# Equivariant formal deformations of Hom-pre-Lie algebras

Shuangjian Guo1, Ripan Saha2111 Corresponding author<EMAIL_ADDRESS>
1\. School of Mathematics and Statistics, Guizhou University of Finance and
Economics
Guiyang 550025, P. R. of China
2\. Department of Mathematics, Raiganj University
Raiganj, 733134, West Bengal, India

ABSTRACT

In this paper, we define a new cohomology theory for multiplicative Hom-pre-
Lie algebras which controls deformations of Hom-pre-Lie algebra structure.
This new cohomology is a natural one considering the structure map $\alpha$.
We show this new cohomology is deformation cohomology of the Hom-pre-Lie
algebra. We also develop cohomology and the associated deformation theory for
Hom-pre-Lie algebras in the equivariant context.

Key words: Group action, Hom-pre-Lie algebra, equivariant cohomology, formal
deformation, rigidity.

2020 MSC: 17A30, 17B56, 17B61, 17D99.

## Introduction

The notion of pre-Lie algebra (also known as left-symmetric algebra) was
introduced by Gerstenhaber [6] while studying the algebraic deformation theory
of associative algebras. The pre-Lie algebras have a close relationship with
Lie algebras. For example, a pre-Lie algebra gives rise to a Lie algebra via
the commutator bracket, which is known as the sub-adjacent Lie algebra. Pre-
Lie algebras have appeared in many areas of mathematics, for example, vertex
algebras, quantum field theories, complex and symplectic structures on Lie
groups and Lie algebras, etc., see [1, 4, 5, 10], and references therein.

The notion of Hom-pre-Lie algebras is a twisted analog of pre-Lie algebras,
where the pre-Lie algebra identity is twisted by a self linear map, called the
structure map. The study of Hom-version of type of algebras is an interesting
and active area of research these days and found some applications in
mathematics as well as in physics. For the various type of Hom-algebras, see
[12, 13, 14, 15].

Gertenhaber [6, 7] initiated the study of algebraic deformation theory for
associative algebra. Later, following his ideas, deformation theory for
various types of algebras has been studied, see [2, 17, 18, 11, 19, 20]. Hurle
and Makhlouf studied cohomology and deformation theory for Hom-associative and
Hom-Lie algebras in [8, 9]. In [16], Mukherjee and Saha studied cohomology and
deformation theory for Hom-Leibniz algebras.

In this paper, we develop a new cohomology theory for Hom-pre-Lie algebras, we
call this new cohomology by $\alpha$-type cohomology of Hom-pre-Lie algebras.
In [11], the authors defined cohomology and a one-parameter formal deformation
for Hom-pre-Lie algebras without considering the structure map $\alpha$. In
[11], they have only deformed the multiplication map, and ignored to deform
the structure map. Thus, the study of cohomology and deformation in [11] is
incomplete Thus, the study of deformation theory is not complete in their
work. This is one of the motivation of writting this paper. In this paper, we
deform both the multiplication operation and the structure map. We also
develop the corresponding deformation cohomology of Hom-pre-Lie algebras. In
the next part, we develop equivariant cohomology for Hom-pre-Lie algebras
equipped with a finite group action and also introduce a one-parameter formal
deformation theory for Hom-pre-Lie algebras in the equivariant contexts. To
develop the equivariant cohomology, we closely follow Bredon’s [3] approach
for equivariant cohomology of topological $G$-spaces. Similar to the non-
equivariant case we show that this equivariant cohomology is deformation
cohomology of the equivariant deformation of Hom-pre-Lie algebras.

This paper is organized as follows: In Section 1, we have discussed the
definition, examples, representation, and a known cohomology of Hom-pre-Lie
algebras which we shall use throughout the paper. In Section 2, we develop a
new cohomology for (multiplicative) Hom-pre-Lie algebras, we call this by
$\alpha$-type cohomology of Hom-pre-lie algebras. In Section 3, we introduce a
one-parameter formal deformation theory of Hom-pre-Lie algebras considering
the structure map $\alpha$. We show that infinitesimal of the deformation is a
$2$-cocycle of the $\alpha$-type cohomology. We also discuss obstructions and
rigidity of Hom-pre-Lie algebra deformation and show that how they are related
to our new cohomology. In Section 4, we develop the equivariant cohomology of
Hom-pre-Lie algebras equipped with an action of a finite group. In Section 5,
we introduce a one-parameter formal deformation theory for Hom-pre-Lie
algberas in the equivariant contexts. We show some results for equivariant
deformation of Hom-pre-Lie algebras analoguous to the non-equivariant case.

##  1 Preliminaries

In this section, we recall the definition, representation, and cohomology of
Hom-pre-Lie algebras following [11, 21]. Throughout the paper, $\mathbb{K}$
denotes a field of characteristics zero.

######  Definition 1.1.

A Hom-Lie algebra is a triple $(g,[~{},~{}],\alpha)$ consisting of a linear
space $g$, a skew- symmetric bilinear map $[~{},~{}]:\wedge^{2}g\rightarrow g$
and an algebra morphism $\alpha:g\rightarrow g$, satisfying:

$\displaystyle[\alpha(x),[y,z]]+[\alpha(y),[z,x]]+[\alpha(z),[x,y]]=0,~{}\text{for
all}~{}x,y,z\in g.$ (1.1)

######  Definition 1.2.

A Hom-pre-Lie algebra $(A,\cdot,\alpha)$ is a vector space $A$ equipped with a
bilinear product $\cdot:A\times A\rightarrow A,~{}\text{and a linear
map}~{}\alpha:A\to A$, satisfying

  1. 1.

($a\cdot b)\cdot\alpha(c)-\alpha(a)\cdot(b\cdot c)=(b\cdot
a)\cdot\alpha(c)-\alpha(b)\cdot(a\cdot c)$, for all $a,b,c\in A$.

  2. 2.

$\alpha(x\cdot y)=\alpha(x)\cdot\alpha(y)$, for all $a,b\in A.$

A Hom-pre-Lie algebra $(A,\cdot,\alpha)$ is called regular if $\alpha$ is
invertible.

######  Definition 1.3.

Let $(A_{1},\cdot_{1},\alpha_{1})$ and $(A_{2},\cdot_{2},\alpha_{2})$ be two
Hom-pre-Lie algebras. A morphism from $A_{1}$ to $A_{2}$ is a
$\mathbb{K}$-linear map $\phi:A_{1}\to A_{2}$ such that

$\phi(a\cdot_{1}b)=\phi(a)\cdot_{2}\phi(b),~{}~{}\phi\circ\alpha_{1}=\alpha_{2}\circ\phi,~{}\text{for
all}~{}a,b\in A_{1}.$

######  Example 1.4.

Let $A$ be a two dimensional vector space with basis $\\{a_{1},a_{2}\\}$.
Define a multiplication on $A$ by

$a_{1}\cdot a_{1}=0,~{}a_{1}\cdot a_{2}=0,~{}a_{2}\cdot
a_{1}=a_{1},~{}a_{2}\cdot a_{2}=a_{1}+a_{2}.$

Let $\alpha$ be a $\mathbb{K}$-linear map on $A$ defined as
$\alpha(a_{1})=a_{1},~{}\alpha(a_{2})=a_{1}+a_{2}$. Then $(A,\cdot,\alpha)$ is
a regular Hom-pre-Lie algebra.

######  Definition 1.5.

A representation of a Hom-Lie algebra $(g,[\cdot,\cdot],\alpha)$ on a vector
space $V$ with respect to $\beta\in gl(V)$ is a linear map $\rho:g\rightarrow
gl(V)$, such that for all $x,y\in g$, the following equalities are satisfied:

$\displaystyle\rho(\alpha(x))\circ\beta=\beta\circ\rho(x),$ (1.2)
$\displaystyle\rho([x,y])\circ\beta=\rho(\alpha(x))\circ\rho(y)-\rho(\alpha(y))\circ\rho(x).$
(1.3)

Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra. We always assume that it is
regular, i.e. $\alpha$ is invertible. The commutator $[x,y]_{C}=x\cdot
y-y\cdot x$ gives a Hom-Lie algebra $(A,[\cdot,\cdot]_{C},\alpha)$, which is
denoted by $A^{C}$ and called the sub-adjacent Hom-Lie algebra of
$(A,\cdot,\alpha)$.

######  Definition 1.6.

A representation of a Hom-pre-Lie algebra $(A,\cdot,\alpha)$ on a vector space
$V$ with respect to $\beta\in gl(V)$ consists of a pair $(\rho,\mu)$, where
$\rho:A\rightarrow gl(V)$ is a representation of the sub-adjacent Hom-Lie
algebra $A^{C}$ on $V$ with respect to $\beta\in gl(V)$, and $\mu:A\rightarrow
gl(V)$ is a linear map, for all $a,b\in A$, satisfying:

$\displaystyle\beta\circ\mu(a)=\mu(\alpha(a))\circ\beta,$ (1.4)
$\displaystyle\mu(\alpha(b))\circ\mu(a)-\mu(a\cdot
b)\circ\beta=\mu(\alpha(b))\circ\rho(a)-\rho(\alpha(a))\circ\mu(b).$ (1.5)

If $V$ is a presentation of a Hom-pre-Lie algebra $A$ as defined above, then
we denote it as $(V,\rho,\mu,\beta).$

Let $(A,\cdot,\alpha)$ be a Hom-pre-Lie algebra. Then $(A,\alpha)$ is a
representation of itself via the maps $L,R:A\to\mathfrak{gl}(A)$ defined as
follows:

$L_{a}(b)=a\cdot b,~{}~{}~{}R_{a}(b)=b\cdot a,~{}\text{for all}~{}a,b\in A.$

It is easy to check that $(A,L,R,\alpha)$ is a representation of
$(A,\cdot,\alpha)$. This representation of $A$ is called the regular
representation. In this paper, we will use regular representation only.

Let $(V,\rho,\mu,\beta)$ be a representation of a Hom-pre-Lie algebra
$(A,\cdot,\alpha)$. The cohomology of $A$ with coefficients in $V$ is the
cohomology of the cochain complex $\\{C^{*}(A,V),\partial\\}$, where
$C^{n}(A,V)=\\{\phi\in\text{Hom}(A^{\otimes
n},V)\mid\beta\circ\phi=\phi\circ\alpha^{\otimes n}\\},(n\geq 0)$ and the
coboundary operator $\partial:C^{n}(A,V)\rightarrow C^{n+1}(A,V)$ given by

$\displaystyle(\partial f)(a_{1},\ldots,a_{n+1})$ $\displaystyle=$
$\displaystyle\sum^{n}_{i=1}(-1)^{i+1}\rho(\alpha^{n-1}(x_{i}))f(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n+1})$
$\displaystyle+\sum^{n}_{i=1}(-1)^{i+1}\mu(\alpha^{n-1}(a_{n+1}))f(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n},a_{i})$
$\displaystyle-\sum^{n}_{i=1}(-1)^{i+1}f(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\alpha(a_{n}),a_{i}\cdot
a_{n+1})$ $\displaystyle+\sum_{1\leq i<j\leq
n}(-1)^{i+j}f([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},\ldots,\alpha(a_{n+1})),$

for $a_{1},\ldots,a_{n+1}\in A$. The corresponding cohomology groups are
denoted by $H^{*}(A,V).$ We will see that this cohomology is not a deformation
cohomology for Hom-pre-Lie algebras.

##  2 $\alpha$-type cohomology for Hom-pre-Lie algebras

In this section, we define an $\alpha$-type cohomology for Hom-pre-Lie
algebras. Let $(V,\rho,\mu,\beta)$ be a representation of a Hom-pre-Lie
algebra $(A,\nu,\alpha)$.

