightarrow A$ is an $A$-bimodule map, we have $f(m\dashv
n)=f(m)f(n)$ and $f(m\vdash n)=f(m)f(n)$ for all $m,n\in M$. Thus,
$f:M\rightarrow A$ is a dialgebra homomorphism. ∎

######  Definition 2.8.

A homomorphism between two associative algebra objects $(M\xrightarrow{f}A)$
and $(M^{\prime}\xrightarrow{f^{\prime}}A^{\prime})$ (with the products
$\bar{\mu}$ and $\bar{\mu}^{\prime}$, respectively) in $\mathcal{LM}$ is a
pair
$\Phi=(\phi^{\prime},\phi):(M\xrightarrow{f}A)\rightarrow(M^{\prime}\xrightarrow{f^{\prime}}A^{\prime})$
such that

$\bar{\mu}^{\prime}\circ(\Phi\otimes_{\mathcal{LM}}\Phi)=\Phi\circ\bar{\mu}$

and the following diagram is commuative

(12) $\begin{CD}M@>{\phi^{\prime}}>{}>M^{\prime}\\\
@V{f}V{}V@V{}V{f^{\prime}}V\\\ <EMAIL_ADDRESS>
Thus, an associative algebra homomorphism between $(M\xrightarrow{f}A)$ and
$(M^{\prime}\xrightarrow{f^{\prime}}A^{\prime})$ is a pair
$(\phi^{\prime},\phi)$ of a dialgebra homomorphism $\phi^{\prime}:M\rightarrow
M^{\prime}$ and an associative algebra homomorphism $\phi:A\rightarrow
A^{\prime}$ such that the diagram (12) commutes.

Let us denote the category of associative algebra objects in $\mathcal{LM}$ by
$\mathsf{As}_{\mathcal{LM}}$. Then, we have a functor
$\mathsf{F}:\mathsf{As}_{\mathcal{LM}}\rightarrow\mathsf{Dias}$ assigning a
dialgebra $M$ for an associative algebra object $M\xrightarrow{f}A$ in
$\mathcal{LM}$. On the other hand, we have a functor
$\mathsf{G}:\mathsf{Dias}\rightarrow\mathsf{As}_{\mathcal{LM}}$ defined as
follows: For a dialgebra $D$, the functor $\mathsf{G}$ sends the dialgebra $D$
to the associative algebra object $D\xrightarrow{p}D_{As}$ where $D_{As}$ is
the associative algebra associated to the dialgebra $D$ and $p$ is the
projection map.

######  Theorem 2.9.

With the above notations, the functor
$\mathsf{G}:\mathsf{Dias}\rightarrow\mathsf{As}_{\mathcal{LM}}$ is left
adjoint to the functor
$\mathsf{F}:\mathsf{As}_{\mathcal{LM}}\rightarrow\mathsf{Dias}$, i.e., there
is a bijection

$\mathrm{Hom}_{\mathsf{As}_{\mathcal{LM}}}(\mathsf{G}(D),M\xrightarrow{f}A)\cong\mathrm{Hom}_{\mathsf{Dias}}(D,\mathsf{F}(M\xrightarrow{f}A)),$

for all $D\in\mathsf{Dias}$ and
$M\xrightarrow{f}A\in\mathsf{As}_{\mathcal{LM}}$ such that the bijection is
functorial in $D\in\mathsf{Dias}$ and
$M\xrightarrow{f}A\in\mathsf{As}_{\mathcal{LM}}$.

###### Proof.

Let $M\xrightarrow{f}A$ be an associative algebra object in ${\mathcal{LM}}$.
Then, there is an induced dialgebra structure on $M$. Let
$\phi^{\prime}\in\mathrm{Hom}_{\mathsf{Dias}}(D,\mathsf{F}(M\xrightarrow{f}A))$,
i.e., we have a dialgebra homomorphism $\phi^{\prime}:D\rightarrow M$. Let us
recall that the associativization functor
$(-)_{As}:\mathsf{Dias}\rightarrow\mathsf{As}$ is left adjoint to the functor
$\mathsf{inc}:\mathsf{As}\rightarrow\mathsf{Dias}$. Also,
$f\circ\phi^{\prime}:D\rightarrow\mathsf{inc}(A)$ is a dialgebra homomorphism,
so there exists a unique algebra homomorphism $\phi:D_{As}\rightarrow A$ such
that the following diagram commutes.

$\begin{CD}D@>{\phi^{\prime}}>{}>M\\\ @V{p}V{}V@V{}V{f}V\\\
<EMAIL_ADDRESS>
It follows that for any morphism
$\phi^{\prime}\in\mathrm{Hom}_{\mathsf{Dias}}(D,\mathsf{F}(M\xrightarrow{f}A))$,
there exists a unique map
$(\phi^{\prime},\phi)\in\mathsf{Hom}_{\mathsf{As}_{\mathcal{LM}}}(\mathsf{G}(D),M\xrightarrow{f}A)$.

Conversely, for any associative algebra morphism
$(\phi^{\prime},\phi)\in\mathrm{Hom}_{\mathsf{As}_{\mathcal{LM}}}(\mathsf{G}(D),M\xrightarrow{f}A)$,
we have a dialgebra homomorphism
$\phi\in\mathrm{Hom}_{\mathsf{Dias}}(D,\mathsf{F}(M\xrightarrow{f}A))$. Again,
the uniqueness follows from the fact that the associativization functor
$\mathsf{(-)_{As}}:\mathsf{Dias}\rightarrow\mathsf{As}$ is left adjoint to the
functor $\mathsf{inc}:\mathsf{As}\rightarrow\mathsf{Dias}$. Hence, we have a
bijection between the sets of homomorphisms

$\mathrm{Hom}_{\mathsf{As}_{\mathcal{LM}}}(\mathsf{G}(D),M\xrightarrow{f}A)\cong\mathrm{Hom}_{\mathsf{Dias}}(D,\mathsf{F}(M\xrightarrow{f}A)).$

For every morphism $\alpha:D\rightarrow D^{\prime}$ in $\mathsf{Dias}$ and
$\beta:(M\xrightarrow{f}A)\rightarrow(M^{\prime}\xrightarrow{f^{\prime}}A^{\prime})$
in $\mathsf{As}_{\mathcal{LM}}$, we have the following commutative diagram:

$\begin{CD}\mathrm{Hom}_{\mathsf{As}_{\mathcal{LM}}}(\mathsf{G}(D^{\prime}),M\xrightarrow{f}A)@>{\mathsf{G}(\alpha)^{*}}>{}>\mathrm{Hom}_{\mathsf{As}_{\mathcal{LM}}}(\mathsf{G}(D),M\xrightarrow{f}A)@>{\beta_{*}}>{}>\mathrm{Hom}_{\mathsf{As}_{\mathcal{LM}}}(\mathsf{G}(D),M^{\prime}\xrightarrow{f^{\prime}}A^{\prime})\\\
@V{\cong}V{}V@V{\cong}V{}V@V{\cong}V{}V\\\
\mathrm{Hom}_{\mathsf{Dias}}(D^{\prime},\mathsf{F}(M\xrightarrow{f}A))@>{\alpha^{*}}>{}>\mathrm{Hom}_{\mathsf{Dias}}(D,\mathsf{F}(M\xrightarrow{f}A))@>{\mathsf{F}(\beta)_{*}}>{}>\mathrm{Hom}_{\mathsf{Dias}}(D,\mathsf{F}(M^{\prime}\xrightarrow{f^{\prime}}A^{\prime})),\end{CD}$

where $\alpha^{*}$ and $\beta_{*}$ are the maps induced by $\alpha$ and
$\beta$. Thus, the bijection is functorial in $D\in\mathsf{Dias}$ and
$M\xrightarrow{f}A\in\mathsf{As}_{\mathcal{LM}}$. ∎

We denote the category of Lie objects in $\mathcal{LM}$ by
$\mathsf{Lie_{\mathcal{LM}}}$. Then let us recall the following result for Lie
algebra objects in the category of linear maps from [11].

######  Theorem 2.10.

For a Lie algebra object $M\xrightarrow{f}L$ in $\mathcal{LM}$, we have a
Leibniz algebra structure on $M$. On the other hand, for any Leibniz algebra
$\mathfrak{g}$ the Liezation $\mathfrak{g}\xrightarrow{p}\mathfrak{g}_{Lie}$
is a Lie algebra object in $\mathcal{LM}$. The two functors
$\mathsf{Leib}\longleftrightarrow\mathsf{Lie}_{\mathcal{LM}}$ are adjoint to
each-other.

###### Proof.

The proof is similar to the proof of the Theorem 2.9. For more details, see
[11]. ∎

##  3\. An associative $2$-algebra associated to a dialgebra

Let us recall that associative $2$-algebras are categorification of
associative algebras. Associative $2$-algebras are equivalent to $2$-term
$A_{\infty}$-algebras [16].

######  Definition 3.1.

A graded vector space $\mathcal{A}=A_{0}\oplus A_{1}$ is called an associative
$2$-algebra, if $\mathcal{A}$ is equipped with linear maps
${\mu_{i}:\otimes^{i}\mathcal{A}\rightarrow\mathcal{A}}$ of degrees $i-2$, for
$i=1,2,3$ and the following identities hold.

  1. (i)

$\mu_{1}\mu_{2}(x,a)=\mu_{2}(x,\mu_{1}(a)),~{}~{}\mu_{1}\mu_{2}(a,x)=\mu_{2}(\mu_{1}(a),x)$,

  2. (ii)

$\mu_{2}(\mu_{1}(a),b)=\mu_{2}(a,\mu_{1}(b))$,

  3. (iii)

$\mu_{2}(\mu_{2}(x,y),z)-\mu_{2}(x,\mu_{2}(y,z))=\mu_{1}\mu_{3}(x,y,z)$,

  4. (iv)

$\mu_{2}(\mu_{2}(a,x),y)-\mu_{2}(a,\mu_{2}(x,y))=\mu_{3}(\mu_{1}(a),x,y)$,

  5. (v)

$\mu_{2}(\mu_{2}(x,a),y)-\mu_{2}(x,\mu_{2}(a,y))=\mu_{3}(x,\mu_{1}(a),y)$,

  6. (vi)

$\mu_{2}(\mu_{2}(x,y),a)-\mu_{2}(x,\mu_{2}(y,a))=\mu_{3}(x,y,\mu_{1}(a))$,

  7. (vii)

$\mu_{3}(\mu_{2}(x,y),z,w)-\mu_{3}(x,\mu_{2}(y,z),w)+\mu_{3}(x,y,\mu_{2}(z,w))=\mu_{2}(\mu_{3}(x,y,z),w)+\mu_{2}(x,\mu_{3}(y,z,w))$,

for all $a,b\in A_{0}$ and $x,y,z,w\in A_{1}$.

Let $(D,\dashv,\vdash)$ be a dialgebra. We consider the following subset of
$D$

$I:=\\{x\in D|~{}~{}x\vdash y=0=y\dashv x,\quad\forall y\in D\\}.$

Note that $I$ is an ideal in $D$, i.e., $x\dashv y\in I$ whenever atleast one
of $x$ and $y$ is in $I$. Now, we define a graded space
$\mathcal{D}:=D_{0}\oplus D_{1}$ with $D_{0}=D$ and $D_{1}=I$.

$\blacktriangleright$ First, define a degree $-1$ map $\mu_{1}:I\rightarrow D$
by the inclusion map.

