$\displaystyle\quad{}-\mathfrak{B}_{d}(a,\partial_{2}(b)\cdot\partial_{1}(c))-\mathfrak{B}_{d}(\eth_{1}^{*}(c^{*}),a\cdot\partial_{2}(b))-\mathfrak{B}_{d}(\eth_{2}^{*}(b^{*}),\partial_{1}(c)\cdot
a)$
$\displaystyle\quad{}-\mathfrak{B}_{d}(a,\eth_{2}^{*}(b^{*})\circ\eth_{1}^{*}(c^{*}))-\mathfrak{B}_{d}(a^{*},\partial_{2}(b)\cdot\partial_{1}(c))-\mathfrak{B}_{d}(\eth_{1}^{*}(c^{*})\circ
a^{*},\partial_{2}(b))$
$\displaystyle\quad{}-\mathfrak{B}_{d}(a^{*}\circ\eth_{2}^{*}(b^{*}),\partial_{1}(c))-\mathfrak{B}_{d}(a^{*},\eth_{2}^{*}(b^{*})\circ\eth_{1}^{*}(c^{*}))$
$\displaystyle=\langle
a,\eth_{1}^{*}(b^{*})\circ\eth_{2}^{*}(c^{*})-\eth_{2}^{*}(b^{*})\circ\eth_{1}^{*}(c^{*})\rangle+\langle
a^{*},\partial_{1}(b)\cdot\partial_{2}(c)-\partial_{2}(b)\cdot\partial_{1}(c)\rangle$
$\displaystyle\quad{}+\langle b,\partial_{1}^{*}(\eth_{2}^{*}(c^{*})\circ
a^{*})-\partial_{2}^{*}(\eth_{1}^{*}(c^{*})\circ a^{*})\rangle+\langle
b^{*},\eth_{1}(\partial_{2}(c)\cdot a)-\eth_{2}(\partial_{1}(c)\cdot
a)\rangle$ $\displaystyle\quad{}+\langle
c,\partial_{2}^{*}(\eth_{1}^{*}(b^{*})\circ
a^{*})-\partial_{1}^{*}(\eth_{2}^{*}(b^{*})\circ a^{*})\rangle+\langle
c^{*},\eth_{2}(\partial_{1}(b)\cdot a)-\eth_{1}(\partial_{2}(b)\cdot
a)\rangle.$

Similarly, we show that

$\displaystyle\mathfrak{B}_{d}([a+a^{*},b+b^{*}],c+c^{*})$
$\displaystyle=\langle a,\partial_{1}^{*}(\eth_{2}^{*}(b^{*})\circ
c^{*})-\partial_{2}^{*}(\eth_{1}^{*}(b^{*})\circ c^{*})\rangle+\langle
a^{*},\eth_{1}(\partial_{2}(b)\cdot c)-\eth_{2}(\partial_{1}(b)\cdot
c)\rangle$ $\displaystyle\quad{}+\langle
b,\partial_{2}^{*}(\eth_{1}^{*}(a^{*})\circ
c^{*})-\partial_{1}^{*}(\eth_{2}^{*}(a^{*})\circ c^{*})\rangle+\langle
b^{*},\eth_{2}(\partial_{1}(a)\cdot c)-\eth_{1}(\partial_{2}(a)\cdot
c)\rangle$ $\displaystyle\quad{}+\langle
c,\eth_{1}^{*}(a^{*})\circ\eth_{2}^{*}(b^{*})-\eth_{2}^{*}(a^{*})\circ\eth_{1}^{*}(b^{*})\rangle+\langle
c^{*},\partial_{1}(a)\cdot\partial_{2}(b)-\partial_{2}(a)\cdot\partial_{1}(b)\rangle.$

Therefore
$\mathfrak{B}_{d}(a+a^{*},[b+b^{*},c+c^{*}])=\mathfrak{B}_{d}([a+a^{*},b+b^{*}],c+c^{*})$
if and only if the following six equations hold:

$\displaystyle\eth_{1}^{*}(b^{*})\circ\eth_{2}^{*}(c^{*})-\eth_{2}^{*}(b^{*})\circ\eth_{1}^{*}(c^{*})=\partial_{1}^{*}(\eth_{2}^{*}(b^{*})\circ
c^{*})-\partial_{2}^{*}(\eth_{1}^{*}(b^{*})\circ c^{*}),$ (5.16)
$\displaystyle\partial_{1}^{*}(\eth_{2}^{*}(c^{*})\circ
a^{*})-\partial_{2}^{*}(\eth_{1}^{*}(c^{*})\circ
a^{*})=\partial_{2}^{*}(\eth_{1}^{*}(a^{*})\circ
c^{*})-\partial_{1}^{*}(\eth_{2}^{*}(a^{*})\circ c^{*}),$ (5.17)
$\displaystyle\partial_{2}^{*}(\eth_{1}^{*}(b^{*})\circ
a^{*})-\partial_{1}^{*}(\eth_{2}^{*}(b^{*})\circ
a^{*})=\eth_{1}^{*}(a^{*})\circ\eth_{2}^{*}(b^{*})-\eth_{2}^{*}(a^{*})\circ\eth_{1}^{*}(b^{*}),$
(5.18)
$\displaystyle\partial_{1}(b)\cdot\partial_{2}(c)-\partial_{2}(b)\cdot\partial_{1}(c)=\eth_{1}(\partial_{2}(b)\cdot
c)-\eth_{2}(\partial_{1}(b)\cdot c),$ (5.19)
$\displaystyle\eth_{1}(\partial_{2}(c)\cdot a)-\eth_{2}(\partial_{1}(c)\cdot
a)=\eth_{2}(\partial_{1}(a)\cdot c)-\eth_{1}(\partial_{2}(a)\cdot c),$ (5.20)
$\displaystyle\eth_{2}(\partial_{1}(b)\cdot a)-\eth_{1}(\partial_{2}(b)\cdot
a)=\partial_{1}(a)\cdot\partial_{2}(b)-\partial_{2}(a)\cdot\partial_{1}(b).$
(5.21)

Note that we have the following relationships:

$\displaystyle{\rm equation~{}\eqref{eq:pf151}}\Longleftrightarrow{\rm
equation~{}\eqref{eq:pf153}},\qquad$ $\displaystyle{\rm
equation~{}\eqref{eq:pf154}}\Longleftrightarrow{\rm
equation~{}\eqref{eq:pf156}},$ $\displaystyle{\rm
equation~{}\eqref{eq:pf151}}\Longrightarrow{\rm
equation~{}\eqref{eq:pf152}},\qquad$ $\displaystyle{\rm
equation~{}\eqref{eq:pf154}}\Longrightarrow{\rm equation~{}\eqref{eq:pf155}}.$

Hence
$\mathfrak{B}_{d}(a+a^{*},[b+b^{*},c+c^{*}])=\mathfrak{B}_{d}([a+a^{*},b+b^{*}],c+c^{*})$
if and only if equations (5.16) and (5.19) hold.

On the other hand, by Lemma 3.5, $\Psi$ is admissible to $(A,\cdot,\Phi)$ and
$\Phi^{*}$ is admissible to $(A^{*},\circ,\Psi^{*})$. By Corollary 2.9, we
have

$\displaystyle\partial_{1}(b)\cdot\partial_{2}(c)-\partial_{2}(b)\cdot\partial_{1}(c)-\eth_{1}(\partial_{2}(b)\cdot
c)+\eth_{2}(\partial_{1}(b)\cdot c)=\eth_{2}(\partial_{1}(b))\cdot
c-\eth_{1}(\partial_{2}(b))\cdot c,$
$\displaystyle\eth_{1}^{*}(b^{*})\circ\eth_{2}^{*}(c^{*})-\eth_{2}^{*}(b^{*})\circ\eth_{1}^{*}(c^{*})-\partial_{1}^{*}(\eth_{2}^{*}(b^{*})\circ
c^{*})+\partial_{2}^{*}(\eth_{1}^{*}(b^{*})\circ c^{*})$
$\displaystyle\qquad{}=\partial_{2}^{*}(\eth_{1}^{*}(b^{*}))\circ
c^{*}-\partial_{1}^{*}(\eth_{2}^{*}(b^{*}))\circ c^{*}.$

Hence equations (5.16) and (5.19) hold if and only if equations (5.5)–(5.6)
hold. By Remark 5.13, the conclusion follows. ∎

With the conditions above, if equations (5.6)–(5.7) hold, then the resulting
Manin triple of Poisson algebras is called the induced Manin triple of Poisson
algebras with respect to $(A\bowtie
A^{*},\star,\Phi+\Psi^{*},\mathfrak{B}_{d})$.

As in the study of matched pairs, given a double construction of commutative
differential Frobenius algebra $(A\bowtie
A^{*},\star,\Phi+\Psi^{*},\mathfrak{B}_{d})$, there is a more general
condition, such that $((A\oplus A^{*},\allowbreak[\ ,\
],\star,\mathfrak{B}_{d}),A,A^{*})$ is a Manin triple of Poisson algebras,
without the requirement that $(A\oplus A^{*},\allowbreak[\ ,\ ],\star)$ is the
induced Poisson algebra of $(A\bowtie A^{*},\star,\Phi+\Psi^{*})$. With a
similar study as the one of Proposition 5.14, we give the following conclusion
omitting the proof.

######  Proposition 5.19.

With the same assumptions in Proposition 5.18. Then there is a Manin triple
$((A\oplus A^{*},[\ ,\ ],\star,\mathfrak{B}_{d}),A,A^{*})$ of Poisson algebras
if and only equations (5.8)–(5.9) hold.

###  5.3 Poisson bialgebras via commutative and cocommutative differential
ASI bialgebras: the general case

Recall that a pair $(A,\delta)$ is called a Lie coalgebra, where $A$ is a
vector space and $\delta\colon A\to A\otimes A$ is a linear map, if $\delta$
is co-antisymmetric, in the sense that $\delta=-\sigma\delta$, and satisfies
the co-Jacobian identity:

$\big{(}\mathrm{id}+\tau+\tau^{2}\big{)}(\mathrm{id}\otimes\delta)\delta=0,$

where $\tau(a\otimes b\otimes c):=c\otimes a\otimes b$ for all $a,b,c\in A$. A
Lie bialgebra is a triple $(A,[\ ,\ ],\delta)$, where $(A,[\ ,\ ])$ is a Lie
algebra and $(A,\delta)$ is a Lie coalgebra, satisfying the following
equation:

$\displaystyle\delta([a,b])=(\mathrm{ad}_{A}(a)\otimes\mathrm{id}+\mathrm{id}\otimes\mathrm{ad}_{A}(a))\delta(b)-(\mathrm{ad}_{A}(b)\otimes\mathrm{id}+\mathrm{id}\otimes\mathrm{ad}_{A}(b))\delta(a),$
(5.22)

for all $a,b\in A$.

