#  Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures
and $\mathrm{\mathcal{O}N}$-structures

Jiefeng Liu  School of Mathematics and Statistics, Northeast Normal
University, Changchun 130024, Jilin, China<EMAIL_ADDRESS>,  Sihan Zhou
School of Mathematics and Statistics, Northeast Normal University, Changchun
130024, Jilin, China<EMAIL_ADDRESS>and  Lamei Yuan∗ School of
Mathematics, Harbin Institute of Technology, Harbin 150001, China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we first study infinitesimal deformations of a Lie conformal
algebra and a Lie conformal algebra with a module (called an $\mathsf{LCMod}$
pair), which lead to the notions of Nijenhuis operator on the Lie conformal
algebra and Nijenhuis structure on the $\mathsf{LCMod}$ pair, respectively.
Then by adding compatibility conditions between Nijenhuis structures and
$\mathcal{O}$-operators, we introduce the notion of an
$\mathrm{\mathcal{O}N}$-structure on an $\mathsf{LCMod}$ pair and show that an
$\mathrm{\mathcal{O}N}$-structure gives rise to a hierarchy of pairwise
compatible $\mathcal{O}$-operators. In particular, we show that compatible
$\mathcal{O}$-operators on a Lie conformal algebra can be characterized by
Nijenhuis operators on Lie conformal algebras. Finally, we introduce the
notions of conformal $r$-matrix-Nijenhuis structure and symplectic-Nijenhuis
structure on the Lie conformal algebra and study their relations.

###### Key words and phrases:

Nijenhuis operator, $\mathrm{\mathcal{O}N}$-structure, conformal $r$-matrix-
Nijenhuis structure, symplectic-Nijenhuis structure

###### 2010 Mathematics Subject Classification:

17A30, 17B69, 17B62

∗ the corresponding author

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries on Lie conformal algebra
  3. 3 Nijenhuis operators and Nijenhuis structures on Lie conformal algebras
    1. 3.1 Nijenhuis operators on Lie conformal algebras
    2. 3.2 Nijenhuis structures on $\mathsf{LCMod}$ pairs
  4. 4 $\mathrm{\mathcal{O}N}$-structures and compatible $\mathcal{O}$-operators on $\mathsf{LCMod}$ pairs
  5. 5 Conformal $r$-matrix-Nijenhuis structures and symplectic-Nijenhuis structures on Lie conformal algebras
    1. 5.1 Conformal $r$-matrix-Nijenhuis structures on Lie conformal algebras
    2. 5.2 Symplectic-Nijenhuis structures on Lie conformal algebras

##  1\. Introduction

The notion of a Lie conformal algebra was introduced by Kac as an axiomatic
description of the operator product expansion of chiral fields in two-
dimensional conformal field theory ([12]). It turns out to be valuable tools
in studying vertex algebras ([3, 4, 12, 17]), Poisson vertex algebras ([2,
21]) and Hamiltonian formalism in the theory of nonlinear evolution equations
([6]). The structure theory ([8]), representation theory ([7]) and cohomology
theory ([5, 9]) of Lie conformal algebras have been well developed.

Motivated by the study of conformal analogues of Lie bialgebras, Liberati
developed the theory of Lie conformal bialgebra in [22]. The notion of
conformal classical Yang-Baxter equation was also introduced to construct
coboundary Lie conformal bialgebras. Explicitly, let $A$ be a Lie conformal
algebra and $r=\sum_{i}a_{i}\otimes b_{i}\in A\otimes A$. Set
$\partial^{\otimes^{3}}=\partial\otimes 1\otimes 1+1\otimes\partial\otimes
1+1\otimes 1\otimes\partial.$ The equation

(1) $\displaystyle\llbracket r,r\rrbracket$ $\displaystyle=$
$\displaystyle\sum_{i,j}([a_{i_{\mu}}a_{j}]\otimes b_{i}\otimes
b_{j}|_{\mu=1\otimes\partial\otimes 1}-a_{i}\otimes[a_{j_{\mu}}b_{i}]\otimes
b_{j}|_{\mu=1\otimes 1\otimes\partial}-a_{i}\otimes
a_{j}\otimes[b_{j_{\mu}}b_{i}]|_{\mu=1\otimes\partial\otimes 1})$
$\displaystyle=$ $\displaystyle 0\quad\mbox{mod}\quad\partial^{\otimes^{3}}$

is called conformal classical Yang-Baxter equation in $A$. In particular, the
skew-symmetric solutions of conformal classical Yang-Baxter equation give rise
to Lie conformal bialgebras. Recently, the authors in [16] showed that $r$ is
a skew-symmetric solution of the conformal classical Yang-Baxter equation in a
finite Lie conformal algebra $A$ if and only if
$r^{\sharp}_{0}=r^{\sharp}_{\lambda}|_{\lambda=0}$ satisfies

(2)
$[r^{\sharp}_{0}(\alpha)_{\lambda}r^{\sharp}_{0}(\beta)]=r^{\sharp}_{0}(\mathrm{ad}^{*}(r^{\sharp}_{0}(\alpha))_{\lambda}(\beta)-\mathrm{ad}^{*}(r^{\sharp}_{0}(\beta))_{-\lambda-\partial}(\alpha)),\quad\forall~{}\alpha,\beta\in
A^{\ast c},$

where $r^{\sharp}\in{\rm Chom}(A^{\ast c},A)$ is defined by

(3)
$r^{\sharp}_{\lambda}(\alpha)=\sum_{i}\langle\alpha,a_{i}\rangle_{-\lambda-\partial}b_{i},\quad\forall~{}\alpha\in
A^{\ast c},$

and $\mathrm{ad}^{*}$ is the coadjoint module over the $A$. More generally,
the authors in [16] introduced the notion of $\mathcal{O}$-operator on a Lie
conformal algebra.

######  Definition 1.1.

A $\mathbb{C}[\partial]$-module homomorphism $T:V\longrightarrow A$ is called
an $\mathcal{O}$-operator on an $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];\rho)$ if it satisfies

(4)
$[T(u)_{\lambda}T(v)]=T(\rho(T(u))_{\lambda}(v)-\rho(T(v))_{-\lambda-\partial}(u)),\quad\forall~{}u,v\in
V.$

Thus $r$ is a skew-symmetric solution of the conformal classical Yang-Baxter
equation in a finite Lie conformal algebra $A$ if and only if
$r^{\sharp}_{0}=r^{\sharp}_{\lambda}|_{\lambda=0}$ is an
$\mathcal{O}$-operator on the $\mathsf{LCMod}$ pair $(A;\mathrm{ad}^{*})$. The
importance of $\mathcal{O}$-operators on Lie conformal algebras is that the
skew-symmetric part of an $\mathcal{O}$-operator gives rise to a skew-
symmetric solution of a conformal classical Yang-Baxter equation and the
symmetric part of an $\mathcal{O}$-operator gives rise to a symmetry solution
of a conformal $S$-equation, which plays an important role in the theory of
left-symmetric conformal bialgebras ([18]). In particular, when the module
$\rho$ is the adjoint module, the $\mathcal{O}$-operator on a Lie conformal
algebra is just the Rota-Baxter operator on the Lie conformal algebra. In the
Lie algebra context, the notion of $\mathcal{O}$-operator was introduced by
Kupershmidt in [15] as a tool to study the classical Yang-Baxter equation. See
[1, 19, 20, 27] for more details and applications of $\mathcal{O}$-operators
on Lie algebras.

Nijenhuis operators on Lie algebras have been introduced in the theory of
integrable systems in the work of Magri, Gelfand and Dorfman (see the book
[10]). In the sense of the theory of deformations of Lie algebras ([25]),
Nijenhuis operators generate trivial deformations naturally. In this paper, we
study the infinitesimal deformation of a Lie conformal algebra, which leads to
the notion of Nijenhuis operator on the Lie conformal algebra and we show that
it generates a trivial deformation of the Lie conformal algebra. See [5] for
more details on deformations of Lie conformal algebras. We find that Nijenhuis
operators play an essential role in certain compatible structures in terms of
$\mathcal{O}$-operators. Roughly speaking, a Nijenhuis operator “connects” two
$\mathcal{O}$-operators on a Lie conformal algebra whose any linear
combination is still an $\mathcal{O}$-operator (they are called “compatible”
$\mathcal{O}$-operators) in some sense.

Poisson-Nijenhuis structures were defined by Magri and Morosi in 1984 in their
study of completely integrable systems [24]. The importance of Poisson-
Nijenhuis structures is that they produce bi-Hamiltonian systems. See [13, 14]
for more details on Poisson-Nijenhuis structures. The aim of this paper is to
study the theory of conformal analogues of Possion-Nijenhuis structures on Lie
conformal algebras. We introduce the notion of an
$\mathrm{\mathcal{O}N}$-structure on a $\mathsf{LCMod}$ pair (a Lie conformal
algebra with a module), which consists of an $\mathcal{O}$-operator and a
Nijenhuis structure on a $\mathsf{LCMod}$ pair satisfying some compatibility
conditions. The notion of a Nijenhuis structure on a $\mathsf{LCMod}$ pair
consists of a Nijenhuis operator $N$ on $A$ and a
$\mathbb{C}[\partial]$-linear map $S$ from $V$ to $V$ satisfying some
compatibility conditions. We point out that the introduction of the linear map
“$S$” provides a general framework for the study of conformal analogues of
Poisson-Nijenhuis structures. See [19, 23] for more details about
$\mathrm{\mathcal{O}N}$-structures on Lie algebras and associative algebras.
Furthermore, as an application of the theory of
$\mathrm{\mathcal{O}N}$-structures on $\mathsf{LCMod}$ pairs, we introduce the
notions of conformal $r$-matrix-Nijenhuis structures and symplectic-Nijenhuis
structures on Lie conformal algebras, which are conformal analogues of
$r$-matrix-Nijenhuis structures ([13, 26]) and symplectic-Nijenhuis structures
on Lie algebras ([10, 14]).

The paper is organized as follows. In Section 2, we recall the module and
cohomology theory of Lie conformal algebras. In Section 3, we study the
infinitesimal deformations of Lie conformal algebras and $\mathsf{LCMod}$
pairs. Then we introduce the notions of Nijenhuis operator on a Lie conformal
algebra and Nijenhuis structure on a $\mathsf{LCMod}$ pair, which give trivial
deformations of the Lie conformal algebra and the $\mathsf{LCMod}$ pair,
respectively. In Section 4, we add compatibility conditions between an
$\mathcal{O}$-operator and a Nijenhuis structure to define the notion of
$\mathrm{\mathcal{O}N}$-structure on a $\mathsf{LCMod}$ pair. Then we study
compatible $\mathcal{O}$-operators on $\mathsf{LCMod}$ pairs, and show that a
Nijenhuis operator connects two compatible $\mathcal{O}$-operators. We also
show that an $\mathrm{\mathcal{O}N}$-structure gives rise to a hierarchy of
pairwise compatible $\mathcal{O}$-operators. In Section 5, we first introduce
the notion of conformal $r$-matrix-Nijenhuis structure on a Lie conformal
algebra $A$, which consists of a skew-symmetric solution $r$ of the conformal
classical Yang-Baxter equation and a Nijenhuis operator $N$ on $A$ such that
some compatibility conditions are satisfied. We show that conformal
$r$-matrix-Nijenhuis structure gives rise to a hierarchy of pairwise
compatible $r$-matrices. Furthermore, we introduce the notion of symplectic-
Nijenhuis structure on a Lie conformal algebra $A$, which consists of a
symplectic structure $\omega$ and a Nijenhuis operator $N$ on $A$ such that
some compatibility conditions are satisfied. The relations between
$\mathrm{\mathcal{O}N}$-structures on $\mathsf{LCMod}$ pairs, conformal
$r$-matrix-Nijenhuis structures and symplectic-Nijenhuis structures on Lie
conformal algebras are given.

##  2\. Preliminaries on Lie conformal algebra

######  Definition 2.1.

A Lie conformal algebra $A$ is a $\mathbb{C}[\partial]$-module endowed with a
$\mathbb{C}$-bilinear map $A\times A\rightarrow A[\lambda]$, denoted by
$a\times b\rightarrow[a_{\lambda}b]$, satisfying

(5) $\displaystyle[\partial a_{\lambda}b]$ $\displaystyle=$
$\displaystyle-\lambda[a_{\lambda}b],$ (6) $\displaystyle{[a_{\lambda}b]}$
$\displaystyle=$ $\displaystyle-[b_{-\lambda-\partial}a],$ (7)
$\displaystyle{[a_{\lambda}[b_{\mu}c]]}$ $\displaystyle=$
$\displaystyle[[a_{\lambda}b]_{\lambda+\mu}c]+[b_{\mu}[a_{\lambda}c]],\quad\forall~{}a,b,c\in
A.$

A Lie conformal algebra is called finite if it is finitely generated as a
$\mathbb{C}[\partial]$-module.

The ordinary examples of finite Lie conformal algebra is the Virasoro and
current Lie conformal algebras.

######  Example 2.2.

The Virasoro Lie conformal algebra ${\rm Vir}$ is the simplest nontrivial
example of Lie conformal algebras, which is defined by

${\rm Vir}=\mathbb{C}[\partial]a,\quad[a_{\lambda}a]=(\partial+2\lambda)a.$

######  Example 2.3.

Let $(\mathfrak{g},[\cdot,\cdot])$ be a Lie algebra. The current Lie conformal
algebra associated with $\mathfrak{g}$ is defined by

${\rm
Cur}~{}\mathfrak{g}=\mathbb{C}[\partial]\otimes\mathfrak{g},\quad[a_{\lambda}b]=[a,b],\quad\forall~{}a,b\in\mathfrak{g}.$

In the following, we recall a class of Lie conformal algebras named quadratic
Lie conformal algebras.

######  Definition 2.4.

If a Lie conformal algebra $A=\mathbb{C}[\partial]V$ with a free
$\mathbb{C}[\partial]$-module and the $\lambda$-bracket is of the following
form:

$[a_{\lambda}b]=\partial u+\lambda v+w,\quad\forall~{}a,b\in V,$

where $u,v,w\in V$, then $A$ is called a quadratic Lie conformal algebra.

Recall that a Novikov algebra is a pair $(A,\circ)$, where $A$ is a vector
space and $\circ:A\otimes A\rightarrow A$ is a bilinear multiplication
satisfying that for all $a,b,c\in A$,

(8) $\displaystyle(a\circ b)\circ c$ $\displaystyle=$ $\displaystyle(a\circ
c)\circ b,$ (9) $\displaystyle(a\circ b)\circ c-a\circ(b\circ c)$
$\displaystyle=$ $\displaystyle(b\circ a)\circ c-b\circ(a\circ c).$

Recall that a Gel’fand-Dorfman bialgebra is a triple $(A,[\cdot,\cdot],\circ)$
such that $(A,[\cdot,\cdot])$ is a Lie algebra, $(A,\circ)$ is a Novikov
algebra and they satisfy the following condition:

(10) $[a\circ b,c]+[a,b]\circ c-a\circ[b,c]-[a\circ c,b]-[a,c]\circ b=0.$

######  Proposition 2.5.

([11, 29]) A Lie conformal algebra $A=\mathbb{C}[\partial]V$ is quadratic if
and only if $V$ is a Gel’fand-Dorfman bialgebra, where the correspondence is
given as follows:

$[a_{\lambda}b]=\partial(b\circ a)+\lambda(a\ast b)+[b,a],$

where $a\ast b=a\circ b+b\circ a$ for $a,b\in V$.

It is not hard to see that Virasoro Lie conformal algebra is a quadratic Lie
conformal algebra, which is corresponding to the Gel’fand-Dorfman bialgebra
$(A,[\cdot,\cdot],\circ)$ induced by a Novikov algebra $(V=\mathbb{C}a,\circ)$
with the multiplication $a\circ a=a$.

######  Definition 2.6.

A module $V$ over a Lie conformal algebra $A$ is a
$\mathbb{C}[\partial]$-module endowed with a $\mathbb{C}$-bilinear map
$A\times V\rightarrow V[\lambda]$, $(a,v)\rightarrow a_{\lambda}v$,satisfying
the following axioms:

(11) $\displaystyle(\partial a)_{\lambda}v$ $\displaystyle=$
$\displaystyle-\lambda a_{\lambda}v,$ (12) $\displaystyle{a_{\lambda}(\partial
v)}$ $\displaystyle=$ $\displaystyle(\partial+\lambda)a_{\lambda}v,$ (13)
$\displaystyle{[a_{\lambda}b]_{\lambda+\mu}v}$ $\displaystyle=$ $\displaystyle
a_{\lambda}(b_{\mu}v)-b_{\mu}(a_{\lambda}v),\quad\forall~{}a,b\in A,v\in V.$

An $A$-module $V$ is called finite if it is finitely generated as a
$\mathbb{C}[\partial]$-module.