$\blacktriangleright$ Also, consider the map $\mu_{2}:\otimes^{2}D\rightarrow
D$ defined by

(13) $\mu_{2}(x,y)=\frac{1}{2}(x\dashv y+x\vdash y),\quad\forall x,y\in D.$

Then, $\mu_{2}(x,y)\in I$ if either of the two elements $x$ and $y$ is in $I$.
More preciesely, for $x\in I$ and $y,w\in D$, we have

$\displaystyle\mu_{2}(x,y)\vdash w$ $\displaystyle=\frac{1}{2}(x\dashv
y+x\vdash y)\vdash w=(x\vdash y)\vdash w=0.\quad(\mbox{since }x\in I)$
$\displaystyle w\dashv\mu_{2}(x,y)$
$\displaystyle=\frac{1}{2}\big{(}w\dashv(x\dashv y+x\vdash
y)\big{)}=w\dashv(x\vdash y)=0.\quad(\mbox{since }x\in I)$

So, $\mu_{2}(x,y)\in I,$ for all $x\in I$ and $y\in D$. Similarly, for all
$x\in D$ and $y\in I$, we have $\mu_{2}(x,y)\in D$. In turn, it follows that
$\mu_{2}$ induces a degre $0$ map
$\mu_{2}:\mathcal{D}\otimes\mathcal{D}\rightarrow\mathcal{D}$.

$\blacktriangleright$ Finally, we consider the map
$\mu_{3}:\otimes^{3}D\rightarrow D$ defined by

(14)
$\mu_{3}(x,y,z)=\mu_{2}(\mu_{2}(x,y),z)-\mu_{2}(x,\mu_{2}(y,z)),\quad\forall
x,y,z\in D.$

Here, $\mu_{3}(x,y,z)$ is the associater with respect to the map $\mu_{2}$. It
follows from (13)-(14) that

(15) $\mu_{3}(x,y,z)=\frac{1}{4}\big{(}(x\dashv y)\vdash z-x\dashv(y\vdash
z)\big{)},\quad\forall x,y,z\in D.$

Let us observe that

$w\dashv\mu_{3}(x,y,z)=0=\mu_{3}(x,y,z)\vdash w,\quad\forall x,y,z,w\in D.$

Therefore, $\mu_{3}(x,y,z)\in I$ for any $x,y,z\in D$.

######  Proposition 3.2.

Let $(D,\dashv,\vdash)$ be a dialgebra. Then $(\mathcal{D}=D\oplus
I,\mu_{1},\mu_{2},\mu_{3})$ is an associative $2$-algebra, where
$\mu_{1}:I\rightarrow D$ is the inclusion map and the maps $\mu_{2}$,
$\mu_{3}$ are given by equations (13) and (15).

###### Proof.

It is straightforward to check that the identities $(i)-(vi)$ in the
definition of an associative $2$-algebra are satisfied for
$(\mathcal{D}=D\oplus I,\mu_{1},\mu_{2},\mu_{3})$. We verify the identity
$(vii)$, i.e.

$\mu_{3}(\mu_{2}(x,y),z,w)-\mu_{3}(x,\mu_{2}(y,z),w)+\mu_{3}(x,y,\mu_{2}(z,w))=\mu_{2}(\mu_{3}(x,y,z),w)+\mu_{2}(x,\mu_{3}(y,z,w)).$

To show that the identity holds, let us rewrite the first term of the Left
hand side using equations (13) and (15)

$\displaystyle\mu_{3}(\mu_{2}(x,y),z,w)=$
$\displaystyle~{}~{}\frac{1}{4}\big{(}(\mu_{2}(x,y)\dashv z)\vdash
w-\mu_{2}(x,y)\dashv(z\vdash w)\big{)}$ $\displaystyle=$
$\displaystyle~{}~{}\frac{1}{8}\big{(}((x\vdash y)\dashv z)\vdash w+((x\dashv
y)\dashv z)\vdash w-(x\vdash y)\dashv(z\vdash w)-(x\dashv y)\dashv(z\vdash
w)\big{)}$

Similarly, second and third terms of the left hand side can be written as

$\displaystyle\mu_{3}(x,\mu_{2}(y,z),w)=$
$\displaystyle~{}~{}\frac{1}{8}\big{(}(x\dashv(y\vdash z))\vdash
w+(x\dashv(y\dashv z))\vdash w-x\dashv((y\vdash z)\vdash w)-x\dashv((y\dashv
z)\vdash w)\big{)}$ $\displaystyle\mu_{3}(x,y,\mu_{2}(z,w))=$
$\displaystyle~{}~{}\frac{1}{8}\big{(}(x\dashv y)\vdash(z\vdash w)+(x\dashv
y)\vdash(z\dashv w)-x\dashv(y\vdash(z\vdash w))-x\dashv(y\vdash(z\dashv
w))\big{)}$

Simplifying the left hand side, one obtains the following expression

$\displaystyle\mu_{3}(\mu_{2}(x,y),z,w)-\mu_{3}(x,\mu_{2}(y,z),w)+\mu_{3}(x,y,\mu_{2}(z,w))$
$\displaystyle=$ $\displaystyle~{}~{}\frac{1}{8}\big{(}((x\vdash y)\dashv
z)\vdash w-(x\vdash y)\dashv(z\vdash w)+(x\dashv y)\vdash(z\dashv
w)-x\dashv(y\vdash(z\dashv w))\big{)}$ $\displaystyle=$
$\displaystyle~{}~{}\frac{1}{8}\big{(}x\vdash((y\dashv z)\vdash
w)-x\vdash(y\dashv(z\vdash w))+((x\dashv y)\vdash z)\dashv w-(x\dashv(y\vdash
z))\dashv w\big{)}$ $\displaystyle=$
$\displaystyle~{}~{}\mu_{2}(x,\mu_{3}(y,z,w))+\mu_{2}(\mu_{3}(x,y,z),w).$

Hence, it follows that $(\mathcal{D}=D\oplus I,\mu_{1},\mu_{2},\mu_{3})$ is an
associative $2$-algebra. ∎

##  4\. Poisson dialgebras

The notion of Poisson dialgebras was first introduced by Loday in [10] as a
generalization of Poisson algebras [1, 6, 7]. We observe that the following
definition is more appropriate one with several supporting examples.

######  Definition 4.1.

A Poisson dialgebra $\mathcal{P}$ is a $\mathbb{K}$-vector space with
dialgebra products $\dashv$ and $\vdash$, and a Leibniz bracket $[-,-]$ such
that the following compatibility relations hold.

(16) $[x,y\dashv z]=y\vdash[x,z]+[x,y]\dashv z=[x,y\vdash z],$ (17) $[x\dashv
y,z]=x\dashv[y,z]+[x,z]\dashv y,$ (18) $[x\vdash y,z]=x\vdash[y,z]+[x,z]\vdash
y,$ (19) $[x,y]\vdash z=-[y,x]\vdash z,$ (20) $x\dashv[y,z]=-x\dashv[z,y],$

for all $x,y,z\in\mathcal{P}$.

We call the last two identities as mixed skew-symmetry of the Leibniz bracket
$[-,-]$.

Let $\mathcal{P}$ and $\mathcal{P}^{\prime}$ be two Poisson dialgebras, then a
map $\varphi:\mathcal{P}\rightarrow\mathcal{P}^{\prime}$ is a homomorphism of
Poisson dialgebras if $\varphi$ preserves both the dialgebra products and the
Leibniz bracket. We denote the category of Poisson dialgebras by
$\mathsf{Pois-di}$.

Note that any Poisson algebra is also a Poisson dialgebra. Thus, we have a
functor $\mathsf{inc}:\mathsf{Pois}\rightarrow\mathsf{Pois-di}$ considering a
Poisson algebra as a Poisson dialgebra.

######  Example 4.2.

Let $(D,\dashv,\vdash)$ be a dialgebra. Then by Proposition 2.4, we have a
Leibniz bracket on $D$ defined by

$[x,y]=x\dashv y-y\vdash x,\quad\forall x,y\in D.$

It follows from identities (7)-(9) that the bracket and the dialgebra products
satisy the compatibility relations given by the equations (16)-(20). Thus, we
have a Poisson dialgebra structure on any dialgebra.

######  Example 4.3.

Let $P$ be a Poisson algebra and $M$ be a Poisson-bimodule over $P$. If
$f:M\rightarrow P$ is a Poisson-bimodule map, then
$(M,\dashv,\vdash,[-,-]_{\mathsf{M}})$ is a Poisson dialgebra, where

$m\dashv n=m\cdot f(n),\quad m\vdash n=f(m)\cdot
n,\quad\mbox{and}\quad[m,n]_{\mathsf{M}}=[m,f(n)],~{}~{}~{}\forall m,n\in M.$

We need to show that compatibility conditions (16)-(20) hold. First, let us
consider the following expression.

$\displaystyle[m,n\dashv p]_{\mathsf{M}}$ $\displaystyle=[m,f(n\cdot f(p))]$
$\displaystyle=[m,f(n)f(p)]\quad\quad\quad\mbox{(since}~{}f~{}\mbox{is a
Poisson-bimodule map)}$
$\displaystyle=f(n)[m,f(p)]+[m,f(n)]f(p)\quad\mbox{(since}~{}M~{}\mbox{is a
Poisson-bimodule) }$
$\displaystyle=n\vdash[m,p]_{\mathsf{M}}+[m,n]_{\mathsf{M}}\dashv p.$

Since $f:M\rightarrow P$ is a Poisson-bimodule map, we have

$[m,n\vdash p]_{\mathsf{M}}=[m,f(f(n)\cdot
p)]=n\vdash[m,p]_{\mathsf{M}}+[m,n]_{\mathsf{M}}\dashv p=[m,n\dashv
p]_{\mathsf{M}}.$

Similarly, one can show that

$[m\dashv n,p]_{\mathsf{M}}=m\dashv[n,p]_{\mathsf{M}}+[m,p]_{\mathsf{M}}\dashv
n,$ $[m\vdash
n,p]_{\mathsf{M}}=m\vdash[n,p]_{\mathsf{M}}+[m,p]_{\mathsf{M}}\vdash
n,\quad\forall m,n,p\in M.$

Also, we obtain the following expression by using the fact that
$f:M\rightarrow P$ is a Poisson-bimodule map

$\displaystyle[m,n]_{\mathsf{M}}\vdash p=f[m,f(n)]\cdot p=[f(m),f(n)]\cdot
p=-[f(n),f(m)]\cdot p=-f[n,f(m)]\cdot p=-[n,m]_{\mathsf{M}}\vdash p.$

Similarly,

$m\dashv[n,p]_{\mathsf{M}}=-m\dashv[p,n]_{\mathsf{M}},\quad\forall m,n,p\in
P.$

Thus, we have a Poisson dialgebra structure
$(M,\dashv,\vdash,[-,-]_{\mathsf{M}})$ on $M$.

######  Example 4.4.

Let $P$ be a Poisson algebra and $d:P\rightarrow P$ be a linear map. Suppose
that $(P,d)$ is a differential Poisson algebra, i.e.,

$\displaystyle d(xy)$ $\displaystyle=x(dy)+(dx)y,$ $\displaystyle d([x,y])$
$\displaystyle=[dx,y]+[x,dy],\quad\forall x,y\in P$

and $d^{2}=0$. Then $\mathcal{P}_{d}:=(P,\dashv,\vdash,[-,-]_{d})$ is a
Poisson dialgebra, where the dialgebra products $\dashv,\vdash$, and the
Leibniz bracket $[-,-]_{d}$ are given by

$x\dashv y=xd(y),\quad x\vdash
y=d(x)y,\quad\mbox{and}\quad[x,y]_{d}=[x,dy],\quad\forall x,y\in P.$

###  4.1. Averaging operators on Poisson algebras

Let us recall the notion of averaging operators from [13, 14]. Let $P$ be a
Poisson algebra with the associative algebra product $\mu$ and the Lie bracket
$[-,-]$. A linear map $\alpha:P\rightarrow P$ is called an averaging operator
if

$\alpha(x\alpha(y))=\alpha(x)\alpha(y)=\alpha(\alpha(x)y)~{}~{}\mbox{and}$
$[\alpha(x),\alpha(y)]=\alpha([\alpha(x),y]),\quad\forall x,y\in P.$

Thus, the map $\alpha$ is an averaging operator on the associative algebra
$(P,\mu)$ as well as on the Lie algebra $(P,[-,-])$. A differential
$d:P\rightarrow P$ of a Poisson algebra $P$ is a particular case of an
averaging operator on $P$.