######  Definition 5.20 ([41]).

A Poisson coalgebra is a triple $(A,\delta,\Delta)$, where $(A,\delta)$ is a
Lie coalgebra and $(A,\Delta)$ is a commutative coalgebra such that the
following equation holds:

$(\mathrm{id}\otimes\Delta)\delta(a)=(\delta\otimes\mathrm{id})\Delta(a)+(\sigma\otimes\mathrm{id})(\mathrm{id}\otimes\delta)\Delta(a),\qquad\forall
a\in A.$

######  Remark 5.21.

The notion of a Poisson coalgebra is the dualization of the notion of a
Poisson algebra, that is, $(A,\delta,\Delta)$ is a Poisson coalgebra if and
only if $(A^{*},\delta^{*},\Delta^{*})$ is a Poisson algebra.

######  Lemma 5.22.

Let $(A,\Delta,\Psi\\!=\\!\\{\eth_{1},\eth_{2}\\})$ be a cocommutative
differential coalgebra. Then $(A,\delta,\Delta)$ is Poisson coalgebra, called
the induced Poisson coalgebra of $(A,\Delta,\Psi)$, where $\delta$ is defined
by

$\delta=(\eth_{1}\otimes\eth_{2}-\eth_{2}\otimes\eth_{1})\Delta.$ (5.23)

Moreover, $(A^{*},\delta^{*},\Delta^{*})$ is exactly the induced Poisson
algebra of the commutative differential algebra $(A^{*},\Delta^{*},\Psi^{*})$.

###### Proof.

By assumption, $(A^{*},\Delta^{*},\Psi^{*})$ is a commutative differential
algebra. Let $(A^{*},[\ ,\ ]_{A^{*}},\Delta^{*})$ be the induced Poisson
algebra of $(A^{*},\Delta^{*},\Psi^{*})$, that is, $[\ ,\ ]_{A^{*}}$ is
defined by

$[a^{*},b^{*}]_{A^{*}}:=\eth_{1}^{*}(a^{*})\circ\eth_{2}^{*}(b^{*})-\eth_{2}^{*}(a^{*})\circ\eth_{1}^{*}(b^{*}),\qquad\forall
a^{*},b^{*}\in A^{*},$

where $\circ=\Delta^{*}$. It is straightforward that $[\ ,\ ]_{A^{*}}$ is the
linear dual of $\delta$. Hence $(A^{*},\delta^{*},\Delta^{*})$ is the induced
Poisson algebra of $(A^{*},\Delta^{*},\Psi^{*})$ and thus $(A,\delta,\Delta)$
is a Poisson coalgebra. ∎

######  Definition 5.23 ([41]).

Let $(A,[\ ,\ ],\cdot)$ be a Poisson algebra. Suppose that it is equipped with
two comultiplications $\delta,\Delta\colon A\to A\otimes A$ such that
$(A,\delta,\Delta)$ is a Poisson coalgebra. If in addition, $(A,[\ ,\
],\delta)$ is a Lie bialgebra, $(A,\cdot,\Delta)$ is a commutative and
cocommutative ASI bialgebra, $\delta$ and $\Delta$ are compatible in the
following sense:

$\displaystyle\delta(a\cdot
b)=(L_{A}(a)\otimes\mathrm{id})\delta(b)+(L_{A}(b)\otimes\mathrm{id})\delta(a)$
$\displaystyle\hphantom{\delta(a\cdot
b)=}{}+(\mathrm{id}\otimes\mathrm{ad}_{A}(a))\Delta(b)+(\mathrm{id}\otimes\mathrm{ad}_{A}(b))\Delta(a),$
(5.24)
$\displaystyle\Delta([a,b])=(\mathrm{ad}_{A}(a)\otimes\mathrm{id}+\mathrm{id}\otimes\mathrm{ad}_{A}(a))\Delta(b)+(L_{A}(b)\otimes\mathrm{id}-\mathrm{id}\otimes
L_{A}(b))\delta(a),$ (5.25)

for all $a,b\in A$, then $(A,[\ ,\ ],\cdot,\delta,\Delta)$ is called a Poisson
bialgebra.

######  Theorem 5.24.

Let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a commutative
differential algebra and $(A,\Delta,\Psi=\\{\eth_{1},\eth_{2}\\})$ be a
cocommutative differential coalgebra. Let $(A,[\ ,\ ],\cdot)$ be the induced
Poisson algebra of $(A,\cdot,\Phi)$ and $(A,\delta,\Delta)$ be the induced
Poisson coalgebra of $(A,\Delta,\Psi)$, that is, $[\ ,\ ]$ is defined by
equation (5.1) and $\delta$ is defined by equation (5.23). Suppose that
$(A,\cdot,\Delta,\Phi,\Psi)$ is a commutative and cocommutative differential
ASI bialgebra. Then $(A,[\ ,\ ],\cdot,\delta,\Delta)$ is a Poisson bialgebra
if and only if equations (5.8)–(5.9) hold. In particular, if equations
(5.6)–(5.7) hold, then $(A,[\ ,\ ],\cdot,\delta,\Delta)$ is a Poisson
bialgebra.

###### Proof.

Let $a,b\in A$. Then we have

$\displaystyle[a,b]$
$\displaystyle\overset{\hphantom{(2.7)}}{=}\partial_{1}(a)\cdot\partial_{2}(b)-\partial_{2}(a)\cdot\partial_{1}(b)$
$\displaystyle\overset{\eqref{eq:qadm1}}{=}-\eth_{2}(\partial_{1}(a)\cdot
b)+\eth_{1}(\partial_{2}(a)\cdot b)+\eth_{2}(\partial_{1}(a))\cdot
b-\eth_{1}(\partial_{2}(a))\cdot b.$

Hence
$\mathrm{ad}_{A}(a)=L_{A}(\partial_{1}(a))\partial_{2}-L_{A}(\partial_{2}(a))\partial_{1}=\eth_{1}L_{A}(\partial_{2}(a))-\eth_{2}L_{A}(\partial_{1}(a))+L_{A}(S(a))$.
Meanwhile,

$\displaystyle\delta$
$\displaystyle\overset{\hphantom{(3.6)}}{=}(\eth_{1}\otimes\eth_{2}-\eth_{2}\otimes\eth_{1})\Delta$
$\displaystyle\overset{\eqref{eq:psadm2}}{=}(\mathrm{id}\otimes\eth_{2})((\mathrm{id}\otimes\partial_{1})\Delta-\Delta\partial_{1})-(\mathrm{id}\otimes\eth_{1})((\mathrm{id}\otimes\partial_{2})\Delta-\Delta\partial_{2})$
$\displaystyle\overset{\hphantom{(3.6)}}{=}(\mathrm{id}\otimes\eth_{1})\Delta\partial_{2}-(\mathrm{id}\otimes\eth_{2})\Delta\partial_{1}+(\mathrm{id}\otimes
S)\Delta.$

Therefore we have

$\displaystyle(L_{A}(a)\otimes\mathrm{id})\delta(b)+(L_{A}(b)\otimes\mathrm{id})\delta(a)+(\mathrm{id}\otimes\mathrm{ad}_{A}(a))\Delta(b)+(\mathrm{id}\otimes\mathrm{ad}_{A}(b))\Delta(a)-\delta(a\cdot
b)$
$\displaystyle\overset{\hphantom{(3.2)}}{=}(L_{A}(a)\otimes\mathrm{id})((\mathrm{id}\otimes\eth_{1})\Delta(\partial_{2}(b))-(\mathrm{id}\otimes\eth_{2})\Delta(\partial_{1}(b))+(\mathrm{id}\otimes
S)\Delta(b))$
$\displaystyle\quad{}+(L_{A}(b)\otimes\mathrm{id})((\mathrm{id}\otimes\eth_{1})\Delta(\partial_{2}(a))-(\mathrm{id}\otimes\eth_{2})\Delta(\partial_{1}(a))+(\mathrm{id}\otimes
S)\Delta(a))$
$\displaystyle\quad{}+(\mathrm{id}\otimes(\eth_{1}L_{A}(\partial_{2}(a))-\eth_{2}L_{A}(\partial_{1}(a))+L_{A}(S(a))))\Delta(b)$
$\displaystyle\quad{}+(\mathrm{id}\otimes(\eth_{1}L_{A}(\partial_{2}(b))-\eth_{2}L_{A}(\partial_{1}(b))+L_{A}(S(b))))\Delta(a)$
$\displaystyle\quad{}-(\mathrm{id}\otimes\eth_{1})\Delta(\partial_{2}(a\cdot
b))+(\mathrm{id}\otimes\eth_{2})\Delta(\partial_{1}(a\cdot
b))-(\mathrm{id}\otimes S)\Delta(a\cdot b)$
$\displaystyle\overset{\eqref{eq:asiifs}}{=}(\mathrm{id}\otimes\eth_{1})\Delta(a\cdot\partial_{2}(b)+\partial_{2}(a)\cdot
b)-(\mathrm{id}\otimes\eth_{2})\Delta(a\cdot\partial_{1}(b)+\partial_{1}(a)\cdot
b)$
$\displaystyle\quad{}-(\mathrm{id}\otimes\eth_{1})\Delta(\partial_{2}(a\cdot
b))+(\mathrm{id}\otimes\eth_{2})\Delta(\partial_{1}(a\cdot b))$
$\displaystyle\quad{}+(-(\mathrm{id}\otimes S)(\mathrm{id}\otimes
L_{A}(b))+L_{A}(b)\otimes S+\mathrm{id}\otimes
L_{A}(S(b)))\Delta(a)+(\mathrm{id}\otimes L_{A}(S(a)))\Delta(b)$
$\displaystyle\overset{\hphantom{(3.2)}}{=}(L_{A}(b)\otimes\mathrm{id}-\mathrm{id}\otimes
L_{A}(b))(\mathrm{id}\otimes S)\Delta(a)+(\mathrm{id}\otimes
L_{A}(S(a)))\Delta(b).$

Hence equation (5.24) holds if and only if equation (5.8) holds. Similarly we
show that equations (5.25) and (5.22) hold if and only if equations (5.8) and
(5.9) hold respectively. So the first conclusion follows. The particular case
is obvious. ∎

With the conditions above, if equations (5.6)–(5.7) hold, then $(A,[\ ,\
],\cdot,\delta,\Delta)$ is called the induced Poisson bialgebra of
$(A,\cdot,\Delta,\Phi,\Psi)$.

######  Remark 5.25.

Note that for a commutative and cocommutative differential ASI bialgebra
$(A,\cdot,\Delta,\Phi,\Psi)$, there is a commutative differential algebra
structure $(A\bowtie A^{*},\star,\Phi+\Psi^{*})$ on $A\oplus A^{*}$, and for a
Poisson bialgebra $(A,[\ ,\ ],\cdot,\delta,\Delta)$, there is a Poisson
algebra structure $(A\bowtie A^{*},[\ ,\ ],\cdot)$ on $A\oplus A^{*}$. Hence
with the conditions above, if equations (5.6)–(5.7) hold, then the resulting
induced Poisson bialgebra $(A,[\ ,\ ],\cdot,\delta,\Delta)$ of
$(A,\cdot,\Delta,\Phi,\Psi)$ satisfies an additional condition that the
Poisson structure on $A\oplus A^{*}$ is the induced Poisson algebra of
$(A\bowtie A^{*},\star,\allowbreak\Phi+\Psi^{*})$. So if one characterizes the
resulting Poisson bialgebra $(A,[\ ,\ ],\cdot,\delta,\Delta)$ of a commutative
and cocommutative differential ASI bialgebra $(A,\cdot,\Delta,\Phi,\Psi)$ to
be “induced” by this additional condition, then the converse still holds, that
is, $(A,[\ ,\ ],\cdot,\delta,\Delta)$ is the “induced” Poisson bialgebra in
this sense if and only if equations (5.6)–(5.7) hold.