Define $\nu:A\to A$ by $\nu(a,b)=a\cdot b$. Using this notation, we denote a
Hom-pre-Lie algebra $(A,\cdot,\alpha)$ as $(A,\nu,\alpha)$. Then the complex
for the cohomology of $A$ with values in $V$ is given by

$\displaystyle\widetilde{C^{n}}(A,V)=\widetilde{C^{n}_{\nu}}(A,V)\oplus\widetilde{C_{\alpha}}^{n-1}(A,V)=\text{Hom}(A^{\otimes
n},V)\oplus\text{Hom}(A^{\otimes n-1},V),~{}\text{for}~{}n\geq 1,$
$\displaystyle\widetilde{C^{n}}(A,V)=\\{0\\},~{}\text{for}~{}n\leq 0.$

Note that for convience, we take
$\widetilde{C^{1}_{\alpha}}(A,V)=\text{Hom}(A^{0},V)=\text{Hom}(\mathbb{K},V)=\\{0\\}$.
We write $(\varphi,\psi)$ with $\varphi\in\widetilde{C^{\bullet}_{\nu}}(A,V)$
and $\psi\in\widetilde{C^{\bullet}_{\alpha}}(A,V)$ for an element in
$\widetilde{C}^{\bullet}(A,V)$.

We define four maps, with domain and range given in the following diagram:

$\textstyle{\widetilde{C_{\nu}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{~{}~{}~{}\widetilde{\partial}_{\alpha\nu}}$$\scriptstyle{\widetilde{\partial}_{\nu\nu}}$$\textstyle{\widetilde{C_{\nu}}^{n+1}}$$\textstyle{\bigoplus}$$\textstyle{\bigoplus}$$\textstyle{\widetilde{C_{\alpha}^{n}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\partial}_{\nu\alpha}~{}}$$\scriptstyle{\widetilde{\partial}_{\alpha\alpha}}$$\textstyle{\widetilde{C_{\alpha}}^{n+1},}$
$\displaystyle(\widetilde{\partial}_{\nu\nu}\varphi)(a_{1},\ldots,a_{n+1})$
$\displaystyle=$
$\displaystyle\sum^{n}_{i=1}(-1)^{i+1}\rho(\alpha^{n-1}(a_{i}))\varphi(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n+1})$
$\displaystyle+\sum^{n}_{i=1}(-1)^{i+1}\mu(\alpha^{n-1}(a_{n+1}))\varphi(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n},a_{i})$
$\displaystyle-\sum^{n}_{i=1}(-1)^{i+1}\varphi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\alpha(a_{n}),a_{i}\cdot
a_{n+1})$ $\displaystyle+\sum_{1\leq i<j\leq
n}(-1)^{i+j}\varphi([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},...,\alpha(a_{n+1})),$
$\displaystyle(\widetilde{\partial}_{\alpha\alpha}\psi)(a_{1},\ldots,a_{n})$
$\displaystyle=$
$\displaystyle\sum^{n}_{i=1}(-1)^{i+1}\rho(\alpha^{n-1}(a_{i}))\psi(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n})$
$\displaystyle+\sum^{n-1}_{i=1}(-1)^{i+1}\mu(\alpha^{n-1}(a_{n}))\psi(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n-1},a_{i})$
$\displaystyle-\sum^{n-1}_{i=1}(-1)^{i+1}\psi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\alpha(a_{n-1}),a_{i}\cdot
a_{n})$ $\displaystyle+\sum_{1\leq i<j\leq
n-1}(-1)^{i+j}\psi([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},\ldots,\alpha(a_{n})),$
$\displaystyle(\widetilde{\partial}_{\nu\alpha}\varphi)(a_{1},\ldots,a_{n})$
$\displaystyle=$
$\displaystyle\beta(\varphi(a_{1},\ldots,a_{n}))-\varphi(\alpha(a_{1}),\ldots,\alpha(a_{n}))$
(2. 3)
$\displaystyle(\widetilde{\partial}_{\alpha\nu}\psi)(a_{1},\ldots,a_{n+1})$
$\displaystyle=$
$\displaystyle\sum^{n}_{i=1}(-1)^{i+1}\mu\circ\alpha^{n-2}(a_{i}\cdot
a_{n+1})\psi(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n})$
$\displaystyle-\sum_{1\leq i<j\leq
n}(-1)^{i+j}\rho([\alpha^{n-2}(a_{i}),\alpha^{n-2}(a_{j})]_{C})\psi(a_{1},\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},\ldots,a_{n+1}),$

where $a_{1},\ldots,a_{n+1}\in A$. The sign given by $(-1)^{\cdot}$ is always
determined by the permutation of the $x_{i\cdot}$. We have the following main
theorem.

######  Theorem 2.1.

Let $(V,\rho,\mu,\beta)$ be a representation of a Hom-pre-Lie algebra
$(A,\nu,\alpha)$ Further, let
$\widetilde{\partial}:\widetilde{C}^{n}(A,V)\rightarrow\widetilde{C}^{n+1}(A,V)$
be a map defined by
$\widetilde{\partial}(\varphi,\psi)=(\widetilde{\partial}_{\nu\nu}\varphi-\widetilde{\partial}_{\alpha\nu}\psi,\widetilde{\partial}_{\nu\alpha}\varphi-\widetilde{\partial}_{\alpha\alpha}\psi)$.
Then the pair $(\widetilde{C}^{\bullet}(A,V),\widetilde{\partial})$ is a
cohomology complex.

###### Proof.

In order to prove that $\widetilde{\partial}^{2}=0$, we will give it here to
some extent. Showing $\widetilde{\partial}^{2}=0$ is same as the showing
following equations:

$\displaystyle\widetilde{\partial}_{\nu\nu}\widetilde{\partial}_{\nu\nu}+\widetilde{\partial}_{\nu\nu}\widetilde{\partial}_{\alpha\nu}-\widetilde{\partial}_{\alpha\nu}\widetilde{\partial}_{\alpha\alpha}-\widetilde{\partial}_{\alpha\nu}\widetilde{\partial}_{\nu\alpha}=0,$
$\displaystyle-\widetilde{\partial}_{\alpha\alpha}\widetilde{\partial}_{\alpha\alpha}+\widetilde{\partial}_{\nu\alpha}\widetilde{\partial}_{\alpha\nu}-\widetilde{\partial}_{\alpha\alpha}\widetilde{\partial}_{\nu\alpha}+\widetilde{\partial}_{\nu\alpha}\widetilde{\partial}_{\nu\nu}=0.$

For this we verify the following equations

$\displaystyle\widetilde{\partial}_{\nu\nu}\widetilde{\partial}_{\nu\nu}=\widetilde{\partial}_{\alpha\nu}\widetilde{\partial}_{\nu\alpha},$
$\displaystyle\widetilde{\partial}_{\nu\nu}\widetilde{\partial}_{\alpha\nu}=\widetilde{\partial}_{\alpha\nu}\widetilde{\partial}_{\alpha\alpha},$
$\displaystyle\widetilde{\partial}_{\alpha\alpha}\widetilde{\partial}_{\alpha\alpha}=\widetilde{\partial}_{\nu\alpha}\widetilde{\partial}_{\alpha\nu},$
$\displaystyle\widetilde{\partial}_{\nu\alpha}\widetilde{\partial}_{\nu\nu}=\widetilde{\partial}_{\alpha\alpha}\widetilde{\partial}_{\nu\alpha}.$

We shall only verify the first equality as computations of such verifications
is very lengthy. The verification of the other three equalities can be done in
a similar manner.