Throughout this paper, we mainly deal with $\mathbb{C}[\partial]$-modules
which are finitely generated.

######  Definition 2.7.

Let $U$ and $V$ be two $\mathbb{C}[\partial]$-modules. A conformal linear map
from $U$ to $V$ is a $\mathbb{C}$-linear map $a:U\rightarrow V[\lambda]$,
denoted by $a_{\lambda}:U\rightarrow V$, such that

$[\partial,a_{\lambda}]=-\lambda a_{\lambda}.$

Denote the $\mathbb{C}$-vector space of all such maps by ${\rm Chom}(U,V)$.

The space of conformal linear maps ${\rm Chom}(U,V)$ has a canonical structure
of a $\mathbb{C}[\partial]$-module:

$(\partial a)_{\lambda}=-\lambda a_{\lambda}.$

Define the conformal dual of a $\mathbb{C}[\partial]$-module $U$ as
$U^{*c}={\rm Chom}(U,\mathbb{C})$, where $\mathbb{C}$ is viewed as the trivial
$\mathbb{C}[\partial]$-module, that is

$U^{*c}=\\{a:U\rightarrow\mathbb{C}[\lambda]\mid\mbox{$a$ is
$\mathbb{C}$-linear and }a_{\lambda}(\partial b)=\lambda a_{\lambda}b\\}.$

Let $U$ and $V$ be finite modules over a Lie conformal algebra $A$. Then, the
$\mathbb{C}[\partial]$-module ${\rm Chom}(U,V)$ has an A-module structure
defined by:

(14)
$(a_{\lambda}\varphi)_{\mu}u=a_{\lambda}(\varphi_{\mu-\lambda}u)-\varphi_{\mu-\lambda}(a_{\lambda}u),\forall~{}a\in
A,\varphi\in{\rm Chom}(U,V),u\in U.$

Let $V$ be a $\mathbb{C}[\partial]$-module and denote by ${\rm gc}(V)={\rm
Chom}(V,V)$. The $\mathbb{C}[\partial]$-module ${\rm gc}(V)$ has a canonical
structure of an associative conformal algebra defined by:

(15)
$(a_{\lambda}b)_{\mu}v=a_{\lambda}(b_{\mu-\lambda}v),\quad\forall~{}a,b\in{\rm
gc}(V),v\in V.$

Therefore, ${\rm gc}(V)$ has a Lie conformal algebra structure given by:

(16)
$[a_{\lambda}b]_{\mu}v=a_{\lambda}(b_{\mu-\lambda}v)-b_{\mu-\lambda}(a_{\lambda}v),\quad\forall~{}a,b\in{\rm
gc}(V),v\in V.$

We call Lie conformal algebra ${\rm gc}(V)$ the general Lie conformal algebra
of $V$.

Note that the structure of a finite module $V$ over a Lie conformal algebra
$A$ is the same as a homomorphism of Lie conformal algebra
$\rho:A\rightarrow{\rm gc}(V)$. It is obvious that $(\mathbb{C};\rho=0)$ is a
module over the Lie conformal algebra $A$, which we call the trivial module.

Let $(V;\rho)$ be a module over a Lie conformal algebra $A$. Define
$\rho^{*}:A\rightarrow{\rm gc}(V^{*c})$ by

$(\rho^{*}(a)_{\lambda}\varphi)_{\mu}u=-\varphi_{\mu-\lambda}(\rho(a)_{\lambda}u)$

for all $a\in A$, $\varphi\in V^{*c}$, $u\in V$. Then $(V^{*};\rho^{*})$ is a
module over $A$.

Assume that $A$ is a finite Lie conformal algebra. Define
$\mathrm{ad}:A\,\rightarrow\,{\rm gc}(A)$ by
$\mathrm{ad}(a)_{\lambda}b=[a_{\lambda}b]$ for all $a,b\in A$. Then
$(A;\mathrm{ad})$ is a module over $A$, which we call the adjoint module.
Furthermore, $(A^{*};\mathrm{ad}^{*})$ is also a module over $A$, which we
call the coadjoint module

The following conclusion is well-known.

######  Proposition 2.8.

Let $A$ be a finite Lie conformal algebra and let $(V;\rho)$ be a finite
module over $A$. Then $A\oplus V$ is endowed with a
$\mathbb{C}[\partial]$-module structure given by:

$\partial(a+v)=\partial a+\partial v,\forall a\in A,v\in V.$

Hence, the $\mathbb{C}[\partial]$-module $A\oplus V$ is endowed with a Lie
conformal algebra structure as follows:

$[(a+u)_{\lambda}(b+v)]=[a_{\lambda}b]+\rho(a)_{\lambda}v-\rho(b)_{-\lambda-\partial}u,\quad\forall~{}a,b\in
A,u,v\in V.$

This Lie conformal algebra is called the semi-direct product of $A$ and $V$,
denoted by $A\ltimes_{\rho}V$.

Let $A$ and $V$ be $\mathbb{C}[\partial]$-modules. For $k\geq 1$, a
$k$-$\lambda$-bracket on $A$ with coefficients in $V$ is a $\mathbb{C}$-linear
map $c:A^{\otimes k}\rightarrow\mathbb{C}[\lambda_{1}...\lambda_{k-1}]\otimes
V$, denoted by

$a_{1}\otimes...\otimes
a_{k}\mapsto\\{a_{1\lambda_{1}}...a_{k-1\lambda_{k-1}}a_{k}\\}_{c}$

satisfying the following conditions:

(17) $\displaystyle\\{a_{1\lambda_{1}}...(\partial
a_{i})_{\lambda_{i}}...a_{k-1\lambda_{k-1}}a_{k}\\}_{c}$ $\displaystyle=$
$\displaystyle-\lambda_{i}\\{a_{1\lambda_{1}}...a_{k-1\lambda_{k-1}}a_{k}\\}_{c},\quad
1\leq i\leq k-1,$ (18)
$\displaystyle\\{a_{1\lambda_{1}}...a_{k-1\lambda_{k-1}}(\partial
a_{k})\\}_{c}$ $\displaystyle=$
$\displaystyle(\lambda_{1}+...+\lambda_{k-1}+\partial)\\{a_{1\lambda_{1}}...a_{k-1\lambda_{k-1}}a_{k}\\}_{c},$
(19) $\displaystyle\\{a_{1\lambda_{1}}...a_{k-1\lambda_{k-1}}a_{k}\\}_{c}$
$\displaystyle=$ $\displaystyle{\rm
sign}(\sigma)\\{a_{\sigma(1)_{\lambda_{\sigma(1)}}}...a_{\sigma(k-1)_{\lambda_{\sigma(k-1)}}}a_{\sigma(k)}\\}_{c}|_{\lambda_{k}\mapsto\lambda_{k}^{+}},$

where the notation $\lambda_{k}\mapsto\lambda_{k}^{+}$ means that
$\lambda_{k}$ is replaced by
$\lambda_{k}^{+}=-\sum_{j=1}^{k-1}\lambda_{j}-\partial$, if it occurs, and
$\partial^{V}$ is moved to the left.

The cohomology complex for a Lie conformal algebra $A$ with a module $V$ in
the language of $\lambda$-brackets is given as follows ([9]). We let
$C^{0}(A,V)=V/\partial V$ and for $k\geq 1$, we denote by $C^{k}(A,V)$ the
space of all $k$-$\lambda$-brackets on $A$ with coefficients in $V$. Define
$C^{\bullet}(A,V)=\oplus_{k\in\mathbb{N}}C^{k}(A,V)$. The corresponding Lie
conformal algebra coboundary operator $\mathrm{d}:C^{k}(A,V)\longrightarrow
C^{k+1}(A,V)$ is given by

$\begin{split}\\{a_{1\lambda_{1}}\cdots
a_{k\lambda_{k}}a_{k+1}\\}_{\mathrm{d}c}&=\sum_{i=1}^{k}(-1)^{i+1}a_{i\lambda_{i}}\\{a_{1\lambda_{1}}\cdots\check{i}\cdots
a_{k\lambda_{k}}a_{k+1}\\}_{c}\\\
&+\sum_{i,j=1,i<j}^{k}(-1)^{k+i+j+1}\\{a_{1\lambda_{1}}\cdots\check{i}\cdots\check{j}\cdots
a_{k\lambda_{k}}a_{k+1_{\lambda_{k+1}^{+}}}[a_{i\lambda_{i}}a_{j}]\\}_{c}\\\
&+(-1)^{k}a_{k+1_{\lambda_{k+1}^{+}}}\\{a_{1\lambda_{1}}\cdots
a_{k-1_{\lambda_{k-1}}}a_{k}\\}_{c}\\\
&+\sum_{i=1}^{k}(-1)^{i}\\{a_{1\lambda_{1}}\cdots\check{i}\cdots
a_{k\lambda_{k}}[a_{i\lambda_{i}}a_{k+1}]\\}_{c},\end{split}$

where $c\in C^{k}(A,V)$,
$\lambda_{k+1}^{+}=-\sum_{j=1}^{k}\lambda_{j}-\partial$ and
$a_{1},\cdots,a_{k+1}\in A$. We denote by
$H^{\bullet}(A,V)=\oplus_{k\in\mathbb{N}}H^{k}(A,V)$ the cohomology of the Lie
conformal algebra $A$ with coefficients in $V$. In particular, we use the
symbol $\mathrm{d}^{T}$ to refer the coboundary operator associated to the
trivial module.

##  3\. Nijenhuis operators and Nijenhuis structures on Lie conformal algebras

In this section, we study infinitesimal deformations of a Lie conformal
algebra and an $\mathsf{LCMod}$ pair. The trivial deformations of the Lie
conformal algebra and the $\mathsf{LCMod}$ pair lead to the notion of
Nijenhuis operator on a Lie conformal algebra and Nijenhuis structure on an
$\mathsf{LCMod}$ pair, respectively.

###  3.1. Nijenhuis operators on Lie conformal algebras

Let $A$ be a Lie conformal algebra with $\lambda$-bracket
$[\cdot_{\lambda}\cdot]$ and $\omega:A\otimes A\rightarrow A[\lambda]$.
Consider a $t$-parameterized family of bilinear $\lambda$-multiplications

(20)
$[a_{\lambda}b]_{t}=[a_{\lambda}b]+t\\{a_{\lambda}b\\}_{\omega},\quad\forall~{}a,b\in
A.$

If $(A,[\cdot_{\lambda}\cdot]_{t})$ is a Lie conformal algebra for all $t$, we
say that $\omega$ generates a infinitesimal deformation of
$(A,[\cdot_{\lambda}\cdot])$.

By a direct calculation, we can deduce that $(A,[\cdot_{\lambda}\cdot]_{t})$
is a Lie conformal algebra for any $t\in\mathbb{\mathbb{R}}$ if and only if

(21) $\displaystyle\\{\partial a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle-\lambda\\{a_{\lambda}b\\}_{\omega},$ (22)
$\displaystyle\\{a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle-\\{b_{-\lambda-\partial}a\\}_{\omega},$ (23)
$\displaystyle\\{a_{\lambda}\\{b_{\mu}c\\}_{\omega}\\}_{\omega}-\\{b_{\mu}\\{a_{\lambda}c\\}_{\omega}\\}_{\omega}$
$\displaystyle=$
$\displaystyle\\{\\{a_{\lambda}b\\}_{\omega_{\lambda+\mu}}c\\}_{\omega},$

and

(24)
$\\{[a_{\lambda}b]_{\lambda+\mu}c\\}_{\omega}+[\\{a_{\lambda}b\\}_{\omega_{\lambda+\mu}}c]-\\{a_{\lambda}[b_{\mu}c]\\}_{\omega}-[a_{\lambda}\\{b_{\mu}c\\}_{\omega}]+\\{b_{\mu}[a_{\lambda}c]\\}_{\omega}+[b_{\mu}\\{a_{\lambda}c\\}_{\omega}]=0.$

By (21) and (22), we have

(25) $\displaystyle\\{a_{\lambda}(\partial b)\\}_{\omega}$ $\displaystyle=$
$\displaystyle(\lambda+\partial)\\{a_{\lambda}b\\}_{\omega}.$

We see that (21)-(23) mean that $(A,\\{\cdot_{\lambda}\cdot\\}_{\omega})$ is a
Lie conformal algebra. (24) means that $\omega$ is a $2$-cocycle of the Lie
conformal algebra $A$ with coefficients in the module $(A;\mathrm{ad})$. Thus
we have

######  Proposition 3.1.

If $\omega$ generates a infinitesimal deformation of the Lie conformal algebra
$(A,[\cdot_{\lambda}\cdot])$, then $\omega$ is a $2$-cocycle of
$(A,[\cdot_{\lambda}\cdot])$.

######  Definition 3.2.

Two infinitesimal deformations $A_{t}=(A,[\cdot_{\lambda}\cdot]_{t})$ and
$A^{\prime}_{t}=(A,[\cdot_{\lambda}\cdot]^{\prime}_{t})$ of a Lie conformal
algebra $(A,[\cdot_{\lambda}\cdot])$, which are generated by $\omega$ and
$\omega^{\prime}$ respectively, are said to be equivalent if there exists a
family of Lie conformal algebra homomorphisms
${\rm{Id}}+tN:A_{t}\longrightarrow A^{\prime}_{t}$. A deformation is said to
be trivial if there exist a family of Lie conformal algebra homomorphisms
${\rm{Id}}+tN:A_{t}\longrightarrow(A,[\cdot_{\lambda}\cdot])$.

By a direct calculation, $A_{t}$ and $A^{\prime}_{t}$ are equivalent
infinitesimal deformations if and only if

(26) $\displaystyle N\circ\partial$ $\displaystyle=$
$\displaystyle\partial\circ N,$ (27)
$\displaystyle\\{a_{\lambda}b\\}_{\omega}-\\{a_{\lambda}b\\}_{\omega^{\prime}}$
$\displaystyle=$
$\displaystyle[N(a)_{\lambda}b]+[a_{\lambda}N(b)]-N[a_{\lambda}b],$ (28)
$\displaystyle N\\{a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle\\{N(a)_{\lambda}b\\}_{\omega^{\prime}}+\\{a_{\lambda}N(b)\\}_{\omega^{\prime}}+[N(a)_{\lambda}N(b)],$
(29) $\displaystyle\\{N(a)_{\lambda}N(b)\\}_{\omega^{\prime}}$
$\displaystyle=$ $\displaystyle 0.$

It is obvious that (26) means that $N$ is a $1$-cochain and
$(\ref{eq:equivalent2})$ means that
$\\{a_{\lambda}b\\}_{\omega}-\\{a_{\lambda}b\\}_{\omega^{\prime}}=\\{a_{\lambda}b\\}_{\mathrm{d}N}$.
We summarize the above discussions into the following conclusion:

######  Proposition 3.3.

If two infinitesimal deformations $A_{t}=(A,[\cdot_{\lambda}\cdot]_{t})$ and
$A^{\prime}_{t}=(A,[\cdot_{\lambda}\cdot]^{\prime}_{t})$ of a Lie conformal
algebra $(A,[\cdot_{\lambda}\cdot])$ generated by $\omega$ and
$\omega^{\prime}$ respectively are equivalent, then $\omega$ and
$\omega^{\prime}$ are in the same cohomology class in $H^{2}(A,A)$.

Now we consider trivial deformations of a Lie conformal algebra
$(A,[\cdot_{\lambda}\cdot])$. Then (26)-(29) reduce to

(30) $\displaystyle N\circ\partial$ $\displaystyle=$
$\displaystyle\partial\circ N,$ (31)
$\displaystyle\\{a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle[N(a)_{\lambda}b]+[a_{\lambda}N(b)]-N[a_{\lambda}b],$ (32)
$\displaystyle N\\{a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle[N(a)_{\lambda}N(b)].$

It follows from (31) and (32) that $N$ should satisfy the following condition

(33)
$N\big{(}[N(a)_{\lambda}(b)]+[a_{\lambda}N(b)]-N[a_{\lambda}b]\big{)}=[N(a)_{\lambda}N(b)].$

The following definition first appears in the study of Nijenhuis operators on
associative algebras.