######  Theorem 5.26 ([41, structure Theorem (II)]).

Let $(A,[\ ,\ ]_{A},\cdot)$ be a Poisson algebra. Suppose that there is a
Poisson algebra structure $(A^{*},[\ ,\ ]_{A^{*}},\circ)$ on $A^{*}$. Let
$\delta,\Delta\colon A\to A\otimes A$ denote the linear duals of the Lie
bracket $[\ ,\ ]_{A^{*}}\colon A^{*}\otimes A^{*}\to A^{*}$ and the
multiplication $\circ\colon A^{*}\otimes A^{*}\to A^{*}$ respectively. Then
the following conditions are equivalent:

  1. $(1)$

$(A,[\ ,\ ]_{A},\cdot,\delta,\Delta)$ is a Poisson bialgebra.

  2. $(2)$

$((A,[\ ,\ ]_{A},\cdot),(A^{*},[\ ,\
]_{A^{*}},\circ),-\mathrm{ad}_{A}^{*},L_{A}^{*},-\mathrm{ad}_{A^{*}}^{*},L_{A^{*}}^{*})$
is a matched pair of Poisson algebras.

  3. $(3)$

There is a Manin triple of Poisson algebras $((A\oplus A^{*},[\ ,\
],\star,\mathfrak{B}_{d}),A,A^{*})$ associated to $(A,[\ ,\ ]_{A},\cdot)$ and
$(A^{*},[\ ,\ ]_{A^{*}},\circ)$, where $\mathfrak{B}_{d}$ is defined by
equation (3.1).

By the equivalences given in Theorems 3.14 and 5.26, Proposition 5.14 and
Corollary 5.12 (in the induced case) are the “matched pair version” of Theorem
5.24 and Remark 5.25, whereas Propositions 5.19 and 5.18 (in the induced case)
are the “Manin triple (double construction) version” of the latter. We
illustrate these relationships in the “induced case” through a commutative
diagram. Explicitly, let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a
commutative differential algebra. Suppose that there is a commutative
differential algebra structure
$(A^{*},\circ,\Psi^{*}=\\{\eth_{1}^{*},\eth_{2}^{*}\\})$ on $A^{*}$. Let
$(A,[\ ,\ ]_{A},\cdot)$ and $(A^{*},[\ ,\ ]_{A^{*}},\circ)$ be the induced
Poisson algebras of $(A,\cdot,\Phi)$ and $(A^{*},\circ,\Psi^{*})$
respectively, that is, both $[\ ,\ ]_{A}$ and $[\ ,\ ]_{A^{*}}$ are defined by
equation (5.1). Let $\delta,\Delta\colon A\to A\otimes A$ be the linear duals
of $[\ ,\ ]_{A^{*}}$ and $\circ$ respectively. So $\delta$ satisfies equation
(5.23). Suppose that equations (5.6)–(5.7) hold. Then we have the following
commutative diagram:

$((A,\cdot,\Phi)$, $(A^{*},\circ,\Psi^{*})$,$L_{A}^{*},L_{A^{*}}^{*})$ a
matched pair of commutative differential algebras $\scriptstyle{{\rm
Cor.}~{}\ref{cor:adjassmat}}$$\scriptstyle{{\rm Thm.}~{}\ref{thm:asieqv}}$
$(A,\cdot,\Delta,\Phi,\Psi)$ a commutative and cocommutative differential ASI
bialgebra $\scriptstyle{{\rm Thm.}~{}\ref{thm:asieqv}}$$\scriptstyle{{\rm
Thm.}~{}\ref{thm:asspoibi}}$ $(A\bowtie A^{*},\star,\mathfrak{B}_{d})$ a
double construction of commutative differential Frobenius algebra
$\scriptstyle{{\rm Prop.}~{}\ref{pro:assFro}}$
$(A,A^{*},-\mathrm{ad}_{A}^{*},L_{A}^{*},-\mathrm{ad}_{A^{*}}^{*},L_{A^{*}}^{*})$
the induced matched pair of Poisson algebras $\scriptstyle{{\rm
Thm.}~{}\ref{thm:poibieq}}$ $(A,[\ ,\ ]_{A},\cdot,\delta,\Delta)$ the induced
Poisson bialgebra $\scriptstyle{{\rm Thm.}~{}\ref{thm:poibieq}}$ $((A\oplus
A^{*},[\ ,\ ],\star,\mathfrak{B}_{d}),A,A^{*})$ the induced Manin triple of
Poisson algebras

###  5.4 Coboundary Poisson bialgebras via coboundary commutative
and cocommutative differential ASI bialgebras

######  Definition 5.27 ([41]).

A Poisson bialgebra $(A,[\ ,\ ],\cdot,\delta,\Delta)$ is called coboundary if
$\delta$ and $\Delta$ are respectively defined by

$\displaystyle\delta(a):=(\mathrm{id}\otimes\mathrm{ad}_{A}(a)+\mathrm{ad}_{A}(a)\otimes\mathrm{id})(r),$
(5.26) $\displaystyle\Delta(a):=(\mathrm{id}\otimes
L_{A}(a)-L_{A}(a)\otimes\mathrm{id})(r),\qquad\forall a\in A,$ (5.27)

for some $r\in A\otimes A$.

######  Definition 5.28 ([41]).

Let $(A,[\ ,\ ],\cdot)$ be a Poisson algebra. $r\in A\otimes A$ is called a
solution of Poisson Yang–Baxter equation (PYBE) in $(A,[\ ,\ ],\cdot)$ if $r$
is a solution of both equation (4.9) and the following equation:

$[r_{12},r_{13}]+[r_{13},r_{23}]+[r_{12},r_{23}]=0,$ (5.28)

where for $r=\sum_{i}a_{i}\otimes b_{i}$, we denote

$\displaystyle[r_{12},r_{13}]=\sum_{i,j}[a_{i},a_{j}]\otimes b_{i}\otimes
b_{j},\qquad[r_{13},r_{23}]=\sum_{i,j}a_{i}\otimes a_{j}\otimes[b_{i},b_{j}],$
$\displaystyle[r_{12},r_{23}]=\sum_{i,j}a_{i}\otimes[b_{i},a_{j}]\otimes
b_{j}.$

######  Proposition 5.29 ([41, Theorem 2]).

Let $(A,[\ ,\ ],\cdot)$ be a Poisson algebra and $r\in A\otimes A$. Let
$\delta\colon A\to A\otimes A$ and $\Delta\colon A\to A\otimes A$ be two
linear maps defined by equations (5.26) and (5.27) respectively. If $r$ is an
antisymmetric solution of PYBE in $(A,[\ ,\ ],\cdot)$, then $(A,[\ ,\
],\cdot,\delta,\Delta)$ is a Poisson bialgebra.

######  Lemma 5.30.

Let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a
$\Psi=\\{\eth_{1},\eth_{2}\\}$-admissible commutative differential algebra and
$(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of $(A,\cdot,\Phi)$. Let
$r\in A\otimes A$ be an antisymmetric solution of $\Psi$-admissible AYBE in
$(A,\cdot,\Phi)$. Suppose that $\Delta$ is defined by equation (5.27). Then we
have the following conclusions.

  1. $(1)$

$(A,\Delta,\Psi)$ is a cocommutative differential coalgebra.

  2. $(2)$

If equation (5.6) holds, then both equation (5.7) and the following equation
hold:

$[a,b]=\eth_{2}(a\cdot\partial_{1}(b))-\eth_{1}(a\cdot\partial_{2}(b)),\qquad\forall
a,b\in A.$ (5.29)

###### Proof.

(1) It is straightforward to show that $(A,\Delta)$ is a cocommutative
coalgebra. For all $a\in A$ and $k=1,2$, we have

$\displaystyle(\eth_{k}\otimes\mathrm{id}+\mathrm{id}\otimes\eth_{k})\Delta(a)=(\eth_{k}\otimes\mathrm{id}+\mathrm{id}\otimes\eth_{k})(\mathrm{id}\otimes
L_{A}(a)-L_{A}(a)\otimes\mathrm{id})(r)$
$\displaystyle\qquad{}\overset{\eqref{eq:pqadm1}}{=}(\mathrm{id}\otimes
L_{A}(a)\partial_{k}+\mathrm{id}\otimes\eth_{k}L_{A}(a)-\eth_{k}L_{A}(a)\otimes\mathrm{id}-L_{A}(a)\partial_{k}\otimes\mathrm{id})(r)$
$\displaystyle\qquad{}\;\overset{\eqref{eq:qadm1}}{=}(\mathrm{id}\otimes
L_{A}(\eth_{k}(a))-L_{A}(\eth_{k}(a))\otimes\mathrm{id})(r)=\Delta(\eth_{k}(a)).$

Hence $\eth_{k}$ is a coderivation on $(A,\Delta)$. Therefore,
$(A,\Delta,\Psi)$ is a cocommutative differential coalgebra.

