# Cohomologies and deformations of pre-Lie-morphism triples

Yibo Wang  School of Mathematics and Statistics, Northeast Normal University,
Changchun 130024, Jilin, China<EMAIL_ADDRESS>,  Shilong Zhang
College of Science, Northwest A&F University, Yangling 712100, Shaanxi, China
<EMAIL_ADDRESS>and  Jiefeng Liu∗ School of Mathematics and Statistics,
Northeast Normal University,
Changchun 130024, Jilin, China<EMAIL_ADDRESS>
###### Abstract.

A (pre-)Lie-morphism triple consists of two (pre-)Lie algebras and a (pre-)Lie
algebra homomorphism between them. We give chomologies of pre-Lie-morphism
triples. As an application, we study the infinitesimal deformations of pre-
Lie-morphism triples. Finally, we show that the cohomology of the pre-Lie-
morphism triple can be deduced from a new cohomology of the Lie-morphism
triple.

###### Key words and phrases:

pre-Lie morphism triple, cohomology, deformation

∗ the corresponding author

00footnotetext: MSC: 17B60, 20G10

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Cohomologies and deformations of pre-Lie-morphism triples
    1. 3.1 Cohomologies of pre-Lie-morphism triples
    2. 3.2 Infinitesimal deformations of pre-Lie-morphism triples
    3. 3.3 Nijenhuis pairs on pre-Lie-morphism triples and compatible structures on pre-Lie algebras
  4. 4 Cohomologies of pre-Lie-morphism triples and Lie-morphism triples

##  1\. Introduction

Pre-Lie algebras are a class of nonassociative algebras coming from the study
of convex homogeneous cones, affine manifolds and affine structures on Lie
groups, and the aforementioned cohomologies of associative algebras. They also
appeared in many fields of mathematics and mathematical physics, such as
symplectic structures on Lie groups and Lie algebras, integrable systems,
Poisson brackets and infinite dimensional Lie algebras, vertex algebras,
quantum field theory, operads and numerical analysis. See the survey [4, 17]
and the references therein for more details.

The cohomology theory is a classical approach associating invariants to a
mathematical structure. Cohomology controls deformations and extension
problems of the corresponding algebraic structures. Cohomology theories of
various kinds of algebras have been developed and studied in [5, 12, 14, 15].
See the review paper [13] for more details. The cohomologies of Lie algebra
homomorphisms were introduced by Nijenhuis and Richardson in [19]. Then the
cohomology of two Lie algebras with a Lie algebra homomorphism between them
were studied by Y. Frégier in [10], which can control simultaneous
deformations of these two Lie algebras with the Lie algebra homomorphism. We
call two Lie algebras with a Lie algebra homomorphism between them a Lie-
morphism triple. See [7] for more details on the cohomology of a Lie-morphism
triple with coefficients in a representation and its applications.

In this paper, following the idea given by Y. Frégier in [10], we study the
cohomologies of pre-Lie-morphism triples. Here a pre-Lie-morphism triple
consists of two pre-Lie algebras and a pre-Lie algebra homomorphism between
them. Pre-Lie-morphism triples arise from Rota-Baxter operators, Nijenhuis
operators, symplectic Lie algebra homomorphisms, $\mathfrak{s}$-matrices,
Cayley maps and $B$-series. As an application of the cohomology of pre-Lie-
morphism triples, we study infinitesimal deformations of pre-Lie-morphism
triples and show that equivalent pre-Lie-morphism triples are in the same
cohomology class. We also introduce the notion of a Nijenhuis pair on a pre-
Lie-morphism triple and show that it can generate a trivial infinitesimal
deformation. We show that compatible ${\mathcal{O}}$-operators on pre-Lie
algebras can give rise to Nijenhuis pairs naturally. See [8, 20] for more
details about Nijenhuis operators on pre-Lie algebras and Lie algebras. Based
on the close relationship between the cohomologies of pre-Lie algebras and the
cohomologies of Lie algebras, we find that the cohomology of a pre-Lie-
morphism triple can be deduced from a new cohomology of the Lie-morphism
triple.

The paper is organized as follows. In Section 2, we recall representations,
cohomologies and homomorphisms of pre-Lie algebras. Then some examples of pre-
Lie algebra homomorphisms are given. In Section 3, we give the cohomology of a
pre-Lie-morphism triple and use it to study infinitesimal deformations of the
pre-Lie-morphism triple. Furthermore, we introduce the notion of a Nijenhuis
pair on the pre-Lie-morphism triple , which generates trivial infinitesimal
deformations. At last, we construct Nijenhuis pairs from compatible
${\mathcal{O}}$-operators on pre-Lie algebras. In Section 4, we study the
relationship between the cohomologies of pre-Lie-morphism triples and the
cohomologies of Lie-morphism triples.

In this paper, we work over an algebraically closed field $\mathbb{K}$ of
characteristic 0 and all the vector spaces are taken over $\mathbb{K}$. Unless
otherwise noted, all vector spaces we consider will be finite dimensional .

##  2\. Preliminaries

In this section, we recall the notion of pre-Lie algebra homomorphisms and
give some examples.

######  Definition 2.1.

A pre-Lie algebra is a pair $(\mathfrak{g},\cdot_{\mathfrak{g}})$, where
$\mathfrak{g}$ is a vector space and
$\cdot_{\mathfrak{g}}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$
is a bilinear multiplication satisfying that for all $x,y,z\in\mathfrak{g}$,
the associator
$(x,y,z)=(x\cdot_{\mathfrak{g}}y)\cdot_{\mathfrak{g}}z-x\cdot_{\mathfrak{g}}(y\cdot_{\mathfrak{g}}z)$
is symmetric in $x,y$, i.e.

$(x,y,z)=(y,x,z),\;\;{\rm or}\;\;{\rm
equivalently,}\;\;(x\cdot_{\mathfrak{g}}y)\cdot_{\mathfrak{g}}z-x\cdot_{\mathfrak{g}}(y\cdot_{\mathfrak{g}}z)=(y\cdot_{\mathfrak{g}}x)\cdot_{\mathfrak{g}}z-y\cdot_{\mathfrak{g}}(x\cdot_{\mathfrak{g}}z).$

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra. The commutator
$[x,y]_{\mathfrak{g}}=x\cdot_{\mathfrak{g}}y-y\cdot_{\mathfrak{g}}x$ defines a
Lie algebra structure on $\mathfrak{g}$, which is called the sub-adjacent Lie
algebra of $(\mathfrak{g},\cdot_{\mathfrak{g}})$ and denoted by
$\mathfrak{g}^{c}$. Furthermore,
$L:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})$ with $x\rightarrow
L_{x}$, where $L_{x}y=x\cdot_{\mathfrak{g}}y$, for all $x,y\in\mathfrak{g}$,
gives a representation of the Lie algebra $\mathfrak{g}^{c}$ on
$\mathfrak{g}$. See [4] for more details.

######  Definition 2.2.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and $V$ a
vector space. A representation of $\mathfrak{g}$ on $V$ consists of a pair
$(\rho,\mu)$, where $\rho:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ is a
representation of the Lie algebra $\mathfrak{g}^{c}$ on $V$ and
$\mu:\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ is a linear map satisfying

(1)
$\displaystyle\rho(x)\mu(y)u-\mu(y)\rho(x)u=\mu(x\cdot_{\mathfrak{g}}y)u-\mu(y)\mu(x)u,\quad\forall~{}x,y\in\mathfrak{g},~{}u\in
V.$

Usually, we denote a representation by $(V;\rho,\mu)$. It is obvious that
$(\mathbb{K};\rho=0,\mu=0)$ is a representation, which is called the trivial
representation. Let $R:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g})$ be
a linear map with $x\longrightarrow R_{x}$, where the linear map
$R_{x}:\mathfrak{g}\longrightarrow\mathfrak{g}$ is defined by
$R_{x}(y)=y\cdot_{\mathfrak{g}}x,$ for all $x,y\in\mathfrak{g}$. Then
$(\mathfrak{g};\rho=L,\mu=R)$ is also a representation, which is called the
regular representation. Define two linear maps
$L^{*},R^{*}:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g}^{*})$ with
$x\longrightarrow L^{*}_{x}$ and $x\longrightarrow R^{*}_{x}$ respectively
(for all $x\in\mathfrak{g}$) by

(2) $\langle L_{x}^{*}(\xi),y\rangle=-\langle\xi,x\cdot y\rangle,\;\;\langle
R_{x}^{*}(\xi),y\rangle=-\langle\xi,y\cdot x\rangle,\;\;\forall
x,y\in\mathfrak{g},\xi\in\mathfrak{g}^{*}.$

Then $(\mathfrak{g}^{*};\rho={\rm ad}^{*}=L^{*}-R^{*},\mu=-R^{*})$ is a
representation of $(\mathfrak{g},\cdot_{\mathfrak{g}})$, which is called the
coregular representation. In fact, it is the dual representation of the
regular representation $(\mathfrak{g};L,R)$.

The cohomology complex for a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ with coefficients in a representation
$(V;\rho,\mu)$ is given as follows ([4]). The cohomology of the pre-Lie
algebra $\mathfrak{g}$ with coefficients in $V$ is the cohomology of the
cochain complex $(\otimes_{n\geq 1}C_{\rm
preLie}^{n}(\mathfrak{g},V),\mathrm{d})$, where $C^{n}_{\rm
preLie}(\mathfrak{g},V)=\mathrm{Hom}(\wedge^{n-1}\mathfrak{g}\otimes\mathfrak{g},V)$
and the coboundary operator $\mathrm{d}:C_{\rm
preLie}^{n}(\mathfrak{g},V)\longrightarrow C_{\rm
preLie}^{n+1}(\mathfrak{g},V)$ is given by

(3) $\displaystyle\mathrm{d}f(x_{1},\cdots,x_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}(-1)^{i+1}\rho(x_{i})f(x_{1},\cdots,\hat{x_{i}},\cdots,x_{n+1})$
$\displaystyle+\sum_{i=1}^{n}(-1)^{i+1}\mu(x_{n+1})f(x_{1},\cdots,\hat{x_{i}},\cdots,x_{n},x_{i})$
$\displaystyle-\sum_{i=1}^{n}(-1)^{i+1}f(x_{1},\cdots,\hat{x_{i}},\cdots,x_{n},x_{i}\cdot_{\mathfrak{g}}x_{n+1})$
$\displaystyle+\sum_{1\leq i<j\leq
n}(-1)^{i+j}f([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\cdots,\hat{x_{i}},\cdots,\hat{x_{j}},\cdots,x_{n+1}),$

for all $x_{i}\in\mathfrak{g},~{}i=1,\cdots,n+1$. We denote the corresponding
$n$-th cohomology group by $H_{\rm preLie}^{k}(\mathfrak{g},V)$.