######  Definition 3.4.

([30]) Let $(A,[\cdot_{\lambda}\cdot])$ be a Lie conformal algebra. A
$\mathbb{C}[\partial]$-module homomorphism $N:A\rightarrow A$ is called a
Nijenhuis operator if (33) holds.

######  Theorem 3.5.

Let $N:A\rightarrow A$ be a Nijenhuis operator. A infinitesimal deformation of
$A$ can be obtained by putting

$\\{a_{\lambda}b\\}_{\omega}=[N(a)_{\lambda}b]+[a_{\lambda}N(b)]-N[a_{\lambda}b],\quad\forall~{}a,b\in
A.$

Moreover, this deformation is trivial.

###### Proof.

By (5) and $N(\partial a)=\partial N(a)$, we have

$\begin{split}\\{\partial a_{\lambda}b\\}_{\omega}&=[\partial
a_{\lambda}N(b)]+[N(\partial a)_{\lambda}b]-N^{2}[\partial a_{\lambda}b]\\\
&=-\lambda[a_{\lambda}N(b)]-\lambda[N(a)_{\lambda}(b)+\lambda
N^{2}[a_{\lambda}b]=-\lambda\\{a_{\lambda}b\\}_{\omega},\end{split}$

which proves (21).

By (6), we have

$\begin{split}\\{a_{\lambda}b\\}_{\omega}&=[a_{\lambda}N(b)]+[N(a)_{\lambda}b]-N^{2}[a_{\lambda}b]\\\
&=-([b_{-\lambda-\partial}N(a)]-[N(b)_{-\lambda-\partial}a]-N^{2}[b_{-\lambda-\partial}a])\\\
&=-\\{b_{-\lambda-\partial}a\\}_{\omega},\end{split}$

which proves (22).

Since $\omega=dN$, we get $d\omega=d(dN)=0$ and thus (24) holds.

Next we prove that the Jacobi identity holds for
$\\{\cdot_{\lambda}\cdot\\}_{\omega}$. Put

$J(a_{1},a_{2},a_{3})=\\{a_{1\lambda_{1}}\\{a_{2\lambda_{2}}a_{3}\\}_{\omega}\\}_{\omega}-\\{a_{2\lambda_{2}}\\{a_{1\lambda_{1}}a_{3}\\}_{\omega}\\}_{\omega}-\\{\\{a_{1\lambda_{1}}a_{2}\\}_{\omega_{\lambda_{1}+\lambda_{2}}}a_{3}\\}_{\omega}.$

Substituting the expression of $\\{\cdot_{\lambda}\cdot\\}_{\omega}$ into this
formula and using the property of Nijenhuis operator on $A$, we have

$\displaystyle J(a_{1},a_{2},a_{3})$ $\displaystyle=$
$\displaystyle[a_{1_{\lambda_{1}}}[Na_{2\lambda_{2}}Na_{3}]]+[Na_{1\lambda_{1}}[a_{2\lambda_{2}}Na_{3}]]-N[a_{1\lambda_{1}}[a_{2\lambda_{2}}Na_{3}]]$
$\displaystyle+[Na_{1\lambda_{1}}[Na_{2\lambda_{2}}a_{3}]]-N[a_{1\lambda_{1}}[Na_{2\lambda_{2}}a_{3}]]-[Na_{1\lambda_{1}}N[a_{2\lambda_{2}}a_{3}]]$
$\displaystyle-[a_{2_{\lambda_{2}}}[Na_{1\lambda_{1}}Na_{3}]]-[Na_{2\lambda_{2}}[a_{1\lambda_{1}}Na_{3}]]+N[a_{2\lambda_{2}}[a_{1\lambda_{1}}Na_{3}]]-[Na_{2\lambda_{2}}[Na_{1\lambda_{1}}a_{3}]]$
$\displaystyle+N[a_{1\lambda_{1}}N[a_{2\lambda_{2}}a_{3}]]+N[a_{2\lambda_{2}}[Na_{1\lambda_{1}}a_{3}]]+[Na_{2\lambda_{2}}N[a_{1\lambda_{1}}a_{3}]]$
$\displaystyle-N[a_{2\lambda_{2}}N[a_{1\lambda_{1}}a_{3}]]-[[a_{1\lambda_{1}}Na_{2}]_{\lambda_{1}+\lambda_{2}}Na_{3}]-[[Na_{1\lambda_{1}}Na_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}]$
$\displaystyle+N[[a_{1\lambda_{1}}Na_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}]-[[Na_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}Na_{3}]+N[[Na_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}]$
$\displaystyle+[N[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}Na_{3}]-N[N[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}].$

Due to the validity of the Jacobi identity of $[\cdot_{\lambda}\cdot]$, we
have

$\displaystyle J(a_{1},a_{2},a_{3})$ $\displaystyle=$
$\displaystyle-N[a_{1\lambda_{1}}[a_{2\lambda_{2}}Na_{3}]]-N[a_{1\lambda_{1}}[Na_{2\lambda_{2}}a_{3}]]-[Na_{1\lambda_{1}}N[a_{2\lambda_{2}}a_{3}]]$
$\displaystyle+N[a_{1\lambda_{1}}N[a_{2\lambda_{2}}a_{3}]]+N[a_{2\lambda_{2}}[a_{1\lambda_{1}}Na_{3}]]+N[a_{2\lambda_{2}}[Na_{1\lambda_{1}}a_{3}]]$
$\displaystyle+[Na_{2\lambda_{2}}N[a_{1\lambda_{1}}a_{3}]]-N[a_{2\lambda_{2}}N[a_{1\lambda_{1}}a_{3}]]+N[[a_{1\lambda_{1}}Na_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}]$
$\displaystyle+N[[Na_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}]+[N[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}Na_{3}]-N[N[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}].$

Since $N$ is a Nijenhuis operator,we have

$\displaystyle-{[Na_{1\lambda_{1}}N[a_{2\lambda_{2}}a_{3}]]}+N[a_{1\lambda_{1}}N[a_{2\lambda_{2}}a_{3}]]$
$\displaystyle=$
$\displaystyle-N[Na_{1\lambda_{1}}[a_{2\lambda_{2}}a_{3}]]+N^{2}[a_{1\lambda_{1}}[a_{2\lambda_{2}}a_{3}]],$
$\displaystyle{[Na_{2\lambda_{2}}N[a_{1\lambda_{1}}a_{3}]]}-N[a_{2\lambda_{2}}N[a_{1\lambda_{1}}a_{3}]]$
$\displaystyle=$ $\displaystyle
N[Na_{2\lambda_{2}}[a_{1\lambda_{1}}a_{3}]]+N^{2}[a_{2\lambda_{2}}[a_{1\lambda_{1}}a_{3}]],$
$\displaystyle{[N[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}Na_{3}]}-N[N[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}]$
$\displaystyle=$ $\displaystyle
N[[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}Na_{3}]-N^{2}[[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}].$

Again using the Jacobi identity of $[\cdot_{\lambda}\cdot]$, we have

$\displaystyle J(a_{1\lambda_{1}}a_{2\lambda_{2}}a_{3})$ $\displaystyle=$
$\displaystyle-N\big{(}[a_{1\lambda_{1}}[a_{2\lambda_{2}}Na_{3}]]-[a_{2\lambda_{2}}[a_{1\lambda_{1}}Na_{3}]]-[[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}Na_{3}]\big{)}$
$\displaystyle+[a_{1\lambda_{1}}[Na_{2\lambda_{2}}a_{3}]]-[a_{2\lambda_{2}}[Na_{1\lambda_{1}}a_{3}]]-[[a_{1\lambda_{1}}Na_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}]$
$\displaystyle+[Na_{1\lambda_{1}}[a_{2\lambda_{2}}a_{3}]]-[Na_{2\lambda_{2}}[a_{1\lambda_{1}}a_{3}]]-[[Na_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}]$
$\displaystyle+N^{2}([a_{1\lambda_{1}}[a_{2\lambda_{2}}a_{3}]]-[a_{2\lambda_{2}}[a_{1\lambda_{1}}a_{3}]]-[[a_{1\lambda_{1}}a_{2}]_{\lambda_{1}+\lambda_{2}}a_{3}])$
$\displaystyle=$ $\displaystyle 0.$

Thus $\omega$ generates a infinitesimal deformation of $A$.

Furthermore, (30)-(32) are satisfied and therefore $\omega$ generates a
trivial deformation of $A$.

######  Corollary 3.6.

Let $(A,[\cdot_{\lambda}\cdot])$ be a Lie conformal algebra and $N$ a
Nijenhuis operator on $A$. Then $(A,[\cdot_{\lambda}\cdot]_{N})$ is a Lie
conformal algebra and $N$ is a Lie conformal algebra homomorphism from
$(A,[\cdot_{\lambda}\cdot]_{N})$ to $(A,[\cdot_{\lambda}\cdot])$, where the
deformed bracket $[\cdot_{\lambda}\cdot]_{N}$ is given by

(34)
$\\{a_{\lambda}b\\}_{N}=[N(a)_{\lambda}b]+[a_{\lambda}N(b)]-N[a_{\lambda}b],\quad\forall~{}a,b\in
A.$

######  Example 3.7.

Let $(A,[\cdot_{\lambda}\cdot])$ be a Lie conformal algebra, which can be
decomposed as a sum of two subalgebras $A=A_{1}\oplus A_{2}$. For any
$k_{1},k_{2}\in\mathbb{C}$, define $N:A\rightarrow A$ by

$N\mid_{A_{i}}=k_{i}\rm{Id}_{A_{i}},\quad i=1,2.$

Then $N$ is a Nijenhuis operator on the Lie conformal algebra $A$.

Recall that a Nijenhuis operator $N$ on a Lie algebra
$(\mathfrak{g},[\cdot,\cdot])$ is a linear operator
$N:\mathfrak{g}\longrightarrow\mathfrak{g}$ satisfying

$N([N(x),y]+[x,N(y)]-N([x,y]))=[N(x),N(y)],\quad\forall~{}x,y\in\mathfrak{g}.$

Then $(\mathfrak{g},[\cdot,\cdot]_{N})$ is a Lie algebra and $N$ is a morphism
from the Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{N})$ to
$(\mathfrak{g},[\cdot,\cdot])$, where the bracket $[\cdot,\cdot]_{N}$ is
defined by

$[x,y]_{N}=[N(x),y]+[x,N(y)]-N([x,y]),\;\;\forall x,y\in\mathfrak{g}.$

######  Example 3.8.

Let $N$ be a Nijenhuis operator on a Lie algebra
$(\mathfrak{g},[\cdot,\cdot])$. Then $\tilde{N}$ defined by

(35)
$\tilde{N}(f(\partial)a)=f(\partial)N(a),\quad\forall~{}f(\partial)\in\mathbb{C}[\partial],a\in\mathfrak{g}$

is a Nijenhuis operator on the current Lie conformal algebra ${\rm
Cur}~{}\mathfrak{g}=\mathbb{C}[\partial]\otimes\mathfrak{g}$.

######  Definition 3.9.

A Nijenhuis operator $N$ on a Novikov algebra $(A,\circ)$ is a linear operator
$N:A\longrightarrow A$ satisfying

$N(N(a)\circ b+a\circ N(b)-N(a\circ b))=N(a)\circ N(b),\quad\forall~{}a,b\in
A.$

######  Proposition 3.10.

Let $N$ be a Nijenhuis operator on a Novikov algebra $(A,\circ)$. Then
$(A,\circ_{N})$ is also a Novikov algebra and furthermore, $N$ is a Novikov
algebra homomorphism from $(A,\circ_{N})$ to $(A,\circ)$.

###### Proof.

It was shown in [28] that $(A,\circ_{N})$ is a left-symmetric algebra, which
means that (9) holds for the operation $\circ_{N}$. Also, by the fact that $N$
is a Nijenhuis operator on the Novikov algebra $(A,\circ)$, we have

$\displaystyle(a\circ_{N}b)\circ_{N}c-(a\circ_{N}c)\circ_{N}c$
$\displaystyle=$ $\displaystyle\big{(}(N(a)\circ N(b))\circ c-(N(a)\circ
c)\circ N(b)\big{)}+\big{(}(N(a)\circ b)\circ N(c)-(N(a)\circ N(c))\circ
b\big{)}$ $\displaystyle+\big{(}(a\circ N(b))\circ N(c)-(a\circ N(c))\circ
N(b)\big{)}-N\big{(}(a\circ b)\circ N(c)-(a\circ N(c))\circ b\big{)}$
$\displaystyle-N\big{(}(N(a)\circ b)\circ c-(N(a)\circ c)\circ
b\big{)}-N\big{(}(a\circ N(b))\circ c-(a\circ c)\circ N(b)\big{)}$
$\displaystyle+N^{2}\big{(}(a\circ b)\circ c-(a\circ c)\circ b\big{)}.$

By (8), the above equality implies that
$(a\circ_{N}b)\circ_{N}c=(a\circ_{N}c)\circ_{N}c$ holds. Thus $(A,\circ_{N})$
is also a Novikov algebra. The rest is direct.

######  Definition 3.11.

Let $(A,[\cdot,\cdot],\circ)$ be a Gel’fand-Dorfman bialgebra. A linear map
$N:A\rightarrow A$ is called a Nijenhuis operator on the Gel’fand-Dorfman
bialgebra $A$ if $N$ is both a Nijenhuis operator on the Lie algebra
$(A,[\cdot,\cdot])$ and a Nijenhuis operator on the Novikov algebra
$(A,\circ)$.

######  Proposition 3.12.

Let $N$ be a Nijenhuis operator on a Gel’fand-Dorfman bialgebra
$(A,[\cdot,\cdot],\circ)$. Then $(A,[\cdot,\cdot]_{N},\circ_{N})$ is also a
Gel’fand-Dorfman bialgebra and furthermore, $N$ is a Gel’fand-Dorfman
bialgebra homomorphism from $(A,[\cdot,\cdot]_{N},\circ_{N})$ to
$(A,[\cdot,\cdot],\circ)$.

###### Proof.

Since $N$ is a Nijenhuis operator on the Lie algebra $(A,[\cdot,\cdot])$,
$(A,[\cdot,\cdot]_{N})$ is a Lie algebra. Similarly, $(A,\circ_{N})$ is also a
Novikov algebra. Furthermore, by the properties of Nijenhuis operator on
Gel’fand-Dorfman bialgebra, by a direct calculation, we have

$\displaystyle[a\circ_{N}b,c]_{N}+[a,b]_{N}\circ_{N}c-a\circ_{N}[b,c]_{N}-[a\circ_{N}c,b]_{N}-[a,c]_{N}\circ_{N}b$
$\displaystyle=$ $\displaystyle\big{(}[N(a)\circ N(b),c]+[N(a),N(b)]\circ
c-N(a)\circ[N(b),c]-[N(a)\circ c,N(b)]-[N(a),c]\circ N(b)\big{)}$
$\displaystyle+\big{(}[N(a)\circ b,N(c)]+[N(a),b]\circ
N(c)-N(a)\circ[b,N(c)]-[N(a)\circ N(c),b]-[N(a),N(c)]\circ b\big{)}$
$\displaystyle+\big{(}[a\circ N(b),N(c)]+[a,N(b)]\circ
N(c)-a\circ[N(b),N(c)]-[a\circ N(c),N(b)]-[a,N(c)]\circ N(b)\big{)}$
$\displaystyle-N\big{(}[N(a)\circ b,c]+[N(a),b]\circ
c-N(a)\circ[b,c]-[N(a)\circ c,b]-[N(a),c]\circ b\big{)}$
$\displaystyle-N\big{(}[a\circ N(b),c]+[a,N(b)]\circ c-a\circ[N(b),c]-[a\circ
c,N(b)]-[a,c]\circ N(b)\big{)}$ $\displaystyle-N\big{(}[a\circ
b,N(c)]+[a,b]\circ N(c)-a\circ[b,N(c)]-[a\circ N(c),b]-[a,N(c)]\circ b\big{)}$
$\displaystyle+N^{2}\big{(}[a\circ b,c]+[a,b]\circ c-a\circ[b,c]-[a\circ
c,b]-[a,c]\circ b\big{)}.$

By (10), the above equality implies that

$[a\circ_{N}b,c]_{N}+[a,b]_{N}\circ_{N}c-a\circ_{N}[b,c]_{N}-[a\circ_{N}c,b]_{N}-[a,c]_{N}\circ_{N}b=0.$

Thus $(A,[\cdot,\cdot]_{N},\circ_{N})$ is a Gel’fand-Dorfman bialgebra.