(2) Let $a,b\in A$. Then we have

$\displaystyle\eth_{1}(\partial_{2}(a\cdot b))$
$\displaystyle\overset{\hphantom{(0.0)}}{=}\eth_{1}(\partial_{2}(a)\cdot
b+a\cdot\partial_{2}(b))$
$\displaystyle\overset{\eqref{eq:qadm1}}{=}\eth_{1}(\partial_{2}(a))\cdot
b-\partial_{2}(a)\cdot\partial_{1}(b)+a\cdot\eth_{1}(\partial_{2}(b))-\partial_{1}(a)\cdot\partial_{2}(b),$
$\displaystyle\eth_{2}(\partial_{1}(a\cdot b))$
$\displaystyle\overset{\hphantom{(0.0)}}{=}\eth_{2}(\partial_{1}(a)\cdot
b+a\cdot\partial_{1}(b))$
$\displaystyle\overset{\eqref{eq:qadm1}}{=}\eth_{2}(\partial_{1}(a))\cdot
b-\partial_{1}(a)\cdot\partial_{2}(b)+a\cdot\eth_{2}(\partial_{1}(b))-\partial_{2}(a)\cdot\partial_{1}(b).$

Hence by assumption, we have $\eth_{1}(\partial_{2}(a\cdot
b))=\eth_{2}(\partial_{1}(a\cdot b))$, i.e.,
$\eth_{1}\partial_{2}L_{A}(a)=\eth_{2}\partial_{1}L_{A}(a)$. Therefore

$\displaystyle(\mathrm{id}\otimes\eth_{1}\partial_{2})\Delta(a)-(\mathrm{id}\otimes\eth_{2}\partial_{1})\Delta(a)=(\mathrm{id}\otimes\eth_{1}\partial_{2}-\mathrm{id}\otimes\eth_{2}\partial_{1})(\mathrm{id}\otimes
L_{A}(a)-L_{A}(a)\otimes\mathrm{id})(r)$
$\displaystyle=-(L_{A}(a)\otimes(\eth_{1}\partial_{2}-\eth_{2}\partial_{1}))(r)\overset{\eqref{eq:pqadm1}}{=}-(L_{A}(a)\eth_{2}\partial_{1}\otimes\mathrm{id})(r)+(L_{A}(a)\eth_{1}\partial_{2}\otimes\mathrm{id})(r)=0.$

That is, equation (5.7) holds. Moreover, we have

$\displaystyle\partial_{1}(a)\cdot\partial_{2}(b)+\eth_{1}(a\cdot\partial_{2}(b))-\partial_{2}(a)\cdot\partial_{1}(b)-\eth_{2}(a\cdot\partial_{1}(b))$
$\displaystyle\qquad{}\overset{\eqref{eq:qadm2}}{=}a\cdot\eth_{1}(\partial_{2}(b))-a\cdot\eth_{2}(\partial_{1}(b))\overset{\eqref{eq:vip1}}{=}0.$

Therefore equation (5.29) holds. ∎

######  Proposition 5.31.

Let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a
$\Psi=\\{\eth_{1},\eth_{2}\\}$-admissible commutative differential algebra and
$(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of $(A,\cdot,\Phi)$.
Suppose that equation (5.6) holds. Then every solution of $\Psi$-admissible
AYBE in $(A,\cdot,\Phi)$ is a solution of PYBE in the Poisson algebra $(A,[\
,\ ],\cdot)$.

###### Proof.

Suppose that $r=\sum_{i}a_{i}\otimes b_{i}$ is a solution of $\Psi$-admissible
AYBE in $(A,\cdot,\Phi)$. Then we have

$\displaystyle\sum_{i,j}[a_{i},a_{j}]\otimes b_{i}\otimes
b_{j}=\sum_{i,j}(\partial_{1}(a_{i})\cdot\partial_{2}(a_{j})-\partial_{2}(a_{i})\cdot\partial_{1}(a_{j}))\otimes
b_{i}\otimes b_{j}$
$\displaystyle\qquad{}\overset{\eqref{eq:pqadm1}}{=}\sum_{i,j}(a_{i}\cdot
a_{j}\otimes\eth_{1}(b_{i})\otimes\eth_{2}(b_{j})-a_{i}\cdot
a_{j}\otimes\eth_{2}(b_{i})\otimes\eth_{1}(b_{j}))$
$\displaystyle\qquad{}\overset{\eqref{eq:aybe}}{=}\sum_{i,j}(a_{i}\otimes\eth_{1}(b_{i}\cdot
a_{j})\otimes\eth_{2}(b_{j})-a_{i}\otimes\eth_{1}(a_{j})\otimes\eth_{2}(b_{i}\cdot
b_{j}))$
$\displaystyle\qquad\qquad{}-\sum_{i,j}(a_{i}\otimes\eth_{2}(b_{i}\cdot
a_{j})\otimes\eth_{1}(b_{j})-a_{i}\otimes\eth_{2}(a_{j})\otimes\eth_{1}(b_{i}\cdot
b_{j}))$
$\displaystyle\qquad{}\overset{\eqref{eq:pqadm1}}{=}\sum_{i,j}(a_{i}\otimes\eth_{1}(b_{i}\cdot\partial_{2}(a_{j}))\otimes
b_{j}-a_{i}\otimes a_{j}\otimes\eth_{2}(b_{i}\cdot\partial_{1}(b_{j})))$
$\displaystyle\qquad\qquad{}-\sum_{i,j}(a_{i}\otimes\eth_{2}(b_{i}\cdot\partial_{1}(a_{j}))\otimes
b_{j}-a_{i}\otimes a_{j}\otimes\eth_{1}(b_{i}\cdot\partial_{2}(b_{j})))$
$\displaystyle\qquad{}\overset{\eqref{eq:liepq}}{=}\sum_{i,j}(-a_{i}\otimes[b_{i},a_{j}]\otimes
b_{j}-a_{i}\otimes a_{j}\otimes[b_{i},b_{j}]).$

Hence $r$ satisfies equation (5.28). Therefore $r$ is a solution of PYBE in
$(A,[\ ,\ ],\cdot)$. ∎

Let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a
$\Psi=\\{\eth_{1},\eth_{2}\\}$-admissible commutative differential algebra and
$(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of $(A,\cdot,\Phi)$.
Suppose that equation (5.6) holds and $r\in A\otimes A$ is an antisymmetric
solution of $\Psi$-admissible AYBE in $(A,\cdot,\Phi)$. On the one hand, by
Corollary 4.9, there is a commutative and cocommutative differential ASI
bialgebra $(A,\cdot,\Delta,\Phi,\Psi)$, where $\Delta$ is defined by equation
(5.27). Furthermore, by Lemma 5.30, $(A,\Delta,\Psi)$ is a cocommutative
differential coalgebra and equation (5.7) holds, and thus by Theorem 5.24,
there is the induced Poisson bialgebra $(A,[\ ,\ ],\cdot,\delta,\Delta)$ of
$(A,\cdot,\Delta,\Phi,\Psi)$, where $\delta$ is defined by equation (5.23). On
the other hand, by Proposition 5.31, $r$ is an antisymmetric solution of PYBE
in the Poisson algebra $(A,[\ ,\ ],\cdot)$ and hence there is a Poisson
bialgebra $(A,[\ ,\ ],\cdot,\delta^{\prime},\Delta)$ by Proposition 5.29,
where $\delta^{\prime}$ is defined by equation (5.26).

######  Corollary 5.32.

With the conditions as above. Then the two Poisson bialgebras $(A,[\ ,\
],\cdot,\delta,\Delta)$ and $(A,[\ ,\ ],\cdot,\delta^{\prime},\Delta)$
coincide. Hence we have the following commutative diagram:

$r$ an antisymmetric solution of $\Psi$-admissible AYBE in $(A,\cdot,\Phi)$
$\scriptstyle{{\rm Prop.}~{}\ref{pro:aybe_to_qybe}}$$\scriptstyle{{\rm
Cor.}~{}\ref{cor:admAYBEdasi}}$ $(A,\cdot,\Delta,\Phi,\Psi)$ a commutative and
cocommutative differential ASI bialgebra $\scriptstyle{{\rm
Thm.}~{}\ref{thm:asspoibi}}$ $r$ an antisymmetric solution of PYBE in $(A,[\
,\ ],\cdot)$ $\scriptstyle{{\rm Prop.}~{}\ref{pro:pybebi}}$ $(A,[\ ,\
],\cdot,\delta,\Delta)$ the induced Poisson bialgebra

###### Proof.

Let $a\in A$. Then we have

$\displaystyle\delta(a)$
$\displaystyle\overset{\hphantom{(0.00)}}{=}(\eth_{1}\otimes\eth_{2}-\eth_{2}\otimes\eth_{1})\Delta(a)=(\eth_{1}\otimes\eth_{2}-\eth_{2}\otimes\eth_{1})(\mathrm{id}\otimes
L_{A}(a)-L_{A}(a)\otimes\mathrm{id})(r)$
$\displaystyle\overset{\eqref{eq:pqadm1}}{=}(\mathrm{id}\otimes\eth_{2}L_{A}(a)\partial_{1}-\mathrm{id}\otimes\eth_{1}L_{A}(a)\partial_{2})(r)-(\eth_{1}L_{A}(a)\partial_{2}\otimes\mathrm{id}-\eth_{2}L_{A}(a)\partial_{1}\otimes\mathrm{id})(r)$
$\displaystyle\overset{\eqref{eq:liepq}}{=}(\mathrm{id}\otimes\mathrm{ad}_{A}(a))(r)+(\mathrm{ad}_{A}(a)\otimes\mathrm{id})(r)=\delta^{\prime}(a).$

Hence the two Poisson bialgebras $(A,[\ ,\ ],\cdot,\delta,\Delta)$ and $(A,[\
,\ ],\cdot,\delta^{\prime},\Delta)$ coincide. ∎

######  Definition 5.33 ([34]).

Let $(A,[\ ,\ ],\cdot)$ be a Poisson algebra and $(V,\rho,\mu)$ be a module of
$(A,[\ ,\ ],\cdot)$. A linear map $T\colon V\to A$ is called an
$\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$ associated to $(V,\rho,\mu)$ if
the following equations hold:

$\displaystyle[T(u),T(v)]=T(\rho(T(u))v-\rho(T(v))u),$ $\displaystyle
T(u)\cdot T(v)=T(\mu(T(u))v+\mu(T(v))u),\qquad\forall u,v\in V.$

In particular, when $(V,\rho,\mu)$ is taken to be $(A,\mathrm{ad}_{A},L_{A})$,
an $\mathcal{O}$-operator $R\colon A\to A$ is called a Rota–Baxter operator
(of weight zero) on $(A,[\ ,\ ],\cdot)$, that is, $R$ satisfies

$\displaystyle R(a)\cdot R(b)=R(R(a)\cdot b+a\cdot R(b)),$
$\displaystyle[R(a),R(b)]=R([R(a),b]+[a,R(b)]),\qquad\forall a,b\in A.$

It is straightforward to get the following conclusion which is similarly to
Lemma 4.14.

######  Lemma 5.34.

Let $(A,[\ ,\ ],\cdot)$ be a Poisson algebra and $(V,\rho,\mu)$ be a module of
$(A,[\ ,\ ],\cdot)$. A linear map $T\colon V\to A$ is an
$\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$ associated to $(V,\rho,\mu)$ if
and only if $\hat{T}\colon A\oplus V\to A\oplus V$ is a Rota–Baxter operator
on the semi-direct product Poisson algebra $(A\ltimes_{\rho,\mu}V,[\ ,\
],\cdot)$, where $\hat{T}$ is defined by equation (4.18).

######  Lemma 5.35.

Let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a commutative
differential algebra and $(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of
$(A,\cdot,\Phi)$. If $R$ is a Rota–Baxter operator on $(A,\cdot)$ commuting
with $\partial_{k}$ for all $k=1,2$, then $R$ is a Rota–Baxter operator on the
Poisson algebra $(A,[\ ,\ ],\cdot)$.