######  Definition 2.3.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ and
$(\mathfrak{h},\cdot_{\mathfrak{h}})$ be two pre-Lie algebras. A homomorphism
between $\mathfrak{g}$ and $\mathfrak{h}$ is a linear map
$\phi:\mathfrak{g}\to\mathfrak{h}$ satisfying

(4)
$\phi(x\cdot_{\mathfrak{g}}y)=\phi(x)\cdot_{\mathfrak{h}}\phi(y),\quad\forall~{}x,y\in\mathfrak{g}.$

Moreover, two pre-Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ with a pre-
Lie algebra homomorphism $\phi$ is call a pre-Lie-morphism triple. We denote
it by
$((\mathfrak{g},\cdot_{\mathfrak{g}}),(\mathfrak{h},\cdot_{\mathfrak{h}}),\phi)$,
or simply $(\mathfrak{g},\mathfrak{h},\phi)$.

It is obvious that if $\phi$ is a homomorphism from a pre-Lie algebra
$\mathfrak{g}$ to a pre-Lie algebra $\mathfrak{h}$, then $\phi$ is a
homomorphism from the sub-adjacent Lie algebra $\mathfrak{g}^{c}$ to the sub-
adjacent Lie algebra $\mathfrak{h}^{c}$. We call the Lie-morphism triple
$(\mathfrak{g}^{c},\mathfrak{h}^{c},\phi)$ a sub-adjacent Lie-morphism triple
of the pre-Lie-morphism triple $(\mathfrak{g},\mathfrak{h},\phi)$.

Let $D$ be a derivation on a commutative associative algebra $(A,\cdot_{A})$.
Then the new product

(5) $x\ast_{D}y=x\cdot_{A}D(y),\quad\forall~{}x,y\in A$

makes $(A,\ast_{D})$ being a pre-Lie algebra by Gel’fand [11].

######  Example 2.4.

Let $D_{1}$ be a derivation on a commutative associative algebra
$(A,\cdot_{A})$ and $D_{2}$ a derivation on a commutative associative algebra
$(B,\cdot_{B})$. Assume that $f:A\to B$ is an associative algebra homomorphism
satisfying $f\circ D_{1}=D_{2}\circ f$. Then $f$ is a pre-Lie algebra
homomorphism from the pre-Lie algebra $(A,\ast_{D_{1}})$ to the pre-Lie
algebra $(B,\ast_{D_{2}})$.

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ be a Lie algebra. Recall
that $\omega\in\wedge^{2}\mathfrak{g}^{*}$ is a 2-cocycle on $\mathfrak{g}$ if

(6)
$\omega([x,y]_{\mathfrak{g}},z)+\omega([z,x]_{\mathfrak{g}},y)+\omega([y,z]_{\mathfrak{g}},x)=0,\quad\forall~{}x,y,z\in\mathfrak{g}.$

A symplectic Lie algebra, denoted by
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\omega)$, is a Lie algebra
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ together with a nondegenerate
$2$-cocycle $\omega\in\wedge^{2}\mathfrak{g}^{*}$ on $\mathfrak{g}$.

######  Lemma 2.5.

([6]) Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\omega)$ be a symplectic
Lie algebra. Then there exists a pre-Lie algebra structure
“$\cdot_{\mathfrak{g}}$” on $\mathfrak{g}$ given by

(7)
$\omega(x\cdot_{\mathfrak{g}}y,z)=-\omega(y,[x,z]_{\mathfrak{g}}),\quad\forall~{}x,y,z\in\mathfrak{g},$

such that the sub-adjacent Lie algebra is exactly
$(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}})$ itself.

######  Example 2.6.

Let $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\omega_{1})$ and
$(\mathfrak{h},[\cdot,\cdot]_{\mathfrak{h}},\omega_{2})$ be two symplectic Lie
algebras. Assume that $\phi:\mathfrak{g}\to\mathfrak{h}$ is a symplectic Lie
algebra homomorphism, i.e.

$\phi([x,y]_{\mathfrak{g}})=[\phi(x),\phi(y)]_{\mathfrak{h}},\quad\omega_{2}(\phi(x),\phi(y))=\omega_{1}(x,y),\quad\forall~{}x,y\in\mathfrak{g}.$

Then $\phi$ is a pre-Lie algebra homomorphism from the pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ to the pre-Lie algebra
$(\mathfrak{h},\cdot_{\mathfrak{h}})$.

######  Definition 2.7.

([1]) Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and
$\mathcal{R}:\mathfrak{g}\longrightarrow\mathfrak{g}$ a linear operator. If
$\mathcal{R}$ satisfies

(8)
$\mathcal{R}(x)\cdot_{\mathfrak{g}}\mathcal{R}(y)=\mathcal{R}(\mathcal{R}(x)\cdot_{\mathfrak{g}}y+x\cdot_{\mathfrak{g}}\mathcal{R}(y)+\lambda
x\cdot_{\mathfrak{g}}y),\quad\forall x,y\in\mathfrak{g},$

then $\mathcal{R}$ is called a Rota-Baxter operator of weight $\lambda$ on
$\mathfrak{g}$.

######  Example 2.8.

Let $\mathcal{R}$ be a Rota-Baxter operator of weight $\lambda$ on a pre-Lie
algebra $(\mathfrak{g},\cdot_{\mathfrak{g}})$. Define a new operation
$\cdot^{\mathcal{R}}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$
by

$x\cdot^{\mathcal{R}}y=\mathcal{R}(x)\cdot_{\mathfrak{g}}y+x\cdot_{\mathfrak{g}}\mathcal{R}(y)+\lambda
x\cdot_{\mathfrak{g}}y,\;\forall x,y\in\mathfrak{g}.$

Then $(\mathfrak{g},\cdot^{\mathcal{R}})$ is a pre-Lie algebra and
$\mathcal{R}$ is a pre-Lie algebra homomorphism from the pre-Lie algebra
$(\mathfrak{g},\cdot^{\mathcal{R}})$ to the pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$.

######  Definition 2.9.

([20]) Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra. A
linear operator $N:\mathfrak{g}\longrightarrow\mathfrak{g}$ is called a
Nijenhuis operator if

(9)
$N(x)\cdot_{\mathfrak{g}}N(y)=N\big{(}N(x)\cdot_{\mathfrak{g}}y+x\cdot_{\mathfrak{g}}N(y)-N(x\cdot_{\mathfrak{g}}y)\big{)},\;\forall
x,y\in\mathfrak{g}.$

######  Example 2.10.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and $N$ a
Nijenhuis operator on $\mathfrak{g}$. Define a new operation
$\cdot_{N}:\otimes^{2}\mathfrak{g}\to\mathfrak{g}$ by

(10) $\displaystyle
x\cdot_{N}y=N(x)\cdot_{\mathfrak{g}}y+x\cdot_{\mathfrak{g}}N(y)-N(x\cdot_{\mathfrak{g}}y),\;\forall
x,y\in\mathfrak{g}.$

Then $(\mathfrak{g},\cdot_{N})$ is a pre-Lie algebra and $N$ is a pre-Lie
algebra homomorphism from $(\mathfrak{g},\cdot_{N})$ to
$(\mathfrak{g},\cdot_{\mathfrak{g}})$.

######  Definition 2.11.

([3]) Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and
$(V;\rho,\mu)$ a representation. A linear map $T:V\longrightarrow\mathfrak{g}$
is called an ${\mathcal{O}}$-operator on a pre-Lie algebra $\mathfrak{g}$
associated to a representation $(V;\rho,\mu)$ if it satisfies

(11)
$T(u)\cdot_{\mathfrak{g}}T(v)=T\Big{(}\rho(T(u))(v)+\mu(T(v))(u)\Big{)},\quad\forall
u,v\in V.$

Note that a Rota-Baxter operator of weight zero on a pre-Lie algebra
$\mathfrak{g}$ is exactly an ${\mathcal{O}}$-operator associated to the
regular representation $(\mathfrak{g};L,R)$.

######  Example 2.12.

Let $T$ be an ${\mathcal{O}}$-operator on a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ associated to a representation
$(V;\rho,\mu)$. Define a new operation
$\cdot^{T}:\otimes^{2}\mathfrak{g}\to\mathfrak{g}$ by

(12) $\displaystyle x\cdot^{T}y=\rho(T(u))(v)+\mu(T(v))(u),\;\forall
x,y\in\mathfrak{g}.$

Then $(\mathfrak{g},\cdot^{T})$ is a pre-Lie algebra and $T$ is a pre-Lie
algebra homomorphism from $(\mathfrak{g},\cdot^{T})$ to
$(\mathfrak{g},\cdot_{\mathfrak{g}})$.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$. We introduce $\llbracket
r,r\rrbracket\in\wedge^{2}\mathfrak{g}\otimes\mathfrak{g}$ as follows

(13) $\llbracket
r,r\rrbracket(\xi,\eta,\zeta)=-\langle\xi,r^{\sharp}(\eta)\cdot_{\mathfrak{g}}r^{\sharp}(\zeta)\rangle+\langle\eta,r^{\sharp}(\xi)\cdot_{\mathfrak{g}}r^{\sharp}(\zeta)\rangle+\langle\zeta,[r^{\sharp}(\xi),r^{\sharp}(\eta)]^{c}\rangle,\quad\forall~{}\xi,\eta,\zeta\in\mathfrak{g}^{*},$

where $r^{\sharp}:\mathfrak{g}^{*}\to\mathfrak{g}$ is given by $\langle
r^{\sharp}(\xi),\eta\rangle=r(\xi,\eta),$ for all
$~{}\xi,\eta\in\mathfrak{g}^{*}$.