######  Proposition 3.13.

Let $N$ be a Nijenhuis operator on a Gel’fand-Dorfman bialgebra
$(A,[\cdot,\cdot],\circ)$. Then $\tilde{N}$ defined by

$\tilde{N}(f(\partial)a)=f(\partial)N(a)$

is a Nijenhuis operator on the quadratic Lie conformal algebra
$A=\mathbb{C}[\partial]V$ with the following $\lambda$-bracket

$[a_{\lambda}b]=\partial(b\circ a)+\lambda(a\ast b)+[b,a].$

Furthermore, $(A,[\cdot_{\lambda}\cdot]_{N})$ is also a quadratic Lie
conformal algebra with the $\lambda$-bracket
$[\cdot_{\lambda}\cdot]_{\tilde{N}}$ given by

$[a_{\lambda}b]_{\tilde{N}}=\partial(b\circ_{\tilde{N}}a)+\lambda(a\ast_{\tilde{N}}b)+[b,a]_{\tilde{N}},$

where $a\ast_{\tilde{N}}b=a\circ_{\tilde{N}}b+b\circ_{\tilde{N}}a$ and this
quadratic Lie conformal algebra corresponds to the Gel’fand-Dorfman bialgebra
$(A,[\cdot,\cdot]_{N},\circ_{N})$.

###### Proof.

It follows by a direct calculation.

######  Example 3.14.

The Virasoro Lie conformal algebra is a quadratic Lie conformal algebra, which
is induced by a Novikov algebra $(V=\mathbb{C}a,\circ)$ with the
multiplication $a\circ a=a$. It is straightforward to check that for any
$k\in\mathbb{C}$, $N(a)=ka$ is a Nijenhuis operator on the Novikov algebra
$(V=\mathbb{C}a,\circ)$. Thus $\tilde{N}$ defined by

$\tilde{N}(f(\partial)a)=kf(\partial)a$

is a Nijenhuis operator on the Virasoro Lie conformal algebra
$\mathbb{C}[\partial]a$. Furthermore, any Nijenhuis operator on the Virasoro
Lie conformal algebra $\mathbb{C}[\partial]a$ is of this form.

The following example says that not all Nijenhuis operators on quadratic Lie
conformal algebra induced by the Nijenhuis operators on Gel’fand-Dorfman
bialgebra.

######  Example 3.15.

Let $A=\mathbb{C}[\partial]a\oplus\mathbb{C}[\partial]b$ be a quadratic Lie
conformal algebra of rank $2$ with the $\lambda$-bracket given by:

$[a_{\lambda}a]=(\partial+2\lambda)a,\quad[a_{\lambda}b]=(\partial+\lambda)b,\quad[b_{\lambda}b]=0,$

which is induced by a Novikov algebra $(V=\mathbb{C}a\oplus\mathbb{C}b,\circ)$
with the multiplication

$a\circ a=a,\quad a\circ b=0,\quad b\circ a=b,\quad b\circ b=0.$

Then $N$ given by

$N(a)=f(\partial)b,\quad N(b)=0,\quad f(\partial)\in\mathbb{C}[\partial]$

is a Nijenhuis operator on the Lie conformal algebra
$A=\mathbb{C}[\partial]a\oplus\mathbb{C}[\partial]b$, which is not induced by
any of the Nijenhuis operator on the Novikov algebra
$(V=\mathbb{C}a\oplus\mathbb{C}b,\circ)$.

###  3.2. Nijenhuis structures on $\mathsf{LCMod}$ pairs

######  Definition 3.16.

An $\mathsf{LCMod}$ pair consists of a Lie conformal algebra
$(A,[\cdot_{\lambda}\cdot])$ and a finite module $(V;\rho)$ over $A$. We
denote an $\mathsf{LCMod}$ pair by $(A,[\cdot_{\lambda}\cdot];\rho)$ or simply
by $(A;\rho)$.

Let $(A;\rho)$ be an $\mathsf{LCMod}$ pair. Let $\omega:A\otimes A\rightarrow
A[\lambda]$ and $\varpi:A\rightarrow{\rm gc}(V)$ be linear maps. Consider a
$t$-parametrized family of bracket operations and linear maps

$\displaystyle[a_{\lambda}b]_{t}$ $\displaystyle=$
$\displaystyle[a_{\lambda}b]+t\omega_{\lambda}(a,b),$
$\displaystyle\rho_{t}(a)_{\mu}(v)$ $\displaystyle=$
$\displaystyle\rho(a)_{\mu}(v)+t\varpi(a)_{\mu}(v).$

If $(A,[\cdot_{\lambda}\cdot]_{t})$ is a Lie conformal algebra and
$(V;\rho_{t})$ is a module for any $t\in\mathbb{R}$, we say that
$(\omega,\varpi)$ generates a infinitesimal deformation of the
$\mathsf{LCMod}$ pair $(A;\rho)$. We denote a infinitesimal deformation of the
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$ by
$(A,[\cdot_{\lambda}\cdot]_{t};\rho_{t})$.

By a direct calculation, we can deduce that
$(A,[\cdot_{\lambda}\cdot]_{t};\rho_{t})$ is an $\mathsf{LCMod}$ pair for any
$t\in\mathbb{\mathbb{R}}$ if and only if $(\omega,\varpi)$ satisfies (21)-(24)
and

$\displaystyle\rho(\\{a_{\lambda}b\\}_{\omega})_{\mu}(v)$ $\displaystyle+$
$\displaystyle\rho(b)_{\mu-\lambda}\varpi(a)_{\lambda}(v)-\rho(a)_{\lambda}\varpi(b)_{\mu-\lambda}(v)$
$\displaystyle=$
$\displaystyle\varpi(a)_{\lambda}\rho(b)_{\mu-\lambda}(v)-\varpi[a_{\lambda}b]_{\mu}(v)-\varpi(b)_{\mu-\lambda}\rho(a)_{\lambda}(v),$
(37) $\displaystyle\varpi(\\{a_{\lambda}b\\}_{\omega})_{\mu}(v)$
$\displaystyle=$
$\displaystyle\varpi(a)_{\lambda}(\varpi(b)_{\mu-\lambda}(v))-\varpi(b)_{\mu-\lambda}(\varpi(a)_{\lambda}(v)),$

where $a,b\in A$ and $v\in V$.

######  Definition 3.17.

Two infinitesimal deformations $(A,[\cdot_{\lambda}\cdot]_{t};\rho_{t})$ and
$(A^{\prime},[\cdot_{\lambda}\cdot];\rho_{t}^{\prime})$ of an $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot];\rho)$ are equivalent if for
$N:A\longrightarrow A$ and $S:V\longrightarrow V$, there exists an isomorphism
$({\rm{Id}}_{A}+tN,{\rm{Id}}_{V}+tS)$ from
$(A^{\prime},[\cdot_{\lambda}\cdot];\rho_{t}^{\prime})$ to
$(A,[-_{\lambda}-]_{t},\rho_{t})$, i.e. for all $a,b\in A$ and $v\in V$,

$\displaystyle({\rm{Id}}_{A}+tN)[a_{\lambda}b]^{\prime}_{t}$ $\displaystyle=$
$\displaystyle[({\rm{Id}}_{A}+tN)(a)_{\lambda}({\rm{Id}}_{A}+tN)(b)],$
$\displaystyle({\rm{Id}}_{V}+tS)\rho^{\prime}_{t}(a)_{\mu}(v)$
$\displaystyle=$
$\displaystyle\rho(({\rm{Id}}_{A}+tN)(a))_{\mu}({\rm{Id}}_{V}+tS)(v).$

A infinitesimal deformation of a $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];\rho)$ is said to be trivial if it is equivalent to
$(A,[-_{\lambda}-];\rho)$.

It is straightforward to check that $(A,[-_{\lambda}-]_{t};\rho_{t})$ is a
trivial deformation if and only if

(38) $\displaystyle N\circ\partial$ $\displaystyle=$
$\displaystyle\partial\circ N,$ (39) $\displaystyle S\circ\partial$
$\displaystyle=$ $\displaystyle\partial\circ S,$ (40)
$\displaystyle\\{a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle[a_{\lambda}Nb]+[Na_{\lambda}b]-N[a_{\lambda}b],$ (41)
$\displaystyle N\\{a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle[Na_{\lambda}Nb],$ (42) $\displaystyle\varpi(a)_{\mu}(v)$
$\displaystyle=$
$\displaystyle\rho(N(a))_{\mu}(v)+\rho(a)_{\mu}S(v)-S\rho(a)_{\mu}(v),$ (43)
$\displaystyle\rho(N(a))_{\mu}S(v)$ $\displaystyle=$ $\displaystyle
S\varpi(a)_{\mu}(v).$

It follows from (38), (40) and (41) that $N$ is a Nijenhuis operator on the
Lie conformal algebra $(A,[\cdot_{\lambda}\cdot])$. It follows from (42) and
(43) that

(44)
$\rho(N(a))_{\mu}(S(v))=S(\rho(N(a))_{\mu}(v))+S(\rho(a)_{\mu}S(v))-S^{2}(\rho(a)_{\mu}(v)).$

In a trivial infinitesimal deformation of an $\mathsf{LCMod}$ pair $(A;\rho)$,
$N$ is a Nijenhuis operator on the Lie conformal algebra $A$ and conditions
(39) and (44) hold. In fact, the converse is also true.

######  Theorem 3.18.

If $N$ is a Nijenhuis operator on the Lie conformal algebra $A$ and $S$
satisfies (39) and (44), then a deformation of the $\mathsf{LCMod}$ pair
$(A;\rho)$ can be obtained by putting

(45) $\displaystyle\\{a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle[a_{\lambda}Nb]+[Na_{\lambda}b]-N[a_{\lambda}b],$ (46)
$\displaystyle\varpi(a)_{\mu}(v)$ $\displaystyle=$
$\displaystyle\rho(N(a))_{\mu}(v)+\rho(a)_{\mu}S(v)-S\rho(a)_{\mu}(v),\quad\forall~{}a,b\in
A,v\in V.$

Furthermore, this deformation is trivial.

###### Proof.

It is a straightforward computation. We omit the details

It is not hard to check that

######  Proposition 3.19.

Let $(A;\rho)$ be an $\mathsf{LCMod}$ pair. Then $N$ is a Nijenhuis operator
on the Lie conformal algebra $A$ and $S$ satisfies (39) and (44) if and only
if $N+S$ is a Nijenhuis operator on the semidirect product Lie conformal
algebra $A\ltimes_{\rho}V.$

######  Definition 3.20.

Let $(A,[\cdot_{\lambda}\cdot];\rho)$ be a $\mathsf{LCMod}$ pair. Let
$N:A\rightarrow A$ and $S:V\rightarrow V$ be $\mathbb{C}[\partial]$-module
homomorphisms. A pair $(N,S)$ is called a Nijenhuis structure on the
$\mathsf{LCMod}$ pair $(A;\rho)$ if $N$ is a Nijenhuis operator on the Lie
conformal algebra $A$ and for all $a\in A,v\in V$, the following conditions
hold:

(47)
$\rho(N(a))_{\lambda}S(v)=S(\rho(N(a))_{\lambda}(v))+\rho(a)_{\lambda}S^{2}(v)-S((\rho(a))_{\lambda}S(v)).$

Let $S:V\rightarrow V$ be an $\mathbb{C}[\partial]$-module homomorphism.
Define

(48)
$S^{*}(\alpha)_{\lambda}v=\alpha_{\lambda}S(v),\quad\forall~{}\alpha\in{V^{*}}^{c},v\in
V.$

Since

$\displaystyle S^{*}(\alpha)_{\lambda}(\partial v)=\alpha_{\lambda}S(\partial
v)=\alpha_{\lambda}\partial S(v)=\lambda S^{*}(\alpha)_{\lambda}v,$

we have $S^{*}(\alpha)\in{V^{*}}^{c}$.

######  Proposition 3.21.

Let $(A,[\cdot_{\lambda}\cdot];\rho)$ be an $\mathsf{LCMod}$ pair. If $(N,S)$
is a Nijenhuis structure on the $\mathsf{LCMod}$ pair $(A;\rho)$, then $N$ and
$S^{*}$ generate a trivial infinitesimal deformation of the $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot];\rho^{*})$.

###### Proof.

By a direct calculation, we have

$\displaystyle
S^{*}(\partial\alpha)_{\lambda}v=(\partial\alpha)_{\lambda}S(v)=-\lambda\alpha_{\lambda}S(v)=(\partial
S^{*}(\alpha))_{\lambda}v,$

which implies that $S^{*}(\partial\alpha)=\partial S^{*}(\alpha)$.

Furthermore, by (47), we have

$\displaystyle(\rho^{*}(N(a))_{\lambda}(S^{*}(\alpha)))_{\mu}v-S^{*}(\rho^{*}(N(a))_{\lambda}(\alpha))_{\mu}v-S^{*}(\rho(a)_{\lambda}S^{*}(\alpha))_{\mu}v+{S^{*}}^{2}(\rho^{*}(a)_{\lambda}(\alpha))_{\mu}v$
$\displaystyle=$
$\displaystyle-\alpha_{\mu-\lambda}(S\rho(N(a))_{\lambda}v)+\alpha_{\mu-\lambda}(\rho(N(a))_{\lambda}S(v))+\alpha_{\mu-\lambda}(S(\rho(a))_{\lambda}S(v))-\alpha_{\mu-\lambda}(\rho(a)_{\lambda}(S^{2}(v)))=0.$

By Theorem 3.18, $N$ and $S^{*}$ generate a trivial infinitesimal deformation
of the $\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho^{*})$

######  Corollary 3.22.

Let $(N,S)$ be a Nijenhuis structure on a $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];\rho)$. Then for any $i\in\mathbb{N}$,
$(N^{i},S^{i})$ is a Nijenhuis structure on the $\mathsf{LCMod}$ pair
$(A;\rho)$.

###### Proof.

It follows by a direct calculation.

######  Example 3.23.

Let $N$ be a Nijenhuis operator on a Lie conformal algebra
$(A,[\cdot_{\lambda}\cdot])$. Then $(N,N^{*})$ is a Nijenhuis structure on the
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\mathrm{ad}^{*})$.

Given a Nijenhuis structure $(N,S)$ on an $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];\rho)$, we consider the associated deformed Lie
conformal algebra $(A,[\cdot_{\lambda}\cdot]_{N})$, and construct a module
over $(A,[\cdot_{\lambda}\cdot]_{N})$ which will be used in the following
sections.

Let $N$ and $S$ be $\mathbb{C}[\partial]$-module homomorphisms. Define
$\tilde{\rho}:A\longrightarrow{\rm gc}(V)$ by

(49)
$\displaystyle\tilde{\rho}(a)_{\lambda}(v)=\rho(Na)_{\lambda}(v)-\rho(a)_{\lambda}S(v)+S(\rho(a)_{\lambda}(v)),\quad\forall~{}a\in
A,v\in V.$

######  Proposition 3.24.

If $(N,S)$ is a Nijenhuis structure on an $\mathsf{LCMod}$ pair $(A;\rho)$,
then $(V;\tilde{\rho})$ is a module over the Lie conformal algebra
$(A,[\cdot_{\lambda}\cdot]_{N})$.