$\displaystyle\widetilde{\partial}_{\nu\nu}\widetilde{\partial}_{\nu\nu}\varphi(a_{1},\ldots,a_{n+2})$
$\displaystyle=$
$\displaystyle\sum^{n+1}_{i=1}(-1)^{i+1}\rho(\alpha^{n}(a_{i}))\widetilde{\partial}_{\nu\nu}\varphi(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n+2})$
$\displaystyle+\sum^{n+1}_{i=1}(-1)^{i+1}\mu(\alpha^{n}(a_{n+2}))\widetilde{\partial}_{\nu\nu}\varphi(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n+1},a_{i})$
$\displaystyle-\sum^{n+1}_{i=1}(-1)^{i+1}\widetilde{\partial}_{\nu\nu}\varphi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\alpha(a_{n+1}),a_{i}\cdot
a_{n+2})$ $\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}\widetilde{\partial}_{\nu\nu}\varphi([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},...,\alpha(a_{n+2})),$
$\displaystyle=$
$\displaystyle\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j<i\end{subarray}}(-1)^{i+j}\rho(\alpha^{n}(a_{i}))\rho(\alpha^{n-1}(a_{j}))\varphi(a_{1},\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots
a_{n+2})$ $\displaystyle+\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j>i\end{subarray}}(-1)^{i+j-1}\rho(\alpha^{n}(a_{i}))\rho(\alpha^{n-1}(a_{j}))\varphi(a_{1},\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots
a_{n+2})$ $\displaystyle+\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j<i\end{subarray}}(-1)^{i+j}\rho(\alpha^{n}(a_{i}))\mu(\alpha^{n-1}(a_{n+2}))\varphi(a_{1},\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots
a_{n+1},a_{j})$
$\displaystyle+\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j>i\end{subarray}}(-1)^{i+j-1}\rho(\alpha^{n}(a_{i}))\rho(\alpha^{n-1}(a_{j}))\varphi(a_{1},\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots
a_{n+1},a_{j})$
$\displaystyle-\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j<i\end{subarray}}(-1)^{i+j}\rho(\alpha^{n}(a_{i}))\varphi(\alpha(a_{1}),\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots\alpha(a_{n+1}),a_{j}\cdot
a_{n+2})$ $\displaystyle-\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j>i\end{subarray}}(-1)^{i+j-1}\rho(\alpha^{n}(a_{i}))\varphi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots\alpha(a_{n+1}),a_{j}\cdot
a_{n+2})$
$\displaystyle-\sum^{n+1}_{i=1}\sum_{j<k<i}(-1)^{i+j+k}\rho(\alpha^{n}(a_{i}))\varphi([a_{j},a_{k}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{j}},\ldots,\widehat{a_{k}},\ldots,\widehat{a_{i}}\ldots,\alpha(a_{n+2}))$
$\displaystyle+\sum^{n+1}_{i=1}\sum_{j<i<k}(-1)^{i+j+k}\rho(\alpha^{n}(a_{i}))\varphi([a_{j},a_{k}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}},\ldots,\widehat{a_{k}}\ldots,\alpha(a_{n+2}))$
$\displaystyle-\sum^{n+1}_{i=1}\sum_{i<j<k}(-1)^{i+j+k}\rho(\alpha^{n}(a_{i}))\varphi([a_{j},a_{k}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},\ldots,\widehat{a_{k}}\ldots,\alpha(a_{n+2}))$
$\displaystyle+\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j<i\end{subarray}}(-1)^{i+j}\mu(\alpha^{n}(a_{n+2}))\rho(\alpha^{n-1}(a_{j}))\varphi(a_{1},\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots
a_{n+1},a_{i})$ (2. 13)
$\displaystyle+\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j>i\end{subarray}}(-1)^{i+j-1}\mu(\alpha^{n}(a_{n+2}))\rho(\alpha^{n-1}(a_{j}))\varphi(a_{1},\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots
a_{n+1},a_{i})~{}~{}~{}~{}$ (2. 14)
$\displaystyle+\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j<i\end{subarray}}(-1)^{i+j}\mu(\alpha^{n}(a_{n+2}))\mu(\alpha^{n-1}(a_{i}))\varphi(a_{1},\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots
a_{n+1},a_{j})$ (2. 15)
$\displaystyle+\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j>i\end{subarray}}(-1)^{i+j-1}\mu(\alpha^{n}(a_{n+2}))\mu(\alpha^{n-1}(a_{i}))\varphi(a_{1},\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots
a_{n+1},a_{j})~{}~{}~{}~{}$ (2. 16)
$\displaystyle-\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j<i\end{subarray}}(-1)^{i+j}\mu(\alpha^{n}(a_{n+2}))\varphi(\alpha(a_{1}),\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots\alpha(a_{n+1}),a_{j}\cdot
a_{i})$ (2. 17)
$\displaystyle-\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j>i\end{subarray}}(-1)^{i+j-1}\mu(\alpha^{n}(a_{n+2}))\varphi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots\alpha(a_{n+1}),a_{j}\cdot
a_{i})$ (2. 18)
$\displaystyle-\sum^{n+1}_{i=1}\sum_{j<k<i}(-1)^{i+j+k}\mu(\alpha^{n}(a_{n+2}))\varphi([a_{j},a_{k}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{j}},\ldots,\widehat{a_{k}},\ldots,\widehat{a_{i}}\ldots,\alpha(a_{n+1}),\alpha(a_{i}))$
(2. 19)
$\displaystyle+\sum^{n+1}_{i=1}\sum_{j<i<k}(-1)^{i+j+k}\mu(\alpha^{n}(a_{n+2}))\varphi([a_{j},a_{k}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}},\ldots,\widehat{a_{k}}\ldots,\alpha(a_{n+1}),\alpha(a_{i}))$
(2. 20)
$\displaystyle-\sum^{n+1}_{i=1}\sum_{i<j<k}(-1)^{i+j+k}\mu(\alpha^{n}(a_{n+2}))\varphi([a_{j},a_{k}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},\ldots,\widehat{a_{k}}\ldots,\alpha(a_{n+1}),\alpha(a_{i}))$
(2. 21) $\displaystyle-\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j<i\end{subarray}}(-1)^{i+j}\rho(\alpha^{n}(a_{j}))\varphi(\alpha(a_{1}),\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots\alpha(a_{n+1}),a_{i}\cdot
a_{n+2})$ (2. 22)
$\displaystyle-\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j>i\end{subarray}}(-1)^{i+j-1}\rho(\alpha^{n}(a_{j}))\varphi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots\alpha(a_{n+1}),a_{i}\cdot
a_{n+2})$ (2. 23)
$\displaystyle-\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j<i\end{subarray}}(-1)^{i+j}\mu(\alpha^{n-1}(a_{i})\cdot\alpha^{n-1}(a_{n+2}))\varphi(\alpha(a_{1}),\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots\alpha(a_{n+1}),\alpha(a_{j}))$
(2. 24) $\displaystyle-\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j>i\end{subarray}}(-1)^{i+j}\mu(\alpha^{n-1}(a_{i})\cdot\alpha^{n-1}(a_{n+2}))\varphi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots\alpha(a_{n+1}),\alpha(a_{j}))$
(2. 25) $\displaystyle+\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j<i\end{subarray}}(-1)^{i+j}\varphi(\alpha^{2}(a_{1}),\ldots,\widehat{a_{j}},\ldots,\widehat{a_{i}}\ldots\alpha^{2}(a_{n+1}),\alpha(x_{j})\cdot(a_{i}\cdot
a_{n+2}))$ (2. 26)
$\displaystyle+\sum^{n+1}_{i=1}\sum^{n}_{\begin{subarray}{c}j=1\\\
j>i\end{subarray}}(-1)^{i+j}\varphi(\alpha^{2}(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots\alpha^{2}(a_{n+1}),\alpha(x_{j})\cdot(a_{i}\cdot
a_{n+2}))$ (2. 27)
$\displaystyle+\sum^{n+1}_{i=1}\sum_{j<k<i}(-1)^{i+j+k}\varphi([\alpha(a_{j}),\alpha(a_{k})]_{C},\alpha^{2}(a_{1}),\ldots,\widehat{a_{j}},\ldots,$
$\displaystyle\widehat{a_{k}},\ldots,\widehat{a_{i}}\ldots,\alpha^{2}(a_{n+1}),\alpha(a_{i})\cdot\alpha(a_{n+2}))$
(2. 28)
$\displaystyle-\sum^{n+1}_{i=1}\sum_{j<i<k}(-1)^{i+j+k}\varphi([\alpha(a_{j}),\alpha(a_{k})]_{C},\alpha^{2}(a_{1}),\ldots,\widehat{a_{j}},\ldots,$
$\displaystyle\widehat{a_{i}},\ldots,\widehat{a_{k}}\ldots,\alpha^{2}(a_{n+1}),\alpha(a_{i})\cdot\alpha(a_{n+2}))$
(2. 29)
$\displaystyle+\sum^{n+1}_{i=1}\sum_{i<j<k}(-1)^{i+j+k}\varphi([\alpha(a_{j}),\alpha(a_{k})]_{C},\alpha^{2}(a_{1}),\ldots,\widehat{a_{i}},\ldots,$
$\displaystyle\widehat{a_{j}},\ldots,\widehat{a_{k}}\ldots,\alpha^{2}(a_{n+1}),\alpha(a_{i})\cdot\alpha(a_{n+2}))$
(2. 30) $\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}\rho([\alpha^{n-1}(a_{i}),\alpha^{n-1}(a_{j})]_{C})\varphi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots\alpha(a_{n+1}),\alpha(a_{n+2}))~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(2. 31) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{k<i}(-1)^{i+j+k+1}\rho(\alpha^{n}(a_{k}))\varphi([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{k}}\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots,\alpha(a_{n+2}))$
(2. 32) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{i<k<j}(-1)^{i+j+k}\rho(\alpha^{n}(a_{k}))\varphi([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{k}},\ldots,\widehat{a_{j}}\ldots,\alpha(a_{n+2}))$
(2. 33) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{j<k}(-1)^{i+j+k+1}\rho(\alpha^{n}(a_{k}))\varphi([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{j}},\ldots,\widehat{a_{k}}\ldots,\alpha(a_{n+2}))$
(2. 34) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}(-1)^{i+j}\mu(\alpha^{n}(a_{n+2}))\varphi(\alpha(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{j}},\ldots,\alpha(a_{n+1}),[a_{i},a_{j}]_{C})$
(2. 35) $\displaystyle-\sum^{n+1}_{1\leq i\leq
j}\sum_{k<i}(-1)^{i+j+k}\mu(\alpha^{n}(a_{n+2}))\varphi([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{k}}\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots,\alpha(a_{k}))$
(2. 36) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{i<k<j}(-1)^{i+j+k}\mu(\alpha^{n}(a_{n+2}))\varphi([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{k}},\ldots,\widehat{a_{j}}\ldots,\alpha(a_{k}))$
(2. 37) $\displaystyle-\sum^{n+1}_{1\leq i\leq
j}\sum_{j<k}(-1)^{i+j+k+1}\mu(\alpha^{n}(a_{n+2}))\varphi([a_{i},a_{j}]_{C},\alpha(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{j}},\ldots,\widehat{a_{k}}\ldots,\alpha(a_{k}))$
(2. 38) $\displaystyle-\sum^{n+1}_{1\leq i\leq
j}(-1)^{i+j}\varphi(\alpha^{2}(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{j}},\ldots,\alpha^{2}(a_{n+1})[a_{i},a_{j}]_{C}\cdot\alpha(a_{n+2}))$
(2. 39) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{k<i}(-1)^{i+j+k}\varphi([\alpha(a_{i}),\alpha(a_{j})]_{C},\alpha^{2}(a_{1}),\ldots,\widehat{a_{k}}\ldots,\widehat{a_{i}},\ldots,$
$\displaystyle\widehat{a_{j}}\ldots,\alpha^{2}(a_{n+1}),\alpha(a_{k})\cdot\alpha(a_{n+2}))$
(2. 40) $\displaystyle-\sum^{n+1}_{1\leq i\leq
j}\sum_{i<k<j}(-1)^{i+j+k}\varphi([\alpha(a_{i}),\alpha(a_{j})]_{C},\alpha^{2}(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{k}},\ldots,$
$\displaystyle\widehat{a_{j}}\ldots,\alpha^{2}(a_{n+1}),\alpha(a_{k})\cdot\alpha(a_{n+2}))$
(2. 41) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{j<k}(-1)^{i+j+k}\varphi([\alpha(a_{i}),\alpha(a_{j})]_{C},\alpha^{2}(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{j}},\ldots,$
$\displaystyle\widehat{a_{k}}\ldots,\alpha^{2}(a_{n+1}),\alpha(a_{k})\cdot\alpha(a_{n+2}))$
(2. 42) $\displaystyle-\sum^{n+1}_{1\leq i\leq
j}\sum_{l<i}(-1)^{i+j+l}\varphi([[a_{i},a_{j}]_{C},\alpha(a_{l})]_{C},\alpha^{2}(a_{1}),\ldots,\widehat{a_{l}}\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots,\alpha^{2}(a_{n+2}))~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(2. 43) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{i<l<j}(-1)^{i+j+l}\varphi([[a_{i},a_{j}]_{C},\alpha(a_{l})]_{C},\alpha^{2}(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{l}},\ldots,\widehat{a_{j}}\ldots,\alpha^{2}(a_{n+2}))~{}~{}~{}$
(2. 44) $\displaystyle-\sum^{n+1}_{1\leq i\leq
j}\sum_{j<l}(-1)^{i+j+l}\varphi([[a_{i},a_{j}]_{C},\alpha(a_{l})]_{C},\alpha^{2}(a_{1}),\ldots,\widehat{a_{i}}\ldots,\widehat{a_{j}},\ldots,\widehat{a_{l}}\ldots,\alpha^{2}(a_{n+2}))~{}~{}~{}$
(2. 45) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{k<l<i}(-1)^{i+j+l+k}\varphi([\alpha(a_{k}),\alpha(a_{l})]_{C},[\alpha(a_{i}),\alpha(a_{j})]_{C},\alpha^{2}(a_{1}),\ldots,$
$\displaystyle\widehat{a_{k}}\ldots,\widehat{a_{l}},\ldots,\widehat{a_{i}}\ldots,\widehat{a_{j}}\ldots,\alpha^{2}(a_{n+2}))$
(2. 46) $\displaystyle-\sum^{n+1}_{1\leq i\leq
j}\sum_{k<i<l<j}(-1)^{i+j+l+k}\varphi([\alpha(a_{k}),\alpha(a_{l})]_{C},[\alpha(a_{i}),\alpha(a_{j})]_{C},\alpha^{2}(a_{1}),\ldots,$
$\displaystyle\widehat{a_{k}}\ldots,\widehat{a_{i}},\ldots,\widehat{a_{l}}\ldots,\widehat{a_{j}}\ldots,\alpha^{2}(a_{n+2}))$
(2. 47) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{k<i<j<l}(-1)^{i+j+l+k}\varphi([\alpha(a_{k}),\alpha(a_{l})]_{C},[\alpha(a_{i}),\alpha(a_{j})]_{C},\alpha^{2}(a_{1}),\ldots,$
$\displaystyle\widehat{a_{k}}\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}}\ldots,\widehat{a_{l}}\ldots,\alpha^{2}(a_{n+2}))$
(2. 48) $\displaystyle-\sum^{n+1}_{1\leq i\leq
j}\sum_{i<k<l<j}(-1)^{i+j+l+k}\varphi([\alpha(a_{k}),\alpha(a_{l})]_{C},[\alpha(a_{i}),\alpha(a_{j})]_{C},\alpha^{2}(a_{1}),\ldots,$
$\displaystyle\widehat{a_{i}}\ldots,\widehat{a_{k}},\ldots,\widehat{a_{l}}\ldots,\widehat{a_{j}}\ldots,\alpha^{2}(a_{n+2}))$
(2. 49) $\displaystyle+\sum^{n+1}_{1\leq i\leq
j}\sum_{i<k<l<j}(-1)^{i+j+l+k}\varphi([\alpha(a_{k}),\alpha(a_{l})]_{C},[\alpha(a_{i}),\alpha(a_{j})]_{C},\alpha^{2}(a_{1}),\ldots,$
$\displaystyle\widehat{a_{i}}\ldots,\widehat{a_{k}},\ldots,\widehat{a_{j}}\ldots,\widehat{a_{l}}\ldots,\alpha^{2}(a_{n+2}))$
(2. 50) $\displaystyle-\sum^{n+1}_{1\leq i\leq
j}\sum_{j<k<l}(-1)^{i+j+l+k}\varphi([\alpha(a_{k}),\alpha(a_{l})]_{C},[\alpha(a_{i}),\alpha(a_{j})]_{C},\alpha^{2}(a_{1}),\ldots,$
$\displaystyle\widehat{a_{i}}\ldots,\widehat{a_{j}},\ldots,\widehat{a_{k}}\ldots,\widehat{a_{l}}\ldots,\alpha^{2}(a_{n+2})).$
(2. 51)

It is clearly to see that

$\displaystyle(2.13)+(2.33)=(2.12)+(2.34)$ $\displaystyle=$
$\displaystyle(2.11)+(2.35)=(2.9)-(2.24)$ $\displaystyle=$
$\displaystyle(2.10)-(2.23)=(2.18)+(2.19)+(2.36)=0,$

and

$\displaystyle(2.20)+(2.39)=(2.21)+(2.38)$ $\displaystyle=$
$\displaystyle(2.22)+(2.37)=(2.29)-(2.43)$ $\displaystyle=$
$\displaystyle(2.30)-(2.42)=(2.31)-(2.41)=0.$

Using the definition of Hom-pre-Lie algebra, we have

$\displaystyle(2.27)+(2.28)-(2.40)=0.$

Using the Hom-Jacobi identity, we have (2.44)+(2.45)+(2.46) = 0. Finally, it
easy to see that (2.47)+$\cdots$+(2.52) = 0.