######  Definition 2.13.

([2]) Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra. If
$r\in\mathrm{Sym}^{2}(\mathfrak{g})$ and satisfies $\llbracket
r,r\rrbracket=0$, then $r$ is called an $\mathfrak{s}$-matrix in
$(\mathfrak{g},\cdot_{\mathfrak{g}})$.

######  Remark 2.14.

An $\mathfrak{s}$-matrix in a pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ is a solution of the $S$-equation in
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ which is an analogue of the classical
Yang-Baxter equation in a Lie algebra. It plays an important role in the
theory of pre-Lie bialgebras, see [2] for more details.

It was shown in [2] that $r\in\mathrm{Sym}^{2}(\mathfrak{g})$ is an
$\mathfrak{s}$-matrix in the pre-Lie algebra $\mathfrak{g}$ if and only if
$r^{\sharp}:\mathfrak{g}^{*}\to\mathfrak{g}$ is an ${\mathcal{O}}$-operator
associated to the coregular representation $(\mathfrak{g}^{*};{\rm
ad}^{*},-R^{*})$.

######  Example 2.15.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and $r$ an
$\mathfrak{s}$-matrix. Then $(\mathfrak{g}^{*},\cdot_{r})$ is a pre-Lie
algebra, where the multiplication
$\cdot^{r}:\otimes^{2}\mathfrak{g}^{*}\longrightarrow\mathfrak{g}^{*}$ is
given by

(14)
$\xi\cdot^{r}\eta=\mathrm{ad}^{*}_{r^{\sharp}(\xi)}\eta-R^{*}_{r^{\sharp}(\eta)}\xi,\quad\forall~{}\xi,\eta\in\mathfrak{g}^{*}.$

Furthermore, $r^{\sharp}:\mathfrak{g}^{*}\longrightarrow\mathfrak{g}$ is a
pre-Lie algebra homomorphism from $(\mathfrak{g}^{*},\cdot^{r})$ to
$(\mathfrak{g},\cdot_{\mathfrak{g}})$.

##  3\. Cohomologies and deformations of pre-Lie-morphism triples

In this section, we give the cohomlogy of the pre-Lie-morphism triple and then
use it to study the deformations of the pre-Lie-morphism triple. In
particular, we show that a Nijenhuis pair on the pre-Lie-morphism triple
generates a trivial deformation. The relations between Nijenhuis pairs and
compatible ${\mathcal{O}}$-operator are discussed.

###  3.1. Cohomologies of pre-Lie-morphism triples

Let $\phi$ be a homomorphism between pre-Lie algebras
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ and
$(\mathfrak{h},\cdot_{\mathfrak{h}})$. Define
$\rho_{\phi}:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{h})$ and
$\mu_{\phi}:\mathfrak{g}\to\mathfrak{gl}(\mathfrak{h})$ by

(15)
$\rho_{\phi}(x)u=\phi(x)\cdot_{\mathfrak{h}}u,\quad\mu_{\phi}(x)u=u\cdot_{\mathfrak{h}}\phi(x),\quad\forall~{}x\in\mathfrak{g},u\in\mathfrak{h}.$

Furthermore, we have

######  Proposition 3.1.

The pair $(\rho_{\phi},\mu_{\phi})$ is a representation of the pre-Lie algebra
$\mathfrak{g}$ on $\mathfrak{h}$.

###### Proof.

Since $\phi$ is a pre-Lie algebra homomorphism from $\mathfrak{g}$ to
$\mathfrak{h}$, $\phi$ is a Lie algebra homomorphism from sub-adjacent Lie
algebra $\mathfrak{g}^{c}$ to $\mathfrak{h}^{c}$. Then we have

$\displaystyle\rho_{\phi}([x,y]_{\mathfrak{g}})u$ $\displaystyle=$
$\displaystyle\phi([x,y]_{\mathfrak{g}})\cdot_{\mathfrak{h}}u$
$\displaystyle=$
$\displaystyle(\phi(x)\cdot_{\mathfrak{h}}\phi(y))\cdot_{\mathfrak{h}}u-(\phi(y)\cdot_{\mathfrak{h}}\phi(x))\cdot_{\mathfrak{h}}u$
$\displaystyle=$
$\displaystyle\phi(x)\cdot_{\mathfrak{h}}(\rho_{\phi}(y)\cdot_{\mathfrak{h}}u)-\phi(y)\cdot_{\mathfrak{h}}(\rho_{\phi}(x)\cdot_{\mathfrak{h}}u)$
$\displaystyle=$
$\displaystyle\rho_{\phi}(x)\rho_{\phi}(y)u-\rho_{\phi}(y)\rho_{\phi}(x)u,$

which implies that
$\rho_{\phi}([x,y]_{\mathfrak{g}})=\rho_{\phi}(x)\rho_{\phi}(y)-\rho_{\phi}(y)\rho_{\phi}(x)$.

Furthermore, since $\mathfrak{h}$ is a pre-Lie algebra, we have

$\displaystyle\rho_{\phi}(x)\mu_{\phi}(y)u-\mu_{\phi}(y)\rho_{\phi}(x)u-\mu_{\phi}(x\cdot_{\mathfrak{g}}y)u+\mu_{\phi}(y)\mu_{\phi}(x)u$
$\displaystyle=$
$\displaystyle\phi(x)\cdot_{\mathfrak{h}}(u\cdot_{\mathfrak{h}}\phi(y))-(\phi(x)\cdot_{\mathfrak{h}}u)\cdot_{\mathfrak{h}}\phi(y)-u\cdot_{\mathfrak{h}}(\phi(x)\cdot_{\mathfrak{h}}\phi(y))+(u\cdot_{\mathfrak{h}}\phi(x))\cdot_{\mathfrak{h}}\phi(y)$
$\displaystyle=$ $\displaystyle 0,$

which implies that
$\rho_{\phi}(x)\mu_{\phi}(y)z-\mu_{\phi}(y)\rho_{\phi}(x)z=\mu_{\phi}(x\cdot_{\mathfrak{g}}y)z-\mu_{\phi}(y)\mu_{\phi}(x)z$.

Thus $(\rho_{\phi},\mu_{\phi})$ is a representation of the pre-Lie algebra
$\mathfrak{g}$ on $\mathfrak{h}$ ∎

Now we give the cohomology complex for pre-Lie-morphism triples. Let $\phi$ be
pre-Lie algebra homomorphism between pre-Lie algebras
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ and
$(\mathfrak{h},\cdot_{\mathfrak{h}})$. Define the set of $k$-cochains by

${\mathcal{C}}_{\rm preLie}^{k}(\phi,\phi)=C_{\rm
preLie}^{k+1}(\mathfrak{g},\mathfrak{g})\oplus C_{\rm
preLie}^{k+1}(\mathfrak{h},\mathfrak{h})\oplus C_{\rm
preLie}^{k}(\mathfrak{g},\mathfrak{h}),k\geq 0,$

in which we set $C_{\rm preLie}^{0}(\mathfrak{g},\mathfrak{h})=0$.

The coboundary operator $\delta:{\mathcal{C}}_{\rm
preLie}^{k}(\phi,\phi)\to{\mathcal{C}}_{\rm preLie}^{k+1}(\phi,\phi)$ is
defined by

$\delta_{\rm
preLie}(f_{1},f_{2},f_{3})=(\mathrm{d}_{\mathfrak{g}}f_{1},\mathrm{d}_{\mathfrak{h}}f_{2},\mathrm{d}_{\phi}f_{3}+(-1)^{k}(\phi\circ
f_{1}-\phi^{*}\circ f_{2})),$

where $\mathrm{d}_{\mathfrak{g}}$ and $\mathrm{d}_{\mathfrak{h}}$ are the
coboundary operators of the pre-Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$
associated to their regular representations respectively, $\mathrm{d}_{\phi}$
is the coboundary operator of the pre-Lie algebra $\mathfrak{g}$ associated to
the representation $(\mathfrak{h};\rho_{\phi},\mu_{\phi})$, and $f_{1}\in
C_{\rm preLie}^{k+1}(\mathfrak{g},\mathfrak{g}),f_{2}\in
C^{k+1}(\mathfrak{h},\mathfrak{h}),f_{3}\in C^{k}(\mathfrak{g},\mathfrak{h})$.
Here $\phi\circ f_{1}\in C_{\rm preLie}^{k+1}(\mathfrak{g},\mathfrak{g})$ and
$\phi^{*}\circ f_{2}\in C^{k+1}(\mathfrak{h},\mathfrak{h})$ are defined by

$\displaystyle(\phi\circ f_{1})(x_{1},\cdots,x_{k+1})$ $\displaystyle=$
$\displaystyle\phi\big{(}f_{1}(x_{1},\cdots,x_{k+1})\big{)},$
$\displaystyle(\phi^{*}\circ f_{2})(x_{1},\cdots,x_{k+1})$ $\displaystyle=$
$\displaystyle f_{2}(\phi(x_{1}),\cdots,\phi(x_{k+1})),$

where $x_{1},\cdots,x_{k+1}\in\mathfrak{g}$.

######  Proposition 3.2.

We have $\delta_{\rm preLie}\circ\delta_{\rm preLie}=0$. Hence
$({\mathcal{C}}^{*}(\phi,\phi)=\otimes_{k\geq 1}C^{n}(\phi,\phi),\delta_{\rm
preLie})$ is a cochain complex.