###### Proof.

Note that for $a\in A,v\in V,\alpha\in V^{*c}$, we have

$(\tilde{\rho}^{*}(a)_{\lambda}(\alpha))_{\mu}v=(\rho^{*}(Na)_{\lambda}\alpha)_{\mu}v-(S^{*}\rho^{*}(a)_{\lambda}\alpha)_{\mu}v+(\rho^{*}(a)_{\lambda}S^{*}(\alpha))_{\mu}v,$

which implies that

$\tilde{\rho}^{*}(a)_{\lambda}(\alpha)=\rho^{*}(Na)_{\lambda}\alpha-S^{*}\rho^{*}(a)_{\lambda}\alpha+\rho^{*}(a)_{\lambda}S^{*}(\alpha).$

Since $N+S^{*}$ is a Nijenhuis operator on the semidirect product Lie
conformal algebra $A\ltimes_{\rho^{*}}V^{*c}$, for $a,b\in A$ and
$\alpha,\beta\in V^{*c}$, the deformed bracket
$[\cdot_{\lambda}\cdot]_{N+S^{*}}$is given by

$\displaystyle[(a+\alpha)_{\lambda}(b+\beta)]_{N+S^{*}}$ $\displaystyle=$
$\displaystyle[(N+S^{*})(a+\alpha)_{\lambda}(b+\beta)]_{\rho}+[(a+\alpha)_{\lambda}(N+S^{*})(b+\beta)]_{\rho}-(N+S^{*})[(a+\alpha)_{\lambda}(b+\beta)]_{\rho}$
$\displaystyle=$
$\displaystyle[N(a)_{\lambda}b]+[a_{\lambda}N(b)]-N[a_{\lambda}b]+\rho^{*}(Na)_{\lambda}(\beta)-\rho^{*}(b)_{-\lambda-\partial}S^{*}(\alpha)+\rho^{*}(a)_{\lambda}S^{*}(\beta)$
$\displaystyle-\rho^{*}(Nb)_{-\lambda-\partial}(\alpha)-S^{*}\rho^{*}(a)_{\lambda}(\beta)+S^{*}\rho^{*}(b)_{-\lambda-\partial}(\alpha)$
$\displaystyle=$
$\displaystyle[a_{\lambda}b]_{N}+\tilde{\rho}^{*}(a)_{\lambda}(\beta)-\tilde{\rho}^{*}(b)_{-\lambda-\partial}(\alpha),$

which implies that $(V^{*c};\tilde{\rho}^{*})$ is a module over the Lie
conformal algebra $(A,[\cdot_{\lambda}\cdot]_{N})$. Since the module $V$ is
finite, $(V;\tilde{\rho})$ is a module over the Lie conformal algebra
$(A,[\cdot_{\lambda}\cdot]_{N})$.

##  4\. $\mathrm{\mathcal{O}N}$-structures and compatible
$\mathcal{O}$-operators on $\mathsf{LCMod}$ pairs

In this section, we give the notion of an $\mathrm{\mathcal{O}N}$-structure on
an $\mathsf{LCMod}$ pair, which is a conformal analogue of the Possion-
Nijenhuis structure.

Recall that a left-symmetric conformal algebra $A$ is a
$\mathbb{C}[\partial]$-module endowed with a $\mathbb{C}$-bilinear map
$A\times A\rightarrow A[\lambda]$, denoted by $a\times b\rightarrow
a\ast_{\lambda}b$, satisfying

(50) $\displaystyle\partial a\ast_{\lambda}b$ $\displaystyle=$
$\displaystyle-\lambda a\ast_{\lambda}b,$ (51)
$\displaystyle{a\ast_{\lambda}\partial b}$ $\displaystyle=$
$\displaystyle(\partial+\lambda)a\ast_{\lambda}b,$ (52)
$\displaystyle(a\ast_{\lambda}b)\ast_{\lambda+\mu}c-a\ast_{\lambda}(b\ast_{\mu}c)$
$\displaystyle=$
$\displaystyle(b\ast_{\mu}a)\ast_{\lambda+\mu}c-b\ast_{\mu}(a\ast_{\lambda}c),\quad\forall~{}a,b,c\in
A.$

Let $(A,\ast_{\lambda})$ be a left-symmetric conformal algebra. Then the
$\lambda$-bracket

(53) $[a_{\lambda}b]=a\ast_{\lambda}b-b\ast_{-\lambda-\partial}a\quad\forall
a,b\in A$

defines a Lie conformal algebra, which is called the sub-adjacent Lie
conformal algebra of $A$. See [4, 17] for more details.

######  Lemma 4.1.

([16]) Let $T:V\longrightarrow A$ be an $\mathcal{O}$-operator on an
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Define a
multiplication $\ast^{T}_{\lambda}$ on $V$ by

(54) $u\ast_{\lambda}^{T}v=\rho(Tu)_{\lambda}(v),\quad\forall u,v\in V.$

Then $(V,\ast^{T}_{\lambda})$ is a left-symmetric conformal algebra.

We denote by $(V,[\cdot_{\lambda}\cdot]^{T})$ the sub-adjacent Lie conformal
algebra of the left-symmetric conformal algebra $(V,\ast_{\lambda}^{T})$,
where $[\cdot_{\lambda}\cdot]^{T}$ is given by

(55)
$[u_{\lambda}v]^{T}=\rho(T(u))_{\lambda}(v)-\rho(T(v))_{-\lambda-\partial}(u),\quad\forall~{}u,v\in
V.$

Furthermore, $T$ is a Lie conformal algebra homomorphism from
$(V,[\cdot_{\lambda}\cdot]^{T})$ to $(A,[\cdot_{\lambda}\cdot])$.

Let $T:V\rightarrow A$ be an $\mathcal{O}$-operator and $(N,S)$ a Nijenhuis
structure on an $\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$. We
define the bracket $[\cdot_{\lambda}\cdot]_{S}^{T}:\otimes^{2}V\rightarrow
V[\lambda]$ to be the deformed bracket of $[\cdot_{\lambda}\cdot]^{T}$ by $S$,
i.e.

(56)
$[u_{\lambda}v]_{S}^{T}=[S(u)_{\lambda}v]^{T}+[u_{\lambda}S(v)]^{T}-S([u_{\lambda}v]^{T}),\quad\forall~{}u,v\in
V.$

Define the bracket
$\\{\cdot_{\lambda}\cdot\\}_{\tilde{\rho}}^{T}:\otimes^{2}V\rightarrow
V[\lambda]$ similar to (55) using the module $(V;\tilde{\rho})$,

(57)
$\\{u_{\lambda}v\\}_{\tilde{\rho}}^{T}=\tilde{\rho}(T(u))_{\lambda}(v)-\tilde{\rho}(T(v))_{-\lambda-\partial}(u),\quad\forall~{}u,v\in
V.$

######  Definition 4.2.

Let $T:V\rightarrow A$ be an $\mathcal{O}$-operator and $(N,S)$ a Nijenhuis
structure on an $\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$. The
triple $(T,N,S)$ is called an $\mathrm{\mathcal{O}N}$-structure on the
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$ if $T$ and $(N,S)$
satisfy the following conditions

(58) $\displaystyle{N\circ T}$ $\displaystyle=$ $\displaystyle{T\circ S},$
(59) $\displaystyle{[u_{\lambda}v]^{N\circ T}}$ $\displaystyle=$
$\displaystyle{[u_{\lambda}v]_{S}^{T}},\quad\forall~{}u,v\in V.$

######  Lemma 4.3.

Let $(T,S,N)$ be an $\mathrm{\mathcal{O}N}$-structure on an $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Then we have

(60) ${[u_{\lambda}v]^{N\circ
T}}=[u_{\lambda}v]_{S}^{T}=\\{u_{\lambda}v\\}_{\tilde{\rho}}^{T},\quad\forall~{}u,v\in
V.$

###### Proof.

By (58), we have
$[u_{\lambda}v]_{S}^{T}+\\{u_{\lambda}v\\}_{\tilde{\rho}}^{T}=2[u_{\lambda}v]^{N\circ
T}$. Furthermore, by (59), we have
$[u_{\lambda}v]_{S}^{T}=\\{u_{\lambda}v\\}_{\tilde{\rho}}^{T}$. The conclusion
follows.

######  Proposition 4.4.

Let $(T,S,N)$ be an $\mathrm{\mathcal{O}N}$-structure on an $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Then $S$ is a Nijenhuis operator on
the sub-adjacent Lie conformal algebra $(V,[\cdot_{\lambda}\cdot]^{T})$. Thus,
the brackets $[\cdot_{\lambda}\cdot]^{T}_{S}$,
$[\cdot_{\lambda}\cdot]_{\tilde{\varrho}}^{T}$ and
$[\cdot_{\lambda}\cdot]^{N\circ T}$ are all Lie conformal brackets.

###### Proof.

By the relation $[u_{\lambda}v]^{T\circ S}=[u_{\lambda}v]_{S}^{T}$, we have

(61)
$\displaystyle\rho(TS(u))_{\lambda}S(v)-\rho(Tv)_{-\lambda-\partial}(S^{2}(u))$
$\displaystyle=$ $\displaystyle
S(\rho(TS(u))_{\lambda}(v))-S\rho(T(v))_{-\lambda-\partial}(S(u)),$ (62)
$\displaystyle
S^{2}\rho(T(u))_{\lambda}(v)-S^{2}\rho(T(v))_{-\lambda-\partial}(u)$
$\displaystyle=$ $\displaystyle
S\rho(Tu)_{\lambda}(S(v))-S\rho(T(v))_{-\lambda-\partial}S(u).$

By (47), we have

$\rho(NT(v))_{\lambda}S(u)=S\rho(NT(v))_{\lambda}(u)+\rho(Tv)_{\lambda}S^{2}(u)-S\rho(Tv)_{\lambda}(S(u)),$

which implies that

(63)
$\rho(T(v))_{\lambda}(S^{2}(u))-\rho(TS(v))_{\lambda}(S(u))=S\rho(T(v))_{\lambda}(S(u))-S\rho(T\circ
S(v))_{\lambda}(u).$

By (61)-(63), we have

$\begin{split}&[S(u)_{\lambda}S(v)]^{T}-S([u_{\lambda}v]_{S}^{T})\\\
=&\rho(TS(u))_{\lambda}(S(v))-\rho(TS(v))_{-\lambda-\partial}(S(u))-S\rho(TS(u))_{\lambda}(v)+S\rho(T(v))_{-\lambda-\partial}(S(u))\\\
&-S\rho(T(u))_{\lambda}S(v)+S\rho(TS(v))_{-\lambda-\partial}(u)+S^{2}\rho(T(u))_{\lambda}(v)-S^{2}\rho(T(v))_{-\lambda-\partial}(u)\\\
=&\rho(TS(u))_{\lambda}(S(v))-\rho(TS(v))_{-\lambda-\partial}(S(u))-\rho(TS(u))_{\lambda}(S(v))+\rho(T(v))_{-\lambda-\partial}(S^{2}(u))\\\
&-S\rho(T(u))_{\lambda}(S(v))+S\rho(TS(v))_{-\lambda-\partial}(u)+S\rho(T(u))_{\lambda}(S(v))-S\rho(T(v))_{-\lambda-\partial}S(u)\\\
=&\rho(T(v))_{-\lambda-\partial}(S^{2}(u))-\rho(TS(v))_{-\lambda-\partial}(S(u))-S\rho(T(v))_{-\lambda-\partial}(S(u))+S\rho(TS(v))_{-\lambda-\partial}(u)\\\
=&0.\end{split}$

Thus $S$ is a Nijenhuis operator on the Lie conformal algebra
$(V,[\cdot_{\lambda}\cdot]^{T})$. By Corollary 3.6,
$[\cdot_{\lambda}\cdot]^{T}_{S}$ is a Lie conformal bracket on $V$. By Lemma
4.3, the brackets $[\cdot_{\lambda}\cdot]^{T}_{S}$,
$[\cdot_{\lambda}\cdot]_{\tilde{\varrho}}^{T}$ and
$[\cdot_{\lambda}\cdot]^{N\circ T}$ are all Lie conformal brackets.

######  Theorem 4.5.

Let $(T,S,N)$ be an $\mathrm{\mathcal{O}N}$-structure on an $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Then we have

  * $\rm(i)$

$T$ is an $\mathcal{O}$-operator on the $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot]_{N};\tilde{\rho})$.

  * $\rm(ii)$

$N\circ T$ is an $\mathcal{O}$-operator on the $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];{\rho})$.

###### Proof.

(i) Since $T$ is an $\mathcal{O}$-operator on the $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];\rho)$ and $T\circ S=N\circ T$, we have

$\begin{split}T\\{u_{\lambda}v\\}_{\tilde{\rho}}^{T}&=T([u_{\lambda}v]_{S}^{T})=T([S(u)_{\lambda}(v)]^{T}+[u_{\lambda}S(v)]^{T}-S[u_{\lambda}v]^{T})\\\
&=[TS(u)_{\lambda}T(v)]+[T(u)_{\lambda}TS(v)]-TS[u_{\lambda}v]^{T}\\\
&=[NT(u)_{\lambda}T(v)]+[T(u)_{\lambda}NT(v)]-NT[u_{\lambda}v]^{T}\\\
&=[T(u)_{\lambda}T(v)]_{N},\end{split}$

which implies that $T$ is an $\mathcal{O}$-operator on the $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot]_{N};\tilde{\rho})$.

(ii) By (59), we have

$NT([u_{\lambda}v]^{N\circ
T})=NT([u_{\lambda}v]_{S}^{T})=N[T(u)_{\lambda}T(v)]_{N}=[NT(u)_{\lambda}NT(v)],$

which implies that $N\circ T$ is an $\mathcal{O}$-operator on the
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];{\rho})$.

In the following, we give the notion of compatible $\mathcal{O}$-operators on
an $\mathsf{LCMod}$ pair, and show that an $\mathrm{\mathcal{O}N}$-structure
gives rise to compatible $\mathcal{O}$-operators.

######  Definition 4.6.

Let $T_{1},T_{2}$:$V\rightarrow A$ be two $\mathcal{O}$-operators on an
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$. If for any
$k_{1},k_{2}\in\mathbb{R}$, $k_{1}T_{1}+k_{2}T_{2}$ is still an
$\mathcal{O}$-operator, then $T_{1}$ and $T_{2}$ are called compatible.

######  Proposition 4.7.

Let $T_{1},T_{2}$:$V\rightarrow A$ be two $\mathcal{O}$-operators on an
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Then $T_{1}$ and
$T_{2}$ are compatible if and only if the following equation holds:

(64)
$\begin{split}[T_{1}(u)_{\lambda}T_{2}(v)]+[T_{2}(u)_{\lambda}T_{1}(v)]&=T_{1}(\rho(T_{2}(u))_{\lambda}(v)-\rho(T_{2}(v))_{-\lambda-\partial}(u))\\\
&+T_{2}(\rho(T_{1}(u))_{\lambda}(v)-\rho(T_{1}(v))_{-\lambda-\partial}(u)),\quad\forall
u,v\in V.\end{split}$

###### Proof.

It follows by a direct calculation.

######  Example 4.8.

Let $A=\mathbb{C}[\partial]a\oplus\mathbb{C}[\partial]b$ be the quadratic Lie
conformal algebra given in Example 3.15. Define $R_{1},R_{2}:A\longrightarrow
A$ by

$\displaystyle R_{1}(a)$ $\displaystyle=-(a+b),\quad$ $\displaystyle R_{1}(b)$
$\displaystyle=(a+b),$ $\displaystyle R_{2}(a)$ $\displaystyle=(a+b),\quad$
$\displaystyle R_{2}(b)$ $\displaystyle=-(a+b).$

Then $R_{1}$ and $R_{2}$ are compatible Rota-Baxter operators.

Using an $\mathcal{O}$-operator and a Nijenhuis operator on a Lie conformal
algebra, we can construct a pair of compatible $\mathcal{O}$-operators.

######  Proposition 4.9.

Let $T:V\rightarrow A$ be an $\mathcal{O}$-operator on an $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot];\rho)$ and $N$ be a Nijenhuis operator on $A$.
Then $N\circ T$ is an $\mathcal{O}$-operator on
$(A,[\cdot_{\lambda}\cdot];\rho)$ if and only if for all $u,v\in V$, the
following equation holds:

(65) $\begin{split}&N([NT(u)_{\lambda}T(v)]+[T(u)_{\lambda}NT(v)])\\\
=&N(T(\rho(NT(u))_{\lambda}(v)-\rho(NT(v))_{-\lambda-\partial}(u))+NT(\rho(T(u))_{\lambda}(v)-\rho(T(v))_{-\lambda-\partial}(u))).\end{split}$

If, in addition, $N$ is invertible, then $T$ and $N\circ T$ are compatible.
More explicitly, for any $\mathcal{O}$-operator $T$, if there exists an
invertible Nijenhuis operator $N$ such that $N\circ T$ is also an
$\mathcal{O}$-operator, then $T$ and $N\circ T$ are compatible.

###### Proof.

Since $N$ is a Nijenhuis operator, we have

$[NT(u)_{\lambda}NT(v)]=N([NT(u)_{\lambda}T(v)]+[T(u)_{\lambda}NT(v)])-N^{2}([T(u)_{\lambda}T(v)]).$

By the fact that $T$ is an $\mathcal{O}$-operator, the following equation

$[NT(u)_{\lambda}NT(v)]=NT\big{(}\rho(NT(u))_{\lambda}(v)-\rho(NT(v))_{-\lambda-\partial}(u)\big{)}$

holds if and only if (65) holds.

