#  Cohomology and deformations of $\mathcal{O}$-operators on Hom-associative
algebras

T. Chtioui, S. Mabrouk, A. Makhlouf Taoufik Chtioui1 †† E-mail:
<EMAIL_ADDRESS>, Sami Mabrouk2 †† E-mail<EMAIL_ADDRESS>,
Abdenacer Makhlouf3 †† E-mail<EMAIL_ADDRESS>(Corresponding
author)
1\. University of Sfax, Faculty of Sciences Sfax, BP 1171, 3038 Sfax, Tunisia
2\. University of Gafsa, Faculty of Sciences Gafsa, 2112 Gafsa, Tunisia
3\. IRIMAS - Département de Mathématiques, 6, rue des frères Lumière, F-68093
Mulhouse, France

###### Abstract

In this paper, we introduce the cohomology theory of $\mathcal{O}$-operators
on Hom-associative algebras. This cohomology can also be viewed as the
Hochschild cohomology of a certain Hom-associative algebra with coefficients
in a suitable bimodule. Next, we study infinitesimal and formal deformations
of an $\mathcal{O}$-operator and show that they are governed by the above-
defined cohomology. Furthermore, the notion of Nijenhuis elements associated
with an $\mathcal{O}$-operator is introduced to characterize trivial
infinitesimal deformations.

MSC(2010): 16E40, 16S80, 16W99.

keywords: $\mathcal{O}$-operator, Maurer-Cartan equation, graded-Lie algebra,
Hochschild cohomology, Formal deformation, Hom-associative r-matrix.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Hom-associative and Hom-dendriform algebras
    2. 2.2 Graded Lie algebras, Lift and Bidegree
  3. 3 Cohomologies of $\mathcal{O}$-operators on Hom-associative algebras
    1. 3.1 Cohomology of $\mathcal{O}$-operator
    2. 3.2 Cohomology of $\mathcal{O}$-operators as Hochschild cohomology
  4. 4 Deformations of $\mathcal{O}$-operators
    1. 4.1 Infinitesimal deformations
    2. 4.2 Formal Deformations
    3. 4.3 Order $n$ deformation of an $\mathcal{O}$-operator

##  1 Introduction

The concept of Rota-Baxter operators on associative algebras was introduced in
the 1960th by G. Baxter [1] in the study of fluctuation theory in Probability
and then the theory was established by G.-C. Rota [2] mainly in connection
with Combinatorics. Motivated by the study of associative analogues of Poisson
structures, Uchino introduced the notion of generalized Rota-Baxter operator
(called also $\mathcal{O}$-operator) on a bimodule over an associative algebra
in [47]. In [48], Uchino also introduced the notion of a strong Maurer-Cartan
equation on a bimodule $(V;\ell,r)$. A solution of the strong Maurer-Cartan
equation can be seen as the associative analogue of a closed $2$-form
complementary to a Poisson structure. Rota-Baxter operators and their
generalization $\mathcal{O}$-operators are becoming rather popular due to
their interest in various domain like nonassociative algebra and applied
algebra. Their deformations and their controlling cohomologies have been
studied for Lie algebras in [46] and for associative algebras in [20], see
also [13] [35].

The first instances of Hom-type algebras appeared in $q$-deformations of
algebras of vector fields in physics contexts. It turns out that the Jacobi
condition is no longer satisfied in the $q$-deformations of Witt algebra and
Virasoro algebra using Jackson derivative. Hartwig, Larsson and Silvestrov
[28] introduced a so called Hom-Lie algebras, generalizing Lie algebras and
where the usual Jacobi identity is twisted by a homomorphism. The pending Hom-
associative algebras were introduced in [42]. Various classical structures
were generalized within this framework. A huge research activity was dedicated
to Hom-type algebras, due in part to the prospect of having a general
framework in which one can produce many types of natural deformations which
are of interest to both mathematicians and physicists. Cohomology and
deformations of Hom-Lie and Hom-associative algebras were studied in [4, 30,
45]. Notice that Cohomology and deformations of $\mathcal{O}$-operators on
Hom-Lie algebras were discussed in [3].

The purpose of this paper is to introduce and study a cohomology controlling
deformations of $\mathcal{O}$-operators on Hom-associative algebras.
Therefore, we consider one parameter formal deformations and provide relevant
results about infinitesimal deformations and general $n$-order deformations.
In addition, infinitesimal deformations are characterized using Nijenhuis
elements. In Section 2, relevant definitions and preliminary results are
presented. They deal with Hom-associative algebras and Hom-dendriform algebra,
bimodule structures, Rota-Baxter structures, $\mathcal{O}$-operators,
Nijenhuis operators, cohomology and graded Lie algebras. Main results of this
paper are given in Section 3 in which we provide Cohomologies of
$\mathcal{O}$-operators on Hom-associative algebras and their deformations are
considered in Section 4.