###### Proof.

For $f_{1}\in C_{\rm preLie}^{k+1}(\mathfrak{g},\mathfrak{g})$, we have

$\displaystyle\mathrm{d}_{\phi}(\phi\circ f_{1})(x_{1},\cdots,x_{k+2})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k+1}(-1)^{i+1}\phi(x_{i})\cdot_{\mathfrak{h}}\phi(f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+2}))$
$\displaystyle+\sum_{i=1}^{k+1}(-1)^{i+1}\phi(f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1},x_{i}))\cdot_{\mathfrak{h}}\phi(x_{k+2})$
$\displaystyle-\sum_{i=1}^{k+1}(-1)^{i+1}\phi(f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1},x_{i}\cdot_{\mathfrak{g}}x_{k+2}))$
$\displaystyle+\sum_{1\leq
i<j\leq{k+1}}(-1)^{i+j}\phi(f_{1}([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\cdots,\hat{x_{i}},\cdots,\hat{x_{j}},\cdots,x_{k+2}))$
$\displaystyle=$
$\displaystyle\phi(\sum_{i=1}^{k+1}(-1)^{i+1}x_{i}\cdot_{\mathfrak{g}}f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+2}))$
$\displaystyle+\phi(\sum_{i=1}^{k+1}(-1)^{i+1}f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1},x_{i}))\cdot_{\mathfrak{g}}x_{k+2})$
$\displaystyle-\phi(\sum_{i=1}^{k+1}(-1)^{i+1}f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1},x_{i}\cdot_{\mathfrak{g}}x_{k+2}))$
$\displaystyle+\phi(\sum_{1\leq
i<j\leq{k+1}}(-1)^{i+j}f_{1}([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\cdots,\hat{x_{i}},\cdots,\hat{x_{j}},\cdots,x_{k+2}))$
$\displaystyle=$
$\displaystyle\phi(\mathrm{d}_{\mathfrak{g}}(f_{1}))(x_{1},\cdots,x_{k+2}),$

which implies that

(16)
$\displaystyle\mathrm{d}_{\phi}(\phi(f_{1}))=\phi(\mathrm{d}_{\mathfrak{g}}(f_{1})).$

For $f_{2}\in C_{\rm preLie}^{k+1}(\mathfrak{h},\mathfrak{h})$, we have

$\displaystyle\mathrm{d}_{\phi}(\phi^{*}f_{2})(x_{1},\cdots,x_{k+2})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k+1}(-1)^{i+1}\phi(x_{i})\cdot_{\mathfrak{h}}f_{2}(\phi(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+2}))$
$\displaystyle+\sum_{i=1}^{k+1}(-1)^{i+1}f_{2}(\phi(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1},x_{i}))\cdot_{\mathfrak{h}}\phi(x_{k+2})$
$\displaystyle-\sum_{i=1}^{k+1}(-1)^{i+1}f_{2}(\phi(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1},x_{i}\cdot_{\mathfrak{g}}x_{k+2}))$
$\displaystyle+\sum_{1\leq
i<j\leq{k+1}}(-1)^{i+j}f_{2}(\phi([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\cdots,\hat{x_{i}},\cdots,\hat{x_{j}},\cdots,x_{k+2}))$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k+1}(-1)^{i+1}\phi(x_{i})\cdot_{\mathfrak{h}}f_{2}(\phi(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+2}))$
$\displaystyle+\sum_{i=1}^{k+1}(-1)^{i+1}f_{2}(\phi(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1},x_{i}))\cdot_{\mathfrak{h}}\phi(x_{k+2})$
$\displaystyle-\sum_{i=1}^{k+1}(-1)^{i+1}f_{2}(\phi(x_{1}),\cdots,\hat{x_{i}},\cdots,\phi(x_{k+1}),\phi(x_{i})\cdot_{\mathfrak{h}}\phi(x_{k+2}))$
$\displaystyle+\sum_{1\leq
i<j\leq{k+1}}(-1)^{i+j}f_{2}([\phi(x_{i}),\phi(x_{j})]_{\mathfrak{h}},\phi(x_{1}),\cdots,\hat{x_{i}},\cdots,\hat{x_{j}},\cdots,\phi(x_{k+2}))$
$\displaystyle=$
$\displaystyle\phi^{*}\mathrm{d}_{\mathfrak{h}}(f_{2})(x_{1},\cdots,x_{k+2}),$

which implies that

(17)
$\displaystyle\mathrm{d}_{\phi}(\phi^{*}f_{2})=\phi^{*}\mathrm{d}_{\mathfrak{h}}(f_{2}).$

Furthermore, by (16), (17) and the fact that
$\mathrm{d}_{\mathfrak{g}}\circ\mathrm{d}_{\mathfrak{g}}=0$ ,
$\mathrm{d}_{\mathfrak{h}}\circ\mathrm{d}_{\mathfrak{h}}=0$ and
$\mathrm{d}_{\phi}\circ\mathrm{d}_{\phi}=0$, we have

$\displaystyle\delta_{\rm preLie}\circ\delta_{\rm preLie}(f_{1},f_{2},f_{3})$
$\displaystyle=$ $\displaystyle\delta_{\rm
preLie}(\mathrm{d}_{\mathfrak{g}}f_{1},\mathrm{d}_{\mathfrak{h}}f_{2},\mathrm{d}_{\phi}f_{3}+(-1)^{k}(\phi\circ
f_{1}-\phi^{*}\circ f_{2}))$ $\displaystyle=$
$\displaystyle(\mathrm{d}_{\mathfrak{g}}\mathrm{d}_{\mathfrak{g}}f_{1},\mathrm{d}_{\mathfrak{h}}\mathrm{d}_{\mathfrak{h}}f_{2},\mathrm{d}_{\phi}(\mathrm{d}_{\phi}f_{3}+(-1)^{k}(\phi\circ
f_{1}-\phi^{*}\circ
f_{2}))+(-1)^{k+1}(\phi\circ\mathrm{d}_{\mathfrak{g}}f_{1}-\phi^{*}\circ\mathrm{d}_{\mathfrak{h}}f_{2}))$
$\displaystyle=$
$\displaystyle\Big{(}0,0,(-1)^{k}\big{(}\mathrm{d}_{\phi}(\phi(f_{1}))-\phi(\mathrm{d}_{\mathfrak{g}}(f_{1})+\phi^{*}\mathrm{d}_{\mathfrak{h}}(f_{2})-\mathrm{d}_{\phi}(\phi^{*}f_{2})\big{)}\Big{)}=(0,0,0),$

which implies that $\delta_{\rm preLie}\circ\delta_{\rm preLie}=0$. ∎

######  Definition 3.3.

The cohomology of the cochain complex
$({\mathcal{C}}^{*}(\phi,\phi),\delta_{\rm preLie})$ is called the cohomology
of the pre-Lie-morphism triple $(\mathfrak{g},\mathfrak{h},\phi)$. The
corresponding $k$-th cohomology group, which we denote by $H^{k}(\phi,\phi)$,
is called the $k$-th cohomology group for the pre-Lie-morphism triple
$(\mathfrak{g},\mathfrak{h},\phi)$.

###  3.2. Infinitesimal deformations of pre-Lie-morphism triples

Let $\phi$ be a pre-Lie algebra homomorphism between pre-Lie algebras
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ and
$(\mathfrak{h},\cdot_{\mathfrak{h}})$,
$\omega\in\mathrm{Hom}(\otimes^{2}\mathfrak{g},\mathfrak{g})$,
$\varpi\in\mathrm{Hom}(\otimes^{2}\mathfrak{h},\mathfrak{h})$ and
$\theta\in\mathrm{Hom}(\mathfrak{g},\mathfrak{h})$. Consider a
$t$-parameterized family of multiplications
$\cdot_{t}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$,
$\ast_{t}:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$ and
linear maps $\phi_{t}:\mathfrak{g}\longrightarrow\mathfrak{h}$ given by

$\displaystyle x\cdot_{t}y$ $\displaystyle=$ $\displaystyle
x\cdot_{\mathfrak{g}}y+t\omega(x,y),$ $\displaystyle u\ast_{t}v$
$\displaystyle=$ $\displaystyle u\cdot_{\mathfrak{h}}v+t\varpi(u,v),$
$\displaystyle\phi_{t}(x)$ $\displaystyle=$ $\displaystyle\phi(x)+t\theta(x),$

where $x,y\in\mathfrak{g}$ and $u,v\in\mathfrak{h}$. If
$\mathfrak{g}_{t}=(\mathfrak{g},\cdot_{t})$ and
$\mathfrak{h}_{t}=(\mathfrak{h},\ast_{t})$ are pre-Lie algebras and

$\phi_{t}:\mathfrak{g}_{t}\to\mathfrak{h}_{t}~{}\mbox{module}~{}t^{3}$

are pre-Lie algebra homomorphisms for all $t$, we say that
$(\omega,\varphi,\theta)$ generates a $1$-parameter infinitesimal deformation
of $(\mathfrak{g},\mathfrak{h},\phi)$. We denote a $1$-parameter infinitesimal
deformation of $(\mathfrak{g},\mathfrak{h},\phi)$ by
$(\mathfrak{g}_{t},\mathfrak{h}_{t},\phi_{t})$.

It is straightforward to check that $(\omega,\varpi,\theta)$ generates a
$1$-parameter infinitesimal deformation of $(\mathfrak{g},\mathfrak{h},\phi)$
if and only if

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
x\cdot_{\mathfrak{g}}\omega(y,z)-y\cdot_{\mathfrak{g}}\omega(x,z)+\omega(y,x)\cdot_{\mathfrak{g}}z-\omega(x,y)\cdot_{\mathfrak{g}}z$
$\displaystyle-\omega(y,x\cdot_{\mathfrak{g}}z)+\omega(x,y\cdot_{\mathfrak{g}}z)+\omega([x,y]_{\mathfrak{g}},z),$
(19) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\omega(\omega(x,y),z)-\omega(x,\omega(y,z))-\omega(\omega(y,x),z)+\omega(y,\omega(x,z)),$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
u\cdot_{\mathfrak{h}}\varpi(v,w)-v\cdot_{\mathfrak{h}}\varpi(u,w)+\varpi(v,u)\cdot_{\mathfrak{h}}w-\varpi(u,v)\cdot_{\mathfrak{h}}w$
$\displaystyle-\varpi(v,u\cdot_{\mathfrak{h}}w)+\varpi(u,v\cdot_{\mathfrak{h}}w)+\varpi([u,v]_{\mathfrak{h}},w),$
(21) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\varpi(\varpi(u,v),w)-\varpi(u,\varpi(v,w))-\varpi(\varpi(v,u),w)+\varpi(v,\varpi(u,w)),$
(22) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\phi(x)\cdot_{\mathfrak{h}}\theta(y)+\theta(x)\cdot_{\mathfrak{h}}\phi(y)-\theta(x\cdot_{\mathfrak{g}}y)-\phi(\omega(x,y))+\varpi(\phi(x),\phi(y)),$
(23) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\theta(\omega(x,y))-\varpi(\phi(x),\theta(y))-\varpi(\theta(x),\phi(y))-\theta(x)\cdot_{\mathfrak{h}}\theta(y).$

Note that $(\ref{2-cocycle g})$, (3.2) and (22) mean that $\delta_{\rm
preLie}(\omega,\varpi,\theta)=0$, $(\ref{omega bracket})$ means that
$(\mathfrak{g},\omega)$ is a pre-Lie algebra, and $(\ref{varpi bracket})$
means that $(\mathfrak{h},\varpi)$ is a pre-Lie algebra.