If $N\circ T$ is an $\mathcal{O}$-operator and $N$ is invertible, then we have

$\displaystyle[NT(u)_{\lambda}T(v)]+[T(u)_{\lambda}NT(v)]$ $\displaystyle=$
$\displaystyle T(\rho(NT(u))_{\lambda}(v)-\rho(NT(v))_{-\lambda-\partial}(u))$
$\displaystyle+NT(\rho(T(u))_{\lambda}(v)-\rho(T(v))_{-\lambda-\partial}(u)).$

This is exactly the condition that $N\circ T$ and $T$ are compatible.

A pair of compatible $\mathcal{O}$-operators can also give rise to a Nijenhuis
operator under an invertible condition.

######  Proposition 4.10.

Let $T_{1},T_{2}:V\rightarrow A$ be two $\mathcal{O}$-operators on an
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Suppose that $T_{2}$
is invertible. If $T_{1}$, $T_{2}$ are compatible, then $N=T_{1}\circ
T_{2}^{-1}$ is a Nijenhuis operator on the Lie conformal algebra $A$.

###### Proof.

Since $T_{1}=N\circ T_{2}$ is an $\mathcal{O}$-operator, we have

$[NT_{2}(u)_{\lambda}NT_{2}(v)]=NT_{2}(\rho(NT_{2}(u)_{\lambda}(v)-\rho(NT_{2}(v)_{-\lambda-\partial}(u)).$

Since $T_{1}$ and $T_{2}$ are compatible, we have

$\begin{split}&[NT_{2}(u)_{\lambda}T_{2}(v)]+[T_{2}(u)_{\lambda}NT_{2}(v)]\\\
=&T_{2}(\rho(NT_{2}(u))_{\lambda}(v)-\rho(NT_{2}(v))_{-\lambda-\partial}(u))+NT_{2}(\rho(T_{2}(u)){\lambda}(v)-\rho(T_{2}(v))_{-\lambda-\partial}(u))\\\
=&T_{2}(\rho(NT_{2}(u))_{\lambda}(v)-\rho(NT_{2}(v))_{-\lambda-\partial}(u))+N([T_{2}(u)_{\lambda}T_{2}(v)]).\end{split}$

Thus we have

$\displaystyle[NT_{2}(u)_{\lambda}NT_{2}(v)]-N[NT_{2}(u)_{\lambda}T_{2}(v)]-N[T_{2}(u)_{\lambda}NT_{2}(v)]+N^{2}([T_{2}(u)_{\lambda}T_{2}(v)])$
$\displaystyle=$ $\displaystyle
NT_{2}(\rho(NT_{2}(u)_{\lambda}(v)-\rho(NT_{2}(v)_{-\lambda-\partial}(u))+T_{2}(\rho(NT_{2}(u))_{\lambda}(v)+\rho(NT_{2}(v))_{-\lambda-\partial}(u))$
$\displaystyle-N^{2}([T_{2}(u)_{\lambda}T_{2}(v)])+N^{2}([T_{2}(u)_{\lambda}T_{2}(v)])$
$\displaystyle=$ $\displaystyle 0.$

Since $T_{2}$ is invertible, for any $a,b\in A$, there exist $u,v\in V$ such
that $T_{2}(u)=a$ and $T_{2}(v)=b$. The above equality implies that

$[Na_{\lambda}Nb]=N[Na_{\lambda}b]+N[a_{\lambda}Nb]-N^{2}[a_{\lambda}b].$

Thus $N$ is a Nijenhuis operator on the Lie conformal algebra $A$.

By Propositions 4.9 and 4.10, we have

######  Corollary 4.11.

Let $T_{1},T_{2}:V\rightarrow A$ be two $\mathcal{O}$-operators on an
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Suppose $T_{1}$ and
$T_{2}$ are invertible. Then $T_{1}$ and $T_{2}$ are compatible if and only if
$N=T_{1}\circ T_{2}^{-1}$ is a Nijenhuis operator on $A$.

The following proposition shows that compatible $\mathcal{O}$-operators can be
obtained from $\mathrm{\mathcal{O}N}$-structures.

######  Proposition 4.12.

Let $(T,S,N)$ be an $\mathrm{\mathcal{O}N}$-structure on an $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Then $T$ and $T\circ S$ are compatible
$\mathcal{O}$-operators.

###### Proof.

It is sufficient to prove that $T+T\circ S$ is an $\mathcal{O}$-operator. It
is not hard to see that

$[u_{\lambda}v]^{T+T\circ S}=[u_{\lambda}v]^{T}+[u_{\lambda}v]^{T\circ
S}=[u_{\lambda}v]^{T}+[u_{\lambda}v]_{S}^{T}.$

Thus we have

$\begin{split}&(T+T\circ S)([u_{\lambda}v]^{T+T\circ S})\\\
=&T([u_{\lambda}v]^{T})+T\circ S([u_{\lambda}v]_{S}^{T})+T\circ
S([u_{\lambda}v]^{T})+T([u_{\lambda}v]_{S}^{T})\\\
=&T([u_{\lambda}v]^{T})+T\circ S([u_{\lambda}v]_{S}^{T})+T\circ
S([u_{\lambda}v]^{T})\\\
&+T([S(u)_{\lambda}v]^{T}+[u_{\lambda}S(v)]^{T}-S[u_{\lambda}v]^{T})\\\
=&T([u_{\lambda}v]^{T})+T\circ
S([u_{\lambda}v]_{S}^{T})+T([S(u)_{\lambda}v]^{T}+[u_{\lambda}S(v)]^{T})\\\
=&[T(u)_{\lambda}T(v)]+[T\circ S(u)_{\lambda}T\circ S(v)]+[T\circ
S(u)_{\lambda}T(v)]+[T(u)_{\lambda}T\circ S(v)]\\\ =&[(T+T\circ
S)(u)_{\lambda}(T+T\circ S)(v)],\end{split}$

which implies that $T+T\circ S$ is an $\mathcal{O}$-operator.

Similar to the properties of $\mathrm{\mathcal{O}N}$-structures on Lie
algebras given in [19], we also have

######  Lemma 4.13.

Let $(T,S,N)$ be an $\mathrm{\mathcal{O}N}$-structure on an $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Then

  * $\rm(i)$

for all $k,i\in\mathbb{N}$, we have

(66) $T_{k}[u_{\lambda}v]_{S^{k+i}}^{T}=[T_{k}(u)_{\lambda}T_{k}(v)]_{N^{i}};$

  * $\rm(ii)$

for all $k,i\in\mathbb{N}$ such that $i\leq k$,

(67)
$[u_{\lambda}v]^{T^{k}}=[u_{\lambda}v]_{S^{k}}^{T}=S^{k-i}[u_{\lambda}v]^{T_{i}},$

where $T_{k}=T\circ S^{k}=N^{k}\circ T$ and set $T_{0}=T$.

###### Proof.

The proof is similar to that of Lemmas 4.8 and 4.9 in [19].

######  Proposition 4.14.

Let $(T,S,N)$ be an $\mathrm{\mathcal{O}N}$-structure on an $\mathsf{LCMod}$
pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Then all $T_{k}=N^{k}\circ T$ are
$\mathcal{O}$-operators on the $\mathsf{LCMod}$ pair $(A;\rho)$ and for all
$k,l\in\mathbb{N}$, $T_{k}$ and $T_{l}$ are compatible.

###### Proof.

By (66) and (67) with $i=0$, we have

$T_{k}[u_{\lambda}v]^{T_{k}}=[T_{k}(u)_{\lambda}T_{k}(v)],$

which implies that $T_{k}$ is an $\mathcal{O}$-operator.

For the second assertion, we only need to prove that $T^{k}+T^{k+i}$ is an
$\mathcal{O}$-operator for all $k,i\in\mathbb{N}$. By (67), we have

$[u_{\lambda}v]^{T_{k}+T_{k+i}}=[u_{\lambda}v]^{T_{k}}+[u_{\lambda}v]^{T_{k+i}}=[u_{\lambda}v]^{T_{k}}+[u_{\lambda}v]_{S^{i}}^{T_{k}}.$

Thus, we have

$\begin{split}&(T_{k}+T_{k+i})([u_{\lambda}v]^{T_{k}+T_{k+i}})\\\
=&T_{k}([u_{\lambda}v]^{T_{k}})+T_{k}([u_{\lambda}v]_{S^{i}}^{T_{k}})+T_{k+i}([u_{\lambda}v]^{T_{k}})+T_{k+i}([u_{\lambda}v]_{S^{i}}^{T_{k}})\\\
=&T_{k}([u_{\lambda}v]^{T_{k}})+T_{k+i}([u_{\lambda}v]^{T_{k}})+T_{k+i}([u_{\lambda}v]_{S^{i}}^{T_{k}})\\\
&+T_{k}([S^{i}(u)_{\lambda}v]^{T_{k}}+[u_{\lambda}S^{i}(v)]^{T_{k}}-S^{i}[u_{\lambda}v]^{T_{k}})\\\
=&T_{k}([u_{\lambda}v]^{T_{k}})+T_{k+i}([u_{\lambda}v]_{S^{i}}^{T_{k}})+T_{k}([S^{i}(u)_{\lambda}v]^{T_{k}})+T_{k}([u_{\lambda}S^{i}(v)]^{T_{k}})\\\
=&[T_{k}(u)_{\lambda}T_{k}(v)]+[T_{k+i}(u)_{\lambda}T_{k+i}(v)]+[T_{k+i}(u)_{\lambda}T_{k}(v)]+[T_{k}(u)_{\lambda}T_{k+i}(v)]\\\
=&[(T_{k}+T_{k+i})(u)_{\lambda}(T_{k}+T_{k+i})(v)],\end{split}$

which implies that $T_{k}+T_{k+i}$ is an $\mathcal{O}$-operator. Thus $T_{k}$
and $T_{l}$ are compatible for all $k,l\in\mathbb{N}$.

Compatible $\mathcal{O}$-operators can give rise to
$\mathrm{\mathcal{O}N}$-structures.

######  Theorem 4.15.

Let $T,T_{1}:V\rightarrow A$ be two $\mathcal{O}$-operators on an
$\mathsf{LCMod}$ pair $(A,[\cdot_{\lambda}\cdot];\rho)$. Suppose that $T$ is
invertible. If $T$ and $T_{1}$ are compatible, then

  * $\rm(i)$

$(T,S=T^{-1}\circ T_{1},N=T_{1}\circ T^{-1})$ is an
$\mathrm{\mathcal{O}N}$-structure on the $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];\rho)$;

  * $\rm(ii)$

$(T_{1},S=T^{-1}\circ T_{1},N=T_{1}\circ T^{-1})$ is an
$\mathrm{\mathcal{O}N}$-structure on the $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];\rho)$.

###### Proof.

By Proposition 4.10, $N=T_{1}\circ T^{-1}$ is a Nijenhuis operator on $A$.

Since $T$ and $T_{1}$ are compatible, we have

$\displaystyle[T(u)_{\lambda}T_{1}(v)]+[T_{1}(u)_{\lambda}T(v)]$
$\displaystyle=$ $\displaystyle
T(\rho(T_{1}(u))_{\lambda}(v)-\rho(T_{1}(v))_{-\lambda-\partial}(u))$
$\displaystyle+T_{1}(\rho(T(u))_{\lambda}(v)-\rho(T(v))_{-\lambda-\partial}(u)).$

Because of $T_{1}=T\circ S$, we get

(68)
$\begin{split}[T(u)_{\lambda}TS(v)]+[TS(u)_{\lambda}T(v)]=&T(\rho(TS(u))_{\lambda}(v)-\rho(TS(v))_{-\lambda-\partial}(u))\\\
&+TS(\rho(T(u))_{\lambda}(v)-\rho(T(v))_{-\lambda-\partial}(u)).\end{split}$

Since $T$ is an $\mathcal{O}$-operator, we have

(69)
$\begin{split}[T(u)_{\lambda}TS(v)]+[TS(u)_{\lambda}T(v)]=&T(\rho(T(u))_{\lambda}(S(v))-\rho(TS(v))_{-\lambda-\partial}(u)\\\
&+T(\rho(TS(u))_{\lambda}(v)-\rho(T(v))_{-\lambda-\partial}(S(u)).\end{split}$

By (68), (69) and the fact that $T$ is invertible, we have

(70)
$S(\rho(T(u))_{\lambda}(v)-\rho(T(v))_{-\lambda-\partial}(u))=\rho(T(u))_{\lambda}(S(v))-\rho(T(v))_{-\lambda-\partial}(S(u)).$

Replacing $v$ with $S(v)$, we get

(71) $S(\rho(T(u))_{\lambda}(S(v))-\rho(T\circ
S(v))_{-\lambda-\partial}(u))=\rho(T(u))_{\lambda}(S^{2}(v))-\rho(T\circ
S(v))_{-\lambda-\partial}(S(u)).$

Since $T$ and $T\circ S$ are $\mathcal{O}$-operators, we have

$TS[u_{\lambda}v]^{T\circ
S}=[TS(u)_{\lambda}TS(v)]=T[S(u)_{\lambda}S(v)]^{T}.$

By the fact that $T$ is invertible, we have

$S[u_{\lambda}v]^{T\circ S}=[S(u)_{\lambda}S(v)]^{T}.$

which implies that

(72)
$S(\rho((TS)(u))_{\lambda}(v)-\rho(TS(v))_{-\lambda-\partial}(u))=\rho((TS)(u))_{\lambda}(S(v))-\rho((TS)(v))_{-\lambda-\partial}(S(u)).$

By (71) and (72), we have

$S(\rho(T(u))_{\lambda}(S(v)))-\rho(T(u))_{\lambda}(S^{2}(v))-S(\rho((T\circ
S)(u))_{\lambda}(v))+\rho((T\circ S)(u))_{\lambda}(S(v))=0.$

By the fact that $N\circ T=(T_{1}\circ T^{-1})\circ T=T_{1}=T\circ S$, we have

$S(\rho(T(u))_{\lambda}(S(v)))-\rho(T(u))_{\lambda}(S^{2}(v))-S(\rho(N\circ
T)(u))_{\lambda}(v))+\rho((N\circ T)(u))_{\lambda}(S(v))=0.$

Since $T$ is invertible and let $a=T(u)$, we have

$S(\rho(a)_{\lambda}(S(v)))-\rho(a)_{\lambda}(S^{2}(v))-S(\rho(N(a))_{\lambda}(v))+\rho(N(a))_{\lambda}(S(v))=0.$

Thus $(N,S)$ is a Nijenhuis structure on an $\mathsf{LCMod}$ pair $(A;\rho)$.

(i) It is obvious that $N\circ T=T\circ S$. By (70), we have

$\begin{split}[u_{\lambda}v]_{S}^{T}-[u_{\lambda}v]^{T\circ
S}&=\rho(T(u))_{\lambda}(S(v))-\rho(T(v))_{-\lambda-\partial}(S(u))-S(\rho(T(u))_{\lambda}(v)-\rho(T(v))_{-\lambda-\partial}(u))\\\
&=0,\end{split}$

which implies that $[u_{\lambda}v]_{S}^{T}=[u_{\lambda}v]^{T\circ S}$.
Therefore, $(T,S=T^{-1}\circ T_{1},N=T_{1}\circ T^{-1})$ is an
$\mathrm{\mathcal{O}N}$-structure on the $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];\rho)$.

(ii) It is obvious that $T_{1}\circ S=N\circ T_{1}$. By (72), we have

$\begin{split}&[u_{\lambda}v]_{S}^{T_{1}}-[u_{\lambda}v]^{T_{1}\circ S}\\\
=&\rho(T_{1}(u))_{\lambda}(S(v))-\rho(T_{1}(v))_{-\lambda-\partial}(S(u))-S(\rho(T_{1}(u))_{\lambda}(v)-\rho(T_{1}(v))_{-\lambda-\partial}(u))\\\
=&\rho((T\circ S)(u))_{\lambda}(S(v))-\rho((T\circ
S)(v))_{-\lambda-\partial}(S(u))-S(\rho((T\circ
S)(u))_{\lambda}(v)-\rho((T\circ S)(v))_{-\lambda-\partial}(u))\\\
=&0,\end{split}$

which implies that $[u_{\lambda}v]_{S}^{T_{1}}=[u_{\lambda}v]^{T_{1}\circ S}$.
Thus $(T_{1},S=T^{-1}\circ T_{1},N=T_{1}\circ T^{-1})$ is an
$\mathrm{\mathcal{O}N}$-structure on the $\mathsf{LCMod}$ pair
$(A,[\cdot_{\lambda}\cdot];\rho)$.