In the next result, we construct a Poisson dialgebra from a Poisson algebra
endowed with an averaging operator.

######  Proposition 4.5.

Let $P$ be a Poisson algebra with the associative algebra product $\mu$, Lie
bracket $[-,-]$, and an averaging operator $\alpha:P\rightarrow P$. Then,
$\mathcal{P}=(P,\dashv,\vdash,[-,-]_{\mathcal{P}})$ is a Poisson dialgebra,
where

$x\dashv y:=x\alpha(y),\quad x\vdash
y:=\alpha(x)y,\quad\mbox{and}\quad[x,y]_{\mathcal{P}}=[x,\alpha(y)],~{}\forall
x,y\in P.$

###### Proof.

It easily follows that $P$ is a dialgebra with the products $\dashv,\vdash$
and $(P,[-,-]_{\mathcal{P}})$ is a right Leibniz algebra. We now show that the
comaptibility relations (16)-(18) hold. For any $x,y,z\in P$, we have

$\displaystyle[x,y\dashv z]_{\mathcal{P}}$
$\displaystyle=[x,\alpha(y\alpha(z))]$ $\displaystyle=[x,\alpha(y)\alpha(z)]$
$\displaystyle=\alpha(y)[x,\alpha(z)]+[x,\alpha(y)]\alpha(z)$
$\displaystyle=y\vdash[x,z]_{\mathcal{P}}+[x,y]_{\mathcal{P}}\dashv z.$

Moreover, $\alpha(y\alpha(z))=\alpha(\alpha(y)z)$ implies that

$[x,y\dashv
z]_{\mathcal{P}}=y\vdash[x,z]_{\mathcal{P}}+[x,y]_{\mathcal{P}}\dashv
z=[x,y\vdash z]_{\mathcal{P}}.$

Similarly,

$[x\dashv
y,z]_{\mathcal{P}}=x\dashv[y,z]_{\mathcal{P}}+[x,z]_{\mathcal{P}}\dashv y,$
$[x\vdash
y,z]_{\mathcal{P}}=x\vdash[y,z]_{\mathcal{P}}+[x,z]_{\mathcal{P}}\vdash
y,\quad\forall x,y,z\in P.$

Next, we consider the following expression

$\displaystyle[x,y]_{\mathcal{P}}\vdash
z=\alpha[x,\alpha(y)]z=[\alpha(x),\alpha(y)]z=-[\alpha(y),\alpha(x)]z=-\alpha[y,\alpha(x)]z=-[y,x]_{\mathcal{P}}\vdash
z.$

A similar calculation shows that

$x\dashv[y,z]_{\mathcal{P}}=-x\dashv[z,y]_{\mathcal{P}},\quad\forall x,y,z\in
P.$

Hence, $\mathcal{P}=(P,\dashv,\vdash,[-,-]_{\mathcal{P}})$ is a Poisson
dialgebra. ∎

######  Definition 4.6.

Let $\mathcal{P}$ be a Poisson dialgebra. Then a Poisson dialgebra bimodule is
a $\mathbb{K}$-vector space $M$ such that it is simultaneously a bimodule over
the dialgebra $(\mathcal{P},\dashv,\vdash)$ and a module over the Leibniz
algebra $(\mathcal{P},[-,-])$ such that the identities (16)-(18) hold whenever
two of the entries are in $\mathcal{P}$ and one entry is in $M$.

######  Definition 4.7.

A graded Poisson dialgebra $\mathcal{P}$ of degree $n$ is a
$\mathbb{K}$-graded vector space with degree $0$ dialgebra products $\dashv$
and $\vdash$, and a Leibniz bracket $[-,-]$ of degree $-n$ such that

(21) $[x,y\dashv z]=(-1)^{|y|(|x|-n)}y\vdash[x,z]+[x,y]\dashv z=[x,y\vdash
z],$ (22) $[x\dashv y,z]=x\dashv[y,z]+(-1)^{|y|(|z|-n)}[x,z]\dashv y,$ (23)
$[x\vdash y,z]=x\vdash[y,z]+(-1)^{|y|(|x|-n)}[x,z]\vdash y,$ (24) $[x,y]\vdash
z=-(-1)^{(|x|-n)(|y|-n)}[y,x]\vdash z,$ (25)
$x\dashv[y,z]=-(-1)^{(|y|-n)(|z|-n)}x\dashv[z,y],$

where $x,y,z\in\mathcal{P}$ and $|r|$ denotes degree of an element
$r\in\mathcal{P}$. We call the graded Poisson dialgebra of degree $0$ the
Poisson dialgebra structure on a graded vector space, and call the graded
Poisson dialgebra of degree $1$ the Gerstenhaber dialgebras.

###  4.2. Poisson dialgebra associated to a Filtered dialgebra

A filtration of a dialgebra $(D,\dashv,\vdash)$ is an increasing sequence of
subspaces $D_{0}\subseteq D_{1}\subseteq D_{2}\subseteq\ldots$ such that

$D=\cup_{n\geq 0}D_{n}\quad\mbox{and }\quad D_{i}\dashv D_{j},~{}D_{i}\vdash
D_{j}\subseteq D_{i+j},\mbox{for all }i,j\geq 0.$

There is a graded vector space $\mathrm{Gr}(D)=\oplus_{i\geq
0}\mathrm{Gr}(D)_{i}$ associated to the filtration of the dialgebra $D$, where
$\mathrm{Gr}(D)_{0}=D_{0}$ and $\mathrm{Gr}(D)_{i}=D_{i}/D_{i-1}$ for any
$i\geq 1$. Moreover, this graded vector space $\mathrm{Gr}(D)$ is a graded
dialgebra, where the dialgebra products (of degree $0$) are given by

$\displaystyle(x+D_{i-1})\dashv(y+D_{j-1})$ $\displaystyle=x\dashv
y+D_{i+j-1},$ $\displaystyle(x+D_{i-1})\vdash(y+D_{j-1})$
$\displaystyle=x\vdash y+D_{i+j-1},$

for all $x\in D_{i}$ and $y\in D_{j}$.

Next, let us define a degree $0$ bracket on $\mathrm{Gr}(D)$ as follows

(26) $[x+D_{i-1},y+D_{j-1}]=x\dashv y-(-1)^{ij}y\vdash x+D_{i+j-1},\forall
x\in D_{i},~{}y\in D_{j}.$

We claim that the above bracket is a graded Leibniz bracket of degree $0$. To
prove our claim, we consider the following expressions.

(27) $\displaystyle[[x+D_{i-1},y+D_{j-1}],z+D_{k-1}]=$ $\displaystyle[x\dashv
y-(-1)^{ij}y\vdash x+D_{i+j-1},z+D_{k-1}]$ $\displaystyle=$
$\displaystyle(x\dashv y-(-1)^{ij}y\vdash x)\dashv
z-(-1)^{k(i+j)}z\vdash(x\dashv y-(-1)^{ij}y\vdash x).$ (28)
$\displaystyle(-1)^{jk}[[x+D_{i-1},z+D_{k-1}],y+D_{j-1}]$ $\displaystyle=$
$\displaystyle(-1)^{jk}[x\dashv z-(-1)^{ik}z\vdash x+D_{i+k-1},y+D_{j-1}]$
$\displaystyle=$ $\displaystyle(-1)^{jk}\\{(x\dashv z-(-1)^{ik}z\vdash
x)\dashv y-(-1)^{j(i+k)}y\vdash(x\dashv z-(-1)^{ik}z\vdash x)\\}.$ (29)
$\displaystyle[x+D_{i-1},[y+D_{j-1},z+D_{k-1}]]=$
$\displaystyle[x+D_{i-1},y\dashv z-(-1)^{jk}z\vdash y+D_{j+k-1}]$
$\displaystyle=$ $\displaystyle x\dashv(y\dashv z-(-1)^{jk}z\vdash
y)-(-1)^{i(j+k)}(y\dashv z-(-1)^{jk}z\vdash y)\vdash x.$

From expressions (27)-(29) and the identities (2)-(6) it follows that

$[[x+D_{i-1},y+D_{j-1}],z+D_{k-1}]=(-1)^{jk}[[x+D_{i-1},z+D_{k-1}],y+D_{j-1}]+[x+D_{i-1},[y+D_{j-1},z+D_{k-1}]].$

Hence, the bracket defined by the equation (26) is a graded Leibniz bracket of
degree $0$. Let us denote $\bar{x}_{i}:=x+D_{i-1},\bar{y}_{j}:=y+D_{j-1},$ and
$\bar{z}_{k}:=z+D_{k-1}$. Then, we can verify the following identities

(30)
$[\bar{x}_{i},\bar{y}_{j}\dashv\bar{z}_{k}]=(-1)^{ij}\bar{y}_{j}\vdash[\bar{x}_{i},\bar{z}_{k}]+[\bar{x}_{i},\bar{y}_{j}]\dashv\bar{z}_{k}=[\bar{x}_{i},\bar{y}_{j}\vdash\bar{z}_{k}],$
(31)
$[\bar{x}_{i}\dashv\bar{y}_{j},\bar{z}_{k}]=\bar{x}_{i}\dashv[\bar{y}_{j},\bar{z}_{k}]+(-1)^{jk}[\bar{x}_{i},\bar{z}_{k}]\dashv\bar{y}_{j},$
(32)
$[\bar{x}_{i}\vdash\bar{y}_{j},\bar{z}_{k}]=\bar{x}_{i}\vdash[\bar{y}_{j},\bar{z}_{k}]+(-1)^{jk}[\bar{x}_{i},\bar{z}_{k}]\vdash\bar{y}_{j},$
(33)
$[\bar{x}_{i},\bar{y}_{j}]\vdash\bar{z}_{k}=-(-1)^{ij}[\bar{y}_{j},\bar{x}_{i}]\vdash\bar{z}_{k},$
(34)
$\bar{x}_{i}\dashv[\bar{y}_{j},\bar{z}_{k}]=-(-1)^{jk}\bar{x}_{i}\dashv[\bar{z}_{k},\bar{y}_{j}].$

Note that the above identities (30)-(34) follows from the fact that any
dialgebra carries a Poisson dialgebra structure where the Leibniz bracket is
given by the Proposition 2.4 (cf. Example 4.2).