######  Definition 3.4.

Two infinitesimal deformations $(\mathfrak{g}_{t},\mathfrak{h}_{t},\phi_{t})$
and $(\mathfrak{g}^{\prime}_{t},\mathfrak{h}^{\prime}_{t},\phi^{\prime}_{t})$
of a pre-Lie-morphism triple $(\mathfrak{g},\mathfrak{h},\phi)$, which are
generated by $(\omega,\varpi,\theta)$ and
$(\omega^{\prime},\varpi^{\prime},\theta^{\prime})$, are said to be equivalent
if there exist a family of pre-Lie algebra isomorphisms
$\rm{Id}_{\mathfrak{g}}+tN:\mathfrak{g}_{t}\to\mathfrak{g}^{\prime}_{t}$ and
$\rm{Id}_{\mathfrak{h}}+tS:\mathfrak{h}_{t}\to\mathfrak{h}^{\prime}_{t}$ with
$N\in\mathfrak{gl}(\mathfrak{g})$ and $S\in\mathfrak{gl}(\mathfrak{h})$, such
that

$\phi^{\prime}_{t}\circ(\rm{Id}_{\mathfrak{g}}+tN)=(\rm{Id}_{\mathfrak{h}}+tS)\circ\phi_{t}.$

A infinitesimal deformation $(\mathfrak{g}_{t},\mathfrak{h}_{t},\phi_{t})$ of
a pre-Lie-morphism triple $(\mathfrak{g},\mathfrak{h},\phi)$ is said to be
trivial if it is equivalent to
$((\mathfrak{g},\cdot_{\mathfrak{g}}),(\mathfrak{h},\cdot_{\mathfrak{h}}),\phi)$.

By a direct calculation, $(\mathfrak{g}_{t},\mathfrak{h}_{t},\phi_{t})$ and
$(\mathfrak{g}^{\prime}_{t},\mathfrak{h}^{\prime}_{t},\phi^{\prime}_{t})$ of a
pre-Lie-morphism triple $(\mathfrak{g},\mathfrak{h},\phi)$ are equivalent if
and only if

(24) $\displaystyle\omega(x,y)-\omega^{\prime}(x,y)$ $\displaystyle=$
$\displaystyle
x\cdot_{\mathfrak{g}}N(y)+N(x)\cdot_{\mathfrak{g}}y-N(x\cdot_{\mathfrak{g}}y),$
(25) $\displaystyle N\omega(x,y)$ $\displaystyle=$
$\displaystyle\omega^{\prime}(x,N(y))+\omega^{\prime}(N(x),y)+N(x)\cdot_{\mathfrak{g}}N(y),$
(26) $\displaystyle\omega^{\prime}(N(x),N(y))$ $\displaystyle=$ $\displaystyle
0,$ (27) $\displaystyle\varpi(u,v)-\varpi^{\prime}(u,v)$ $\displaystyle=$
$\displaystyle
u\cdot_{\mathfrak{h}}S(v)+S(u)\cdot_{\mathfrak{h}}v-S(u\cdot_{\mathfrak{h}}v),$
(28) $\displaystyle S\varpi(u,v)$ $\displaystyle=$
$\displaystyle\varpi^{\prime}(u,S(v))+\varpi^{\prime}(S(u),v)+S(u)\cdot_{\mathfrak{h}}S(v),$
(29) $\displaystyle\varpi^{\prime}(S(u,S(v))$ $\displaystyle=$ $\displaystyle
0,$ (30) $\displaystyle\theta-\theta^{\prime}$ $\displaystyle=$
$\displaystyle\phi\circ N-S\circ\phi,$ (31) $\displaystyle\theta^{\prime}\circ
N$ $\displaystyle=$ $\displaystyle S\circ\theta.$

Note that $(\ref{2-exact})$, (27) and (30) mean that
$(\omega,\varpi,\theta)-(\omega^{\prime},\varpi^{\prime},\theta^{\prime})=\delta_{\rm
preLie}(N,S,0)$.

We summarize the above discussion into the following theorem:

######  Theorem 3.5.

Let $(\mathfrak{g}_{t},\mathfrak{h}_{t},\phi_{t})$ be a infinitesimal
deformation of a pre-Lie-morphism triple $(\mathfrak{g},\mathfrak{h},\phi)$
generated by $(\omega,\varpi,\theta)$. Then $(\omega,\varpi,\theta)$ is
closed, i.e. $\delta_{\rm preLie}(\omega,\varpi,\theta)=0$. Furthermore, if
two infinitesimal deformations $(\mathfrak{g}_{t},\mathfrak{h}_{t},\phi_{t})$
and $(\mathfrak{g}^{\prime}_{t},\mathfrak{h}^{\prime}_{t},\phi^{\prime}_{t})$
of the pre-Lie-morphism triple $(\mathfrak{g},\mathfrak{h},\phi)$ generated by
$(\omega,\varpi,\theta)$ and
$(\omega^{\prime},\varpi^{\prime},\theta^{\prime})$ are equivalent, then
$(\omega,\varpi,\theta)$ and
$(\omega^{\prime},\varpi^{\prime},\theta^{\prime})$ are in the same cohomology
class in $H^{1}(\phi,\phi)$.

Now we consider trivial deformations of a pre-Lie-morphism triple
$(\mathfrak{g},\mathfrak{h},\phi)$. Then (24)-(31) reduce to

(32) $\displaystyle\omega(x,y)$ $\displaystyle=$ $\displaystyle
x\cdot_{\mathfrak{g}}N(y)+N(x)\cdot_{\mathfrak{g}}y-N(x\cdot_{\mathfrak{g}}y),$
(33) $\displaystyle N\omega(x,y)$ $\displaystyle=$ $\displaystyle
N(x)\cdot_{\mathfrak{g}}N(y),$ (34) $\displaystyle\varpi(u,v)$
$\displaystyle=$ $\displaystyle
u\cdot_{\mathfrak{h}}S(v)+S(u)\cdot_{\mathfrak{h}}v-S(u\cdot_{\mathfrak{h}}v),$
(35) $\displaystyle S\varpi(u,v)$ $\displaystyle=$ $\displaystyle
S(u)\cdot_{\mathfrak{h}}S(v),$ (36) $\displaystyle\theta$ $\displaystyle=$
$\displaystyle\phi\circ N-S\circ\phi,$ (37) $\displaystyle S\circ\theta$
$\displaystyle=$ $\displaystyle 0.$

It follows from (32) and (33) that $N$ must satisfy the following condition:

(38) $\displaystyle
N(x)\cdot_{\mathfrak{g}}N(y)-N(x\cdot_{\mathfrak{g}}N(y)+N(x)\cdot_{\mathfrak{g}}y-N(x\cdot_{\mathfrak{g}}y))=0,$

which means that $N$ is a Nijenhuis operator on the pre-Lie algebra
$(\mathfrak{g},\cdot_{\mathfrak{g}})$.

It follows from (34) and (35) that $S$ must satisfy the following condition:

(39) $\displaystyle
S(u)\cdot_{\mathfrak{h}}S(v)-S(u\cdot_{\mathfrak{h}}S(v)+S(u)\cdot_{\mathfrak{h}}v-S(u\cdot_{\mathfrak{h}}v))=0,$

which means that $S$ is a Nijenhuis operator on the pre-Lie algebra
$(\mathfrak{h},\cdot_{\mathfrak{h}})$.

It follows from (36) and (37) that $N$ and $S$ must satisfy the following
condition:

(40) $\displaystyle S\circ\phi\circ N=S^{2}\circ\phi.$

######  Definition 3.6.

Let
$((\mathfrak{g},\cdot_{\mathfrak{g}}),(\mathfrak{h},\cdot_{\mathfrak{h}}),\phi)$
be a pre-Lie-morphism triple. A pair $(N,S)$, where
$N\in\mathfrak{gl}(\mathfrak{g})$ and $S\in\mathfrak{gl}(\mathfrak{h})$, is
called a Nijenhuis pair on the pre-Lie-morphism triple
$(\mathfrak{g},\mathfrak{h},\phi)$ if $N$ is a Nijenhuis operator on the pre-
Lie algebra $(\mathfrak{g},\cdot_{\mathfrak{g}})$, $S$ is a Nijenhuis operator
on the pre-Lie algebra $(\mathfrak{h},\cdot_{\mathfrak{h}})$ and (40) holds.

We have seen that a trivial deformation of a pre-Lie-morphism triple
$(\mathfrak{g},\mathfrak{h},\phi)$ could give rise to a Nijenhuis pair. In
fact, the converse is also true.

######  Theorem 3.7.

Let $(N,S)$ be a Nijenhuis pair on the pre-Lie-morphism triple
$(\mathfrak{g},\mathfrak{h},\phi)$. Then a deformation of the pre-Lie-morphism
triple $\phi$ can be obtained by putting

(41) $\displaystyle\omega(x,y)$ $\displaystyle=$
$\displaystyle\mathrm{d}_{\mathfrak{g}}N(x,y),$ (42)
$\displaystyle\varpi(u,v)$ $\displaystyle=$
$\displaystyle\mathrm{d}_{\mathfrak{h}}S(u,v),$ (43) $\displaystyle\theta(x)$
$\displaystyle=$
$\displaystyle\phi(N(x))-S(\phi(x)),\quad\forall~{}x,y\in\mathfrak{g},u,v\in\mathfrak{h}.$

Furthermore, this deformation is trivial.