###### Proof.

For all $a,b\in A$, we have

$\displaystyle R([R(a),b]+[a,R(b)])$
$\displaystyle\qquad{}\overset{\eqref{eq:lie}}{=}R\big{(}\partial_{1}(R(a))\cdot\partial_{2}(b)-\partial_{2}(R(a))\cdot\partial_{1}(b)+\partial_{1}(a)\cdot\partial_{2}(R(b))-\partial_{2}(a)\cdot\partial_{1}(R(b))\big{)}$
$\displaystyle\qquad{}\overset{\hphantom{(5.1)}}{=}R\big{(}R(\partial_{1}(a))\cdot\partial_{2}(b)-R(\partial_{2}(a))\cdot\partial_{1}(b)+\partial_{1}(a)\cdot
R(\partial_{2}(b))-\partial_{2}(a)\cdot R(\partial_{1}(b))\big{)}$
$\displaystyle\qquad{}\overset{\hphantom{(5.1)}}{=}R(\partial_{1}(a))\cdot
R(\partial_{2}(b))-R(\partial_{2}(a))\cdot R(\partial_{1}(b))$
$\displaystyle\qquad{}\overset{\hphantom{(5.1)}}{=}\partial_{1}(R(a))\cdot\partial_{2}(R(b))-\partial_{2}(R(a))\cdot\partial_{1}(R(b))=[R(a),R(b)].$

We finish the proof. ∎

######  Proposition 5.36 ([7, Proposition A.6, Theorem 2.4.7]).

Let $(A,[\ ,\ ],\cdot)$ be a Poisson algebra and $r\in A\otimes A$ be
antisymmetric. Then $r$ is an antisymmetric solution of PYBE in $(A,[\ ,\
],\cdot)$ if and only if $r^{\sharp}$ is an $\mathcal{O}$-operator of $(A,[\
,\ ],\cdot)$ associated to the module
$(A^{*},-\mathrm{ad}_{A}^{*},L_{A}^{*})$.

######  Proposition 5.37.

Let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a commutative
differential algebra and $(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of
$(A,\cdot,\Phi)$. Let $T\colon V\to A$ be an $\mathcal{O}$-operator of
$(A,\cdot,\Phi)$ associated to a module
$(V,\mu,\Omega=\\{\alpha_{1},\alpha_{2}\\})$. Then $T$ is an
$\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$ associated to
$(V,\rho_{\mu},\mu)$, where $(V,\rho_{\mu},\mu)$ is the induced module of
$(A,[\ ,\ ],\cdot)$ with respect to $(V,\mu,\Omega)$. In particular, when
taking
$(V,\mu,\Omega)=(A^{*},L_{A}^{*},\Psi^{*}=\\{\eth_{1}^{*},\eth_{2}^{*}\\})$,
if in addition equation (5.6) holds, then $T$ is an $\mathcal{O}$-operator of
$(A,[\ ,\ ],\cdot)$ associated to $(A^{*},-\mathrm{ad}_{A}^{*},L_{A}^{*})$.

###### Proof.

Since $T\colon V\to A$ is an $\mathcal{O}$-operator of $(A,\cdot,\Phi)$
associated to $(V,\mu,\Omega)$, by Lemma 4.14, the linear map $\hat{T}$
defined by equation (4.18) is a Rota–Baxter operator on the semi-direct
algebra $(A\ltimes_{\mu}V,\cdot)$. Moreover, since $\partial_{k}T=T\alpha_{k}$
for all $k=1,2$, we have $\hat{T}$ commutes with $\partial_{k}+\alpha_{k}$ for
all $k=1,2$. By Lemma 5.35, $\hat{T}$ is a Rota–Baxter operator on the induced
Poisson algebra of $(A\ltimes_{\mu}V,\cdot,\Phi+\Omega)$. By Corollary 5.11,
the induced Poisson algebra is exactly the semi-direct product Poisson algebra
by $(A,[\ ,\ ],\cdot)$ and $(V,\rho_{\mu},\mu)$. Hence $\hat{T}$ is a
Rota–Baxter operator on the semi-direct product Poisson algebra by $(A,[\ ,\
],\cdot)$ and $(V,\rho_{\mu},\mu)$. Therefore $T$ is an $\mathcal{O}$-operator
of $(A,[\ ,\ ],\cdot)$ associated to $(V,\rho_{\mu},\mu)$ by Lemma 5.34. The
particular case follows from Proposition 5.7. ∎

Combining Corollary 4.15, Propositions 5.31, 5.36 and 5.37 together, we have
the following conclusion.

######  Corollary 5.38.

Let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a
$\Psi=\\{\eth_{1},\eth_{2}\\}$-admissible commutative differential algebra and
$(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of $(A,\cdot,\Phi)$.
Suppose that equation (5.6) holds and $r$ is an antisymmetric solution of
$\Psi$-admissible AYBE in $(A,\cdot,\Phi)$. On the one hand, $r$ is a solution
of PYBE in the Poisson algebra $(A,[\ ,\ ],\cdot)$ by Proposition 5.31 and
hence $r^{\sharp}$ is an $\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$
associated to $(A^{*},-\mathrm{ad}_{A}^{*},L_{A}^{*})$ by Proposition 5.36. On
the other hand, $r^{\sharp}$ is an $\mathcal{O}$-operator of $(A,\cdot,\Phi)$
associated to $(A^{*},L_{A}^{*},\Psi^{*})$ by Corollary 4.15 and hence
$r^{\sharp}$ is an $\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$ associated to
$(A^{*},-\mathrm{ad}_{A}^{*},L_{A}^{*})$ by Proposition 5.37. That is, the two
approaches to get $r^{\sharp}$ as an $\mathcal{O}$-operator of $(A,[\ ,\
],\cdot)$ coincide, and thus we have the following commutative diagram:

$r$ an antisymmetric solution of $\Psi$-admissible AYBE in $(A,\cdot,\Phi)$
$\scriptstyle{{\rm Prop.}~{}\ref{pro:aybe_to_qybe}}$$\scriptstyle{{\rm
Cor.}~{}\ref{cor:aybe_o}}$ $r^{\sharp}$ an $\mathcal{O}$-operator of
$(A,\cdot,\Phi)$ associated to $(A^{*},L_{A}^{*},\Psi^{*})$ $\scriptstyle{{\rm
Prop.}~{}\ref{pro:assopp}}$ $r$ an antisymmetric solution of PYBE in $(A,[\ ,\
],\cdot)$ $\scriptstyle{{\rm Prop.}~{}\ref{pro:pybe_o}}$ $r^{\sharp}$ an
$\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$ associated to
$(A^{*},-\mathrm{ad}_{A}^{*},L_{A}^{*})$

###  5.5 Poisson bialgebras via $\boldsymbol{\mathcal{O}}$-operators of
commutative differential algebras
and differential Zinbiel algebras

######  Proposition 5.39.

Let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a commutative
differential algebra and $(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of
$(A,\cdot,\Phi)$. Suppose that $(V,\mu,\Omega=\\{\alpha_{1},\alpha_{2}\\})$ is
a module of $(A,\cdot,\Phi)$ and $(V,\rho_{\mu},\mu)$ is the induced module of
$(A,[\ ,\ ],\cdot)$ with respect to $(V,\mu,\Omega)$. Then
$(A\ltimes_{\mu^{*}}V^{*},\cdot,\Phi-\Omega^{*})$ is a
$(-\Phi+\Omega^{*})$-admissible commutative differential algebra. Furthermore,
$(A\ltimes_{-\rho_{\mu}^{*},\mu^{*}}V^{*},[\ ,\ ],\cdot)$ is the induced
Poisson algebra of $(A\ltimes_{\mu^{*}}V^{*},\cdot,\Phi-\Omega^{*})$ and
equation (5.6) holds, where $\eth_{k}$ is replaced by
$-\partial_{k}+\alpha_{k}^{*}$ and $\partial_{k}$ is replaced by
$\partial_{k}-\alpha_{k}^{*}$ for all $k=1,2$.

###### Proof.

By Example 2.10 and the proof of Corollary 4.19,
$(A\ltimes_{\mu^{*}}V^{*},\cdot,\Phi-\Omega^{*})$ is a
$(-\Phi+\Omega^{*})$-admissible commutative differential algebra. By Corollary
5.11, $(A\ltimes_{\rho_{\mu^{*}},\mu^{*}}V^{*},[\ ,\ ],\cdot)$ is the induced
Poisson algebra of $(A\ltimes_{\mu^{*}}V^{*},\cdot,\Phi-\Omega^{*})$, where
$(V^{*},\rho_{\mu^{*}},\mu^{*})$ is the induced module of $(A,[\ ,\ ],\cdot)$
with respect to the module $(V^{*},\mu^{*},-\Omega^{*})$. For all $a\in A,v\in
V$, we have

$\displaystyle(-\alpha_{2})(\mu(\partial_{1}(a))v)-(-\alpha_{1})(\mu(\partial_{2}(a))v)+\mu(\partial_{1}(a))\alpha_{2}(v)-\mu(\partial_{2}(a))\alpha_{1}(v)$
$\displaystyle\qquad{}=(\mu(\partial_{1}(a))\alpha_{2}(v)-\alpha_{2}(\mu(\partial_{1}(a))v))-(\mu(\partial_{2}(a))\alpha_{1}(v)-\alpha_{1}(\mu(\partial_{2}(a))v))$
$\displaystyle\qquad{}=-\mu(\partial_{2}(\partial_{1}(a)))v+\mu(\partial_{1}(\partial_{2}(a)))v=0.$

Therefore, by Proposition 5.7, we have $\rho_{\mu^{*}}=-\rho_{\mu}^{*}$ and
hence $(A\ltimes_{-\rho_{\mu}^{*},\mu^{*}}V^{*},[\ ,\ ],\cdot)$ is exactly the
induced Poisson algebra of $(A\ltimes_{\mu^{*}}V^{*},\cdot,\Phi-\Omega)$.
Moreover, we have

$(-\partial_{2}+\alpha_{2}^{*})((\partial_{1}-\alpha_{1}^{*})(a+v^{*}))\cdot(b+w^{*})=(-\partial_{1}+\alpha_{1}^{*})((\partial_{2}-\alpha_{2}^{*})(a+v^{*}))\cdot(b+w^{*}),$

for all $a,b\in A$, $v^{*},w^{*}\in V^{*}$. Hence equation (5.6) holds. ∎

######  Remark 5.40.

One may consider extending the above study to the case that
$\big{(}A\ltimes_{\mu^{*}}V^{*},\cdot,\Phi+\\{-\alpha_{k}^{*}+\theta_{k}\mathrm{id}_{V^{*}}\\}_{k=1}^{2}\big{)}$
is a
$\big{(}\\{-\partial_{k}+\theta_{k}\mathrm{id}_{A}\\}_{k=1}^{2}+\Omega^{*}\big{)}$-admissible
commutative differential algebra for any $\theta_{1},\theta_{2}\in\mathbb{F}$
as in Corollary 4.19. Then by a similar argument,
$(A\ltimes_{-\rho_{\mu}^{*},\mu^{*}}V^{*},[\ ,\ ],\cdot)$ is the induced
Poisson algebra of
$\big{(}A\ltimes_{\mu^{*}}V^{*},\cdot,\Phi+\\{-\alpha_{k}^{*}+\theta_{k}\mathrm{id}_{V^{*}}\\}_{k=1}^{2}\big{)}$,
where $(V,\rho_{\mu},\mu)$ is the induced module of $(A,[\ ,\ ],\cdot)$ with
respect to
$(V,\mu,\\{\alpha_{1}-\theta_{1}\mathrm{id}_{V},\alpha_{2}-\theta_{2}\mathrm{id}_{V}\\})$.
Furthermore, it is straightforward to show that equation (5.6) holds where
$\eth_{k}$ is replaced by
$-\partial_{k}+\theta_{k}\mathrm{id}_{A}+\alpha_{k}^{*}$ and $\partial_{k}$ is
replaced by $\partial_{k}-\alpha_{k}^{*}+\theta_{k}\mathrm{id}_{V^{*}}$ for
all $k=1,2$ if and only if