######  Proposition 4.16.

Let $(T,N,S)$ be an $\mathrm{\mathcal{O}N}$-structure on an $\mathsf{LCMod}$
pair $(A;\rho)$. If $T$ is invertible, then $(T,N^{k},S^{k})$ and
$(T_{k}=N^{k}\circ T,N^{k},S^{k})$ are $\mathrm{\mathcal{O}N}$-structures for
all $k\in\mathbb{N}$.

###### Proof.

Since $(T,N,S)$ is an $\mathrm{\mathcal{O}N}$-structure on the
$\mathsf{LCMod}$ pair $(A;\rho)$, by Proposition 4.14, $T$ and
$T_{k}=N^{k}\circ T$ are compatible $\mathcal{O}$-operators. Then by the
condition that $T$ is invertible and Theorem 4.15, the conclusion follows.

##  5\. Conformal $r$-matrix-Nijenhuis structures and symplectic-Nijenhuis
structures on Lie conformal algebras

In this section, we introduce the notions of conformal $r$-matrix-Nijenhuis
structure and symplectic-Nijenhuis structure on a Lie conformal algebra.
Furthermore, we study their connections with
$\mathrm{\mathcal{O}N}$-structures on $\mathsf{LCMod}$ pairs.

###  5.1. Conformal $r$-matrix-Nijenhuis structures on Lie conformal algebras

Let $V$ be free and of finite rank as a $\mathbb{C}[\partial]$-module. Then
$V\cong{({V^{*}}^{c})^{*}}^{c}$ through the $\mathbb{C}[\partial]$-module
homomorphism $v\mapsto v^{**},$ where $v^{**}$ is given by

$v^{**}_{\lambda}(\alpha)=\alpha_{-\lambda}(v),\quad\forall~{}\alpha\in{V^{*}}^{c}.$

Suppose $A$ is a finite Lie conformal algebra which is free as a
$\mathbb{C}[\partial]$-module. Then we have

$A\otimes A\cong A^{\ast c\ast c}\otimes A\cong{\rm Chom}(A^{\ast c},A).$

Define

$\langle\alpha,a\rangle_{\lambda}=\alpha_{\lambda}(a),\quad
and\quad\langle\alpha\otimes\beta,a\otimes
b\rangle_{(\lambda,\mu)}=\langle\alpha,a\rangle_{\lambda}\langle\beta,b\rangle_{\mu},$

where $a,b\in A$ and $\alpha,\beta\in A^{*c}$. Then for any $r\in A\otimes A$
with $r=\sum_{i}a_{i}\otimes b_{i}\in A\otimes A$, we associate a conformal
linear map $r^{\sharp}\in{\rm Chom}(A^{\ast c},A)$ as follows:

(73)
$r^{\sharp}_{\lambda}(\alpha)=\sum_{i}\langle\alpha,a_{i}\rangle_{-\lambda-\partial}b_{i},\quad\forall~{}\alpha\in
A^{\ast c},$

which is equivalent to

(74)
$\langle\beta,r^{\sharp}_{-\mu-\partial}(\alpha)\rangle_{\lambda}=\langle\alpha\otimes\beta,r\rangle_{(\mu,\lambda)},\quad\forall~{}\alpha,\beta\in
A^{\ast c}.$

Set $r^{21}=\sum_{i}b_{i}\otimes a_{i}$. We say $r$ is skew-symmetric if
$r=-r^{21}$.

We say that $r$ is non-degenerate if
$r_{\lambda}^{\sharp}=r^{\sharp}_{\lambda}|_{\lambda=0}$ and $r_{0}^{\sharp}$
is a $\mathbb{C}[\partial]$-module isomorphism from $A^{\ast c}$ to $A$. In
this case, (74) becomes

(75)
$\langle\beta,r_{0}^{\sharp}(\alpha)\rangle_{\lambda}=\langle\alpha\otimes\beta,r\rangle_{(-\lambda,\lambda)}\quad\forall~{}\alpha,\beta\in
A^{\ast c}.$

######  Definition 5.1.

([22]) Let $A$ be a Lie conformal algebra. An element $r\in A\otimes A$ is
called a conformal $r$-matrix  if $\llbracket r,r\rrbracket\equiv 0~{}{\rm
mod}~{}\partial^{\otimes^{3}}$ in $A\otimes A\otimes A$, where
$\partial^{\otimes^{3}}=\partial\otimes 1\otimes 1+1\otimes\partial\otimes
1+1\otimes 1\otimes\partial.$

There is a close relationship between conformal $r$-matrices and
$\mathcal{O}$-operators.

######  Lemma 5.2.

([16]) Let $A$ be a finite Lie conformal algebra. Then, $r$ is a skew-
symmetric conformal $r$-matrix if and only if
$r^{\sharp}_{0}=r^{\sharp}_{\lambda}|_{\lambda=0}$ is an
$\mathcal{O}$-operator on the $\mathsf{LCMod}$ pair $(A;\mathrm{ad}^{*})$.

######  Definition 5.3.

Let $r$ be a skew-symmetric conformal $r$-matrix and $N:A\rightarrow A$ a
Nijenhuis operator on a finite Lie conformal algebra A. A pair $(r,N)$ is
called a conformal $r$-matrix-Nijenhuis structure on the Lie conformal algebra
$A$ if for all $\alpha,\beta\in A^{\ast c}$, the following conditions are
satisfied:

(76) $\displaystyle{N\circ r_{\lambda}^{\sharp}}$ $\displaystyle=$
$\displaystyle{r_{\lambda}^{\sharp}\circ N^{\ast}},$ (77)
$\displaystyle{[\alpha_{\lambda}\beta]^{N\circ r_{0}^{\sharp}}}$
$\displaystyle=$
$\displaystyle{[\alpha_{\lambda}\beta]_{N^{\ast}}^{r_{0}^{\sharp}}}.$

There is a close relationship between conformal $r$-matrix-Nijenhuis
structures and $\mathrm{\mathcal{O}N}$-structures on Lie conformal algebras.

######  Theorem 5.4.

Let $r$ be a skew-symmetric conformal $r$-matrix and $N:A\rightarrow A$ a
Nijenhuis operator on a Lie conformal algebra $A$. If $(r,N)$ is a conformal
$r$-matrix-Nijenhuis structure on $A$, then $(r_{0}^{\sharp},N,S=N^{\ast})$ is
an $\mathrm{\mathcal{O}N}$-structure on the $\mathsf{LCMod}$ pair
$(A;\mathrm{ad}^{\ast})$.

###### Proof.

It is straightforward.

The following proposition shows that if the conformal $r$-matrix $r$ is
determined by $r_{0}^{\sharp}$, the conformal $r$-matrix-Nijenhuis structure
is equivalent to the $\mathrm{\mathcal{O}N}$-structure on the $\mathsf{LCMod}$
pair $(A;\mathrm{ad}^{\ast})$.

######  Proposition 5.5.

Let $r$ be a skew-symmetric conformal $r$-matrix with
$r_{\lambda}^{\sharp}=r_{0}^{\sharp}$ and $N:A\rightarrow A$ a Nijenhuis
operator on a Lie conformal algebra $A$. Then $(r,N)$ is a conformal
$r$-matrix-Nijenhuis structure on $A$ if and only if
$(r_{0}^{\sharp},N,S=N^{\ast})$ is an $\mathrm{\mathcal{O}N}$-structure on the
$\mathsf{LCMod}$ pair $(A;\mathrm{ad}^{\ast})$.

###### Proof.

Since $r_{\lambda}^{\sharp}=r_{0}^{\sharp}$, ${N\circ
r_{\lambda}^{\sharp}}={r_{\lambda}^{\sharp}\circ N^{\ast}}$ holds if and only
if ${N\circ r_{0}^{\sharp}}={r_{0}^{\sharp}\circ N^{\ast}}$ holds. The rest is
straightforward.

######  Proposition 5.6.

Assume that $(r,N)$ is skew-symmetric conformal $r$-matrix-Nijenhuis structure
on $A$ and set $r=\sum_{i}a_{i}\otimes b_{i}$. Then
$r_{N}=\sum_{i}a_{i}\otimes N(b_{i})$ is skew-symmetric. Furthermore, $r$ and
$r_{N}$ are compatible conformal $r$-matrices in the sense that any linear
combination of $r$ and $r_{N}$ is still a conformal $r$-matrix.

###### Proof.

By the fact that the Lie conformal algebra $A$ is finite, $r$ is skew-
symmetric if and only if

$\langle\beta\otimes\alpha,r\rangle_{(\mu,\lambda)}=-\langle\alpha\otimes\beta,r\rangle_{(\lambda,\mu)},\quad\forall~{}\alpha,\beta\in
A^{*c},$

which is also equivalent to

$\langle\alpha,r^{\sharp}_{-\mu-\partial}(\beta)\rangle_{\lambda}=-\langle\beta,r^{\sharp}_{-\lambda-\partial}(\alpha)\rangle_{\mu}.$

Furthermore, by (74) and ${N\circ
r_{\lambda}^{\sharp}}={r_{\lambda}^{\sharp}\circ N^{\ast}}$, we have

$\displaystyle\langle\alpha\otimes\beta,r_{N}\rangle_{(\lambda,\mu)}$
$\displaystyle=$ $\displaystyle\langle\alpha\otimes
N^{*}\beta,r\rangle_{(\lambda,\mu)}=\langle
N^{*}\beta,r^{\sharp}_{-\mu-\partial}(\alpha)\rangle_{\lambda}$
$\displaystyle=$
$\displaystyle-\langle\alpha,r^{\sharp}_{-\lambda-\partial}(N^{*}\beta)\rangle_{\mu}=-\langle\alpha,Nr^{\sharp}_{-\lambda-\partial}(\beta)\rangle_{\mu}$
$\displaystyle=$ $\displaystyle-\langle
N^{*}\alpha,r^{\sharp}_{-\lambda-\partial}(\beta)\rangle_{\mu}=-\langle\beta\otimes
N^{*}\alpha,r\rangle_{(\lambda,\mu)}$ $\displaystyle=$
$\displaystyle-\langle\beta\otimes\alpha,r_{N}\rangle_{(\mu,\lambda)},$

which implies that $r_{N}$ is skew-symmetric.

By Theorem 5.4 and Proposition 4.12,
$(r_{N}^{\sharp})_{0}=(r_{N}^{\sharp})_{\lambda}|_{\lambda=0}$ is an
$\mathcal{O}$-operator on the $\mathsf{LCMod}$ pair $(A;\mathrm{ad}^{*})$ and,
furthermore, $r^{\sharp}_{0}$ and $(r_{N}^{\sharp})_{0}$ are compatible. By
Lemma 5.2, $r_{N}$ is a conformal $r$-matrix. It is straightforward to check
that the $\mathcal{O}$-operators $r^{\sharp}_{0}$ and $(r_{N}^{\sharp})_{0}$
are compatible if and only if the conformal $r$-matrices $r$ and $r_{N}$ are
compatible. The conclusion follows.

######  Proposition 5.7.

Let $(r,N)$ be a conformal $r$-matrix-Nijenhuis structure on a conformal
algebra $A$ and and set $r=\sum_{i}a_{i}\otimes a_{j}$. Then for all
$k\in\mathbb{N}$, $r_{N^{k}}=\sum_{i}a_{i}\otimes N^{k}(b_{i})\in A\otimes A$
are pairwise compatible skew-symmetric conformal $r$-matrices.

###### Proof.

Similar to the proof of Proposition 5.6, the conclusion follows from Theorem
5.4 and Proposition 4.14.

###  5.2. Symplectic-Nijenhuis structures on Lie conformal algebras

Recall that a $2$-form on a Lie conformal algebra $A$ is a
$\mathbb{C}$-bilinear map $\omega:A\otimes A\rightarrow\mathbb{C}[\lambda]$
satisfies the following conditions:

(78) $\displaystyle\\{\partial a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle-\lambda\\{a_{\lambda}b\\}_{\omega},$ (79)
$\displaystyle\\{a_{\lambda}\partial b\\}_{\omega}$ $\displaystyle=$
$\displaystyle\lambda\\{a_{\lambda}b\\}_{\omega},$ (80)
$\displaystyle\\{a_{\lambda}b\\}_{\omega}$ $\displaystyle=$
$\displaystyle-\\{b_{\lambda}a\\}_{\omega},\forall~{}a,b,c\in A.$

A $2$-form $\omega$ on a finite Lie conformal algebra $A$ is called non-
degenerate if $\omega^{\natural}$ defined by

(81)
$\\{a_{\lambda}b\\}_{\omega}=\langle\omega^{\natural}(a),b\rangle_{\lambda}$

is a $\mathbb{C}[\partial]$-module isomorphism from $A$ to $A^{\ast c}$.

Recall that a $2$-cocycle on a Lie conformal algebra $A$ with coefficients in
the trivial module is a $2$-form $\omega$ on $A$ satisfying
$\mathrm{d}^{T}\omega=0$, i.e.

(82)
$\displaystyle\\{a_{\lambda}[b_{\mu}c]\\}_{\omega}-\\{b_{\mu}[a_{\lambda}c]\\}_{\omega}$
$\displaystyle=$
$\displaystyle\\{[a_{\lambda}b]_{\lambda+\mu}c\\}_{\omega},\quad\forall~{}a,b,c\in
A.$

######  Definition 5.8.

([18]) A symplectic structure on a Lie conformal algebra
$(A,[\cdot_{\lambda}\cdot])$ is a non-degenerate $2$-cocycle $\omega$ on $A$.

######  Lemma 5.9.

([16]) Let $A$ be a Lie conformal algebra. Then $r\in A\otimes A$ is a skew-
symmetric and non-degenerate $r$-matrix if and only if the bilinear form
$\omega$ defined by

(83)
$\\{a_{\lambda}b\\}_{\omega}=\langle(r_{0}^{\sharp})^{-1}(a),b\rangle_{\lambda},\quad
a,b\in A$

is a symplectic structure on $A$, where $r_{0}^{\sharp}\in Chom(A^{\ast c},A)$
is given by (75).

By Lemmas 5.2 and 5.9, we have

######  Corollary 5.10.

Let $A$ be a Lie conformal algebra and $\omega$ a $2$-form on $A$. Then
$\omega$ is a symplectic structure on the Lie conformal algebra $A$ if and
only if $(\omega^{\natural})^{-1}:A^{\ast c}\rightarrow A$ is an
$\mathcal{O}$-operator on the $\mathsf{LCMod}$ pair $(A;\mathrm{ad}^{\ast})$.

######  Definition 5.11.

Let $\omega$ be a symplectic structure and $N$ a Nijenhuis operator on a Lie
conformal algebra $A$. Then $(\omega,N)$ is called a symplectic-Nijenhuis
structure  on the Lie conformal algebra $A$ if for all $a,b\in A$,

(84) $\\{N(a)_{\lambda}b\\}_{\omega}=\\{a_{\lambda}N(b)\\}_{\omega}$

and $\omega_{N}:A\otimes A\rightarrow\mathbb{C}[\lambda]$ defined by

(85)
$\\{a_{\lambda}b\\}_{\omega_{N}}=\langle\omega^{\natural}(N(a)),b\rangle_{\lambda}$

is also a $2$-cocycle.

Note that (84) implies that $\omega_{N}$ is a $2$-form on the Lie conformal
algebra $A$.

Recall that a symplectic structure on a Lie algebra
$(\mathfrak{g},[\cdot,\cdot])$ is a nondegenerate $2$-cocycle
$\omega\in\wedge^{2}\mathfrak{g}^{*}$, i.e.

$\omega([x,y],z)+\omega([y,z],x)+\omega([z,x],y)=0,\quad\forall~{}x,y,z\in\mathfrak{g}.$

Let $\omega$ be a symplectic structure and $N$ a Nijenhuis operator on a Lie
algebra $\mathfrak{g}$. Recall that a symplectic-Nijenhuis structure on a Lie
algebra $\mathfrak{g}$ is a pair $(\omega,N)$ satisfying that for
$x,y\in\mathfrak{g}$,

$\omega(N(x),y)=\omega(x,N(y))$

and $\omega_{N}:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$
defined by

$\omega_{N}(x,y)=\omega(N(x),y)$

is also a $2$-cocycle on the Lie algebra $\mathfrak{g}$.