###### Proof.

By (41)-(43), we have $(\omega,\varpi,\theta)=\delta_{\rm preLie}(N,S,0)$.
Then $(\omega,\varpi,\theta)$ is closed naturally , which implies that
$(\ref{2-cocycle g})$, (3.2) and (22) hold. $(\ref{omega bracket})$ and
$(\ref{varpi bracket})$ follow from that $N$ and $S$ are Nijenhuis operators
on pre-Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$, respectively. We will
show that (23) holds. By the fact that $\phi$ is a pre-Lie homomorphism and
(40) , we have

$\displaystyle\theta(\omega(x,y))-\varpi(\phi(x),\theta(y))-\varpi(\theta(x),\phi(y))-\theta(x)\cdot_{\mathfrak{h}}\theta(y)$
$\displaystyle=$ $\displaystyle
S(\phi(x))\cdot_{\mathfrak{h}}S(\phi(y))-S(S(\phi(x))\cdot_{\mathfrak{h}}\phi(y)+\phi(x)\cdot_{\mathfrak{h}}S(\phi(y))-S(\phi(x)\cdot_{\mathfrak{h}}\phi(y)))$
$\displaystyle=$ $\displaystyle 0.$

The last equality follows from $S$ is a Nijenhuis operator on the pre-Lie
algebra $\mathfrak{g}$.

At last, it is obvious that

$\displaystyle(\rm{Id}_{\mathfrak{g}}+tN)(x\cdot_{t}y)$ $\displaystyle=$
$\displaystyle(\rm{Id}_{\mathfrak{g}}+tN)(x)\cdot_{\mathfrak{g}}(\rm{Id}_{\mathfrak{g}}+tN)(y),$
$\displaystyle(\rm{Id}_{\mathfrak{h}}+tS)(u\ast_{t}v)$ $\displaystyle=$
$\displaystyle(\rm{Id}_{\mathfrak{h}}+tS)(u)\cdot_{\mathfrak{h}}(\rm{Id}_{\mathfrak{h}}+tS)(v),$
$\displaystyle\phi((\rm{Id}_{\mathfrak{g}}+tN)(x))$ $\displaystyle=$
$\displaystyle(\rm{Id}_{\mathfrak{h}}+tS)((\phi+t\theta)(x)),\quad\forall~{}x,y\in\mathfrak{g},u,v\in\mathfrak{h}.$

Thus the deformation is trivial. ∎

###  3.3. Nijenhuis pairs on pre-Lie-morphism triples and compatible
structures on pre-Lie algebras

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and
$(V;\rho,\mu)$ a representation. Let
$T_{1},T_{2}:V\longrightarrow\mathfrak{g}$ be two ${\mathcal{O}}$-operators on
the pre-Lie algebra $\mathfrak{g}$ associated to the representation
$(V;\rho,\mu)$. Recall that $T_{1}$ and $T_{2}$ are compatible if for any
$k_{1},k_{2}$, $k_{1}T_{1}+k_{2}T_{2}$ is still an ${\mathcal{O}}$-operator.

A pair of compatible ${\mathcal{O}}$-operators can give rise to a Nijenhuis
operator under some conditions.

######  Proposition 3.8.

([20]) Let $T_{1},T_{2}:V\longrightarrow\mathfrak{g}$ be two
${\mathcal{O}}$-operators on a pre-Lie algebra $\mathfrak{g}$ associated to a
representation $(V;\rho,\mu)$. Suppose that $T_{2}$ is invertible. If $T_{1}$
and $T_{2}$ are compatible, then $N=T_{1}\circ T_{2}^{-1}$ is a Nijenhuis
operator on the pre-Lie algebra $(\mathfrak{g},\cdot_{\mathfrak{g}})$.

Given two compatible ${\mathcal{O}}$-operators $T_{1}$ and $T_{2}$ on the pre-
Lie algebra $\mathfrak{g}$ associated to the representation $(V;\rho,\mu)$, by
Example 2.12, $((V,\cdot^{T_{1}}),(\mathfrak{g},\cdot_{\mathfrak{g}}),T_{1})$
and $((V,\cdot^{T_{2}}),(\mathfrak{g},\cdot_{\mathfrak{g}}),T_{2})$ are pre-
Lie-morphism triples.

######  Lemma 3.9.

Let $T_{1},T_{2}:V\longrightarrow\mathfrak{g}$ be two
${\mathcal{O}}$-operators on the pre-Lie algebra $\mathfrak{g}$ associated to
the representation $(V;\rho,\mu)$. Suppose that $T_{2}$ is invertible. If
$T_{1}$ and $T_{2}$ are compatible, then

  * (i)

$N\circ T_{1}=T_{1}\circ S$ and $S$ is a Nijenhuis operator on the pre-Lie
algebra $(V,\cdot^{T_{1}})$, where $N=T_{1}\circ T_{2}^{-1}$ and
$S=T^{-1}_{2}\circ T_{1}$;

  * (ii)

$N\circ T_{2}=T_{2}\circ S$ with $S=T^{-1}_{2}\circ T_{1}$ and $S$ is a
Nijenhuis operator on the pre-Lie algebra $(V,\cdot^{T_{2}})$, where
$N=T_{1}\circ T_{2}^{-1}$ and $S=T^{-1}_{2}\circ T_{1}$.

###### Proof.

It follows from Theorem 4.4 and Proposition 4.21 in [16]. ∎

######  Proposition 3.10.

Let $T_{1},T_{2}:V\longrightarrow\mathfrak{g}$ be two
${\mathcal{O}}$-operators on the pre-Lie algebra $\mathfrak{g}$ associated to
the representation $(V;\rho,\mu)$. Suppose that $T_{2}$ is invertible. If
$T_{1}$ and $T_{2}$ are compatible, then $(N,S)$ is both a Nijenhuis pair on
the pre-Lie-morphism triples
$((V,\cdot_{T_{1}}),(\mathfrak{g},\cdot_{\mathfrak{g}}),T_{1})$ and
$((V,\cdot_{T_{2}}),(\mathfrak{g},\cdot_{\mathfrak{g}}),T_{2})$, where
$N=T_{1}\circ T_{2}^{-1}$ and $S=T^{-1}_{2}\circ T_{1}$.

Furthermore, $((V,\cdot^{T_{1}}_{S}),(\mathfrak{g},\cdot_{N}),T_{1})$ and
$((V,\cdot^{T_{2}}_{S}),(\mathfrak{g},\cdot_{N}),T_{2})$ are pre-Lie-morphism
triples, where the multiplication
$\cdot^{T_{i}}_{S}:\otimes^{2}V\longrightarrow V$ for $i=1,2$ is given by

(44) $\displaystyle u\cdot^{T_{i}}_{S}v$ $\displaystyle=$ $\displaystyle
S(u)\cdot^{T_{i}}v+u\cdot^{T_{i}}S(v)-S(u\cdot^{T_{i}}v),\quad\forall~{}u,v\in
V.$

###### Proof.

By Lemma 3.9, $(N,S)$ is the Nijenhuis pair on the pre-Lie-morphism triples
$((V,\cdot_{T_{1}}),(\mathfrak{g},\cdot_{\mathfrak{g}}),T_{1})$ and
$((V,\cdot_{T_{2}}),(\mathfrak{g},\cdot_{\mathfrak{g}}),T_{2})$. By the fact
that $T_{1}$ is an ${\mathcal{O}}$-operators on the pre-Lie algebra
$\mathfrak{g}$, we have
$T_{1}(u\cdot^{T_{1}}v)=T_{1}(u)\cdot_{\mathfrak{g}}T_{1}(v)$ for all
$u,v\in\mathfrak{g}$. Furthermore, by $N\circ T_{1}=T_{1}\circ S$, we have

$\displaystyle T_{1}(u\cdot^{T_{1}}_{S}v)$ $\displaystyle=$ $\displaystyle
T_{1}(S(u)\cdot^{T_{1}}v+u\cdot^{T_{1}}S(v)-S(u\cdot^{T_{1}}v))$
$\displaystyle=$ $\displaystyle
N(T_{1}(u))\cdot_{\mathfrak{g}}T_{1}(v)+T_{1}(u)\cdot^{T_{1}}N(T_{1}(v))-N(T_{1}(u)\cdot_{\mathfrak{g}}T_{1}(v))$
$\displaystyle=$ $\displaystyle T_{1}(u)\cdot_{N}T_{1}(v).$

Thus $((V,\cdot^{T_{1}}_{S}),(\mathfrak{g},\cdot_{N}),T_{1})$ is a pre-Lie-
morphism triple. Similarly,
$((V,\cdot^{T_{2}}_{S}),(\mathfrak{g},\cdot_{N}),T_{2})$ is also a pre-Lie-
morphism triple. ∎

By Example 2.15 and Proposition 3.10, we have

######  Corollary 3.11.

Let $r_{1},r_{2}\in\mathrm{Sym}^{2}(\mathfrak{g})$ be two
$\mathfrak{s}$-matrices on the pre-Lie algebra $\mathfrak{g}$. Assume that
$r_{2}$ is invertible. If $r_{1}$ and $r_{2}$ are compatible in the sense that
any linear combination of $r_{1}$ and $r_{2}$ is still an
$\mathfrak{s}$-matrix, then
$(N=r_{1}^{\sharp}\circ(r_{2}^{\sharp})^{-1},S=(r_{2}^{\sharp})^{-1}\circ
r_{1}^{\sharp})$ is both a Nijenhuis pair on the pre-Lie-morphism triples
$((\mathfrak{g}^{*},\cdot^{r_{1}}),(\mathfrak{g},\cdot_{\mathfrak{g}}),r_{1}^{\sharp})$
and
$((\mathfrak{g}^{*},\cdot^{r_{2}}),(\mathfrak{g},\cdot_{\mathfrak{g}}),r_{2}^{\sharp})$,
where the operation $\cdot^{r_{i}}$ for $i=1,2$ is given by (14).