Furthermore, we have

$\displaystyle\widetilde{\partial}_{\alpha\nu}\widetilde{\partial}_{\nu\alpha}\varphi(a_{1},\cdots,a_{n+2})$
(2. 55) $\displaystyle=$
$\displaystyle\sum^{n+1}_{i=1}(-1)^{i+1}\mu(\alpha^{n-1}(a_{i}\cdot
a_{n+2}))\widetilde{\partial}_{\nu\alpha}\psi(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n+1})$
$\displaystyle-\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}\rho([\alpha^{n-1}(a_{i}),\alpha^{n-1}(a_{j})]_{C})\widetilde{\partial}_{\nu\alpha}\psi(a_{1},\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},\ldots,a_{n+2})$
$\displaystyle=$
$\displaystyle\sum^{n+1}_{i=1}(-1)^{i+1}\mu(\alpha^{n-1}(a_{i}\cdot
a_{n+2}))\beta\psi(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n+1})$
$\displaystyle-\sum^{n+1}_{i=1}(-1)^{i+1}\mu(\alpha^{n-1}(a_{i}\cdot
a_{n+2}))\psi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\alpha(a_{n+1}))$
$\displaystyle-\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}\rho([\alpha^{n-1}(a_{i}),\alpha^{n-1}(a_{j})]_{C})\beta\psi(a_{1},\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},\ldots,a_{n+2})$
$\displaystyle+\sum_{1\leq i<j\leq
n+1}(-1)^{i+j}\rho([\alpha^{n-1}(a_{i}),\alpha^{n-1}(a_{j})]_{C})\psi(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\widehat{a_{j}},\ldots,\alpha(a_{n+2})).~{}~{}~{}~{}~{}~{}~{}~{}~{}$

Thus, we have (2.5)+(2.6)+(2.32)-(2.55)-(2.56) = 0 and
(2.7)+(2.14)+(2.16)-(2.25) -(2.53)-(2.54) =
(2.8)+(2.15)+(2.17)-(2.26)-(2.53)-(2.54) = 0.

Thus,
$\widetilde{\partial_{\nu\nu}}\widetilde{\partial_{\nu\nu}}=\widetilde{\partial}_{\alpha\nu}\widetilde{\partial}_{\nu\alpha}$.
∎

In this paper, we will only consider cohomology of $A$ with coefficients over
itself and denote it as $\widetilde{H^{\ast}}(A,A)$.

######  Remark 2.2.

Observe that the $\alpha$-type cohomology for multiplicative Hom-pre-Lie
algebras generalizes the cohomology defined in the Section 1 (cf. 1). For this
onsider only those elements in $\widetilde{C^{n}}(A,A)$ where second summand
is zero, that is,

$\widetilde{C}_{\alpha}^{n}(A,A)=\\{0\\}.$

Therefore, we have elements of the form $(\phi,0)$. We define a subcomplex of
$\widetilde{C^{n}}(A,A)$ as follows:

$\displaystyle{C}_{\alpha}^{n}(A,A)$
$\displaystyle=\\{(\phi,0)\in\widetilde{C^{n}}(A,A)\mid\widetilde{\partial}_{\nu\alpha}\phi=0\\}$
$\displaystyle=\\{\phi\in\widetilde{C}_{\nu}^{n}(A,A)\mid\beta\circ\phi=\phi\circ\alpha^{\otimes
n}\\}.$

The map $\widetilde{\partial}_{\nu\nu}$ defines a diffential on this complex
and this complex is same as the complex defined in the Section 1. Thus,
$\alpha$-type cohomology generalizes the cohomology developed in Section 1.

##  3 Formal deformation theory of Hom-pre-Lie algebras

In this section, we show that the $\alpha$-type cohomology for Hom-pre-Lie
algebras control one-parameter formal deformation of Hom-pre-Lie algebras.
Unlike [11], our formal deformation takes into account the structure map
$\alpha$.

######  Definition 3.1.

_Let $(A,\nu,\alpha)$ be a Hom-pre-Lie algebra over $\mathbb{K}$. A one-
parameter formal deformation of $A$ is given by a $\mathbb{K}[[t]]$-bilinear
map $\nu_{t}:A[[t]]\times A[[t]]\to A[[t]]$, and a $\mathbb{K}[[t]]$-linear
map $\alpha_{t}:A[[t]]\to A[[t]]$ of the forms_

$\nu_{t}=\sum_{i\geq 0}\nu_{i}t^{i},~{}~{}~{}\alpha_{t}=\sum_{i\geq
0}\alpha_{i}t^{i}$

_such that_

  1. 1.

For all $i\geq 0$, $\nu_{i}:A\times A\to A$ is a $\mathbb{K}$-bilinear map,
and $\alpha_{i}:A\to A$ is a $\mathbb{K}$-linear map.

  2. 2.

$\nu_{0}(a,b)=\nu(a,b)=a\cdot b$, is the the multiplication of $A$ and
$\alpha_{0}=\alpha$ is the given structure map on $A$.

  3. 3.

$\nu_{t}(\nu_{t}(a,b),\alpha_{t}(c))-\nu_{t}(\alpha_{t}(a),\nu_{t}(b,c))=\nu_{t}(\nu_{t}(b,a),\alpha_{t}(c))-\nu_{t}(\alpha_{t}(b),\nu_{t}(a,c))$
for all $a,b,c\in A.$

  4. 4.

The map $\alpha_{t}$ is multiplicative, that is,
$\nu_{t}(\alpha_{t}(a),\alpha_{t}(b))=\alpha_{t}(\nu_{t}(a,b))$, for all
$a,b\in A$.

From the Condition 3, we have the following equations for all $n\geq 0$.

$\displaystyle\sum_{\begin{subarray}{c}i+j+k=n\\\ i,j,k\geq
0\end{subarray}}\nu_{i}(\nu_{j}(a,b),\alpha_{k}(c))-\nu_{i}(\alpha_{j}(a),\nu_{k}(b,c))-\nu_{i}(\nu_{j}(b,a),\alpha_{k}(c))+\nu_{i}(\alpha_{j}(b),\nu_{k}(a,c))=0.$
(3. 56)

The Condition 4 is equivalent to the following equation:

$\displaystyle\sum_{\begin{subarray}{c}i+j+k=n\\\ i,j,k\geq
0\end{subarray}}\nu_{i}(\alpha_{j}(a),\alpha_{k}(b))-\sum_{\begin{subarray}{c}i+j=n\\\
i,j\geq 0\end{subarray}}\alpha_{i}(\nu_{j}(a,b))=0.$ (3. 57)

For a Hom-pre-Lie algebra $(A,\nu,\alpha)$, an $\alpha_{j}$-associator is
defined as

$\displaystyle\nu_{i}\circ_{\alpha_{k}}\nu_{j}(a,b,c)=\nu_{i}(\nu_{j}(a,b),\alpha_{k}(c))-\nu_{i}(\alpha_{k}(a),\nu_{j}(b,c)).$
(3. 58)

Thus, the Equation 3\. 56 may be written in terms of associator $\alpha_{j}$
as follows:

$\displaystyle\sum_{\begin{subarray}{c}i+j+k=n\\\ i,j,k\geq
0\end{subarray}}(\nu_{i}\circ_{\alpha_{j}}\nu_{k})(a,b,c)-\sum_{\begin{subarray}{c}i+j+k=n\\\
i,j,k\geq 0\end{subarray}}(\nu_{i}\circ_{\alpha_{k}}\nu_{j})(b,a,c)=0.$ (3.
59)

We can re-write the Equation 3\. 59 as follows:

$\displaystyle(\widetilde{\partial}_{\nu\nu}\nu_{n}-\widetilde{\partial}_{\alpha\nu}\alpha_{n})(a,b,c)=\sum_{\begin{subarray}{c}i+j+k=n\\\
0\leq i,j,k\leq
n-1\end{subarray}}(\nu_{i}\circ_{\alpha_{k}}\nu_{j})(a,b,c)-\sum_{\begin{subarray}{c}i+j+k=n\\\
0\leq i,j,k\leq n-1\end{subarray}}(\nu_{i}\circ_{\alpha_{k}}\nu_{j})(b,a,c).$
(3. 60)

Observe that

$\displaystyle(\widetilde{\partial}_{\nu\nu}\nu_{n})(a,b,c)=$
$\displaystyle\alpha(a)\cdot\nu_{n}(b,c)-\alpha(b)\cdot\nu_{n}(a,c)+\nu_{n}(b,a)\cdot\alpha(c)-\nu_{n}(a,b)\cdot\alpha(c)-\nu_{n}(\alpha(b),a\cdot
c)$ $\displaystyle+\nu_{n}(\alpha(a),b\cdot c)-\nu_{n}(a\cdot
b,\alpha(c))+\nu_{n}(b\cdot a,\alpha(c)),$

and

$\displaystyle(\widetilde{\partial}_{\alpha\nu}\alpha_{n})(a,b,c)=\alpha_{n}(b)\cdot(a\cdot
c)-\alpha_{n}(a)(b\cdot c)+(a\cdot b)\cdot\alpha_{n}(c)-(b\cdot
a)\cdot\alpha_{n}(c)$

The Equation 3\. 57 is same as the following equation involving differentials.

$\displaystyle(\widetilde{\partial}_{\nu\alpha}\nu_{n}-\widetilde{\partial}_{\alpha\alpha}\alpha_{n})(a,b)=\sum_{\begin{subarray}{c}i+j+k=n\\\
0\leq i,j,k\leq
n-1\end{subarray}}\nu_{i}(\alpha_{j}(a),\alpha_{k}(b))-\sum_{\begin{subarray}{c}i+j=n\\\
0\leq i,j\leq n-1\end{subarray}}\alpha_{i}(\nu_{j}(a,b)).$ (3. 61)

Note that

$\displaystyle(\widetilde{\partial}_{\alpha\alpha}\alpha_{n})(a,b)=\alpha(a)\cdot\alpha_{n}(b)+\alpha_{n}(a)\cdot\alpha(b)-\alpha_{n}(a\cdot
b).$

and

$\displaystyle(\widetilde{\partial}_{\nu\alpha}\nu_{n})(a,b)=\alpha(\nu_{n}(a,b))-\nu_{n}(\alpha(a),\alpha(b)).$

For $n=0$, from the Equation 3\. 56, we have

$\displaystyle(a.b).\alpha(c)-\alpha(a).(b.c)-(b.a).\alpha(c)+\alpha(b).(a.c)=0.$

This equation is same as Hom-pre-Lie identity.