######  Example 5.12.

Let $(\omega,N)$ be a symplectic-Nijenhuis structure on a Lie algebra
$\mathfrak{g}$. Then $(\tilde{\omega},\tilde{N})$ is a symplectic-Nijenhuis
structure on the current Lie conformal algebra ${\rm
Cur}~{}\mathfrak{g}=\mathbb{C}[\partial]\otimes\mathfrak{g}$, where
$\tilde{\omega}$ is defined by

(86)
$\\{f(\partial)x_{\lambda}g(\partial)y\\}_{\tilde{\omega}}=f(-\lambda)g(\lambda)\omega(x,y),\quad
x,y\in\mathfrak{g},f(\partial),g(\partial)\in\mathbb{C}[\partial],$

and $\tilde{N}$ is given by (35).

######  Example 5.13.

Given a Lie conformal algebra
$A=\mathbb{C}[\partial]a\oplus\mathbb{C}[\partial]b$ with the
$\lambda$-bracket given by

$[a_{\lambda}a]=(\partial+2\lambda)a,\quad[a_{\lambda}b]=(\partial+k\lambda+l)b,\quad[b_{\lambda}b]=0,\quad
k,l\in\mathbb{C}.$

Define $\rho:A\rightarrow{\rm gc}(A)$ by

$\rho(a)_{\lambda}(a)=(\partial+\lambda+m)a,\quad\rho(a)_{\lambda}(b)=(\partial+k\lambda+l)b,\quad\rho(b)_{\lambda}(a)=\rho(b)_{\lambda}(b)=0,$

where $m\in\mathbb{C}$. It is straightforward to check that $\rho$ gives a
module of $A$ over $A$. Consider the semi-direct Lie conformal algebra
$D=A\ltimes_{\rho^{*}}A^{*c}$, $\omega$ defined by

$\displaystyle\\{a_{\lambda}b\\}_{\omega}=0,\quad\\{a_{\lambda}a^{*}\\}_{\omega}=-1,\quad\\{a_{\lambda}b^{*}\\}_{\omega}=0,\quad\\{b_{\lambda}a^{*}\\}_{\omega}=0,\quad\\{b_{\lambda}b^{*}\\}_{\omega}=-1$

is a symplectic structure on the Lie conformal algebra $D$, where
$\\{a^{*},b^{*}\\}$ is the dual $\mathbb{C}[\partial]$-basis of $A^{*c}$. Then
$N:D\rightarrow D$ defined by

$N(a)=k_{1}a,\quad N(b)=k_{2}b,\quad N(a^{*})=k_{1}a^{*},\quad
N(b^{*})=k_{2}b^{*}$

is a Nijenhuis operator on the Lie conformal algebra $D$. Furthermore,
$(\omega,N)$ is a symplectic-Nijenhuis structure on the Lie conformal algebra
$D$.

######  Example 5.14.

Given a Lie conformal algebra
$A=\mathbb{C}[\partial]a\oplus\mathbb{C}[\partial]b$ with the
$\lambda$-bracket given by

$[a_{\lambda}a]=0,\quad[a_{\lambda}b]=\lambda a,\quad[b_{\lambda}b]=\partial
b+2\lambda b.$

Then $\omega$ defined by

$\displaystyle\\{a_{\lambda}b\\}_{\omega}$ $\displaystyle=\lambda^{2}b$

is a symplectic structure on the Lie conformal algebra $A$. It is obvious that
$N:A\rightarrow A$ defined by

$N(a)=k_{1}a,\quad N(b)=k_{1}b$

is a Nijenhuis operator on the Lie conformal algebra $A$. Furthermore,
$(\omega,N)$ is a symplectic-Nijenhuis structure on the Lie conformal algebra
$A$.

######  Proposition 5.15.

If $(\omega,N)$ is a symplectic-Nijenhuis structure on $A$, then for any
$k\in\mathbb{N}$, $\omega_{N^{k}}$ is closed, i.e.
$\mathrm{d}^{T}\omega_{N^{k}}=0$, where $\omega_{N^{k}}:A\otimes
A\rightarrow\mathbb{C}[\lambda]$ is defined by

(87)
$\\{a_{\lambda}b\\}_{\omega_{N^{k}}}=\langle\omega^{\natural}(N^{k}(a)),b\rangle_{\lambda}.$

###### Proof.

By a direct calculation, for $a,b,c\in A$, we have

$\displaystyle\\{a_{\lambda_{1}}b_{\lambda_{2}}c\\}_{\mathrm{d}^{T}\omega_{N^{k+1}}}$
$\displaystyle=$
$\displaystyle\\{N(a)_{\lambda_{1}}b_{\lambda_{2}}c\\}_{\omega_{N^{k}}}+\\{a_{\lambda_{1}}N(b)_{\lambda_{2}}c\\}_{\omega_{N^{k}}}-\\{N(a)_{\lambda_{1}}N(b)_{\lambda_{2}}c\\}_{\omega_{N^{k-1}}}$
$\displaystyle+\\{\big{(}N\big{(}[N(a)_{\lambda_{1}}(b)]+[a_{\lambda_{1}}N(b)]-N[a_{\lambda_{1}}b]\big{)}-[N(a)_{\lambda_{1}}N(b)]\big{)}_{\lambda_{1}+\lambda_{2}}N^{k-1}(c)\\}_{\omega}.$

Since $N$ is a Nijenhuis operator on the Lie conformal algebra $A$, the above
equality implies

$\\{a_{\lambda_{1}}b_{\lambda_{2}}c\\}_{\mathrm{d}^{T}\omega_{N^{k+1}}}=\\{N(a)_{\lambda_{1}}b_{\lambda_{2}}c\\}_{\omega_{N^{k}}}+\\{a_{\lambda_{1}}N(b)_{\lambda_{2}}c\\}_{\omega_{N^{k}}}-\\{N(a)_{\lambda_{1}}N(b)_{\lambda_{2}}c\\}_{\omega_{N^{k-1}}}.$

By induction on $k$, the conclusion follows.

There is a close relationship between symplectic-Nijenhuis structures and
$\mathrm{\mathcal{O}N}$-structures.

######  Theorem 5.16.

Let $\omega:A\otimes A\rightarrow\mathbb{C}[\lambda]$ be a non-degenerate
$2$-form and $N:A\rightarrow A$ a Nijenhuis operator on a finite Lie conformal
algebra $A$. Then $(\omega,N)$ is a symplectic-Nijenhuis structure on $A$ if
and only if $((\omega^{\natural})^{-1},N,S=N^{\ast})$ is an
$\mathrm{\mathcal{O}N}$-structure on the $\mathsf{LCMod}$ pair
$(A;\mathrm{ad}^{\ast})$.

###### Proof.

Assume that $(\omega,N)$ is a symplectic-Nijenhuis structure on $A$. By (80),
we have

$\langle\omega^{\natural}(a),b\rangle_{\lambda}=-\langle\omega^{\natural}(b),a\rangle_{\lambda}.$

Since $\omega$ is non-degenerate, letting $a=(\omega^{\natural})^{-1}(\alpha)$
and $b=(\omega^{\natural})^{-1}(\beta)$, we also have

(88)
$\langle\alpha,(\omega^{\natural})^{-1}(\beta)\rangle_{\lambda}=-\langle\beta,(\omega^{\natural})^{-1}(\alpha)\rangle_{\lambda}.$

By (84), letting $a=(\omega^{\natural})^{-1}(\alpha)$ and
$b=(\omega^{\natural})^{-1}(\beta)$, we have

$\langle\omega^{\natural}(N(\omega^{\natural})^{-1}(\alpha)),(\omega^{\natural})^{-1}(\beta)\rangle_{\lambda}=-\langle\alpha,N((\omega^{\natural})^{-1}(\beta))\rangle_{\lambda}.$

Furthermore, by (88), we have

$\langle\alpha,(\omega^{\natural})^{-1}(N^{*}\beta)\rangle_{\lambda}=\langle\alpha,N((\omega^{\natural})^{-1}(\beta))\rangle_{\lambda},$

which implies that $(\omega^{\natural})^{-1}\circ
N^{*}=N\circ(\omega^{\natural})^{-1}$ holds.

By Corollary 5.10, $\omega$ is a symplectic structure on the Lie conformal
algebra $A$ if and only if $(\omega^{\natural})^{-1}:A^{\ast c}\rightarrow A$
is an $\mathcal{O}$-operator on the $\mathsf{LCMod}$ pair
$(A;\mathrm{ad}^{\ast})$. In the following, we need to show that

(89)
$[\alpha_{\lambda}\beta]^{N\circ(\omega^{\natural})^{-1}}=[\alpha_{\lambda}\beta]^{(\omega^{\natural})^{-1}}_{N^{*}},\quad\forall~{}\alpha,\beta\in
A^{*c}.$

By the fact that $\omega$ is a $2$-cocycle, letting
$a=(\omega^{\natural})^{-1}(\alpha)$ and $b=(\omega^{\natural})^{-1}(\beta)$,
we have

$\displaystyle\langle[\alpha_{\lambda}\beta]^{N\circ(\omega^{\natural})^{-1}}-[\alpha_{\lambda}\beta]^{(\omega^{\natural})^{-1}}_{N^{*}},c\rangle_{\mu}$
$\displaystyle=$
$\displaystyle\langle\mathrm{ad}^{*}((\omega^{\natural})^{-1}(\alpha))_{\lambda}N^{*}(\beta)-\mathrm{ad}^{*}((\omega^{\natural})^{-1}(\beta))_{-\lambda-\partial}N^{*}(\alpha)-N^{*}(\mathrm{ad}^{*}((\omega^{\natural})^{-1}(\alpha))_{\lambda}\beta)$
$\displaystyle+N^{*}(\mathrm{ad}^{*}((\omega^{\natural})^{-1}(\beta))_{-\lambda-\partial}\alpha),c\rangle_{\mu}$
$\displaystyle=$
$\displaystyle\langle\mathrm{ad}^{*}(a)_{\lambda}N^{*}\omega^{\natural}(b)-\mathrm{ad}^{*}(b)_{-\lambda-\partial}N^{*}\omega^{\natural}(a)-N^{*}(\mathrm{ad}^{*}(a)_{\lambda}\omega^{\natural}(b))+N^{*}(\mathrm{ad}^{*}(b)_{-\lambda-\partial}\omega^{\natural}(a)),c\rangle_{\mu}$
$\displaystyle=$
$\displaystyle\langle\mathrm{ad}^{*}(a)_{\lambda}N^{*}\omega^{\natural}(b)-\mathrm{ad}^{*}(b)_{\mu-\lambda}N^{*}\omega^{\natural}(a)-N^{*}(\mathrm{ad}^{*}(a)_{\lambda}\omega^{\natural}(b))+N^{*}(\mathrm{ad}^{*}(b)_{\mu-\lambda}\omega^{\natural}(a)),c\rangle_{\mu}$
$\displaystyle=$
$\displaystyle-\\{b_{\mu-\lambda}N[a_{\lambda}c]\\}_{\omega}+\\{a_{\lambda}N[b_{\mu-\lambda}c]\\}_{\omega}+\\{b_{\mu-\lambda}[a_{\lambda}N(c)]\\}_{\omega}-\\{a_{\lambda}[b_{\mu-\lambda}N(c)]\\}_{\omega}$
$\displaystyle=$
$\displaystyle-\\{b_{\mu-\lambda}N[a_{\lambda}c]\\}_{\omega}+\\{a_{\lambda}N[b_{\mu-\lambda}c]\\}_{\omega}-\\{[a_{\lambda}b]_{\mu}N(c)\\}_{\omega}$
$\displaystyle=$
$\displaystyle-\\{N(b)_{\mu-\lambda}[a_{\lambda}c]\\}_{\omega}+\\{N(a)_{\lambda}[b_{\mu-\lambda}c]\\}_{\omega}-\\{N[a_{\lambda}b]_{\mu}c\\}_{\omega}$
$\displaystyle=$
$\displaystyle-\\{b_{\mu-\lambda}[a_{\lambda}c]\\}_{\omega_{N}}+\\{a_{\lambda}[b_{\mu-\lambda}c]\\}_{\omega_{N}}-\\{[a_{\lambda}b]_{\mu}c\\}_{\omega_{N}}.$

Since $\omega_{N}$ is a $2$-cocycle, (89) follows. Thus
$((\omega^{\natural})^{-1},N,S=N^{\ast})$ is an
$\mathrm{\mathcal{O}N}$-structure on the $\mathsf{LCMod}$ pair
$(A;\mathrm{ad}^{\ast})$.

The converse can be proved similarly. We omit the details.

By Proposition 5.5 and Theorem 5.16, we have

######  Corollary 5.17.

Let $\omega$ be a nondegenerate $2$-form and $N:A\rightarrow A$ a Nijenhuis
operator on a finite Lie conformal algebra $A$. Then $(\omega,N)$ is
symplectic-Nijenhuis structure on $A$ if and only if $(r,N)$ is a conformal
$r$-matrix-Nijenhuis structure on $A$, where $r\in A\otimes A$ is defined by

(90)
$\langle\beta,(\omega^{\natural})^{-1}(\alpha)\rangle_{\lambda}=\langle\alpha\otimes\beta,r\rangle_{(-\lambda,\lambda)}\quad\forall~{}\alpha,\beta\in
A^{\ast c}.$

By Proposition 5.6 and Corollary 5.17, we have

######  Corollary 5.18.

If $(\omega,N)$ is a symplectic-Nijenhuis structure on a Lie conformal algebra
$A$, then $r$ and $r_{N}$ are compatible $r$-matrices, where $r$ is given by
(90) and $r_{N}$ is given by

(91)
$\langle\beta,N(\omega^{\natural})^{-1}(\alpha)\rangle_{\lambda}=\langle\alpha\otimes\beta,r_{N}\rangle_{(-\lambda,\lambda)}\quad\forall~{}\alpha,\beta\in
A^{\ast c}.$

By Proposition 5.7 and Corollary 5.17, we have

######  Corollary 5.19.

If $(\omega,N)$ is a symplectic-Nijenhuis structure on a Lie conformal algebra
$A$, then for any $k,l\in\mathbb{N}$, $r_{N^{k}}$ and $r_{N^{l}}$ defined by

$\displaystyle\langle\beta,N(\omega^{\natural})^{-1}(\alpha)\rangle_{\lambda}=\langle\alpha\otimes\beta,r_{N}\rangle_{(-\lambda,\lambda)},\quad\langle\beta,N(\omega^{\natural})^{-1}(\alpha)\rangle_{\lambda}=\langle\alpha\otimes\beta,r_{N}\rangle_{(-\lambda,\lambda)},\quad\forall~{}\alpha,\beta\in
A^{\ast c}$

are pairwise compatible $r$-matrices.

By Theorems 4.15 and 5.16, we have

######  Corollary 5.20.

Let $r$ be a conformal $r$-matrix and $\omega$ a symplectic structure on a Lie
conformal algebra $A$. Define $\tilde{r}\in A\otimes A$ by

$\langle\beta,(\omega^{\natural})^{-1}(\alpha)\rangle_{\lambda}=\langle\alpha\otimes\beta,\tilde{r}\rangle_{(-\lambda,\lambda)}\quad\forall~{}\alpha,\beta\in
A^{\ast c}.$

If $r$ and $\tilde{r}$ are compatible $r$-matrices, then
$(\omega,N=r_{0}^{\sharp}\circ\omega^{\natural})$ is a symplectic-Nijenhuis
structure on the Lie conformal algebra $A$.

By Proposition 4.16 and Theorem 5.16, we have

######  Corollary 5.21.

If $(\omega,N)$ is a symplectic-Nijenhuis structure on a Lie conformal algebra
$A$, then for any $k\in\mathbb{N}$, $(\omega,N^{k})$ is a symplectic-Nijenhuis
structure on the Lie conformal algebra $A$.

Acknowledgements.  This research was supported by the National Key Research
and Development Program of China (2021YFA1002000), the National Natural
Science Foundation of China (11901501), the China Postdoctoral Science
Foundation (2021M700750) and the Fundamental Research Funds for the Central
Universities (2412022QD033).

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