$\displaystyle\theta_{2}\partial_{1}(a)b=\theta_{1}\partial_{2}(a)b,\qquad\\!\theta_{2}\mu(\partial_{1}(a))v=\theta_{1}\mu(\partial_{2}(a))v,\qquad\\!\theta_{2}\alpha_{1}(\mu(a)v)=\theta_{1}\alpha_{2}(\mu(a)v),\\!\\!\\!\\!$
(5.30)

for all $a\in A,v\in V$. Suppose that equation (5.30) holds for some
$\theta_{k}\neq 0$. We assume that $\theta_{1}\neq 0$ without loss of
generality. Let $a,b\in A$. Then $\partial_{2}(a)\cdot
b=\theta_{1}^{-1}\theta_{2}\partial_{1}(a)\cdot b$ and thus we have

$[a,b]=\partial_{1}(a)\cdot\partial_{2}(b)-\partial_{2}(a)\cdot\partial_{1}(b)=\theta_{1}^{-1}\theta_{2}\partial_{1}(a)\cdot\partial_{1}(b)-\theta_{1}^{-1}\theta_{2}\partial_{1}(a)\cdot\partial_{1}(b)=0.$

Similarly, the Lie bracket in $(A\ltimes_{-\rho_{\mu}^{*},\mu^{*}}V^{*},[\ ,\
],\cdot)$ is also trivial. So in this sense, we only need to consider the case
that $\theta_{1}=\theta_{2}=0$ as we have done in Proposition 5.39.

######  Theorem 5.41 ([34, Theorem 5.24]).

Let $(V,\rho,\mu)$ be a module of a Poisson algebra $(A,[\ ,\ ],\cdot)$. Let
$T\colon V\to A$ be a linear map. Then $r=T-\sigma(T)$ is an antisymmetric
solution of PYBE in $(A\ltimes_{-\rho^{*},\mu^{*}}V^{*},[\ ,\ ],\cdot)$ if and
only if $T$ is an $\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$ associated to
$(V,\rho,\mu)$.

By Proposition 5.39, combining Corollary 4.19, Propositions 5.31 and 5.37 and
Theorem 5.41 together, we have the following conclusion.

######  Corollary 5.42.

With the conditions in Proposition 5.39. Suppose that $T\colon V\to A$ is an
$\mathcal{O}$-operator of $(A,\cdot,\Phi)$ associated to $(V,\mu,\Omega)$. On
the one hand, $T$ is an $\mathcal{O}$-operator of the induced Poisson algebra
$(A,[\ ,\ ],\cdot)$ associated to $(V,\rho_{\mu},\mu)$ by Proposition 5.37 and
hence $r=T-\sigma(T)$ is an antisymmetric solution of PYBE in
$(A\ltimes_{-\rho_{\mu}^{*},\mu^{*}}V^{*},[\ ,\ ],\cdot)$ by Theorem 5.41. On
the other hand, $r=T-\sigma(T)$ is an antisymmetric solution of
$(-\Phi+\Omega^{*})$-admissible AYBE in
$(A\ltimes_{\mu^{*}}V^{*},\cdot,\Phi-\Omega^{*})$ by Corollary 4.19. Due to
Proposition 5.39, $r$ is an antisymmetric solution of PYBE in the induced
Poisson algebra $(A\ltimes_{-\rho_{\mu}^{*},\mu^{*}}V^{*},[\ ,\ ],\cdot)$ by
Proposition 5.31. That is, the two approaches to get $r=T-\sigma(T)$ as an
antisymmetric solution of PYBE in $(A\ltimes_{-\rho_{\mu}^{*},\mu^{*}}V^{*},[\
,\ ],\cdot)$ coincide, and thus we have the following commutative diagram:

$T$ an $\mathcal{O}$-operator of $(A,\cdot,\Phi)$ associated to
$(V,\mu,\Omega)$ $\scriptstyle{{\rm
Prop.}~{}\ref{pro:assopp}}$$\scriptstyle{{\rm Cor.}~{}\ref{cor:qminp}}$
$r=T-\sigma(T)$ an antisymmetric solution of $(-\Phi+\Omega^{*})$-admissible
AYBE in $(A\ltimes_{\mu^{*}}V^{*},\cdot,\Phi-\Omega^{*})$ $\scriptstyle{{\rm
Prop.}~{}\ref{pro:aybe_to_qybe}}$ $T$ an $\mathcal{O}$-operator of $(A,[\ ,\
],\cdot)$ associated to $(V,\rho_{\mu},\mu)$ $\scriptstyle{{\rm
Thm.}~{}\ref{thm:oopsemi}}$ $r=T-\sigma(T)$ an antisymmetric solution of PYBE
in $(A\ltimes_{-\rho_{\mu}^{*},\mu^{*}}V^{*},[\ ,\ ],\cdot)$

######  Definition 5.43 ([22]).

A $($left$)$ pre-Lie algebra $(A,\diamond)$ is a vector space $A$ together
with a bilinear multiplication $\diamond\colon A\otimes A\to A$ satisfying the
following equation:

$a\diamond(b\diamond c)-(a\diamond b)\diamond c=b\diamond(a\diamond
c)-(b\diamond a)\diamond c,\qquad\forall a,b,c\in A.$

######  Definition 5.44 ([36]).

A $($left$)$ Zinbiel algebra $(A,\ast)$ is a vector space $A$ together with a
bilinear multiplication $\ast\colon A\otimes A\to A$ satisfying the following
equation:

$a\ast(b\ast c)=(a\ast b)\ast c+(b\ast a)\ast c,\qquad\forall a,b,c\in A.$

A Zinbiel algebra $(A,\ast)$ is equivalently defined as a dendriform algebra
$(A,\succ,\prec)$ in which

$a\succ b=b\prec a=a\ast b,\qquad\forall a,b\in A.$

Hence for a Zinbiel algebra $(A,\ast)$, the associated algebra $(A,\cdot)$ is
commutative, where

$a\cdot b=a\ast b+b\ast a,\qquad\forall a,b\in A.$ (5.31)

Furthermore, we give the notion of a differential Zinbiel algebra:
$(A,\ast,\Phi)$ is called a differential Zinbiel algebra if $(A,\ast,\ast^{\rm
op},\Phi)$ is a differential dendriform algebra, where $a\ast^{\rm op}b=b\ast
a$ for all $a,b\in A$. It is obvious that the associated differential algebra
of a differential Zinbiel algebra is commutative.

######  Definition 5.45 ([2]).

A $($left$)$ pre-Poisson algebra is a triple $(A,\diamond,\ast)$, where
$(A,\diamond)$ is a pre-Lie algebra and $(A,\ast)$ is a Zinbiel algebra such
that the following conditions hold:

$\displaystyle(a\diamond b-b\diamond a)\ast c=a\diamond(b\ast
c)-b\ast(a\diamond c),\qquad(a\ast b+b\ast a)\diamond c=a\ast(b\diamond
c)+b\ast(a\diamond c),$

for all $a,b,c\in A$.

######  Proposition 5.46 ([2, Proposition 2.2]).

Let $(A,\diamond,\ast)$ be a pre-Poisson algebra. Define two bilinear
multiplications $\cdot,[\ ,\ ]\colon A\otimes A\rightarrow A$ respectively by

$a\cdot b:=a\ast b+b\ast a\qquad{\rm and}\qquad[a,b]:=a\diamond b-b\diamond
a,\qquad\forall a,b\in A.$ (5.32)

Then $(A,[\ ,\ ],\cdot)$ is a Poisson algebra, called the associated Poisson
algebra of $(A,\diamond,\ast)$.

Recall that a perm-algebra is a vector space $P$ with a bilinear
multiplication $(a,b)\to ab$, such that $P$ is an algebra which is left-
commutative in the following sense:

$(ab)c=(ba)c,\qquad\forall a,b,c\in P,$

that is, a perm-algebra is associative and left-commutative.

######  Lemma 5.47 ([23]).

Let $A$ be a vector space with two bilinear multiplications $\diamond$ and
$*$. Then $(A,\diamond,*)$ is a pre-Poisson algebra if and only if for every
perm-algebra $P$, $(P\otimes A,[\ ,\ ],\cdot)$ is a Poisson algebra, where $[\
,\ ]$ and $\cdot$ are respectively defined by

$\displaystyle[p\otimes a,q\otimes b]=pq\otimes a\diamond b-qp\otimes
b\diamond a,$ $\displaystyle(p\otimes a)\cdot(q\otimes b)=pq\otimes
a*b+qp\otimes b*a,\qquad\forall p,q\in P,a,b\in A.$

######  Proposition 5.48.

Let $(A,\ast,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a differential Zinbiel
algebra. Define a bilinear multiplication $\diamond\colon A\otimes A\to A$ by

$a\diamond
b:=\partial_{1}(a)\ast\partial_{2}(b)-\partial_{2}(a)\ast\partial_{1}(b),\qquad\forall
a,b\in A.$ (5.33)

Then $(A,\diamond,\ast)$ is a pre-Poisson algebra, called the induced pre-
Poisson algebra of $(A,\ast,\Phi)$.