Again, for $n=0$, from the Equation 3\. 57, we have

$\displaystyle\alpha(a).\alpha(b)-\alpha(a.b)=0.$

This shows $\alpha$ is multiplicative. Thus, for $n=0$ case, deformed Hom-pre-
Lie algebra is same as the original Hom-pre-Lie algebra $A$.

For $n=1$, from Equation 3\. 56, we have

$\displaystyle\alpha(a)\cdot\nu_{1}(b,c)-\alpha(b)\cdot\nu_{1}(a,c)+\nu_{1}(b,a)\cdot\alpha(c)-\nu_{1}(a,b)\cdot\alpha(c)-\nu_{1}(\alpha(b),a.c)-\alpha_{1}(b)\cdot(a\cdot
c)$ $\displaystyle+\alpha_{1}(a)(b\cdot c)-(a\cdot
b)\cdot\alpha_{1}(c)+(b\cdot a)\cdot\alpha_{n}(c)=0.$

This is same as
$(\tilde{\partial_{\nu\nu}}\nu_{1}-\tilde{\partial_{\alpha\nu}}\alpha_{1})(a,b,c)=0$.

For $n=1$, from Equation 3\. 57, we also have

$\alpha(a)\cdot\alpha_{1}(b)+\alpha_{1}(a)\cdot\alpha(b)-\alpha_{1}(a\cdot
b)-\alpha(\nu_{1}(a,b))+\nu_{1}(\alpha(a),\alpha(b)).$

Again, this is same as
$(\widetilde{\partial}_{\nu\alpha}\nu_{1}-\widetilde{\partial}_{\alpha\alpha}\alpha_{1})(a,b)=0.$

######  Definition 3.2.

The pair $(\nu_{1},\alpha_{1})$ is called the infinitesimal of the given
deformation $(\nu_{t},\alpha_{t})$. More generally, assume that
$(\nu_{n},\alpha_{n})$ is the first non-zero term after $(\nu_{0},\alpha_{0})$
of $(\nu_{t},\alpha_{t})$. Then such pair $(\nu_{n},\alpha_{n})$ is called the
$n$th infinitesimal of $(\nu_{t},\alpha_{t})$.

Therefore, from the above discussion we have the following theorem.

######  Theorem 3.3.

Let $(\nu_{t},\alpha_{t})$ be a one-parameter formal deformation of the Hom-
pre-Lie algebra $A$. Then the inifinitesimal of a given deformation of $A$ is
a $2$-cocycle of the $\alpha$-type cohomology of $A$.

###  3.1 Obstructions of deformation

In this section, we show how $\alpha$-type cohomology control obstructions of
given deformations.

######  Definition 3.4.

_A deformation of order $n$ of $A$ is given by a $\mathbb{K}[[t]]$-bilinear
map $\nu_{t}:A[[t]]\times A[[t]]\to A[[t]]$ and a $\mathbb{K}[[t]]$-linear map
$\alpha_{t}:A[[t]]\to A[[t]]$ of the forms_

$\nu_{t}=\sum^{n}_{i=0}\nu_{i}t^{i},~{}~{}~{}\alpha_{t}=\sum^{n}_{i=0}\alpha_{i}t^{i},$

_such that the pair $(\nu_{t},\alpha_{t})$ satisfies the definition of the
formal deformation of $A$ (mod $t^{n+1}$). _

We say a deformation $(\nu_{t},\alpha_{t})$ of order $n$ is extendable to a
deformation of order $(n+1)$ if there exists
$\nu_{n+1}\in\widetilde{C_{\nu}}^{n+1}(A,A)$ and
$\alpha_{n+1}\in\widetilde{C_{\alpha}}^{n+1}(A,A)$ such that

$\bar{\nu_{t}}=\nu_{t}+\nu_{n+1}t^{n+1},~{}~{}~{}\bar{\alpha_{t}}=\alpha_{t}+\alpha_{n+1}t^{n+1},$

and the pair $(\bar{\nu_{t}},\bar{\alpha_{t}})$ satisfies the Definition 3.1
mod($t^{n+2}$). From a deformation $(\nu_{t},\alpha_{t})$ of order $(n+1)$, we
have the following equations:

$\displaystyle\sum_{\begin{subarray}{c}i+j+k=n+1\\\ i,j,k\geq
0\end{subarray}}\nu_{i}(\nu_{j}(a,b),\alpha_{k}(c))-\nu_{i}(\alpha_{j}(a),\nu_{k}(b,c))-\nu_{i}(\nu_{j}(b,a),\alpha_{k}(c))+\nu_{i}(\alpha_{j}(b),\nu_{k}(a,c))=0,$
$\displaystyle\sum_{\begin{subarray}{c}i+j+k=n+1\\\ i,j,k\geq
0\end{subarray}}\nu_{i}(\alpha_{j}(a),\alpha_{k}(b))-\sum_{\begin{subarray}{c}i+j=n+1\\\
i,j\geq 0\end{subarray}}\alpha_{i}(\nu_{j}(a,b))=0.$

We can re-write the above equations as follows:

$\displaystyle(\widetilde{\partial}_{\nu\nu}\nu_{n}-\widetilde{\partial}_{\alpha\nu}\alpha_{n})(a,b,c)=\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq
n\end{subarray}}(\nu_{i}\circ_{\alpha_{j}}\nu_{k})(a,b,c)-\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq n\end{subarray}}(\nu_{i}\circ_{\alpha_{j}}\nu_{k})(b,a,c),$
$\displaystyle(\widetilde{\partial}_{\nu\alpha}\nu_{n}-\widetilde{\partial}_{\alpha\alpha}\alpha_{n})(a,b)=\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq
n\end{subarray}}\nu_{i}(\alpha_{j}(a),\alpha_{k}(b))-\sum_{\begin{subarray}{c}i+j=n+1\\\
0\leq i,j\leq n\end{subarray}}\alpha_{i}(\nu_{j}(a,b)).$

We define the $n$th obstruction to extend a deformation of order $n$ to $n+1$
as $O^{n}(A):=(O^{n}_{\nu}(A),O^{n}_{\alpha}(A))$, where

$\displaystyle O^{n}_{\nu}(A)(a,b,c)=\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq
n\end{subarray}}(\nu_{i}\circ_{\alpha_{j}}\nu_{k})(a,b,c)-\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq n\end{subarray}}(\nu_{i}\circ_{\alpha_{j}}\nu_{k})(b,a,c),$
(3. 62) $\displaystyle
O^{n}_{\alpha}(A)(a,b,c)=\sum_{\begin{subarray}{c}i+j+k=n+1\\\ 0\leq i,j,k\leq
n\end{subarray}}\nu_{i}(\alpha_{j}(a),\alpha_{k}(b))-\sum_{\begin{subarray}{c}i+j=n+1\\\
0\leq i,j\leq n\end{subarray}}\alpha_{i}(\nu_{j}(a,b)).$ (3. 63)

Note that $O^{n}(A)\in\widetilde{C}^{3}(A,A)$.

######  Theorem 3.5.

A deformation $(\nu_{t},\alpha_{t})$ of order $n$ is extendable to a
deformation of order $n+1$ if and only if the cohomology class of $O^{n}(A)$
vanishes.

###### Proof.

Suppose a deformation $(\nu_{t},\alpha_{t})$ of order $n$ extends a
deformation of order $n+1$. From the definition of obstruction, we have

$O^{n}(A):=(O^{n}_{\nu}(A),O^{n}_{\alpha}(A))=\partial(\nu_{n+1},\alpha_{n+1})$

Thus, $\partial(O^{n}(A))=0$, that is, the cohomology class of the obstruction
vanishes.

Conversely, suppose the cohomology class of $O^{n}(A)$ vanishes, that is,

$O^{n}(A)=\partial(\nu_{n+1},\alpha_{n+1}),$

for some $2$-cochain $(\nu_{n+1},\alpha_{n+1})$. We define an
$(n+1)$-deformation extending an $n$-deformation $(\nu_{t},\alpha_{t})$ as
follows:

$\displaystyle\bar{\nu_{t}}=\nu_{t}+\nu_{n+1}t^{n+1},$
$\displaystyle\bar{\alpha_{t}}=\alpha_{t}+\alpha_{n+1}t^{n+1}.$

It is easy to check that $(\bar{\nu_{t}},\bar{\alpha_{t}})$ defines an
$(n+1)$-deformation. ∎

######  Corollary 3.6.

The obstruction of extending an infinitesimal deformation lies in
$\widetilde{H^{3}}(A,A)$.

###  3.2 Equivalent deformation and cohomology

Let $(\nu_{t},\alpha_{t})$ and
$(\nu^{{}^{\prime}}_{t},\alpha^{{}^{\prime}}_{t})$ be two formal deformations
of $A$, where $\nu^{{}^{\prime}}_{t}=\sum_{i\geq
0}\nu^{{}^{\prime}}_{i}t^{i},~{}~{}~{}\alpha^{{}^{\prime}}_{t}=\sum_{i\geq
0}\alpha^{{}^{\prime}}_{i}t^{i}$.

######  Definition 3.7.

Two deformations $(\nu_{t},\alpha_{t})$ and
$(\nu^{{}^{\prime}}_{t},\alpha^{{}^{\prime}}_{t})$ are said to be equivalent
if there exists $\mathbb{K}[[t]]$-isomorphism $\Psi_{t}:A[[t]]\to A[[t]]$ of
the form $\Psi_{t}=\sum_{i\geq 0}\psi_{i}t^{i}$ such that
$\psi_{0}=\text{id}$, and $\psi_{i}:A\to A$ are $\mathbb{K}$-linear maps for
all $i\geq 0$, satisfying the following conditions:

$\displaystyle\Psi_{t}\circ\nu^{{}^{\prime}}_{t}=\nu_{t}\circ(\Psi_{t}\otimes\Psi_{t}),~{}\text{and}~{}\alpha_{t}\circ\Psi_{t}=\Psi_{t}\circ\alpha^{{}^{\prime}}_{t}.$
(3. 64)

######  Definition 3.8.

A deformation $(\nu_{t},\alpha_{t})$ of $A$ is called trivial if
$(\nu_{t},\alpha_{t})$ is equivalent to $(\nu_{0},\alpha_{0})$. A Hom-pre-Lie
algebra $A$ is called rigid if it has only trivial deformation upto
equivalence.

The Condition 3\. 64 is equivalent to the following equations:

$\displaystyle\sum_{i\geq 0}\psi_{i}\bigg{(}\sum_{j\geq
0}\nu^{\prime}_{j}(a,b)t^{j}\bigg{)}t^{i}=\sum_{i\geq
0}\nu_{i}\bigg{(}\sum_{j\geq 0}\psi_{j}(a)t^{j},\sum_{k\geq
0}\psi_{k}(b)t^{k}\bigg{)}t^{i},$ (3. 65) $\displaystyle\sum_{i\geq
0}\alpha_{i}\bigg{(}\sum_{j\geq 0}\psi_{j}(a)t^{j}\bigg{)}t^{i}=\sum_{i\geq
0}\psi_{i}\bigg{(}\sum_{j\geq 0}\alpha^{\prime}_{j}(a)t^{j}\bigg{)}t^{i}.$ (3.
66)

This is same as the following equations:

$\displaystyle\sum_{i,j\geq
0}\psi_{i}(\nu^{\prime}_{j}(a,b))t^{i+j}=\sum_{i,j,k\geq
0}\nu_{i}(\psi_{j}(a),\psi_{k}(b))t^{i+j+k},$ (3. 67)
$\displaystyle\sum_{i,j\geq 0}\alpha_{i}(\psi_{j}(a))t^{i+j}=\sum_{i,j\geq
0}\psi_{i}(\alpha^{\prime}_{j}(a))t^{i+j}.$ (3. 68)

Comparing constant terms on both sides of the above equations, we have

$\displaystyle\nu^{\prime}_{0}(a,b)=\nu_{0}(a,b)=a\cdot
b,\,\,\,\text{as}\,\,\,\psi_{0}=Id,$
$\displaystyle\alpha_{0}(a)=\alpha^{\prime}_{0}(a)=\alpha(a).$

Now comparing coefficients of $t$, we have

$\displaystyle\nu^{\prime}_{1}(a,b)+\psi_{1}(\nu^{\prime}_{0}(a,b))=\nu_{1}(a,b)+\nu_{0}(\psi_{1}(a),b)+\nu_{0}(a,\psi_{1}(b)),$
(3. 69)
$\displaystyle\alpha_{1}(a)+\alpha_{0}(\psi_{1}(a))=\alpha^{\prime}_{1}(a)+\psi_{1}(\alpha^{\prime}_{0}(a)).$
(3. 70)

The Equations (3\. 69) and (3\. 70) are same as

$\displaystyle\nu^{\prime}_{1}(a,b)-\nu_{1}(a,b)=a\cdot\psi_{1}(b)+\psi_{1}(a)\cdot
b-\psi_{1}(a\cdot b)=\widetilde{\partial}_{\nu\nu}\psi_{1}(a,b).$
$\displaystyle\alpha_{1}^{\prime}(a)-\alpha_{1}(a)=\alpha(\psi_{1}(a))-\psi_{1}(\alpha(a))=\widetilde{\partial}_{\nu\alpha}\psi_{1}(a).$

Thus, we have the following proposition.