##  4\. Cohomologies of pre-Lie-morphism triples and Lie-morphism triples

In this section, we show that the cohomology for a pre-Lie-morphism triple can
be deduced from a new cohomology of the sub-adjacent Lie-morphism triple.

The Chevalley-Eilenberg cohomology theory for a Lie algebra
$(\mathfrak{g},[-,-]_{\mathfrak{g}})$ associated to a representation
$(V;\rho)$ is given as follows. Denote by
$C^{n}_{\mathrm{CE}}(\mathfrak{g},V):=\mathrm{Hom}(\wedge^{n}\mathfrak{g},V)$,
the space of $n$-cochains. The corresponding Chevalley-Eilenberg coboundary
operator $\partial:C_{\mathrm{CE}}^{n}(\mathfrak{g},V)\longrightarrow
C_{\mathrm{CE}}^{n+1}(\mathfrak{g},V)$ is given by

$\displaystyle\partial f(x_{1},\ldots,x_{n+1})$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i+1}\rho({x_{i}})f(x_{1},\ldots,\hat{x_{i}},\ldots,x_{n+1})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}^{n+1}(-1)^{i+j}f([x_{i},x_{j}]_{\mathfrak{g}},\ldots,\hat{x_{i}},\ldots,\hat{x_{j}},\ldots,x_{n+1})$

for all $f\in C_{\mathrm{CE}}^{n}(\mathfrak{g},V)$ and
$x_{1},x_{2},\cdots,x_{n+1}\in\mathfrak{g}$. We denote the corresponding
$n$-th cohomology group by $H_{\rm CE}^{n}(\mathfrak{g},V)$.

Let $(\mathfrak{g},\cdot_{\mathfrak{g}})$ be a pre-Lie algebra and
$(V;\rho,\mu)$ a representation. Define
$\varrho:\mathfrak{g}\to\mathfrak{gl}(\mathrm{Hom}(\mathfrak{g},V))$ by

(46)
$\varrho(x)(f)(y)=\rho(x)f(y)-\mu(y)f(x)-f(x\cdot_{\mathfrak{g}}y),\quad\forall~{}f\in\mathrm{Hom}(\mathfrak{g},V),x,y\in\mathfrak{g}.$

Then it was shown in [9] that $\varrho$ is a representation of the sub-
adjacent Lie algebra $\mathfrak{g}^{c}$ on the vector space
$\mathrm{Hom}(\mathfrak{g},V)$. Furthermore, we define
$\Phi:C^{k}_{\mathrm{CE}}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},V))\to C_{\rm
preLie}^{k+1}(\mathfrak{g},V)$ by

(47)
$\Phi(f)(x_{1},\cdots,x_{n},y)=f(x_{1},\cdots,x_{n})(y),\quad\forall~{}x_{1},\cdots,x_{n},y\in\mathfrak{g}.$

######  Theorem 4.1.

([9, 18]) The map $\Phi$ is a cochain map from the cochain complex
$(C^{*}_{\mathrm{CE}}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},V)),\partial_{\varrho})$
to the cochain complex $(C_{\rm preLie}^{*}(\mathfrak{g},V),\mathrm{d})$,
where $\partial_{\varrho}$ is the coboundary operator of the sub-adjacent Lie
algebra $\mathfrak{g}^{c}$ associated to representation $\varrho$ and
$\mathrm{d}$ is the coboundary operator of the pre-Lie algebra $\mathfrak{g}$
associated to representation $(\rho,\mu)$.

Moreover, $\Phi$ induces an isomorphism between the corresponding cohomology
groups:

(48) $H^{n}_{\rm preLie}(\mathfrak{g},V)\cong H^{n-1}_{\rm
CE}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},V)),\quad n=1,2\cdots.$

Let $\phi$ be a homomorphism between pre-Lie algebras
$(\mathfrak{g},\cdot_{\mathfrak{g}})$ and
$(\mathfrak{h},\cdot_{\mathfrak{h}})$. Define
$\varrho_{\mathfrak{g}}:\mathfrak{g}\to\mathfrak{gl}(\mathrm{Hom}(\mathfrak{g},\mathfrak{g}))$,
$\varrho_{\mathfrak{h}}:\mathfrak{h}\to\mathfrak{gl}(\mathrm{Hom}(\mathfrak{h},\mathfrak{h}))$
and
$\varrho_{\phi}:\mathfrak{g}\to\mathfrak{gl}(\mathrm{Hom}(\mathfrak{g},\mathfrak{h}))$
by

(49) $\displaystyle\varrho_{\mathfrak{g}}(x)(f_{1})(y)$ $\displaystyle=$
$\displaystyle
x\cdot_{\mathfrak{g}}f_{1}(y)-f_{1}(x)\cdot_{\mathfrak{g}}y-f_{1}(x\cdot_{\mathfrak{g}}y),$
(50) $\displaystyle\varrho_{\mathfrak{h}}(u)(f_{2})(v)$ $\displaystyle=$
$\displaystyle
u\cdot_{\mathfrak{h}}f_{2}(v)-f_{2}(u)\cdot_{\mathfrak{h}}v-f_{2}(u\cdot_{\mathfrak{h}}v),$
(51) $\displaystyle\varrho_{\phi}(x)(f_{3})(y)$ $\displaystyle=$
$\displaystyle\rho_{\phi}(x)f_{3}(y)-\mu_{\phi}(y)f_{3}(x)-f_{3}(x\cdot_{\mathfrak{g}}y),$

where $f_{1}\in\mathrm{Hom}(\mathfrak{g},\mathfrak{g})$,
$f_{2}\in\mathrm{Hom}(\mathfrak{h},\mathfrak{h})$,
$f_{3}\in\mathrm{Hom}(\mathfrak{g},\mathfrak{h})$ and
$(\rho_{\phi},\mu_{\phi})$ is given by (15). Then $\varrho_{\mathfrak{g}}$,
$\varrho_{\mathfrak{h}}$ and $\varrho_{\phi}$ are presentations of the
corresponding sub-adjacent Lie algebras.

In the following, we generalize Theorem 4.1 to the case of pre-Lie-morphism
triples. Define the set of $k$-cochains by

${\mathcal{C}}_{\mathrm{CE}}^{k}(\phi,\phi)=C_{\mathrm{CE}}^{k+1}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{g}))\oplus
C_{\mathrm{CE}}^{k+1}(\mathfrak{h},\mathrm{Hom}(\mathfrak{h},\mathfrak{h}))\oplus
C_{\mathrm{CE}}^{k}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{h})),k\geq
0,$

in which we set
$C_{\mathrm{CE}}^{0}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{h}))=0$.

The coboundary operator
$\delta_{\mathrm{CE}}:{\mathcal{C}}_{\mathrm{CE}}^{k}(\phi,\phi)\to{\mathcal{C}}_{\mathrm{CE}}^{k+1}(\phi,\phi)$
is defined by

$\delta_{\mathrm{CE}}(f_{1},f_{2},f_{3})=(\partial_{\mathfrak{g}}f_{1},\partial_{\mathfrak{h}}f_{2},\partial_{\phi}f_{3}+(-1)^{k}(\phi\circ
f_{1}-\phi^{*}\circ f_{2})),$

for all $f_{1}\in
C_{\mathrm{CE}}^{k+1}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{g})),f_{2}\in
C_{\mathrm{CE}}^{k+1}(\mathfrak{h},\mathrm{Hom}(\mathfrak{h},\mathfrak{h})),f_{3}\in
C_{\mathrm{CE}}^{k}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{h}))$,
where $\partial_{\mathfrak{g}}$, $\partial_{\mathfrak{h}}$ and
$\partial_{\phi}$ are Chevalley-Eilenberg coboundary operators of the sub-
adjacent Lie algebras $\mathfrak{g}^{c}$, $\mathfrak{h}^{c}$ and
$\mathfrak{g}^{c}$ associated to representations $\varrho_{\mathfrak{g}}$,
$\varrho_{\mathfrak{h}}$ and $\varrho_{\phi}$, respectively. Here $\phi\circ
f_{1}\in
C_{\mathrm{CE}}^{k+1}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{h}))$
and $\phi^{*}\circ f_{2}\in
C_{\mathrm{CE}}^{k+1}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{h}))$
are defined by

$\displaystyle(\phi\circ f_{1})(x_{1},\cdots,x_{k+1})(y)$ $\displaystyle=$
$\displaystyle\phi\big{(}f_{1}(x_{1},\cdots,x_{k+1})(y)\big{)},$
$\displaystyle(\phi^{*}\circ f_{2})(x_{1},\cdots,x_{k+1})(y)$ $\displaystyle=$
$\displaystyle f_{2}(\phi(x_{1}),\cdots,\phi(x_{k+1}))(\phi(y)),$

where $x_{1},\cdots,x_{k+1},y\in\mathfrak{g}$.

######  Proposition 4.2.

With the above notations, we have
$\delta_{\mathrm{CE}}\circ\delta_{\mathrm{CE}}=0$. Hence
$({\mathcal{C}}_{\mathrm{CE}}^{*}(\phi,\phi)=\otimes_{k\geq
1}C^{n}(\phi,\phi),\delta_{\mathrm{CE}})$ is a cochain complex. We denote the
corresponding $k$-th cohomology group by $H_{\mathrm{CE}}^{k}(\phi,\phi)$.