###  4.3. Gerstenhaber dialgebra associated to a Filtered dialgebra

Let $(D,\dashv,\vdash)$ be a filtered dialgebra with the filtration
$D=\cup_{n\geq 0}D_{n}$ and $D_{0}\subseteq D_{1}\subseteq
D_{2}\subseteq\ldots$ (increasing sequence of subspaces). Let the associated
dialgebra structure on the graded vector space $\mathrm{Gr}(D)$ is
commutative, i.e.,

$(x+D_{i-1})\dashv(y+D_{j-1})=(y+D_{j-1})\vdash(x+D_{i-1}),\quad\forall x\in
D_{i},y\in D_{j}.$

The commuativity condition implies that $x\dashv y-y\vdash x\in D_{i+j-1}$.
Thus, the expression

$[x+D_{i-1},y+D_{j-1}]=x\dashv y-(-1)^{(i-1)(j-1)}y\vdash
x+D_{i+j-2},\quad\forall x\in D_{i},~{}y\in D_{j}$

defines a degree $-1$ bracket on the graded vector space $\mathrm{Gr}(D)$. A
similar calculation as in the previous subsection shows that the above bracket
is a degree $-1$ Leibniz bracket on $\mathrm{Gr}(D)$. Furthermore,
$\mathrm{Gr}(D)$ is a graded Poisson dialgebra of degree $1$.

###  4.4. Poisson algebra associated to a Poisson dialgebra

Let $\mathcal{P}$ be a Poisson dialgebra with dialgebra products
$\dashv,\vdash$ and Leibniz bracket $[-,-]$. We consider the quotient space of
$\mathcal{P}$ by the subspace spanned by the elements of the form $x\dashv
y-x\vdash y$ and $[x,y]-[y,x]$ for all $x,y\in\mathcal{P}$. We denote this
quotient space by $\mathcal{P}_{\mathrm{Poiss}}$. The left and right dialgebra
products are same in the quotient space $\mathcal{P}_{\mathrm{Poiss}}$ and the
Leibniz bracket induces a Lie bracket on $\mathcal{P}_{\mathrm{Poiss}}$. Thus,
the quotient space $\mathcal{P}_{\mathrm{Poiss}}$ becomes a Poisson algebra.

This construction gives a functor $\mathrm{(-)_{Poiss}}:\mathsf{Pois-
di}\rightarrow\mathsf{Pois}$. We call this functor the Poissonization functor.
It is left adjoint to the functor
$\mathsf{inc}:\mathsf{Pois}\rightarrow\mathsf{Pois-di}$.

###  4.5. Poisson algebra objects in the category $\mathcal{LM}$

######  Definition 4.8.

A Poisson algebra object in the category $\mathcal{LM}$ is an object
$M\xrightarrow{f}P$ with two morphisms

$\bar{\mu}=(\mu^{\prime},\mu):(M\xrightarrow{f}P)\otimes_{\mathcal{LM}}(M\xrightarrow{f}P)\rightarrow(M\xrightarrow{f}P)~{}~{}\mbox{and}$
$\bar{\nu}=(\nu^{\prime},\nu):(M\xrightarrow{f}P)\otimes_{\mathcal{LM}}(M\xrightarrow{f}P)\rightarrow(M\xrightarrow{f}P)$

such that

(35)
$\bar{\mu}(\mathsf{1}\otimes_{\mathcal{LM}}\bar{\mu})=\bar{\mu}(\bar{\mu}\otimes_{\mathcal{LM}}\mathsf{1}),$
(36) $\bar{\nu}(1~{}2)=-\bar{\nu},$ (37)
$\bar{\nu}(\mathsf{1}\otimes_{\mathcal{LM}}\bar{\nu})+\bar{\nu}(\mathsf{1}\otimes_{\mathcal{LM}}\bar{\nu})(1~{}2~{}3)+\bar{\nu}(\mathsf{1}\otimes_{\mathcal{LM}}\bar{\nu})(1~{}3~{}2)=0,$
(38)
$\bar{\nu}(\mathsf{1}\otimes_{\mathcal{LM}}\bar{\mu})=\bar{\mu}(\mathsf{1}\otimes_{\mathcal{LM}}\bar{\nu})(1~{}2)+\bar{\mu}(\bar{\nu}\otimes_{\mathcal{LM}}\mathsf{1}).$

Here, $(1~{}2),~{}(1~{}2~{}3),~{}(1~{}3~{}2)$ are cycles in symmetric group
$S_{3}$.

The first condition (35) implies that $M\xrightarrow{f}P$ is an associative
algebra object in $\mathcal{LM}$. The identities (36) and (37) implies that
$M\xrightarrow{f}P$ is a Lie algebra object in $\mathcal{LM}$. The last
identity (38) is the compatibility condition between the two structures in
$\mathcal{LM}$.

######  Proposition 4.9.

A Poisson algebra object $M\xrightarrow{f}P$ in the category $\mathcal{LM}$ is
equivalent to a Poisson-bimodule map $f:M\rightarrow P$, where $P$ is a
Poisson algebra and $M$ is a $P$-bimodule.

###### Proof.

The downstairs of the above conditions (35)-(37) is equivalent to the fact
that $P$ is an associative algebra with the product $\mu$ and a Lie algebra
with the bracket $\nu$. The downstairs of the identity (38) is equivalent to
the statement that the Lie bracket $\nu$ acts as a derivation of the
associative product $\mu$. Thus, $P$ is a Poisson algebra.

The upstairs of the identity (35) implies that there is an $A$-bimodule
structure on $M$ (given by the map $\mu^{\prime}$). Moreover, the upstairs of
the identities (36)-(37) yields a Lie module structure on $M$ (via the map
$\nu^{\prime}$). In the end, the upstairs of the equation (38) implies that
the Poisson identity (1) holds whenever two of the entries are in $P$ and one
of them is in $M$. So, $M$ is a Poisson-bimodule over $P$.

The map $\bar{\mu}$ and $\bar{\nu}$ are morphisms in $\mathcal{LM}$, which
equivalently means that

$f(\mu^{\prime}(m,x))=\mu(f(m),x),~{}\quad
f(\mu^{\prime}(x,m))=\mu(x,f(m)),\quad\mbox{and}\quad
f(\nu^{\prime}(x,m))=\nu(x,f(m)),\quad\forall m\in M,x\in P.$

Hence, Poisson algebra object $M\xrightarrow{f}P$ in the category
$\mathcal{LM}$ is equivalent to a Poisson-bimodule map $f:M\rightarrow P$.

∎

######  Theorem 4.10.

For a Poisson algebra object $M\xrightarrow{f}P$ in $\mathcal{LM}$, we have a
Poisson dialgebra structure on $M$. On the other hand, for any Poisson
dialgebra $\mathcal{P}$ the projection map
$\mathcal{P}\xrightarrow{p}\mathcal{P}_{\mathrm{Poiss}}$ is a Poisson algebra
object in $\mathcal{LM}$. The two functors $\mathsf{Pois-
di}\longleftrightarrow\mathsf{Pois}_{\mathcal{LM}}$ are adjoint to each-other.

###### Proof.

Let $M\xrightarrow{f}P$ be a Poisson algebra object in $\mathcal{LM}$.
Proposition 4.9 implies that the Poisson object $M\xrightarrow{f}P$ is
equivalent to a Poisson-bimodule map $f:M\rightarrow P$, where $P$ is a
Poisson algebra and $M$ is a $P$-bimodule. By Example 4.3, we have a Poisson
dialgebra structure $(M,\dashv,\vdash,[-,-]_{\mathsf{M}})$ on $M$ with the
dialgebra products

$m\dashv n=m\cdot f(n)\quad\mbox{and}\quad m\vdash n=f(m)\cdot n,\quad\forall
m,n\in M,$

and the (right) Leibniz bracket given by

$[m,n]_{\mathsf{M}}=[m,f(n)],\quad\forall m,n\in M.$

Here, the bracket $[-,-]$ on the right hand side is the Lie action of $P$ on
$M$. We denote by
$\mathsf{F}:\mathsf{Pois}_{\mathcal{LM}}\rightarrow\mathsf{Pois-di}$, the
functor that associates the Poisson dialgebra
$(M,\dashv,\vdash,[-,-]_{\mathsf{M}})$ to a Poisson object $M\xrightarrow{f}P$
in $\mathcal{LM}$. Also, denote $\mathsf{G}:\mathsf{Pois-
di}\rightarrow\mathsf{Pois}_{\mathcal{LM}}$, the functor associating the
Poisson object $\mathcal{P}\xrightarrow{p}\mathcal{P}_{\mathrm{Poiss}}$ to a
Poisson dialgebra $\mathcal{P}$.

Next, we need to show that there is a bijection between the sets of
homomorphisms

$\mathrm{Hom}_{\mathsf{Pois}_{\mathcal{LM}}}(\mathcal{P}\xrightarrow{p}\mathcal{P}_{\mathrm{Poiss}},M\xrightarrow{f}P)\cong\mathrm{Hom}_{\mathsf{Pois-
di}}(\mathcal{P},M).$

Since the Poissonization functor $\mathrm{(-)_{Poiss}}:\mathsf{Pois-
di}\rightarrow\mathsf{Pois}$ is left adjoint to the functor
$\mathsf{inc}:\mathsf{Pois}\rightarrow\mathsf{Pois-di}$, the bijection follows
from a similar argument as in the proof of Theorem 2.9.

Moreover, the above bijection is functorial in $\mathcal{P}\in\mathsf{Pois-
di}$ and $M\xrightarrow{f}P\in\mathsf{Pois}_{\mathcal{LM}}$ since for any
morphism $\alpha:\mathcal{P}\rightarrow\mathcal{P}^{\prime}$ in $\mathsf{Pois-
di}$ and
$\beta:(M\xrightarrow{f}P)\rightarrow(M^{\prime}\xrightarrow{f^{\prime}}P^{\prime})$
in $\mathsf{Pois}_{\mathcal{LM}}$, we have the following commutative diagram:

$\begin{CD}\mathrm{Hom}_{\mathsf{Pois}_{\mathcal{LM}}}(\mathsf{G}(\mathcal{P}^{\prime}),M\xrightarrow{f}P)@>{\mathsf{G}(\alpha)^{*}}>{}>\mathrm{Hom}_{\mathsf{Pois}_{\mathcal{LM}}}(\mathsf{G}(\mathcal{P}),M\xrightarrow{f}P)@>{\beta_{*}}>{}>\mathrm{Hom}_{\mathsf{Pois}_{\mathcal{LM}}}(\mathsf{G}(\mathcal{P}),M^{\prime}\xrightarrow{f^{\prime}}P^{\prime})\\\
@V{\cong}V{}V@V{\cong}V{}V@V{\cong}V{}V\\\ \mathrm{Hom}_{\mathsf{Pois-
di}}(\mathcal{P}^{\prime},\mathsf{F}(M\xrightarrow{f}P))@>{\alpha^{*}}>{}>\mathrm{Hom}_{\mathsf{Pois-
di}}(\mathcal{P},\mathsf{F}(M\xrightarrow{f}P))@>{\mathsf{F}(\beta)_{*}}>{}>\mathrm{Hom}_{\mathsf{Pois-
di}}(\mathcal{P},\mathsf{F}(M^{\prime}\xrightarrow{f^{\prime}}P^{\prime})),\end{CD}$

where $\alpha^{*}$ and $\beta_{*}$ are the maps induced by $\alpha$ and
$\beta$.

∎

##  5\. Homotopy algebra structures associated to Poisson dialgebras

In Section $3$, we discussed that there is an associative $2$-algebra
$(D\oplus I,\mu_{1},\mu_{2},\mu_{3})$ associated to a dialgebra $D$. In this
section, we show that there is a graded vector space associated to a Poisson
dialgebra, which carries both a Lie $2$-algebra structure and an associative
$2$-algebra structure. In the end, we consider the particular case when the
left and right product of the Poisson dialgebra becomes identical, and we
obtain an associated homotopy Poisson algebra of finite type. We first recall
the definition of a Lie $2$-algebra.