######  Proposition 3.9.

Two equivalent formal deformations of $A$ have cohomologous infinitesimals.

######  Theorem 3.10.

Let $(\nu_{t},\alpha_{t})$ be a non-trivial formal deformation of the Hom-pre-
Lie algebra $A$. Then $(\nu_{t},\alpha_{t})$ is equivalent to a deformation
whose infinitesimal is not a coboundary.

###### Proof.

Let $(\nu_{t},\alpha_{t})$ be a deformation of the Hom-pre-Lie algebra $A$ and
$(\nu_{n},\alpha_{n})$ be the $n$-infinitesimal of the deformation for some
$n\geq 1$. Then by Theorem (3.3), $(\nu_{n},\alpha_{n})$ is a $2$-cocycle,
that is, $\widetilde{\partial}(\nu_{n},\alpha_{n})=0$. Suppose
$\nu_{n}=-\widetilde{\partial}_{\nu\nu}\phi_{n}$ and
$\alpha_{n}=-\widetilde{\partial}_{\nu\alpha}\phi_{n}$for some
$\phi_{n}\in\widetilde{C_{\nu}}^{1}(A,A)$, that is, $(\nu_{n},\alpha_{n})$ is
a coboundary. We define a formal isomorphism $\Psi_{t}$ of $A[[t]]$ as
follows:

$\Psi_{t}(a)=a+\phi_{n}(a)t^{n}.$

We set

$\bar{\nu_{t}}=\Psi^{-1}_{t}\circ\nu_{t}\circ(\Psi_{t}\otimes\Psi_{t})\,\,\,\text{and}\,\,\,\bar{\alpha_{t}}=\Psi^{-1}_{t}\circ\alpha_{t}\circ\Psi_{t}$

Thus, we have a new deformation $(\bar{\nu_{t}},\bar{\alpha_{t}})$ which is
equivalent to $(\nu_{t},\alpha_{t})$. By expanding the above equations and
comparing coefficients of $t^{n}$, we get

$\bar{\nu_{n}}-\nu_{n}=\widetilde{\partial}_{\nu\nu}\phi_{n},~{}~{}~{}\bar{\alpha_{n}}-\alpha=\widetilde{\partial}_{\nu\alpha}\phi_{n}.$

Hence, $\bar{\nu_{n}}=0,\bar{\alpha_{n}}=0$. By repeating this argument, we
can kill off any infinitesimal which is a coboundary. Thus, the process must
stop if the deformation is non-trivial. ∎

######  Corollary 3.11.

If $\widetilde{H^{2}}(A,A)=0$ then every deformation of $A$ is equivalent to a
trivial deformation, that is, $A$ is rigid.

##  4 Equivariant cohomology of Hom-pre-Lie algebras

In this section, we develop an equivariant cohomology for Hom-pre-Lie algebras
equipped with a group action. To construct such an equivariant cohomology, we
will follow Bredon’s [3] approach for equivarint cohomology of topological
$G$-spaces.

######  Definition 4.1.

_Let $G$ be a finite group and $A$ be a Hom-pre-Lie algebra. The group $G$ is
said to act on $A$ from left if there exists a map_

$\phi:G\times A\to A,~{}~{}~{}(g,a)\mapsto ga,$

_satisfying_

  1. 1.

For each $g\in G$, the map $\phi(g,-)=\psi_{g}:A\to A,~{}a\mapsto ga$ is
linear.

  2. 2.

For all $a\in A,~{}ea=a$, where $e$ denotes the identity element of $G$.

  3. 3.

$(g_{1}g_{2})a=g_{1}(g_{2}a)$ for all $g_{1},g_{2}\in G$ and $a\in A$.

  4. 4.

$g(a\cdot b)=ga\cdot gb$ and $\alpha(g\cdot a)=g\alpha(a)$, for all $g\in
G,a,b\in A.$

######  Remark 4.2.

We defined finite group actions on Hom-pre-Lie algebras for simplicity, but
one can do it for any discrete groups.

Let $G$ is acting on the Hom-pre-Lie algebra $A$. Let $H$ be a subgroup of
$G$. We define the $H$-fixed point set of $A$ as follows:

$A^{H}=\\{a\in A\mid ha=a,\forall h\in H\\}.$

######  Lemma 4.3.

For all subgroup $H$ of $G$, the $H$-fixed point set $A^{H}$ is a sub Hom-pre-
Lie algebra of $A$.

###### Proof.

It is enough to show that $A^{H}$ is closed under multiplication and if $a\in
A^{H}$ then $\alpha(a)\in A^{H}$.

Let $a,b\in A^{H}$. Observe that

$h(a\cdot b)=ha\cdot hb=a\cdot b,~{}~{}~{}h(\alpha(a))=\alpha(h\cdot
a)=\alpha(a),~{}\text{for all}~{}h\in H.$

Thus, $A^{H}$ is a sub Hom-pre-Lie algebra of $A$. ∎

An $O_{G}$-module is a contravariant functor

$\mathcal{M}:O_{G}\to{\bf{Mod}}.$

Note that ${\bf{Mod}}$ denotes a category of $\mathbb{K}$-modules, the
category whose objects are $O_{G}$-modules with morphisms the natural
transformations between $O_{G}$-modules is an abelian category denoted by
$C_{G}$.

######  Definition 4.4.

An $O_{G}$-Hom-pre-Lie algebra is a contravariant functor

$\mathcal{A}:O_{G}\to\bf{HpLie}.$

$\bf{HpLie}$ denotes a category of Hom-pre-Lie algebras over $\mathbb{K}$ with
Hom-pre-Lie algebra morphisms between them.

######  Example 4.5.

Suppose $A$ is a Hom-pre-Lie algebra equipped with an action of $G$, then we
have a contravariant functor

$\chi_{A}:O_{G}\to\bf{HpLie},$

given by $\chi_{A}(G/H)=A^{H}$ and for a morphism $\hat{g}:G/H\to G/K$
corresponding to the subconjugacy relation $g^{-1}Hg\subseteq K$,

$\chi_{A}(\hat{g})=\psi_{g}:A^{K}\to A^{H}.$

Thus, $\chi_{A}$ is an $O_{G}$-Hom-pre-Lie algebra.

A representation of an $O_{G}$-Hom-pre-Lie algebra $\mathcal{A}$ is an
$O_{G}$-module $\mathcal{M}$ together with a natural transformation
$\mathcal{\beta}$ on $\mathcal{M}$, and the following natural transformations

$\displaystyle\mathcal{\rho}:\mathcal{A}\otimes\mathcal{M}\to\mathcal{M},~{}\mathcal{\mu}:\mathcal{M}\otimes\mathcal{A}\to\mathcal{M}$

such that for all $G/H\in\text{Ob}(O_{G})$, $\mathcal{M}(G/H)$ is a non-
equivariant representation of $\mathcal{A}(G/H)$ associated with natural
transformations $\eta_{l},\eta_{r}$.

######  Remark 4.6.

From the above definition of the representation in an equivariant setting we
have the following relations coming from the naturality of
$\eta_{l},\eta_{r}$.

$\displaystyle\mathcal{\rho}(G/H)\circ(\mathcal{A}(\hat{g})\otimes\mathcal{M}(\hat{g}))=\mathcal{M}(\hat{g})\circ\mathcal{\rho}(G/K),$
(4. 71)
$\displaystyle\mathcal{\mu}(G/H)\circ(\mathcal{A}(\hat{g})\otimes\mathcal{M}(\hat{g}))=\mathcal{M}(\hat{g})\circ\mathcal{\mu}(G/K).$
(4. 72)

######  Remark 4.7.

It is well-known that any Hom-pre-Lie algebra $A$ is a representation over
itself. In the equivariant setting also one can easily see that any
$O_{G}$-Hom-pre-Lie algebra $\mathcal{A}$ can be thought of as a
representation over itself.

We set

$S^{n}(\mathcal{A};\mathcal{M}):=\bigoplus_{H\leq
G}\widetilde{C^{n}}(\mathcal{A}(G/H),\mathcal{M}(G/H)).$

We define

$\displaystyle\delta^{n}:$ $\displaystyle S^{n}(\mathcal{A};\mathcal{M})\to
S^{n+1}(\mathcal{A};\mathcal{M}),$ $\displaystyle\delta^{n}:=\bigoplus_{H\leq
G}\widetilde{\partial}^{n}_{H}.$

Here
$\widetilde{\partial}^{H}_{n}:\widetilde{C^{n}}(\mathcal{A}(G/H),\mathcal{M}(G/H))\to\widetilde{C^{n+1}}(\mathcal{A}(G/H),\mathcal{M}(G/H))$
is a non-equivariant coboundary map.

Clearly,$\\{S^{\sharp}(\mathcal{A};\mathcal{M}),\delta\\}$ is a cochain
complex. Throughout this paper, we take $O_{G}$-Hom-pre-Lie algebra
$\mathcal{A}$ as $\chi_{A}$ and consider the cohomology of $\chi_{A}$ with
coefficients in $\chi_{A}$. We define a subcomplex of this cochain complex as
follows:

######  Definition 4.8.

A cochain $c=\\{(c^{H}_{\nu},c^{H}_{\alpha})\mid H\leq G\\}$ is said to be
invariant if for every morphism $\hat{g}:G/H\to G/K$ corresponding to a
subconjugacy relation $g^{-1}Hg\subseteq K$, following relation holds,

$\displaystyle c^{H}_{\nu}\circ\mathcal{A}(\hat{g})^{\otimes
n}=\mathcal{A}(\hat{g})\circ c^{K}_{\nu},$ $\displaystyle
c^{H}_{\alpha}\circ\mathcal{A}(\hat{g})^{\otimes{n-1}}=\mathcal{A}(\hat{g})\circ
c^{H}_{\alpha}.$

As $\chi_{A}$ is an $O_{G}$-Hom-pre-Lie algebra corresponding to $A$. This is
same as,

$\displaystyle c^{H}_{\nu}\circ\psi_{g}^{\otimes n}=\psi_{g}\circ
c^{K}_{\nu},$ $\displaystyle
c^{H}_{\alpha}\circ\psi_{g}^{\otimes{n-1}}=\psi_{g}\circ c^{K}_{\alpha}.$

######  Lemma 4.9.

The set of all invariant $n$-cochains is a subgroup
$S_{G}^{n}(\chi_{A};\chi_{A})$ of $S^{n}(\chi_{A};\chi_{A})$. If
$c=\\{(c^{H}_{\nu},c^{H}_{\alpha})\\}\in S^{n}(\chi_{A};\chi_{A})$ is an
invariant cochain then
$\delta^{n}(c)=\\{\widetilde{\partial}^{n}_{H}(c^{H}_{\nu},c^{H}_{\alpha})\\}\in
S^{n+1}(\chi_{A};\chi_{A})$ is an invariant $(n+1)$-cochain.