###### Proof.

The conclusion can be verified by a direct proof or the following approach in
terms of perm-algebras. Let $P$ be a perm-algebra. Define a bilinear
multiplication $\cdot$ on $P\otimes A$ by

$(p\otimes a)\cdot(q\otimes b):=pq\otimes a*b+qp\otimes b*a,\qquad\forall
p,q\in P,a,b\in A.$

For all $p,q,r\in P$ and $a,b,c\in A$, we have

$\displaystyle(p\otimes a)\cdot((q\otimes b)\cdot(r\otimes c))$
$\displaystyle\qquad{}=pqr\otimes a\ast(b\ast c)+qrp\otimes(b\ast c)\ast
a+prq\otimes a\ast(c\ast b)+rqp\otimes(c\ast b)\ast a$
$\displaystyle\qquad{}=pqr\otimes a\ast(b\ast c)+rqp\otimes(b\ast c)\ast
a+prq\otimes a\ast(c\ast b)+rqp\otimes(c\ast b)\ast a$
$\displaystyle\qquad{}=pqr\otimes a\ast(b\ast c)+prq\otimes a\ast(c\ast
b)+rqp\otimes c\ast(b\ast a),$ $\displaystyle((p\otimes a)\cdot(q\otimes
b))\cdot(r\otimes c)$ $\displaystyle\qquad{}=pqr\otimes(a\ast b)\ast
c+rpq\otimes c\ast(a\ast b)+qpr\otimes(b\ast a)\ast c+rqp\otimes c\ast(b\ast
a)$ $\displaystyle\qquad{}=pqr\otimes(a\ast b)\ast c+rpq\otimes c\ast(a\ast
b)+pqr\otimes(b\ast a)\ast c+rqp\otimes c\ast(b\ast a)$
$\displaystyle\qquad{}=pqr\otimes a\ast(b\ast c)+rpq\otimes c\ast(a\ast
b)+rqp\otimes c\ast(b\ast a).$

Hence $(P\otimes A,\cdot)$ is an algebra. Obviously $(P\otimes A,\cdot)$ is
commutative. Define linear maps $d_{1},d_{2}\colon P\otimes A\to P\otimes A$
respectively by

$d_{i}:P\otimes A\to P\otimes A,\qquad p\otimes a\mapsto
p\otimes\partial_{i}(a),\qquad\forall a\in A,\quad p\in P,\quad i=1,2.$

It is straightforward to verify that $(P\otimes A,\cdot,\\{d_{1},d_{2}\\})$ is
a commutative differential algebra. Let $(P\otimes A,[\ ,\ ],\cdot)$ be the
induced Poisson algebra of $(P\otimes A,\cdot,\\{d_{1},d_{2}\\})$, where $[\
,\ ]$ is defined by equation (5.1), i.e.,

$\displaystyle[p\otimes a,q\otimes b]=d_{1}(p\otimes a)\cdot d_{2}(q\otimes
b)-d_{2}(p\otimes a)\cdot d_{1}(q\otimes b)$
$\displaystyle\qquad{}=(p\otimes\partial_{1}(a))\cdot(q\otimes\partial_{2}(b))-(p\otimes\partial_{2}(a))\cdot(q\otimes\partial_{1}(b))\cdot$
$\displaystyle\qquad{}=(pq\otimes\partial_{1}(a)\ast\partial_{2}(b)+qp\otimes\partial_{2}(b)\ast\partial_{1}(a))-(pq\otimes\partial_{2}(a)\ast\partial_{1}(b)+qp\otimes\partial_{1}(b)\ast\partial_{2}(a))$
$\displaystyle\qquad{}=pq\otimes a\diamond b-qp\otimes b\diamond
a,\qquad\forall p,q\in P,\quad a,b\in A.$

Hence the conclusion follows from Lemma 5.47. ∎

######  Theorem 5.49.

Let $(A,\ast,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a differential Zinbiel
algebra. Let $(A,\cdot,\Phi)$ be the associated commutative differential
algebra of $(A,\ast,\Phi)$, where $\cdot$ is defined by equation (5.31). Let
$(A,\diamond,\ast)$ be the induced pre-Poisson algebra of $(A,\ast,\Phi)$,
where $\diamond$ is defined by equation (5.33). Then the associated Poisson
algebra $(A,[\ ,\ ],\cdot)$ of the pre-Poisson algebra $(A,\diamond,\ast)$,
where $[\ ,\ ]$ and $\cdot$ are respectively defined by equation (5.32), is
exactly the induced Poisson algebra $(A,[\ ,\ ],\cdot)$ of the differential
algebra $(A,\cdot,\Phi)$, where $[\ ,\ ]$ is defined by equation (5.1). Hence
we have the following commutative diagram:

differential Zinbiel algebra $(A,\ast,\Phi)$
$\scriptstyle{equation~{}\eqref{eq:dzpp}}$$\scriptstyle{equation~{}\eqref{eq:Zin}}$
commutative differential algebra $(A,\cdot,\Phi)$
$\scriptstyle{equation~{}\eqref{eq:lie}}$ pre-Poisson algebra
$(A,\diamond,\ast)$ $\scriptstyle{equation~{}\eqref{eq:pptp}}$ Poisson algebra
$(A,[\ ,\ ],\cdot)$

###### Proof.

Let $P$ be a one-dimensional vector space with a basis $\\{e\\}$ with a
bilinear multiplication whose product is given by $ee=e$. Obviously $P$ is a
perm-algebra which is isomorphic to the field $\mathbb{F}$. Substituting such
$P$ into the proof of Proposition 5.48 and noting that in this case, for any
vector space $A$, $P\otimes A$ and $A$ are isomorphic as vector spaces, we get
the conclusion immediately. ∎

######  Lemma 5.50 ([34, Theorem 5.18]).

Let $(A,[\ ,\ ],\cdot)$ be a Poisson algebra. Suppose that $T$ is an
$\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$ associated to $(V,\rho,\mu)$.
Define two bilinear multiplications $\ast,\circ\colon V\otimes V\to V$
respectively by

$u\diamond v:=\rho(T(u))v,\qquad u\ast v:=\mu(T(u))v,\qquad\forall u,v\in V.$
(5.34)

Then $(V,\diamond,\ast)$ is a pre-Poisson algebra.

Let $(A,\cdot,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a commutative
differential algebra. Suppose that $T$ is an $\mathcal{O}$-operator of
$(A,\cdot,\Phi)$ associated to a module
$(V,\mu,\Omega=\\{\alpha_{1},\alpha_{2}\\})$. Then $(V,\ast,\Omega)$ is a
differential Zinbiel algebra by Proposition 4.26, where $\ast$ is defined by
equation (4.27) in the commutative case, that is,

$u\ast v:=\mu(T(u))v,\qquad\forall u,v\in V.$ (5.35)

Furthermore, there is the induced pre-Poisson algebra $(V,\diamond,\ast)$ of
$(V,\ast,\Omega)$ by Proposition 5.48, where $\diamond$ is defined by equation
(5.33). On the other hand, let $(A,[\ ,\ ],\cdot)$ be the induced Poisson
algebra of $(A,\cdot,\Phi)$, where $[\ ,\ ]$ is defined by equation (5.1).
Then by Proposition 5.37, $T$ is an $\mathcal{O}$-operator on $(A,[\ ,\
],\cdot)$ associated to $(V,\rho_{\mu},\mu)$ which is the induced module of
$(A,[\ ,\ ],\cdot)$ with respect to $(V,\mu,\Omega)$. Hence there is a pre-
Poisson algebra $(V,\diamond^{\prime},\ast^{\prime})$ by Lemma 5.50, where
$\diamond^{\prime}$ and $\ast^{\prime}$ are respectively defined by equation
(5.34).

######  Proposition 5.51.

With the conditions above. Then the two pre-Poisson algebras
$(V,\diamond,\ast)$ and $(V,\diamond^{\prime},\ast^{\prime})$ coincide. Hence
we have the following commutative diagram:

$T$ an $\mathcal{O}$-operator of $(A,\cdot,\Phi)$ associated to
$(V,\mu,\Omega)$ $\scriptstyle{{\rm Prop.}~{}\ref{pro:opd}}$$\scriptstyle{{\rm
Prop.}~{}\ref{pro:assopp}}$ $(V,\ast,\Omega)$ a differential Zinbiel algebra
$\scriptstyle{{\rm Prop.}~{}\ref{pro:dzpp}}$ $T$ an $\mathcal{O}$-operator of
$(A,[\ ,\ ],\cdot)$ associated to $(V,\rho_{\mu},\mu)$ $\scriptstyle{{\rm
Lem.}~{}\ref{lem:otpp}}$ $(V,\diamond,\ast)$ a pre-Poisson algebra

###### Proof.

Let $u,v\in V$. Then by equations (5.33) and (5.35), we have

$\displaystyle u\ast v=\mu(T(u))v,$ $\displaystyle u\diamond
v=\alpha_{1}(u)\ast\alpha_{2}(v)-\alpha_{2}(u)\ast\alpha_{1}(v)=\mu(T(\alpha_{1}(u)))\alpha_{2}(v)-\mu(T(\alpha_{2}(u)))\alpha_{1}(v).$

On the other hand, by equations (5.34) and (5.2), we have

$\displaystyle u\ast^{\prime}v$
$\displaystyle\overset{\hphantom{(0.00)}}{=}\mu(T(u))v,$ $\displaystyle
u\diamond^{\prime}v$
$\displaystyle\overset{\hphantom{(0.00)}}{=}\rho_{\mu}(T(u))v=\mu(\partial_{1}(T(u)))\alpha_{2}(v)-\mu(\partial_{2}(T(u)))\alpha_{1}(v)$
$\displaystyle\overset{\eqref{eq:oop2}}{=}\mu(T(\alpha_{1}(u)))\alpha_{2}(v)-\mu(T(\alpha_{2}(u)))\alpha_{1}(v).$

Hence $(V,\diamond,\ast)$ and $(V,\diamond^{\prime},\ast^{\prime})$ coincide.
∎

######  Proposition 5.52 ([34, Example 5.16]).

Let $(A,\diamond,\ast)$ be a pre-Poisson algebra and $(A,[\ ,\ ],\cdot)$ be
the associated Poisson algebra of $(A,\diamond,\ast)$, where $[\ ,\ ]$ and
$\cdot$ are respectively defined by equation (5.32). Then
$(A,L_{\diamond},L_{\ast})$ is a module of $(A,[\ ,\ ],\cdot)$, where
$L_{\diamond},L_{\ast}\colon A\to\mathrm{End}(A)$ are respectively defined by

$L_{\diamond}(a)(b):=a\diamond b,\qquad L_{\ast}(a)(b):=a\ast b,\qquad\forall
a,b\in A.$

Furthermore, the identity map $\mathrm{id}\colon A\to A$ is an
$\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$ associated to
$(A,L_{\diamond},L_{\ast})$.