######  Definition 5.1 ([2]).

A graded vector space $\mathcal{G}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ is
called a Lie $2$-algebra, if $\mathcal{G}$ is equipped with linear maps
${l_{i}:\wedge^{i}\mathcal{G}\rightarrow\mathcal{G}}$ of degrees $i-2$, for
$i=1,2,3$ and the following identities hold.

  1. (i)

$l_{1}l_{2}(x,a)=l_{2}(x,l_{1}(a))$,

  2. (ii)

$l_{2}(l_{1}(a),b)=l_{2}(a,l_{1}(b))$,

  3. (iii)

$l_{2}(x,l_{2}(y,z))+l_{2}(y,l_{2}(z,x))+l_{2}(z,l_{2}(x,y))=l_{1}l_{3}(x,y,z)$,

  4. (iv)

$l_{2}(x,l_{2}(y,a))+l_{2}(y,l_{2}(a,x))+l_{2}(a,l_{2}(x,y))=l_{3}(x,y,l_{1}(a))$,

  5. (v)

$l_{3}(l_{2}(x,y),z,w)+l_{3}(l_{2}(y,z),w,x)+l_{3}(l_{2}(z,w),x,y)+l_{3}(l_{2}(w,x),y,z)=l_{2}(l_{3}(x,y,z),w)+l_{2}(l_{3}(y,z,w),x)+l_{2}(l_{3}(z,w,x),y)$,

for all $a,b\in\mathfrak{g}_{0}$ and $x,y,z,w\in\mathfrak{g}_{1}$.

Let $\mathfrak{g}$ be a Leibniz algebra and the set

$Z(\mathfrak{g})=\\{x\in\mathfrak{g}~{}|~{}[y,x]=0,\quad\forall
y\in\mathfrak{g}\\}$

denotes the right center of the Leibniz algebra $\mathfrak{g}$. We consider
the graded vector space $\mathfrak{g}\oplus Z(\mathfrak{g})$, where
$\mathfrak{g}$ is degree $0$ and the center is of degree $1$. Then let us
recall the following result from [15].

######  Proposition 5.2.

The tuple $(\mathfrak{g}\oplus Z(\mathfrak{g}),l_{1},l_{2},l_{3})$ is a Lie
$2$-algebra, where the map $l_{1}:Z(\mathfrak{g})\rightarrow\mathfrak{g}$ is
the inclusion map and the maps $l_{2}$ and $l_{3}$ are given as follows:

$\displaystyle l_{2}(x,y)$
$\displaystyle=\frac{1}{2}([x,y]-[y,x]),\quad\forall x,y\in\mathfrak{g},$
$\displaystyle l_{3}(x,y,z)$
$\displaystyle=\frac{1}{4}\big{(}[[z,y],x]+[[x,z],y]+[[y,x],z]\big{)},\quad\forall
x,y,z\in\mathfrak{g}$

Let $\mathcal{P}:=(P,\dashv,\vdash,[-,-])$ be a Poisson dialgebra. We have
seen that the set

$I=\\{x\in P~{}|~{}x\vdash y=0=y\dashv x,\quad\forall y\in P\\}$

becomes an ideal in the underlying dialgebra structure $(P,\dashv,\vdash)$.
Denote by $Z(P)$, the right center of the underlying Leibniz algebra structure
$(P,[-,-])$ of $\mathcal{P}$. Say, $J$ be the intersection of the two ideals
$I$ and $Z(P)$. Then, we associate a graded vector space $P\oplus J$
associated to the Poisson dialgebra $\mathcal{P}$, where $P$ is degree $0$ and
$J$ is degree $1$. With the above notations, we have the following theorem.

######  Theorem 5.3.

For a Poisson dialgebra $\mathcal{P}:=(P,\dashv,\vdash,[-,-])$, the associated
graded vector space $P\oplus J$ has the following homotopy algebra structures.

  1. (1)

$(P\oplus J,\mu_{1},\mu_{2},\mu_{3})$ becomes an associative $2$-algebra,
where $\mu_{1}:J\rightarrow P$ is the inclusion map and the maps $\mu_{2}$,
$\mu_{3}$ are given by equations (13) and (15).

  2. (2)

$(P\oplus J,l_{1},l_{2},l_{3})$ is a Lie $2$-algebra, where
$l_{1}:J\rightarrow P$ is the inclusion map and the maps $l_{2}$, $l_{3}$ are
given as in Proposition 5.2.

###### Proof.

$(1).$ To prove the first statement, we only need to show that

$\mu_{2}(x,c),\mu_{2}(c,x)\in J,\quad\forall c\in J,x\in P$

and

$\mu_{3}(x,y,z)\in J,\quad\forall x,y,z\in P.$

By the definition of the map $\mu_{2}$, we have

$\mu_{2}(x,y)=\frac{1}{2}(x\dashv y+x\vdash y),\quad\forall x,y\in P.$

For any $c\in J=I\cap Z(P)$,

$[z,\mu_{2}(x,c)]=\frac{1}{2}[z,x\vdash c]=x\vdash[z,c]+[z,x]\dashv
c=0,\quad\forall z,x\in P.$

Similarly, $[z,\mu_{2}(c,x)]=0$ for all $z,x\in P$. Therefore,

$\mu_{2}(x,c),\mu_{2}(c,x)\in J,\quad\forall c\in J,x\in P$

Next, for all $x,y,z,w\in P$, we have the following expression

$\displaystyle[w,\mu_{3}(x,y,z)]$ $\displaystyle=$
$\displaystyle\frac{1}{4}[w,(x\dashv y)\vdash z-x\dashv(y\vdash z)]$
$\displaystyle=$ $\displaystyle\frac{1}{4}\Big{(}(x\dashv
y)\vdash[w,z]+[w,x\dashv y]\dashv z-[w,x]\dashv(y\vdash z)-x\vdash[w,y\vdash
z]\Big{)}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\Big{(}(x\dashv
y)\vdash[w,z]+([w,x]\dashv y)\dashv z+(x\vdash[w,y])\dashv
z-[w,x]\dashv(y\vdash z)-x\vdash(y\vdash[w,z])-x\vdash([w,y]\dashv z)\Big{)}$
$\displaystyle=$ $\displaystyle 0.$

It implies that $\mu_{3}(x,y,z)\in Z(P)$ for all $x,y,z\in P$. By Proposition
3.2, we know that $\mu_{3}(x,y,z)\in I$ for all $x,y,z\in P$. Hence,

$\mu_{3}(x,y,z)\in J,\quad\forall x,y,z\in P.$

$(2).$ Again, to prove the second statement, we need to simply verify that

$l_{2}(x,c)\in J,\quad l_{3}(x,y,z)\in J\quad\forall c\in J,~{}~{}x,y,z\in P.$

For any $c\in J$, we have the following expression

$l_{2}(c,x)\vdash z=\frac{1}{2}[c,x]\vdash z=[c\vdash z,x]-c\vdash[z,x]=0$

Also,

$z\dashv l_{2}(c,x)=\frac{1}{2}z\dashv[c,x]=[z\dashv c,x]-[z,x]\dashv c=0$

Thus, it is clear that $l_{2}(x,y)\in J$ whenever atleast one of the elements
$x$ and $y$ is in $J$.

Now, by the definition of the map $l_{3}$ and mixed skew-symmetry of the
bracket $[-,-]$, we get

$\displaystyle l_{3}(x,y,z)\vdash w$
$\displaystyle=\frac{1}{4}\big{(}[[z,y],x]+[[x,z],y]+[[y,x],z]\big{)}\vdash w$
$\displaystyle=\frac{1}{4}\big{(}[[z,y],x]-[[z,x],y]-[z,[y,x]]\big{)}\vdash w$
$\displaystyle=0,\quad\quad\forall x,y,z,w\in\mathfrak{g}.$

Similar calculation shows that $w\dashv l_{3}(x,y,z)=0$.

∎

###  5.1. Homotopy Poisson algebra associated to a reduced Poisson dialgebra

In this subsection, we consider the particular case of a Poisson dialgebra
$\mathcal{P}:=(P,\dashv,\vdash,[-,-])$ when the left and right dialgebra
products become identical, i.e.,

$x\dashv y=x\vdash y,\quad\forall x,y\in P.$

We denote $\mu(x,y):=x\dashv y=x\vdash y$. In this case, we call
$\mathcal{P}=(P,\mu,[-,-])$ a reduced Poisson dialgebra.

######  Remark 5.4.

Let $(P,\mu)$ is a unital associative algebra. With the mixed skew-symmetry
property, it follows that the reduced Poisson dialgebra simply becomes a
Poisson algebra. Therefore, we only consider non-unital associative algebras
in this section.

There are two homotopy algebra structures on the graded space $P\oplus J$ for
a Poisson dialgebra $\mathcal{P}$. It would be interesting to find a
compatibility relation between them, which is subject to further
investigation. However, we observe that if the Poisson dialgebra is reduced,
then we can obtain a homotopy Poisson algebra structure of finite type on
$P\oplus J$. We first recall the definition of a homotopy Poisson algebra from
[9, 12]. These algebras are closely related to homotopy Poisson manifolds [5]
and higher Poisson manifolds [4, 17].

######  Definition 5.5.

A homotopy Poisson algebra (of degree $0$) is a graded associative algebra
$\mathcal{A}$ with an $L_{\infty}$ algebra structure $\\{l_{k}\\}_{k\geq 1}$
on $\mathcal{A}$ such that the map $\phi:\mathcal{A}\rightarrow\mathcal{A}$
given by

$\phi(x)=l_{k}(x_{1},\ldots,x_{k-1},x),\quad\mbox{for
}x_{1},\ldots,x_{k-1},x\in\mathcal{A}$

is a derivation of degree $\varepsilon:=2-k+\sum_{i=1}^{k-1}|x_{i}|$, which
means that

$l_{k}(x_{1},\ldots,x_{k-1},xy)=l_{k}(x_{1},\ldots,x_{k-1},x)y+(-1)^{\varepsilon.|x|}xl_{k}(x_{1},\ldots,x_{k-1},y),\quad\forall
x,y\in\mathfrak{A}.$

A homotopy Poisson algebra is called of finite type if there exists some $m$
such that for $k>m$, $l_{k}=0$. A $2$-term homotopy Poisson algebra is a
homotopy Poisson algebra structure on a $2$-term graded associative algebra
$\mathcal{A}=A_{0}\oplus A_{1}$.

######  Proposition 5.6.

For a reduced Poisson dialgebra $(P,\mu,[-,-])$, the tuple $(P\oplus
J,\mu,l_{1},l_{2},l_{3})$ becomes a $2$-term homotopy Poisson algebra (of
degree $0$).

###### Proof.

The product $\mu$ is given by $\mu(x,y):=x\dashv y=x\vdash y$, then the graded
vector space $P\oplus J$ becomes a graded associative algebra with the product
$\mu$ concentrated in degree $0$. Moreover, the tuple $(P\oplus
J,l_{1},l_{2},l_{3})$ is a Lie $2$-algebra, where $l_{1}:J\rightarrow P$ is
the inclusion map and the maps $l_{2}$, $l_{3}$ are given as in Proposition
5.2.