###### Proof.

Let $c=\\{(c^{H}_{\nu},c^{H}_{\alpha})\\}\in S^{n}(\chi_{A};\chi_{A})$ is an
invariant cochain. This implies

$\displaystyle
c^{H}_{\nu}(ga_{1},ga_{2},\ldots,ga_{n})=gc^{K}_{\nu}(a_{1},a_{2},\ldots,a_{n}),$
$\displaystyle
c^{H}_{\alpha}(ga_{1},ga_{2},\ldots,ga_{n-1})=gc^{K}_{\alpha}(a_{1},a_{2},\ldots,a_{n-1}).$

It is enough to show that four component differentials
$\widetilde{\partial}_{\nu\nu},\widetilde{\partial}_{\nu\alpha},\widetilde{\partial}_{\alpha\alpha},~{}\text{and}~{}\widetilde{\partial}_{\alpha\nu}$
respect the group action. Note that

$\displaystyle\widetilde{\partial}_{\nu\nu}(c^{H}_{\nu})(\psi_{g}(a_{1}),\ldots,\psi_{g}(a_{n+1}))$
$\displaystyle=\widetilde{\partial}_{\nu\nu}(c^{H}_{\nu})(ga_{1},\ldots,ga_{n+1})$
$\displaystyle\sum^{n}_{i=1}(-1)^{i+1}\rho(\alpha^{n-1}(ga_{i}))c^{H}_{\nu}(ga_{1},\ldots,\widehat{ga_{i}},\ldots,ga_{n+1})$
$\displaystyle+\sum^{n}_{i=1}(-1)^{i+1}\mu(\alpha^{n-1}(ga_{n+1}))c^{H}_{\nu}(ga_{1},\ldots,\widehat{ga_{i}},\ldots,ga_{n},ga_{i})$
$\displaystyle-\sum^{n}_{i=1}(-1)^{i+1}c^{H}_{\nu}(\alpha(ga_{1}),\ldots,\widehat{ga_{i}},\ldots,\alpha(ga_{n}),ga_{i}\cdot
ga_{n+1})$
$\displaystyle=g\big{(}\sum^{n}_{i=1}(-1)^{i+1}\rho(\alpha^{n-1}(a_{i}))c^{H}_{\nu}(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n+1})\big{)}$
$\displaystyle+g\big{(}\sum^{n}_{i=1}(-1)^{i+1}\mu(\alpha^{n-1}(a_{n+1}))c^{H}_{\nu}(a_{1},\ldots,\widehat{a_{i}},\ldots,a_{n},a_{i})\big{)}$
$\displaystyle-g\big{(}\sum^{n}_{i=1}(-1)^{i+1}c^{H}_{\nu}(\alpha(a_{1}),\ldots,\widehat{a_{i}},\ldots,\alpha(a_{n}),a_{i}\cdot
a_{n+1})\big{)}$
$\displaystyle=g\widetilde{\partial}_{\nu\nu}(c^{H}_{\nu})(a_{1},\ldots,a_{n+1})$

In a similar way, it is a routine work to check that other three components of
the differential also preserves the group action. Thus,
$\delta^{n}(c)=\\{\widetilde{\partial}^{n}_{H}(c^{H}_{\nu},c^{H}_{\alpha})\\}\in
S^{n+1}(\chi_{A};\chi_{A})$ is an invariant $(n+1)$-cochain. ∎

Thus, $(S^{\sharp}_{G}(\chi_{A};\chi_{A}),\delta)$ is a cochain complex and
homology of this complex is called equivariant cohomology of the Hom-pre-Lie
algebra $A$. We denote $n$th cohomology of $A$ by
$\widetilde{H}_{G}^{n}(A,A)$.

##  5 Equivariant formal deformation of Hom-pre-Lie algebras

In this section, we study a one-parameter formal deformation theory for Hom-
pre-Lie algebras equipped with a group action. We also show that equivariant
cohomology defined in Section 4 is the deformation cohomology of such formal
deformations.

Let $\chi_{A}$ is an $O_{G}$-Hom-pre-Lie algebra. For any subgroup $H$ of $G$,
we denote $A^{H}[[t]]$ as a formal power series ring in variable $t$ over
$H$-fixed point sets $A^{H}$.

######  Definition 5.1.

A one-parameter formal deformation of $\chi_{A}$ is a family consisting of
pair of natural transformation $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ such that
for every $G/H\in\text{Ob}(O_{G})$, components of $\mathfrak{v}_{t}$ are
bilinear maps, and components of $\mathfrak{a}_{t}$ are (invertible) linear
maps

$\displaystyle\mathfrak{v}_{t}(G/H):\chi_{A}(G/H)[[t]]\times\chi_{A}(G/H)[[t]]\to\chi_{A}(G/H)[[t]],$
$\displaystyle\mathfrak{a}_{t}(G/H):\chi_{A}(G/H)[[t]]\to\chi_{A}(G/H)[[t]].$

This is same as

$\displaystyle\nu^{H}_{t}:A^{H}[[t]]\times A^{H}[[t]]\to
A^{H}[[t]],~{}~{}~{}\alpha^{H}_{t}:A^{H}[[t]]\to A^{H}[[t]]$

which can be expressed in the following form

$\displaystyle\nu^{H}_{t}(a,b)=\nu^{H}_{0}(a,b)+\nu^{H}_{1}(a,b)t+\nu^{H}_{2}(a,b)t^{2}+\cdots,$
$\displaystyle\alpha^{H,}_{t}(a)=\alpha^{H}_{0}(a)+\alpha_{1}^{H}(a)t+\alpha^{H}_{2}(a)t^{2}+\cdots,$

such that

  1. 1.

$\nu^{H}_{0}(a,b)=a\cdot b$ and $\alpha^{H}_{0}(a)=\alpha(a)$ for $a,b\in A$.

  2. 2.

For $i\geq 0$, $\nu^{H}_{i}$ and $\alpha^{H}_{i}$ are $\mathbb{K}$-bilinear
and $\mathbb{K}$-linear maps respectively on $A^{H}$, and satisfies the
following Hom-pre-Lie algebra relations.

$\nu^{H}_{t}(\nu^{H}_{t}(a,b),\alpha^{H}_{t}(c))-\nu^{H}_{t}(\alpha_{t}(a),\nu^{H}_{t}(b,c))=\nu^{H}_{t}(\nu^{H}_{t}(b,a),\alpha^{H}_{t}(c))-\nu^{H}_{t}(\alpha^{H}_{t}(b),\nu^{H}_{t}(a,c)),$

for all $a,b,c\in A^{H}$ and $H\leq G$.

  3. 3.

For all the subgroups $H$ of $G$, the map $\alpha^{H}_{t}$ is multiplicative,
that is,
$\nu^{H}_{t}(\alpha^{H}_{t}(a),\alpha^{H}_{t}(b))=\alpha^{H}_{t}(\nu^{H}_{t}(a,b))$.

  4. 4.

For every morphism $\hat{g}:G/H\to G/K$ and $g\in G$ satisfying
$g^{-1}Hg\subseteq K$ the following relations hold-

$\displaystyle\nu^{H}_{t}\circ(\chi_{A}(\hat{g})\otimes\chi_{A}(\hat{g}))=\chi_{A}(\hat{g})\circ\nu^{K}_{t},$
$\displaystyle\alpha_{t}^{H}\circ\chi_{A}(\hat{g})=\alpha_{t}^{K}\circ\chi_{A}(\hat{g}).$

From the Condition 2, we have the following equations for all $n\geq 0$.

$\displaystyle\sum_{\begin{subarray}{c}i+j+k=n\\\ i,j,k\geq
0\end{subarray}}\nu^{H}_{i}(\nu^{H}_{j}(a,b),\alpha^{H}_{k}(c))-\nu^{H}_{i}(\alpha^{H}_{j}(a),\nu^{H}_{k}(b,c))-\nu^{H}_{i}(\nu^{H}_{j}(b,a),\alpha^{H}_{k}(c))+\nu^{H}_{i}(\alpha^{H}_{j}(b),\nu^{H}_{k}(a,c))=0.$
(5. 73)

The Condition 3 is equivalent to the following equation:

$\displaystyle\sum_{\begin{subarray}{c}i+j+k=n\\\ i,j,k\geq
0\end{subarray}}\nu^{H}_{i}(\alpha^{H}_{j}(a),\alpha^{H}_{k}(b))-\sum_{\begin{subarray}{c}i+j=n\\\
i,j\geq 0\end{subarray}}\alpha^{H}_{i}(\nu^{H}_{j}(a,b))=0.$ (5. 74)
$\displaystyle(\widetilde{\partial}_{\nu\nu}\nu^{H}_{n}-\widetilde{\partial}_{\alpha\nu}\alpha^{H}_{n})(a,b,b)$
(5. 75) $\displaystyle=\sum_{\begin{subarray}{c}i+j+k=n\\\ i,j,k\geq
0\end{subarray}}\nu^{H}_{i}(\nu^{H}_{j}(a,b),\alpha^{H}_{k}(c))-\nu^{H}_{i}(\alpha^{H}_{j}(a),\nu^{H}_{k}(b,c))-\nu^{H}_{i}(\nu^{H}_{j}(b,a),\alpha^{H}_{k}(c))+\nu^{H}_{i}(\alpha^{H}_{j}(b),\nu^{H}_{k}(a,c)).$
(5. 76)

Note that

$\displaystyle(\widetilde{\partial}_{\nu\nu}\nu^{H}_{n})(a,b,c)=$
$\displaystyle\alpha(a)\cdot\nu^{H}_{n}(b,c)-\alpha(b)\cdot\nu^{H}_{n}(a,c)+\nu^{H}_{n}(b,a)\cdot\alpha(c)-\nu^{H}_{n}(a,b)\cdot\alpha(c)-\nu^{H}_{n}(\alpha(b),a\cdot
c)$ $\displaystyle+\nu^{H}_{n}(\alpha(a),b\cdot c)-\nu^{H}_{n}(a\cdot
b,\alpha(c))-\nu^{H}_{n}(b\cdot a,\alpha(c)),~{}\text{for all}~{}a,b,c\in
A^{H}.$

and

$\displaystyle(\widetilde{\partial}_{\alpha^{H}\nu^{H}}\alpha^{H}_{n})(a,b,c)=\alpha^{H}_{n}(b)\cdot(a\cdot
c)-\alpha^{H}_{n}(a)(b\cdot c)+(a\cdot b)\cdot\alpha^{H}_{n}(c)-(b\cdot
a)\cdot\alpha^{H}_{n}(c)$

The Equation 5\. 74 is same as the following equation involving differentials.

$\displaystyle(\widetilde{\partial}_{\nu\alpha}\nu^{H}_{n}-\widetilde{\partial}_{\alpha\alpha}\alpha^{H}_{n})(a,b)=\sum_{\begin{subarray}{c}i+j+k=n\\\
0\leq i,j,k\leq
n-1\end{subarray}}\nu^{H}_{i}(\alpha^{H}_{j}(a),\alpha^{H}_{k}(b))-\sum_{\begin{subarray}{c}i+j=n\\\
0\leq i,j\leq n-1\end{subarray}}\alpha^{H}_{i}(\nu^{H}_{j}(a,b)).$ (5. 77)

Observe that

$\displaystyle(\widetilde{\partial}_{\alpha\alpha}\alpha^{H}_{n})(a,b)=\alpha(a)\cdot\alpha^{H}_{n}(b)+\alpha^{H}_{n}(a)\cdot\alpha(b)-\alpha^{H}_{n}(a\cdot
b).$

and

$\displaystyle(\tilde{\partial_{\nu\alpha}}\nu^{H}_{n})(a,b)=\alpha(\nu^{H}_{n}(a,b))-\nu^{H}_{n}(\alpha(a),\alpha(b)).$

######  Definition 5.2.

We define

$\nu^{G}_{1}=\oplus_{H\leq G}\nu_{1}^{H},~{}~{}~{}\alpha^{G}_{1}=\oplus_{H\leq
G}\alpha^{H}_{1}$

The pair $(\nu^{G}_{1},\alpha^{G}_{1})$ is called the equivariant
infinitesimal of the given deformation $(\mathfrak{v}_{t},\mathfrak{a}_{t})$.
More generally, assume that $(\nu^{H}_{n},\alpha^{H}_{n})$ is the first non-
zero term after $(\nu^{H}_{0},\alpha^{H}_{0})$ of
$(\mathfrak{v}_{t},\mathfrak{a}_{t})$, for all subgroups $H$ of $G$ and for
some $n\geq 0$. We define the $n$th equivariant infinitesimal of the
equivariant deformation $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ as the pair
$(\nu^{G}_{n},\alpha^{G}_{n})$, where

$\nu^{G}_{n}=\oplus_{H\leq G}\nu_{n}^{H},~{}~{}~{}\alpha^{G}_{n}=\oplus_{H\leq
G}\alpha^{H}_{n}$

Similar to the non-equivariant case, we have the following theorem.