###### Proof.

For $f_{1}\in
C_{\mathrm{CE}}^{k}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{g}))$, we
have

$\displaystyle\partial_{\phi}(\phi\circ f_{1})(x_{1},\cdots,x_{k+1})(y)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k+1}(-1)^{i+1}\phi(x_{i})\cdot_{\mathfrak{h}}\phi(f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1})(y))$
$\displaystyle\sum_{i=1}^{k+1}(-1)^{i+1}\phi(f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1})(x_{i}))\cdot_{\mathfrak{h}}\phi(y)$
$\displaystyle-\sum_{i=1}^{k+1}(-1)^{i+1}\phi(f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1})(x_{i}\cdot_{\mathfrak{g}}y))$
$\displaystyle+\sum_{1\leq
i<j\leq{k+1}}(-1)^{i+j}\phi(f_{1}([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\cdots,\hat{x_{i}},\cdots,\hat{x_{j}},\cdots,x_{k+1})(y))$
$\displaystyle=$
$\displaystyle\phi(\sum_{i=1}^{k+1}(-1)^{i+1}x_{i}\cdot_{\mathfrak{g}}f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1})(y))$
$\displaystyle-\phi(\sum_{i=1}^{k+1}(-1)^{i+1}f_{1}(x_{1},\cdots,\hat{x_{i}},\cdots,x_{k+1})(x_{i})\cdot_{\mathfrak{g}}y)$
$\displaystyle-\phi(f_{1}([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\cdots,\hat{x_{i}},\cdots,\hat{x_{j}},\cdots,x_{k+1})(x_{i}\cdot_{\mathfrak{g}}y))$
$\displaystyle+\phi(\sum_{1\leq
i<j\leq{k+1}}(-1)^{i+j}f_{1}([x_{i},x_{j}]_{\mathfrak{g}},x_{1},\cdots,\hat{x_{i}},\cdots,\hat{x_{j}},\cdots,x_{k+1})(y))$
$\displaystyle=$
$\displaystyle\phi(\partial_{\mathfrak{g}}(f_{1})(x_{1},\cdots,x_{k+1})(y)),$
$\displaystyle=$
$\displaystyle(\phi\circ\partial_{\mathfrak{g}}(f_{1}))(x_{1},\cdots,x_{k+1})(y)$

which implies that

(52) $\displaystyle\partial_{\phi}(\phi\circ
f_{1})=\phi\circ\partial_{\mathfrak{g}}(f_{1}).$

Similarly, for $f_{2}\in
C_{\mathrm{CE}}^{k+1}(\mathfrak{h},\mathrm{Hom}(\mathfrak{h},\mathfrak{h}))$,
we have

(53) $\displaystyle\partial_{\phi}(\phi^{*}\circ
f_{2})=\phi^{*}\circ\partial_{\mathfrak{h}}(f_{2}).$

Furthermore, by (52), (53) and the fact that
$\partial_{\mathfrak{g}}\circ\partial_{\mathfrak{g}}=0$ ,
$\partial_{\mathfrak{h}}\circ\partial_{\mathfrak{h}}=0$ and
$\partial_{\phi}\circ\partial_{\phi}=0$, we have

$\displaystyle\delta_{\mathrm{CE}}\circ\delta_{\mathrm{CE}}(f_{1},f_{2},f_{3})$
$\displaystyle=$
$\displaystyle\delta_{\mathrm{CE}}(\partial_{\mathfrak{g}}f_{1},\partial_{\mathfrak{h}}f_{2},\partial_{\phi}f_{3}+(-1)^{k}(\phi\circ
f_{1}-\phi^{*}\circ f_{2}))$ $\displaystyle=$
$\displaystyle(\partial_{\mathfrak{g}}\circ\partial_{\mathfrak{g}}f_{1},\partial_{\mathfrak{h}}\circ\partial_{\mathfrak{h}}f_{2},\partial_{\phi}\circ(\partial_{\phi}f_{3}+(-1)^{k}(\phi\circ
f_{1}-\phi^{*}\circ
f_{2}))+(-1)^{k+1}(\phi\circ\partial_{\mathfrak{g}}f_{1}-\phi^{*}\circ\partial_{\mathfrak{h}}f_{2}))$
$\displaystyle=$
$\displaystyle\Big{(}0,0,(-1)^{k}\big{(}\partial_{\phi}(\phi\circ
f_{1})-\phi\circ\partial_{\mathfrak{g}}(f_{1})+\phi^{*}\circ\partial_{\mathfrak{h}}(f_{2})-\partial_{\phi}(\phi^{*}\circ
f_{2})\big{)}\Big{)}=(0,0,0),$

which implies that $\delta_{\mathrm{CE}}\circ\delta_{\mathrm{CE}}=0$. ∎

######  Remark 4.3.

In [7], the author gives a cohomology of a Lie-morphism triple with
coefficients in a representation. Our cohomology for the sub-adjacent Lie-
morphism triple in this paper is not fit in the mentioned above cohomology
theory.

Furthermore, we define $\Phi:C^{k}_{\mathrm{CE}}(\phi,\phi)\to C_{\rm
preLie}^{k+1}(\phi,\phi)$ by

(54)
$\Phi(f_{1},f_{2},f_{3})=(\Phi_{1}(f_{1}),\Phi_{2}(f_{2}),\Phi_{3}(f_{3})),$

where
$\Phi_{1}:C_{\mathrm{CE}}^{k+1}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{g}))\to
C_{\rm preLie}^{k+2}(\mathfrak{g},\mathfrak{g})$,
$\Phi_{2}:C_{\mathrm{CE}}^{k+1}(\mathfrak{h},\mathrm{Hom}(\mathfrak{h},\mathfrak{h}))\to
C_{\rm preLie}^{k+2}(\mathfrak{h},\mathfrak{h})$ and
$\Phi_{3}:C_{\mathrm{CE}}^{k}(\mathfrak{g},\mathrm{Hom}(\mathfrak{g},\mathfrak{h}))\to
C_{\rm preLie}^{k+1}(\mathfrak{g},\mathfrak{h})$ are given by

$\displaystyle\Phi_{1}(f_{1})(x_{1},\cdots,x_{k+1},y)$ $\displaystyle=$
$\displaystyle f_{1}(x_{1},\cdots,x_{k+1})(y),$
$\displaystyle\Phi_{2}(f_{2})(u_{1},\cdots,u_{k+1},v)$ $\displaystyle=$
$\displaystyle f_{2}(u_{1},\cdots,u_{k+1})(v),$
$\displaystyle\Phi_{3}(f_{3})(x_{1},\cdots,x_{k},y)$ $\displaystyle=$
$\displaystyle f_{3}(x_{1},\cdots,x_{k})(y)$

for all $x_{1},\cdots,x_{k+1},y\in\mathfrak{g}$,
$u_{1},\cdots,u_{k+1},v\in\mathfrak{h}$.

######  Theorem 4.4.

The map $\Phi$ is an isomorphic cochain map from the cochain complex
$(C^{*}_{\mathrm{CE}}(\phi,\phi),\delta_{\mathrm{CE}})$ to the cochain complex
$(C_{\rm preLie}^{*}(\phi,\phi),\delta_{\rm preLie})$.

Moreover, $\Phi$ induces an isomorphism between the corresponding cohomology
groups:

(55) $H^{k}_{\rm preLie}(\phi,\phi)\cong H^{k-1}_{\rm CE}(\phi,\phi),\quad
k=1,2,\cdots.$

###### Proof.

It is obvious that $\Phi$ is an isomorphic map. In the following, we need to
show that $\delta_{\mathrm{CE}}\circ\Phi=\Phi\circ\delta_{\rm preLie}$. By a
direct calculation, we have

$\displaystyle\delta_{\mathrm{CE}}\circ\Phi(f_{1},f_{2},f_{3})$
$\displaystyle=$
$\displaystyle(\partial_{\mathfrak{g}}\Phi_{1}(f_{1}),\partial_{\mathfrak{h}}\Phi_{2}(f_{2}),\partial_{\phi}\Phi_{3}(f_{3})+(-1)^{k}(\phi\circ\Phi_{1}(f_{1})-\phi^{*}\circ\Phi_{2}(f_{2})));$
$\displaystyle\Phi\circ\delta_{\rm preLie}(f_{1},f_{2},f_{3})$
$\displaystyle=$
$\displaystyle(\Phi_{1}(\mathrm{d}_{\mathfrak{g}}f_{1}),\Phi_{2}(\mathrm{d}_{\mathfrak{h}}f_{2}),\Phi_{3}(\mathrm{d}_{\phi}f_{3})+(-1)^{k}\Phi_{3}(\phi\circ
f_{1}-\phi^{*}\circ f_{2})).$

By Theorem 4.1, we have

$\displaystyle\partial_{\mathfrak{g}}\Phi_{1}(f_{1})=\Phi_{1}(\mathrm{d}_{\mathfrak{g}}f_{1}),\quad\partial_{\mathfrak{h}}\Phi_{2}(f_{2})=\Phi_{2}(\mathrm{d}_{\mathfrak{h}}f_{2}),\quad\partial_{\phi}\Phi_{3}(f_{3})=\Phi_{3}(\mathrm{d}_{\phi}f_{3}).$

Thus $\delta_{\mathrm{CE}}\circ\Phi=\Phi\circ\delta_{\rm preLie}$ holds if and
only if

$\phi\circ\Phi_{1}(f_{1})-\phi^{*}\circ\Phi_{2}(f_{2})=\Phi_{3}(\phi\circ
f_{1}-\phi^{*}\circ f_{2}).$

By a direct calculation, we have

$\displaystyle\phi\circ\Phi_{1}(f_{1})(x_{1},\cdots,x_{k+1},y)$
$\displaystyle=$ $\displaystyle\phi(f_{1}(x_{1},\cdots,x_{k+1})(y)),$
$\displaystyle=$ $\displaystyle(\phi\circ f_{1})(x_{1},\cdots,x_{k+1})(y)$
$\displaystyle=$ $\displaystyle\Phi_{3}(\phi\circ
f_{1})(x_{1},\cdots,x_{k+1},y),$

which implies that

$\phi\circ\Phi_{1}(f_{1})=\Phi_{3}(\phi\circ f_{1}).$

Similarly, we also have

$\phi^{*}\circ\Phi_{2}(f_{2})=\Phi_{3}(\phi^{*}\circ f_{2}).$

Thus $\delta_{\mathrm{CE}}\circ\Phi=\Phi\circ\delta_{\rm preLie}$ holds, i.e.
the map $\Phi$ is an isomorphic cochain map from the cochain complex
$(C^{*}_{\mathrm{CE}}(\phi,\phi),\delta_{\mathrm{CE}})$ to the cochain complex
$(C_{\rm preLie}^{*}(\phi,\phi),\delta_{\rm preLie})$.

The second conclusion follows directly. ∎

Acknowledgements.  This research is supported by NSFC (11901501).

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