A straightforward calculation shows that for $x,y\in P\oplus J$, the map
$z\mapsto l_{2}(x,z)$ is a derivation of degree $|x|$ and the map $z\mapsto
l_{3}(x,y,z)$ is derivation of degree $|x|+|y|-1$. Hence, the tuple $(P\oplus
J,\mu,l_{1},l_{2},l_{3})$ becomes a $2$-term homotopy Poisson algebra (degree
$0$). ∎

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# Search for environment-dependent dilatons
_Preprint Version_

Hauke Fischer, Christian Käding, René I.P. Sedmik, Hartmut Abele
Atominstitut, Technische Universität Wien, Stadionallee 2, A-1020 Vienna,
Austria  Philippe Brax  Institut de Physique Théorique, Université Paris-
Scalay, CEA, CNRS, F-91191 Gif/Yvette Cedex, France  Mario Pitschmann
<EMAIL_ADDRESS>Atominstitut, Technische Universität Wien,
Stadionallee 2, A-1020 Vienna, Austria

###### Abstract

The environment-dependent dilaton field is a well-motivated candidate for dark
energy and naturally arises in the strong coupling limit of string theory. In
this article, we present the very first experimental constraints on the
parameters of this model. For this, we employ data obtained from the qBounce
collaboration and the Lunar Laser Ranging (LLR) experiment. Furthermore, we
forecast expected exclusion plots for the Casimir And Non Newtonian force
EXperiment (Cannex) soon to be realised in an improved setup. Finally, we
provide a detailed analysis of the screening mechanism and additional
symmetries of the dilaton field theory.

###### pacs:

98.80.-k, 04.80.Cc, 04.50.Kd, 95.36.+x

##  I Introduction

The origin of dark energy is one of the greatest puzzles in modern physics.
Unexpectedly, type Ia supernovae data have revealed that our Universe is
currently expanding at an accelerated rate [1, 2, 3]. This has been confirmed
by many other cosmological probes.

The theoretical framework describing the Universe on cosmological scales is
general relativity (GR). As GR is a crucial ingredient in the interpretation
of cosmological observations, it seems natural that modifying GR could be at
the heart of the observed accelerated expansion of the Universe. While a
modification at short distances is indeed easily realisable by extending the
Einstein-Hilbert action with quantities invariant under general coordinate
transformations and containing higher derivatives of the metric (see e.g.
[4]), a modification for large distance scales by making the theory massive is
very intricate [5]. Amending GR by the so-called cosmological constant
$\Lambda$ allows one to describe the accelerated expansion. However, such a
procedure would lead to a severe fine-tuning problem [6]. Consequently, the
existence of new hypothetical scalar fields has been postulated, which couple
to gravity and can account for dark energy [7]. Those new scalars generically
lead to new interactions, so-called fifth forces and are theoretically well-
motivated irrespective of their role for dark energy. As they have avoided
detection in past fifth force experiments, they must be subject to a screening
mechanism. Several such screening mechanisms have been devised, such as the
chameleon [8, 9], K-mouflage [10, 11], Vainshtein [12] and Damour-Polyakov
mechanisms [13].

In this article, we investigate the dilaton model with a Damour-Polyakov
mechanism. This is a screened scalar field model whose behaviour in local
tests of gravity has been less studied so far [13, 14, 15, 16, 17, 18, 19].
This model has been proposed as a possible candidate for dark energy [20, 21].
Its potential naturally arises in the strong coupling limit of string theory
and gives rise to a screening mechanism in connection with the Damour-Polyakov
mechanism. Due to its origin in string theory this model is particularly well-
motivated in comparison to similar models such as chameleons and symmetrons
(for a related investigation concerning symmetrons we refer to [22, 23, 24]).

Herein, we provide a brief summary of this model, discuss its screening
mechanism and parameter symmetries, followed by succinct descriptions of the
corresponding experiments and methods that we employ in order to constrain the
parameters of the dilaton. This article complements the theoretical analysis
presented in [25].

##  II The dilaton with Damour-Polyakov mechanism

The effective potential of the dilaton is given by [26]

$\displaystyle
V_{\text{eff}}(\phi;\rho)=V_{0}\,e^{-\lambda\phi/m_{\text{pl}}}+\beta(\phi)\,\frac{\rho}{2m_{\text{pl}}}\,\phi\>,$
(1)

where $V_{0}$ is a constant energy density, $\lambda$ a dimensionless
constant, $\beta(\phi)=A_{2}\phi/m_{\text{pl}}$ the full coupling to the
matter density $\rho$, $A_{2}$ a dimensionless coupling constant and
$m_{\text{pl}}$ the reduced Planck mass. Inside matter with density $\rho$,
the dilaton field approaches its minimum value given by

$\displaystyle\phi_{\rho}=\frac{m_{\text{pl}}}{\lambda}\,W\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho}\right)\>,$
(2)

where $W(x)$ is the Lambert $W$ function, which is the inverse function of
$xe^{x}$.

This potential is motivated from the string dilaton $\chi$ and the condition
$V(\chi)\rightarrow 0$ for $\chi\rightarrow\infty$, which is associated with
the strong coupling limit of string theory [14]. Hence, an asymptotic
expansion $V(\chi)=\tilde{V}_{0}\,e^{-\chi}+\tilde{V}_{1}\,e^{-2\chi}\ldots$
is applied. Furthermore, in Ref. [13] it has been assumed that the coupling to
matter has a minimum at some large value $\chi=\chi_{0}$. Consequently, near
the minimum the coupling is proportional to $(\chi-\chi_{0})^{2}$. Redefining
$\phi:=\frac{m_{\text{pl}}}{\lambda}(\chi-\chi_{0})$ leads to Eq. (1) (for a
full derivation see e.g. [27]). For the derivation of experimental limits we
demand the condition

$\displaystyle A_{2}\phi^{2}/(2m_{\text{pl}}^{2})\ll 1$ (3)

to hold in order to ensure that couplings to matter of higher order in $\phi$
can be neglected.

The parameter space of this model can naturally be divided into three regions
(see Appendix A.1). A large enough $\lambda$ (at fixed $V_{0}$ and $A_{2}$)
guarantees $e^{-\lambda\phi/m_{\text{pl}}}\ll 1$ and condition (3). Inside
this region the dilaton field primarily screens by increasing its mass in
dense environments. Additionally, there is an approximate symmetry between
$A_{2}$ and $V_{0}$; physical effects mainly depend on the product
$A_{2}\,\text{ln}(V_{0}/\rho)$, but not on the individual values of $V_{0}$
and $A_{2}$ (see Appendix A.2). This is evident in the obtained experimental
limits in Fig. (2) that shift systematically towards lower values of $A_{2}$
for increasing $V_{0}$. Condition (3) results in an ever stronger cut in the
parameter space for increasing values of $V_{0}$ (for the calculation of
limits a second cut-off was set to ensure that treating the experimental
setups as 1D is appropriate).

Figure 2: The exclusion plots typically separate into two distinct regimes;
Left: here limits for small values of the parameter $\lambda$ are plotted;
Right: exclusion limits for large $\lambda$ are depicted (for further
explanations we refer to the main text);
LLR: exclusion plots are filled areas in the bottom left in each region;
limits obtained from violations of the equivalence principle are surrounded by
solid lines, while limits from the precession of the lunar perigee are
encompassed by dashed lines;
qBounce: exclusion plots are filled areas in the top right in each region;
lighter areas correspond to micron screening and darker ones to fermi
screening;
Cannex: prospective limits are surrounded by dotted lines; the two areas
plotted right correspond to $\log_{10}(V_{0}/{\rm MeV}^{4})=1$ and $10^{24}$,
respectively.

If $\lambda$ is small enough then $e^{-\lambda\phi/m_{\text{pl}}}\simeq 1$ and
(3) holds. The dilaton field has a functional dependence only on the product
of parameters $V_{0}\lambda$ rather than on the individual parameters $V_{0}$
or $\lambda$, and the screening of the field in this region is primarily due
to the decrease of its matter coupling $\beta(\phi)$ in dense environments
(see Appendix A.1). Hence, computed limits in Fig. (2) simply shift towards
lower values of $\lambda$ for increasing values of $V_{0}$ without changing
their shapes, as long as $\lambda V_{0}$ is kept constant.

In between these two regions, for intermediate values of $\lambda$, there is a
region where $A_{2}\phi^{2}/(2m_{\text{pl}}^{2})\gg 1$ and, consequently, this
effective dilaton model is outside its range of applicability. However, for
$V_{0}\ll 1$ MeV4 the distinct experimental limits in Fig. (2) merge. The
merging point depends on the specific experiment and on the vacuum density
employed, but is qualitatively at $V_{0}\sim 10^{-20}$ MeV4. For much lower
values of $V_{0}$ physical effects become weak and all experimental limits
quickly disappear.

Tabletop experiments in a vacuum chamber play an important role in the search
for screened scalar fields such as the dilaton. This follows from the low
matter density within the vacuum chamber ensuring that the scalar field is
less suppressed there than in dense environments, while sufficiently thick
chamber walls effectively shield any influences from the outside world for a
large region of parameter space. The same techniques have been utilised
previously for experimental searches for chameleons [28, 29] and symmetrons
[22]. Furthermore, screened scalar fields with comparably small interaction
ranges can be probed better with tabletop experiments than via astrophysical
searches.

##  III The qBounce experiment

In qBounce [30, 31, 28] ultracold neutrons, which are totally reflected from
most materials, are bouncing in the gravitational field of the Earth. The
discrete energy levels are not equidistant, which allows to perform resonance
spectroscopy in the gravitational field. In its realization corresponding to a
Rabi setup [29], neutrons pass through three regions: The first region acts
effectively as a state selector and has a length of around 15 cm. A polished
mirror at the bottom and a rough scatterer at a height of 20 $\mu$m on top
ensure that only neutrons in the lowest few states can pass. Unwanted higher
energy states are scattered out of the system. In the second region, neutrons
pass a vibrating mirror with tunable frequency $\omega$ that can drive the
neutron towards a higher energy state. This region has a length of 20 cm. The
final region is identical to the first region (see Fig. 1 for a schematic
setup).

Figure 1: Sketch of qBounce

If the energy $\hbar\omega$ associated with the frequency of the mirror is
close to the energy $\Delta E_{n}=E_{n}-E_{1}$ needed to drive the neutron to
a specific higher energy state, the system enters a coherent superposition of
the ground state and this excited state. If the neutron is not in one of the
lowest $\sim 2$ states anymore when entering the last region, a loss in
transmission is observed.

Since neutrons are electrically neutral and have very low polarizability, they
are very insensitive to experimental background disturbances. Hence, qBounce
is a highly sensitive probe for new physics and has already been used to probe
and set stringent limits on many hypothetical new interactions [32]. Here,
qBounce is employed for the first time to set limits on the dilaton field. The
presence of the latter would induce energy shifts that can directly be
obtained from the stationary Schrödinger equation. Due to the comparatively
large extension of the mirrors, the setup can safely be approximated as one
dimensional, in which case the stationary Schrödinger equation reads

$\displaystyle\bigg{[}-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
z^{2}}+mgz+\mathfrak{Q}\,\frac{A_{2}}{2}\frac{m}{m_{\text{pl}}^{2}}\,\phi^{2}(z)\bigg{]}\,\Psi_{n}(z)$
$\displaystyle=E_{n}\Psi_{n}(z)\>.$ (4)

In general, this is a two-body problem since the mirror as well as the neutron
interact with the dilaton field. We approximate this problem by treating the
neutron as a sphere and extracting a “screening charge” $\mathfrak{Q}$, which
multiplies the dilaton potential and approximately describes the interaction
of the neutron with the dilaton. For further details and an explicit
expression for $\mathfrak{Q}$ we refer to the accompanying article [25]. Two
limiting cases are considered depending on whether the neutron is described as
a sphere of radius 0.5 fm in agreement with QCD (“fermi screening”) or 5.9
$\mu$m corresponding to the natural extend of the wave function (“micron
screening”). We assume that the true coupling lies within the boundaries
provided by these two limiting cases.