######  Theorem 5.3.

Let $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ be an equivariant one-parameter
formal deformation of the Hom-pre-Lie algebra $A$. Then the equivariant
infinitesimal of a given deformation of $A$ equipped with an action of $G$ is
a $2$-cocycle of the equivariant $\alpha$-type cohomology of $A$.

Now, we show how equivariant $\alpha$-type cohomology control obstructions of
given equivariant formal deformations.

######  Definition 5.4.

_An equivariant deformation of order $n$ of $A$ eqipped with action of $G$ is
given by a $\mathbb{K}[[t]]$-bilinear map $\nu^{H}_{t}:A^{H}[[t]]\times
A^{H}[[t]]\to A^{H}[[t]]$ and a $\mathbb{K}[[t]]$-linear map
$\alpha^{H}_{t}:A^{H}[[t]]\to A^{H}[[t]]$ of the forms_

$\nu^{H}_{t}=\sum^{n}_{i=0}\nu^{H}_{i}t^{i},~{}~{}~{}\alpha^{H}_{t}=\sum^{n}_{i=0}\alpha^{H}_{i}t^{i},$

_such that the pair $(\nu^{H}_{t},\alpha^{H}_{t})$ satisfies the definition of
the formal deformation of $A^{H}$ (mod $t^{n+1}$), for all subgroups $H$ of
$G$. _

We say an equivariant deformation $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ of
order $n$ is extendable to a deformation of order $(n+1)$ if for all subgroups
$H$, there exists $\nu^{H}_{n+1}\in\widetilde{C_{\nu}^{n+1}}(A,A)$ and
$\alpha^{H}_{n+1}\in\widetilde{C_{\alpha}^{n+1}}(A,A)$ such that

$\bar{\nu^{H}_{t}}=\nu^{H}_{t}+\nu^{H}_{n+1}t^{n+1},~{}~{}~{}\bar{\alpha^{H}_{t}}=\alpha^{H}_{t}+\alpha^{H}_{n+1}t^{n+1},$

and the pair $(\bar{\nu^{H}_{t}},\bar{\alpha^{H}_{t}})$ satisfies the
Definition 5.1 mod($t^{n+2}$). From an equivariant deformation
$(\mathfrak{v}_{t},\mathfrak{a}_{t})$ of order $(n+1)$, we have the following
equations:

$\displaystyle\sum_{\begin{subarray}{c}i+j+k=n+1\\\ i,j,k\geq
0\end{subarray}}\nu^{H}_{i}(\nu^{H}_{j}(a,b),\alpha^{H}_{k}(c))-\nu^{H}_{i}(\alpha^{H}_{j}(a),\nu^{H}_{k}(b,c))-\nu^{H}_{i}(\nu^{H}_{j}(b,a),\alpha^{H}_{k}(c))+\nu^{H}_{i}(\alpha^{H}_{j}(b),\nu^{H}_{k}(a,c))=0,$
$\displaystyle\sum_{\begin{subarray}{c}i+j+k=n+1\\\ i,j,k\geq
0\end{subarray}}\nu^{H}_{i}(\alpha^{H}_{j}(a),\alpha^{H}_{k}(b))-\sum_{\begin{subarray}{c}i+j=n+1\\\
i,j\geq 0\end{subarray}}\alpha^{H}_{i}(\nu^{H}_{j}(a,b))=0.$

We can re-write the above equations as follows:

$\displaystyle(\widetilde{\partial}_{\nu\nu}\nu^{H}_{n}-\widetilde{\partial}_{\alpha\nu}\alpha^{H}_{n})(a,b,c)=\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq
n\end{subarray}}(\nu^{H}_{i}\circ_{\alpha^{H}_{j}}\nu^{H}_{k})(a,b,c)-\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq
n\end{subarray}}(\nu^{H}_{i}\circ_{\alpha^{H}_{j}}\nu^{H}_{k})(b,a,c),$
$\displaystyle(\widetilde{\partial}_{\alpha\alpha}\alpha^{H}_{n}-\widetilde{\partial}_{\nu\alpha}\nu^{H}_{n})(a,b)=\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq
n\end{subarray}}\nu^{H}_{i}(\alpha^{H}_{j}(a),\alpha^{H}_{k}(b))-\sum_{\begin{subarray}{c}i+j=n+1\\\
0\leq i,j\leq n\end{subarray}}\alpha^{H}_{i}(\nu^{H}_{j}(a,b)).$

We define the $n$th equivariant obstruction to extend an equivariant
deformation of order $n$ to $n+1$ as $O_{G}^{n}(A):=(\oplus_{H\leq
G}O^{n,H}_{\nu}(A),\oplus_{H\leq G}O^{n,H}_{\alpha}(A))$, where

$\displaystyle O^{n,H}_{\nu}(A)=\sum_{\begin{subarray}{c}i+j+k=n+1\\\ 0\leq
i,j,k\leq
n\end{subarray}}(\nu^{H}_{i}\circ_{\alpha^{H}_{k}}\nu^{H}_{j})(a,b,c)-\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq
n\end{subarray}}(\nu^{H}_{i}\circ_{\alpha^{H}_{k}}\nu^{H}_{j})(b,a,c),$ (5.
78) $\displaystyle O^{n,H}_{\alpha}(A)=\sum_{\begin{subarray}{c}i+j+k=n+1\\\
0\leq i,j,k\leq
n\end{subarray}}\nu^{H}_{i}(\alpha^{H}_{j}(a),\alpha^{H}_{k}(b))-\sum_{\begin{subarray}{c}i+j=n+1\\\
0\leq i,j\leq n\end{subarray}}\alpha^{H}_{i}(\nu^{H}_{j}(a,b)).$ (5. 79)

The proofs of the following results are similar to the proofs of the
corresponding non-equivariant results.

######  Theorem 5.5.

An equivariant deformation $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ of order $n$
is extendable to a deformation of order $n+1$ if and only if the cohomology
class of $O_{G}^{n}(A)$ vanishes.

######  Corollary 5.6.

The obstruction of extending an equivariant infinitesimal deformation lies in
$\widetilde{H^{3}}_{G}(A,A)$.

Let $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ and
$(\mathfrak{v}^{{}^{\prime}}_{t},\mathfrak{a}^{{}^{\prime}}_{t})$ be two
equivariant formal deformations of $A$, where
$(\nu^{{}^{\prime}})^{H}_{t}=\sum_{i\geq
0}\nu^{{}^{\prime}}_{i}t^{i},~{}~{}~{}(\alpha^{{}^{\prime}})^{H}_{t}=\sum_{i\geq
0}\alpha^{{}^{\prime}}_{i}t^{i}$, for all subgroups $H$ of $G$.

######  Definition 5.7.

Two equivariant deformations $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ and
$(\mathfrak{v}^{{}^{\prime}}_{t},\mathfrak{a}^{{}^{\prime}}_{t})$ are said to
be equivalent if for all subgroups $H$ of $G$, there exists
$\mathbb{K}[[t]]$-isomorphism $\Psi^{H}_{t}:A[[t]]\to A[[t]]$ of the form
$\Psi^{H}_{t}=\sum_{i\geq 0}\psi^{H}_{i}t^{i}$ such that for all $i\geq 0$,

$\displaystyle 1.~{}~{}~{}\psi^{H}_{0}=\text{id}$ (5. 80) $\displaystyle
2.~{}~{}~{}\Psi^{H}_{t}\circ(\nu^{{}^{\prime}}_{t})^{H}=\nu^{H}_{t}\circ(\Psi^{H}_{t}\otimes\Psi^{H}_{t}),~{}\text{and}~{}\alpha^{H}_{t}\circ\Psi^{H}_{t}=\Psi^{H}_{t}\circ(\alpha^{{}^{\prime}}_{t})^{H}.$
(5. 81)

######  Definition 5.8.

An equivariant deformation $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ of $A$ is
called trivial if $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ is equivalent to
$(\mathfrak{v}_{0},\mathfrak{a}_{0})$. A Hom-pre-Lie algebra $A$ equipped with
a group action is called rigid if it has only trivial deformation upto
equivalence.

This is same as the following equations:

$\displaystyle\sum_{i,j\geq
0}\psi^{H}_{i}((\nu^{\prime}_{j})^{H}(a,b))t^{i+j}=\sum_{i,j,k\geq
0}\nu_{i}(\psi^{H}_{j}(a),\psi^{H}_{k}(b))t^{i+j+k},$ (5. 82)
$\displaystyle\sum_{i,j\geq
0}\alpha^{H}_{i}(\psi^{H}_{j}(x))t^{i+j}=\sum_{i,j\geq
0}\psi^{H}_{i}((\alpha^{\prime}_{j})^{H}(x))t^{i+j}.$ (5. 83)

Now comparing coefficients of $t$, we have

$\displaystyle(\nu^{\prime}_{1})^{H}(a,b)+\psi^{H}_{1}((\nu^{\prime}_{0})^{H}(a,b))=\nu^{H}_{1}(a,b)+\nu^{H}_{0}(\psi^{H}_{1}(a),b)+\nu^{H}_{0}(a,\psi^{H}_{1}(b)),$
(5. 84)
$\displaystyle\alpha^{H}_{1}(a)+\alpha^{H}_{0}(\psi^{H}_{1}(a))=(\alpha^{\prime}_{1})^{H}(a)+\psi^{H}_{1}((\alpha^{\prime}_{0})^{H}(a)).$
(5. 85)

The Equations (5\. 84) and (5\. 85) are same as

$\displaystyle(\nu^{\prime}_{1})^{H}(a,b)-\nu^{H}_{1}(a,b)=a\cdot\psi^{H}_{1}(b)+\psi^{H}_{1}(a)\cdot
b-\psi^{H}_{1}(a\cdot b)=\widetilde{\partial}_{\nu\nu}\psi^{H}_{1}(a,b).$
$\displaystyle(\alpha_{1}^{\prime})^{H}(a)-\alpha^{H}_{1}(a)=\alpha(\psi^{H}_{1}(a))-\psi^{H}_{1}(\alpha(a))=\widetilde{\partial}_{\nu\alpha}\psi^{H}_{1}(a).$

Similar to the non-equivariant case, one can show the following results.

######  Proposition 5.9.

Two equivalent formal deformations of $A$ equipped with an action of $G$ have
cohomologous infinitesimals.

######  Theorem 5.10.

Let $(\mathfrak{v}_{t},\mathfrak{a}_{t})$ be a non-trivial equivariant formal
deformation of the Hom-pre-Lie algebra $A$. Then
$(\mathfrak{v}_{t},\mathfrak{a}_{t})$ is equivalent to a deformation whose
infinitesimal is not a coboundary.

######  Corollary 5.11.

If $\widetilde{H^{2}}_{G}(A,A)=0$ then every equivariant deformation of $A$ is
equivalent to a trivial deformation, that is, $A$ is equivariantly rigid.

ACKNOWLEDGEMENT

The paper is supported by the NSF of China (No. 11761017), Guizhou Provincial
Science and Technology Foundation (No. [2020]1Y005).

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