######  Proposition 5.53.

Let $(A,\ast,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a differential Zinbiel
algebra. Let $(A,\cdot,\Phi)$ be the associated commutative differential
algebra of $(A,\ast,\Phi)$, where $\cdot$ is defined by equation (5.31), and
$(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of $(A,\cdot,\Phi)$, where
$[\ ,\ ]$ is defined by equation (5.1). Let $(A,\diamond,\ast)$ be the induced
pre-Poisson algebra of $(A,\ast,\Phi)$ given in Proposition 5.48, where
$\diamond$ is defined by equation (5.33). Then the module
$(A,L_{\diamond},L_{\ast})$ of the Poisson algebra $(A,[\ ,\ ],\cdot)$ is
exactly the induced module $(A,\rho_{L_{\ast}},L_{\ast})$ of $(A,[\ ,\
],\cdot)$ with respect to $(A,L_{\ast},\Phi)$, that is,
${L_{\diamond}=\rho_{L_{\ast}}}$. Furthermore, on the one hand, the identity
map $\mathrm{id}$ is an $\mathcal{O}$-operator of the commutative differential
algebra $(A,\cdot,\Phi)$ associated to $(A,L_{\ast},\Phi)$ by Proposition 4.24
and hence $\mathrm{id}$ is an $\mathcal{O}$-operator of the Poisson algebra
$(A,[\ ,\ ],\cdot)$ associated to $(A,\rho_{L_{\ast}},L_{\ast})$ by
Proposition 5.37. On the other hand, $\mathrm{id}$ is an
$\mathcal{O}$-operator of $(A,[\ ,\ ],\cdot)$ associated to
$(A,L_{\diamond},L_{\ast})$ by Proposition 5.52. Therefore the two approaches
for $\mathrm{id}$ as an $\mathcal{O}$-operator of the Poisson algebra $(A,[\
,\ ],\cdot)$ coincide and hence we have the following commutative diagram:

$(A,\ast,\Phi)$ a differential Zinbiel algebra $\scriptstyle{{\rm
Prop.}~{}\ref{pro:dzpp}}$$\scriptstyle{{\rm Prop.}~{}\ref{pro:assda}}$
$\mathrm{id}$ an $\mathcal{O}$-operator of $(A,\cdot,\Phi)$ associated to
$(A,L_{\ast},\Phi)$ $\scriptstyle{{\rm Prop.}~{}\ref{pro:assopp}}$
$(A,\diamond,\ast)$ a pre-Poisson algebra $\scriptstyle{{\rm
Prop.}~{}\ref{pro:ppto}}$ $\mathrm{id}$ an $\mathcal{O}$-operator of $(A,[\ ,\
],\cdot)$ associated to $(A,L_{\diamond},L_{\ast})$

###### Proof.

In fact, we have

$\rho_{L_{\ast}}(a)(b)=L_{\ast}(\partial_{1}(a))\partial_{2}(b)-L_{\ast}(\partial_{2}(a))\partial_{1}(b)\overset{\eqref{eq:dzpp}}{=}a\diamond
b=L_{\diamond}(a)(b),\qquad\forall a,b\in A.$

Hence the conclusion follows from Propositions 4.24, 5.37, 5.48 and 5.52. ∎

At the end of this paper, we give the following construction of Poisson
bialgebras from differential Zinbiel algebras.

######  Proposition 5.54.

Let $(A,\ast,\Phi=\\{\partial_{1},\partial_{2}\\})$ be a differential Zinbiel
algebra. Let $(A,\cdot,\Phi)$ be the associated commutative differential
algebra of $(A,\ast,\Phi)$, where $\cdot$ is defined by equation (5.31), and
$(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of $(A,\cdot,\Phi)$, where
$[\ ,\ ]$ is defined by equation (5.1). Let $(A,\diamond,\ast)$ be the induced
pre-Poisson algebra of $(A,\ast,\Phi)$, where $\diamond$ is defined by
equation (5.33). Let $\\{e_{1},e_{2},\dots,e_{n}\\}$ be a basis of $A$ and
$\\{e_{1}^{*},e_{2}^{*},\dots,e_{n}^{*}\\}$ be the dual basis. Set

$r=\sum_{i=1}^{n}(e_{i}\otimes e_{i}^{*}-e_{i}^{*}\otimes e_{i}).$

Then there is a Poisson bialgebra
$(A\ltimes_{-L^{*}_{\diamond},L^{*}_{\ast}}A^{*},[\ ,\
],\cdot,\delta,\Delta)$, where $\delta$ and $\Delta$ are defined by equations
(5.26) and (5.27) with the above $r$ respectively. Moreover,
$(A\ltimes_{L^{*}_{\ast}}A^{*},\cdot,\Delta,\Phi-\Phi^{*},-\Phi+\Phi^{*})$ is
a commutative and cocommutative differential ASI bialgebra whose induced
Poisson bialgebra is exactly
$(A\ltimes_{-L^{*}_{\diamond},L^{*}_{\ast}}A^{*},[\ ,\
],\cdot,\delta,\Delta)$.

###### Proof.

Note that $\mathrm{id}=\sum^{n}_{i=1}e_{i}\otimes e_{i}^{*}$ and
$r=\mathrm{id}-\sigma(\mathrm{id})$. The first conclusion follows from
Proposition 5.53, Theorem 5.41 and Proposition 5.29. The second conclusion
follows from Proposition 5.53, Corollaries 5.42 and 5.32. Note that here
equation (5.6) holds, where $\eth_{k}$ is replaced by
$-\partial_{k}+\partial_{k}^{*}$ and $\partial_{k}$ is replaced by
$\partial_{k}-\partial_{k}^{*}$ for all $k=1,2$. ∎

We give the following example to illustrate the above construction explicitly.
Note that the classification of complex differential Zinbiel algebras in
dimension $\leq 4$ is given in [5].

######  Example 5.55.

Let $(A,\ast)$ be a Zinbiel algebra in dimension 3 with a basis
$\\{e_{1},e_{2},e_{3}\\}$ whose non-zero product is given by

$e_{1}\ast e_{1}=e_{3},\qquad e_{2}\ast e_{1}=e_{3},\qquad e_{2}\ast
e_{2}=e_{3}.$

Let $\partial_{1},\partial_{2}\colon A\to A$ be two liner maps respectively
given by

$\displaystyle\partial_{1}(e_{1})=e_{1},\qquad\partial_{1}(e_{2})=e_{2},\qquad\partial_{1}(e_{3})=2e_{3};$
$\displaystyle\partial_{2}(e_{1})=e_{2},\qquad\partial_{2}(e_{2})=-e_{1}+e_{2},\qquad\partial_{2}(e_{3})=e_{3}.$

Then $(A,\ast,\Phi=\\{\partial_{1},\partial_{2}\\})$ is a differential Zinbiel
algebra. Let $\\{e_{1}^{*},e_{2}^{*},e_{3}^{*}\\}$ be the dual basis of
$\\{e_{1},e_{2},e_{3}\\}$. Then we have

$\displaystyle\partial_{1}^{*}(e_{1}^{*})=e_{1}^{*},\qquad\partial_{1}^{*}(e_{2}^{*})=e_{2}^{*},\qquad\partial_{1}^{*}(e_{3}^{*})=2e_{3}^{*};$
$\displaystyle\partial_{2}^{*}(e_{1}^{*})=-e_{2}^{*},\qquad\partial_{2}^{*}(e_{2}^{*})=e_{1}^{*}+e_{2}^{*},\qquad\partial_{2}^{*}(e_{3}^{*})=e_{3}^{*}.$

Let $(A,\cdot,\Phi)$ be the associated commutative differential algebra of
$(A,\ast,\Phi)$, where $\cdot$ is defined by equation (5.31) whose non-zero
product is given by

$e_{1}\cdot e_{1}=e_{2}\cdot e_{2}=2e_{3},\qquad e_{1}\cdot e_{2}=e_{2}\cdot
e_{1}=e_{3}.$

Let $(A,[\ ,\ ],\cdot)$ be the induced Poisson algebra of $(A,\cdot,\Phi)$,
where $[\ ,\ ]$ is defined by equation (5.1) whose non-zero product is given
by

$[e_{1},e_{2}]=-[e_{2},e_{1}]=-3e_{3}.$

Let $(A,\diamond,\ast)$ be the induced pre-Poisson algebra of $(A,\ast,\Phi)$,
where $\diamond$ is defined by equation (5.33) whose non-zero product is given
by

$e_{1}\diamond e_{1}=-e_{3},\qquad e_{1}\diamond e_{2}=-2e_{3},\qquad
e_{2}\diamond e_{1}=e_{3},\qquad e_{2}\diamond e_{2}=-e_{3}.$

Hence the non-zero product of $[\ ,\ ]$ and $\cdot$ on the Poisson algebra
$(A\ltimes_{-L^{*}_{\diamond},L^{*}_{\ast}}A^{*},[\ ,\ ],\cdot)$ is given
respectively by

$\displaystyle[e_{1},e_{2}]=-[e_{2},e_{1}]=-3e_{3},\qquad[e_{1},e_{3}^{*}]=-[e_{3}^{*},e_{1}]=e_{1}^{*}+2e_{2}^{*},$
$\displaystyle[e_{2},e_{3}^{*}]=-[e_{3}^{*},e_{2}]=-e_{1}^{*}+e_{2}^{*};$
$\displaystyle e_{1}\cdot e_{1}=e_{2}\cdot e_{2}=2e_{3},\qquad e_{1}\cdot
e_{2}=e_{2}\cdot e_{1}=e_{3},\qquad e_{1}\cdot e_{3}^{*}=e_{3}^{*}\cdot
e_{1}=e_{1}^{*},$ $\displaystyle e_{2}\cdot e_{3}^{*}=e_{3}^{*}\cdot
e_{2}=e_{1}^{*}+e_{2}^{*}.$

Set $r=\sum_{i=1}^{3}(e_{i}\otimes e_{i}^{*}-e_{i}^{*}\otimes e_{i})$. Define
$\delta$ and $\Delta$ by equations (5.26) and (5.27) respectively. Thus we
have

$\displaystyle\delta(e_{1})=e_{3}\otimes e_{1}^{*}-e_{1}^{*}\otimes
e_{3}-e_{3}\otimes e_{2}^{*}+e_{2}^{*}\otimes e_{3},$
$\displaystyle\delta(e_{2})=2e_{3}\otimes e_{1}^{*}-2e_{1}^{*}\otimes
e_{3}^{*}+e_{3}\otimes e_{2}^{*}-e_{2}^{*}\otimes e_{3},$
$\displaystyle\delta(e_{3})=\delta(e_{1}^{*})=\delta(e_{2}^{*})=0,\qquad\delta(e_{3}^{*})=3e_{1}^{*}\otimes
e_{2}^{*}-3e_{2}^{*}\otimes e_{1}^{*};$
$\displaystyle\Delta(e_{1})=-e_{3}\otimes e_{1}^{*}-e_{1}^{*}\otimes
e_{3}-e_{2}^{*}\otimes e_{3}-e_{3}\otimes e_{2}^{*},$
$\displaystyle\Delta(e_{2})=-e_{3}\otimes e_{2}^{*}-e_{2}^{*}\otimes e_{3},$
$\displaystyle\Delta(e_{3})=\Delta(e_{1}^{*})=\Delta(e_{2}^{*})=0,\qquad\Delta(e_{3}^{*})=-2e_{1}^{*}\otimes
e_{1}^{*}-2e_{2}^{*}\otimes e_{2}^{*}-e_{2}^{*}\otimes
e_{1}^{*}-e_{1}^{*}\otimes e_{2}^{*}.$

Therefore $(A\ltimes_{-L^{*}_{\diamond},L^{*}_{\ast}}A^{*},[\ ,\
],\cdot,\delta,\Delta)$ is a Poisson bialgebra. At the same time, we obtain a
commutative and cocommutative differential ASI bialgebra
$(A\ltimes_{L^{*}_{\ast}}A^{*},\cdot,\Delta,\Phi-\Phi^{*},-\Phi+\Phi^{*})$
whose induced Poisson bialgebra is exactly
$(A\ltimes_{-L^{*}_{\diamond},L^{*}_{\ast}}A^{*},[\ ,\
],\cdot,\delta,\Delta)$.

### Acknowledgements

This work is supported by NSFC (11931009, 12271265, 12261131498), the
Fundamental Research Funds for the Central Universities and Nankai Zhide
Foundation. The authors thank the referees for valuable suggestions to improve
the paper.

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