For the calculation of the dilaton-induced energy shift, perturbation theory,
as has been detailed in [25], is not applicable for a large part of the
parameter space since the computed effects of the dilaton field can be very
large. Therefore, the eigenvalue problem associated with the stationary
Schrödinger equation has been solved numerically to allow for a non-
perturbative treatment. Details on this procedure can be found in Appendix
A.5.

The experimental sensitivity achieved in the Rabi-like setup corresponds to an
energy resolution of $\Delta E=2\times 10^{-15}$ eV in a vacuum chamber with a
pressure of $2\times 10^{-4}$ mbar. This sensitivity allows us to exclude a
large part of the 3D parameter space of the dilaton field as shown in Fig. 2.

##  IV Lunar Laser Ranging

Lunar Laser Ranging (LLR) measures the distance between the surfaces of the
Earth and the Moon with high precision. This method involves firing a laser
beam at a retroreflector array installed on the lunar surface during the
Apollo missions. The retroreflectors consist of a series of small mirrors that
reflect the laser beam back to Earth [33].

Measuring the time it takes for the laser pulse to propagate to the Moon and
back provides the distance between the two bodies with an accuracy of a few
centimeters. This data has been used to measure the Moon’s orbit to high
experimental precision, which allows to test GR and set stringent limits on
any alternative theories. To date, the data is compatible with GR, which
necessitates that scalar fields with a non-minimal coupling to matter, if they
exist, must have a screening mechanism.

Lunar Laser Ranging has been used to test the equivalence principle.
Similarly, deviations from the inverse-square law of gravity would induce
shifts in the precession of the lunar perigee. The experimental constraint for
equivalence principle violations of the Earth (♁) and Moon (☾) in the field of
the Sun (☼) is given by [25, 34]

$\displaystyle\delta_{\text{em}}\simeq\frac{|\vec{a}_{\phi\earth}-\vec{a}_{\phi\leftmoon}|}{|\vec{a}_{G}|}\leq
2\times 10^{-13}\>,$ (5)

where $\vec{a}_{\phi}$ refers to the dilaton-induced acceleration towards the
Sun in addition to the Newtonian acceleration $\vec{a}_{G}$. A second
constraint is placed on any shift of the precession of the lunar perigee given
by

$\displaystyle\left|\frac{\delta\Omega}{\Omega}\right|$
$\displaystyle\simeq\left|\frac{R^{2}}{GM_{\earth}}\,\big{(}\delta
f(R)+\frac{R}{2}\,\delta f^{\prime}(R)\big{)}\right|$ $\displaystyle\leq
6.23833\times 10^{-12}\>,$ (6)

where $\delta f$ is the centripetal dilaton force per mass. For the numerical
generation of the corresponding dilaton limits we used the analytical results
from Ref. [25]. The obtained exclusion volume is shown in Fig. 2.

##  V The Cannex experiment

The Casimir And Non-Newtonian force EXperiment (Cannex) is currently being
rebuilt at the Conrad Observatory in Austria [35]. It is especially designed
to measure the Casimir force with unprecedented accuracy as well as fifth
forces due to hypothetical new interactions, and gravity. The experimental
setup consists of two plane parallel plates in close proximity, and allows to
measure induced forces and their gradients between these plates in direct or
Cavendish configuration (see Fig. 3 for a schematic setup).

Figure 3: Schematic cut view of the Cannex setup in direct configuration
(without electrostatic shield between the plates). Forces are detected using
Fabry Pérot interferometers sensing the extension of the mass-spring system
created by the helical springs and the upper plate. The insert on the left
defines the material and thickness of the various layers. Note that the upper
plate and the springs are coated in addition on all sides with a thin
($50\text{\,}\mathrm{nm}$) layer of gold.

Due to the geometry of two truly parallel plates, force generation by any
interaction is maximized. With an effective area of
$1\text{\,}\mathrm{c}\mathrm{m}^{2}$ and a targeted sensitivity of
$0.1\text{\,}\mathrm{n}\mathrm{N}\mathrm{/}\mathrm{m}^{2}$ at separations
between 3 and $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$, the Casimir effect
as well as several hypothetical interactions could be measured at
unprecedented accuracy [35]. By varying the pressure of Xe gas, the vacuum
density surrounding the plates can be tuned between 5.3$\times 10^{-12}$
kg/$\text{m}^{3}$ and 0.0026 kg/$\text{m}^{3}$. This variability allows for
relative measurements triggering the distinctive feature of hypothetical new
scalar fields with non-minimal coupling to matter – their strong sensitivity
to ambient densities. Cannex therefore will be a powerful tool in the search
for such interactions. In one dimension, the setup can approximately be
modeled as a half space with density
$\rho_{M}=$2514\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$$ for
$z\leq-d$, a vacuum region with density $\rho_{V}$ for $-d<z<d$, an upper
plate with density $\rho_{M}$ for $d<z<d+D$, and a vacuum region with density
$\rho_{V}$ for $z>d+D$. The upper plate has a thickness of
$D=$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and is movable, such that
$$1.5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$<d<$15\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. If dilatons indeed exist,
they would induce an additional pressure between the plates. To compute this
pressure, the corresponding differential equation for the dilaton field

$\displaystyle\frac{d^{2}\phi}{dz^{2}}+\frac{\lambda
V_{0}}{m_{\text{pl}}}\,e^{-\lambda\phi/m_{\text{pl}}}-\frac{A_{2}\rho(z)}{m_{\text{pl}}^{2}}\,\phi=0\>,$
(7)

has been solved numerically for all parameters of interest.

For further details on the simulations and the pressure calculation we refer
to Appendices A.3 and A.4. An example of a simulated dilaton field for the
Cannex setup is provided in Fig. 4.

Figure 4: Simulated dilaton field in between the parallel plates of the Cannex
setup for $\lambda=10^{31},A_{2}=10^{45}$ and $V_{0}=10$ MeV4; the lower
(yellow) and upper (blue) plates are indicated in color.

##  VI Dilaton dark energy

Requiring that the dilaton provides the vacuum energy accounting for dark
energy results in a reduction of parameter space to two dimensions, where the
condition
$V_{\text{eff}}(\phi_{V};\rho_{V})=3\Omega_{\Lambda_{0}}m_{\text{pl}}^{2}H_{0}^{2}$
holds for the cosmological vacuum density $\rho_{V}$ with the corresponding
field minimum $\phi_{V}$. This idea has been detailed in Ref. [25], where it
has been shown that $V_{0}$ can then be expressed in analytically closed form
as a function of $\lambda$ and $A_{2}$. The numerical analysis shows that such
dark energy dilatons violate condition (3) inside the entire parameter region
where $e^{-\lambda\phi_{V}/m_{\text{pl}}}\ll 1$ for the cosmological vacuum
density $\rho_{V}$. Interestingly, $A_{2}\phi_{V}^{2}/(2m_{\text{pl}}^{2})\sim
1$ is roughly constant in this region. The larger part of the experimentally
feasible parameter space where $e^{-\lambda\phi_{V}/m_{\text{pl}}}\simeq 1$
also violates condition (3). This is the reason why there are only comparably
small excluded areas for lunar laser ranging (see Fig. 5), while there are no
other limits for the tabletop experiments considered herein.

Figure 5: Limits for the dilaton field as the source of dark energy. Only LLR
can set limits in this case. Inside the plotted region $V_{0}$ takes the value
$V_{0}\simeq 3\Omega_{\Lambda_{0}}m_{\text{pl}}^{2}H_{0}^{2}$.

However, if the dilaton field were to contribute only 10% or less to the dark
energy, condition (3) would pose no strong restrictions any more, which would
allow to exclude large areas of the 2D parameter space for all investigated
experiments in this case.

##  VII Discussion

The analysis provided herein shows that LLR is sensitive to the dilaton field
for interaction ranges in vacuum of approximately 1 AU and larger, while the
tabletop experiments considered herein can probe the field for ranges as low
as 1 $\mu$m in agreement with expectations. In the future, Cannex will be able
to access a large part of the parameter space which is left open by qBounce
and LLR. If the dilaton is the only source of dark energy, only minor
measurable effects are expected.

The code that has been used to generate all obtained results is available at
[36].

##  VIII Acknowledgments

This article was supported by the Austrian Science Fund (FWF): P 34240-N,
P-33279-N, P36577-N, and is based upon work from COST Action COSMIC WISPers
CA21106, supported by COST (European Cooperation in Science and Technology).
We thank Tobias Jenke, measurements with qBounce were performed at the ultra-
cold beam position PF2@Institut Laue-Langevin, Grenoble.

##  Appendix A Supplementary materials

###  A.1 Derivation of the three parameter regions, the screening mechanisms
and the parameter symmetry

In this section, we describe the three regions of the parameter space obtained
by varying the magnitude of $\lambda$. Increasing $\lambda$ while keeping the
other parameters fixed eventually leads to

$\displaystyle\frac{\lambda^{2}V_{0}}{A_{2}\rho}\gg 1\>.$ (8)

Using $W(x)\simeq\text{ln}(x)-\text{ln}(\text{ln}(x))$ for large $x$ we can
approximate

$\displaystyle\phi_{\rho}\simeq\frac{m_{\text{pl}}}{\lambda}\left\\{\ln\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho}\right)-\ln\left[\ln\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho}\right)\right]\right\\},$
(9)

which shows that

$\displaystyle
e^{-\lambda\phi_{\rho}/m_{\text{pl}}}\simeq\text{ln}\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho}\right)\bigg{/}\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho}\right)\ll
1\>.$ (10)

The mass $\mu_{\rho}$ of the dilaton is given by [25]

$\displaystyle\mu_{\rho}$
$\displaystyle=\frac{1}{m_{\text{pl}}}\,\sqrt{\lambda^{2}V_{0}\,e^{-\lambda\phi_{\rho}/m_{\text{pl}}}+A_{2}\rho}$
$\displaystyle\simeq\frac{\sqrt{A_{2}\rho}}{m_{\text{pl}}}\sqrt{1+\ln\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho}\right)}$
$\displaystyle\simeq\frac{1}{m_{\text{pl}}}\sqrt{A_{2}\rho\,\ln\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho}\right)}\>.$
(11)

Then, the full coupling to matter is approximately

$\displaystyle\beta(\phi_{\rho})$
$\displaystyle=\frac{A_{2}\phi_{\rho}}{m_{\text{pl}}}$
$\displaystyle\simeq\frac{A_{2}}{\lambda}\left\\{\ln\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho}\right)-\ln\left[\ln\left(\frac{\lambda^{2}V_{0}}{A_{2}\rho}\right)\right]\right\\}.$
(12)

Since $\rho$ effects $\beta(\phi_{\rho})$ only logarithmically (as long as Eq.
(8) holds), while the mass has a square root dependence, increasing the
density primarily leads to an increase in the mass of the field but only a
negligible decrease of $\beta(\phi_{\rho}).$

Decreasing $\lambda$ inside this region increases $\phi_{\rho}$ according to
Eq. (9), which eventually leads to a violation of the condition
$A_{2}\phi^{2}/(2m_{\text{pl}}^{2})\ll 1$. Eventually, however, $\lambda$ gets
small enough such that $\lambda^{2}V_{0}/(A_{2}\rho)\ll 1$ holds. Hence, using
$W(x)\simeq x$ for small $x$, we obtain in this second region

$\displaystyle\phi_{\rho}$ $\displaystyle\simeq m_{\text{pl}}\,\frac{\lambda
V_{0}}{A_{2}\rho}\>,$ (13) $\displaystyle
e^{-\lambda\phi_{\rho}/m_{\text{pl}}}$ $\displaystyle\simeq
e^{-\frac{\lambda^{2}V_{0}}{A_{2}\rho}}\simeq 1\>,$ (14)
$\displaystyle\mu_{\rho}$
$\displaystyle\simeq\frac{\sqrt{A_{2}\rho}}{m_{\text{pl}}}\>,$ (15)
$\displaystyle\beta(\phi_{\rho})$ $\displaystyle\simeq\frac{\lambda
V_{0}}{\rho}\>.$ (16)

Decreasing $\lambda$ inside this second region decreases $\phi_{\rho}$ (in
contrast to the behaviour in the first region) and hence the condition
$A_{2}\phi^{2}/(2m_{\text{pl}}^{2})\ll 1$ is eventually fulfilled again.
Inside this parameter region, $\beta(\phi_{\rho})$ decreases considerably by
increasing $\rho$. Finally, since

$\displaystyle V_{\text{eff}}(\phi)$
$\displaystyle=V_{0}\,e^{-\lambda\phi/m_{\text{pl}}}+\frac{A_{2}\rho}{2m_{\text{pl}}^{2}}\,\phi^{2}$
$\displaystyle\simeq V_{0}-\lambda
V_{0}\,\frac{\phi}{m_{\text{pl}}}+\frac{A_{2}\rho}{2m_{\text{pl}}^{2}}\,\phi^{2}\>,$
(17)

only the product of $\lambda V_{0}$ enters the equations of motion, which
explains the parameter symmetry that was observed also numerically, i.e.
changing the parameters $\lambda$ and $V_{0}$ whilst keeping their product
$\lambda V_{0}$ fixed preserves the constraints on the parameter space for
small enough $\lambda$.

###  A.2 Additional explanation for the exclusion plots in the
$e^{-\lambda\phi/m_{\text{pl}}}\ll 1$ region

There is another approximate symmetry inside the
$e^{-\lambda\phi/m_{\text{pl}}}\ll 1$ region, which explains why the exclusion
plots shift systematically towards lower values of $A_{2}$ when increasing
$V_{0}$. To leading order the parameters $A_{2}$ and $V_{0}$ enter the full
coupling to matter (12) and the dilaton mass (11) via the same functional
dependence $A_{2}\ln\left(\lambda^{2}V_{0}/A_{2}\rho\right)$. In the excluded
regions in the main paper
$\ln\left(V_{0}/\rho\right)\gg\ln\left(\lambda^{2}/A_{2}\right)$ holds for
essentially all of the displayed parameter space. Hence,
$A_{2}\ln\left(\lambda^{2}V_{0}/A_{2}\rho\right)\simeq
A_{2}\ln\left(V_{0}/\rho\right)$. Therefore, the full coupling as well as the
dilaton mass essentially depend only the product
$A_{2}\ln\left(V_{0}/\rho\right)$, which is why there is an approximate
symmetry between these two parameters. Hence, increasing $V_{0}$ can
effectively be compensated by a corresponding decrease of $A_{2}$ as has been
observed in the excluded regions. In contrast, the precession of the lunar
perigee does not follow that symmetry. This is due to the sum of two physical
effects with opposite signs that cancel each other for larger values of
$V_{0}$ in this case.

###  A.3 Derivation of the pressure in the Cannex experiment

For numerical calculations we made use of the formula for the pressure $P_{z}$
on the upper plate of the Cannex setup

$\displaystyle
P_{z}=\frac{\rho_{M}}{\rho_{M}-\rho_{V}}\,\big{(}V_{\text{eff}}(\phi_{V},\rho_{V})-V_{\text{eff}}(\phi_{0},\rho_{V})\big{)}\>,$
(18)

where $\phi_{0}=\phi(0)$ is the dilaton field value at the center between both
plates and the effective potential is given by

$\displaystyle V_{\text{eff}}(\phi;\rho)=V(\phi)+\rho A(\phi)\>.$ (19)

In [25] the relation

$\displaystyle P_{z}$ $\displaystyle=\rho_{M}\,\big{(}\ln A(\phi(d))-\ln
A(\phi(d+D)\big{)}$
$\displaystyle\simeq\rho_{M}\,\big{(}A(\phi(d))-A(\phi(d+D)\big{)}\>,$ (20)

has been obtained, where in the second line $A(\phi)\simeq 1$ has been used,
which holds for all models of interest as e.g. dilatons, symmetrons and
chameleons. However, this relation has been found challenging to work with
numerically due to extreme slopes of the dilaton field near the mirror
surfaces.

Therefore, it turns out that the relation for the pressure Eq. (18) is more
convenient for numerical simulations. We detail its derivation in what
follows. Due to the screening mechanism, the field assumedly takes on its
minimum value $\phi_{M}$ inside the upper mirror of thickness $D$ (this has
been checked explicitly for all parameter values where limits have been set)
and the value of $\phi(d)$ is therefore to a very good approximation given by
the value at the surface of a two-mirror setup, where both mirrors are
infinitely extended with a vacuum region in between them. Analogously, the
value $\phi(d+D)$ is given by the value at the surface of the setup where one
mirror is infinitely extended with a vacuum region above. In [25] the
integrated equation of motion

$\displaystyle\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}-\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}\bigg{|}_{z=z_{0}}=V_{\text{eff}}(\phi;\rho)-V_{\text{eff}}(\phi;\rho)\big{|}_{z=z_{0}}\>,$
(21)

has been derived. For the one-mirror case we take the boundary conditions
$\phi(z)\rightarrow\phi_{M}$ for $z\rightarrow-\infty$ and
$\phi(z)\rightarrow\phi_{V}$ for $z\rightarrow\infty$. In the limit
$z\rightarrow\infty$ we get

$\displaystyle-\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}\bigg{|}_{z=z_{0}}=V_{\text{eff}}(\phi_{V};\rho_{V})-V_{\text{eff}}(\phi;\rho)\big{|}_{z=z_{0}}\>.$
(22)

Subtracting Eq. (22) from Eq. (21) gives inside the vacuum

$\displaystyle\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}=V_{\text{eff}}(\phi;\rho_{V})-V_{\text{eff}}(\phi_{V};\rho_{V})\>.$
(23)

Similarly, inside the mirror we find

$\displaystyle\frac{1}{2}\left(\frac{d\phi}{dz}\right)^{2}=V_{\text{eff}}(\phi;\rho_{M})-V_{\text{eff}}(\phi_{M};\rho_{M})\>.$
(24)

By continuity of the derivative at $z=d+D$ we straightforwardly obtain

$\displaystyle
A\big{(}\phi(d+D)\big{)}=\frac{1}{\rho_{M}-\rho_{V}}\,\big{(}V_{\text{eff}}(\phi_{M};\rho_{M})-V_{\text{eff}}(\phi_{V};\rho_{V})\big{)}\>.$
(25)

In case of the two infinitely extended mirrors we can use analogous reasoning,
using that $\partial\phi/\partial z|_{z=0}=0$ due to the symmetry of the setup
with $\phi_{0}:=\phi(0)$ being the value at the center between both mirrors.
This results in

$\displaystyle
A(\phi(d))=\frac{1}{\rho_{M}-\rho_{V}}\,\big{(}V_{\text{eff}}(\phi_{M};\rho_{M})-V_{\text{eff}}(\phi_{0};\rho_{V})\big{)}\>.$
(26)

Substituting these results into Eq. (20) proves Eq. (18).

###  A.4 Details of numerical simulations of the dilaton field for Cannex

For our numerical simulations we used Mathematica 13.1. We found that the
built-in NDSolve function for solving differential equations numerically does
not work well for simulating the dilaton field, or solving the Schrödinger
equation in the presence of a dilaton field. Therefore, we wrote our own code
adapted to solving these equations. We work with a non-uniform finite
difference method to approximate the second derivative of $\phi$ occurring in
both differential equations, namely [37]

$\displaystyle\phi_{i}^{\prime\prime}\approx\frac{2(\phi_{i+1}-\phi_{i})}{h_{i}(h_{i}+h_{i-1})}-\frac{2(\phi_{i}-\phi_{i-1})}{h_{i-1}(h_{i}+h_{i-1})}$
(27)

with $h_{i}:=x_{i+1}-x_{i}$ and the non-uniform approximation of the
simulation interval $x_{1},...,x_{N}$. For the one dimensional dilaton field
this results in the discretized differential equation

$\displaystyle\frac{2(\phi_{i+1}-\phi_{i})}{h_{i}(h_{i}+h_{i-1})}-\frac{2(\phi_{i}-\phi_{i-1})}{h_{i-1}(h_{i}+h_{i-1})}+$
$\displaystyle\frac{\lambda
V_{0}}{m_{\text{pl}}}\,e^{-\lambda\phi_{i}/m_{\text{pl}}}-\frac{A_{2}}{m_{\text{pl}}^{2}}\,\rho_{i}\phi_{i}=0\>.$
(28)

This is a non-linear system of equations on $\mathbb{R}^{N}$ that we solved
with a self-programmed Newton’s method. Boundary conditions were implemented
by setting $\phi_{0}=\phi_{N+1}=\phi_{M}$. This allowed us to use an arbitrary
mesh, which we fine-tuned for the dilaton field profiles. Unlike the built-in
finite element method that also allows arbitrary meshes, our algorithm is not
restricted to machine precision, but works with arbitrary precision, which is
a major advantage for the dilaton field. Furthermore, we found that the non-
linear FEM algorithms in Mathematica often fail to converge to the correct
solution without returning any error messages and are therefore unreliable.
Our code is freely available for investigation of any further details at [36].

###  A.5 Details for computing the energy shifts for qBounce

We used perturbation theory when applicable. In all other cases we discretized
the Hamilton operator using the same discretization method as explained for
Cannex. The corresponding discretized version of the stationary Schrödinger
equation is hence given by

$\displaystyle-\frac{1}{2m}\left[\frac{2(\Psi_{i+1}-\Psi_{i})}{h_{i}(h_{i}+h_{i-1})}-\frac{2(\Psi_{i}-\Psi_{i-1})}{h_{i-1}(h_{i}+h_{i-1})}\right]+V_{i}\Psi_{i}=E\Psi_{i}$
(29)

with $\displaystyle
V_{i}=\mathfrak{Q}\,\frac{A_{2}}{2}\frac{m_{N}}{m_{\text{pl}}^{2}}\,\phi^{2}(x_{i})$.
This results in a discrete approximation of the Hamilton operator given by

$\displaystyle
H_{ij}=\begin{cases}-\displaystyle\frac{1}{2m}\frac{2}{h_{i}(h_{i}+h_{i-1})}&\text{,
if }j=i+1\\\
\displaystyle-\frac{1}{2m}\frac{2}{h_{i-1}(h_{i}+h_{i-1})}&\text{, if
}j=i-1\\\ -H_{ii}-H_{i,i-1}+V_{i}&\text{, if }j=i\\\ 0&\text{,
else}\>.\end{cases}$ (30)

Boundary conditions can be implemented analogously to the dilaton field
simulation. Since the resulting approximation for the Hamilton operator is not
symmetric on non-uniform grids, in our code we applied a transformation to
restore symmetry following [37]. This procedure results in an eigenvalue
problem for a $N\times N$ matrix that can easily be solved numerically, and
returns all possible eigenstates and eigenvalues obtainable with the fineness
of the grid, from which we can safely extract the first and fourth energy
state, and the corresponding energies. Due to the high computational cost of
this procedure, we only computed around 10 points for the remaining non-
trivial edge (which does not come from a cut-off or can be obtained from
perturbation theory) of the exclusion area and fitted the result with a linear
function, which approximates the contour well. This procedure is justified
because the difference between fermi and micron screening, which is our error
guess of the edge of the exclusion area, is much larger than the error
introduced by our fit.

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