##  2 Preliminaries

In this section, we recall the relevant definitions and the notion of
$\mathcal{O}$-operator on Hom-associative algebra. We also recall Hom-
dendriform structures induced from $\mathcal{O}$-operators. Finally, we
construct a graded Lie algebra with explicit graded Lie bracket whose Maurer-
Cartan elements are given by $\mathcal{O}$-operators.

###  2.1 Hom-associative and Hom-dendriform algebras

######  Definition 2.1.

A Hom-associative algebra is a Hom-module $(A,\alpha)$, consisting of a
$\mathbb{K}$-vector space $A$ and a linear map $\alpha$, together with a
bilinear map $\mu:A\times A\rightarrow A,~{}(a,b)\mapsto a\cdot b$, that
satisfies

$\displaystyle\alpha(a)\cdot(b\cdot c)=(a\cdot
b)\cdot\alpha(c),~{}~{}~{}\text{ for all }~{}a,b,c\in A.$

A Hom-associative algebra is called multiplicative if $\alpha(a\cdot
b)=\alpha(a)\cdot\alpha(b)$. By a Hom-associative algebra, we shall always
mean a multiplicative one, unless otherwise specified.

######  Definition 2.2.

Let $(A,\mu,\alpha)$ be a Hom-associative algebra and $M$ a vector space. Let
$\phi\in gl(M)$ and $l:A\otimes M\to M,\ r:M\otimes A\to M$ be two linear
maps. We say that the tuple $(M,l,r,\phi)$ is a bimodule of $A$ if for any
$x,y\in A$ and $v\in M$

$\displaystyle\phi l(x,v)=l(\alpha(x),\phi(v)),$ $\displaystyle\ \phi
r(v,x)=r(\phi(v),\alpha(x)),$ $\displaystyle l(xy,\phi(v))=$ $\displaystyle
l(\alpha(x),l(y,v)),$ $\displaystyle r(\phi(v),xy)=$ $\displaystyle
r(r(v,x),\alpha(y)),$ $\displaystyle l(\alpha(x),r(v,y))=$ $\displaystyle
r(l(x,v),\alpha(y)).$

Unless otherwise stated, $l(x,v)$ (resp. $r(x,v)$) may be denoted, depending
on the situation, by $xv$ or $l(x)v$ (resp. $vx$ or $r(x)v$).

Recall that if we consider a Hom-associative algebra $(A,\mu,\alpha)$. Then
$(M,l,r,\phi)$ is a bimodule of $A$ if and only if the vector space $A\oplus
M$ (the split extension of $A$ by $M$) carries a Hom-associative structure
with product given by

$\displaystyle(x,u)\cdot_{l,r}(y,v)=$ $\displaystyle(x\cdot y,l(x,u)+r(v,y)),\
\forall x,y\in A,u,v\in M,$ $\displaystyle(\alpha,\phi)(x,u)=$
$\displaystyle(\alpha(x),\phi(u)).$

This is called the semi-direct product of $A$ with $M$.

In [4, 44] the authors define a Hochschild-type cohomology of a Hom-
associative algebra and study the one parameter formal deformation theory for
these type of algebras. Let $(A,\alpha,\mu)$ be a Hom-associative algebra and
$(M,l,r,\phi)$ be a bimodule. For each $n\geq 1$, the group of $n$-cochains is
defined as

$\displaystyle C^{n}_{\alpha,\phi}(A,M):=\\{f:A^{\otimes n}\rightarrow
M|~{}\phi\circ f(a_{1},\ldots,a_{n})=f(\alpha(a_{1}),\ldots,\alpha(a_{n}))\\}$

and the differential $\delta_{\alpha,\phi}:C^{n}_{\alpha,\phi}(A,M)\rightarrow
C^{n+1}_{\alpha,\phi}(A,M)$ is given by

$\displaystyle(\delta_{\alpha,\phi}f)(a_{1},\ldots,a_{n+1})=~{}$
$\displaystyle\alpha^{n-1}(a_{1})f(a_{2},\ldots,a_{n+1})$ (2.1)
$\displaystyle+\sum_{i=1}^{n}(-1)^{i}f(\alpha(a_{1}),\ldots,\alpha(a_{i-1}),\mu(a_{i},a_{i+1}),\alpha(a_{i+2}),\ldots,\alpha(a_{n+1}))$
$\displaystyle+(-1)^{n+1}f(a_{1},\ldots,a_{n})\alpha^{n-1}(a_{n+1}).$

for $a_{1},\ldots,a_{n+1}\in A$. The cohomology of this complex is called the
Hochschild cohomology of $A$ with coefficients in the bimodule $(M,l,r,\phi)$
and the cohomology groups are denoted by $H^{\bullet}_{\alpha,\phi}(A,M)$.

In [47], Uchino gave a characterization of an $\mathcal{O}$-operator on an
associative algebra. This result is still true for Hom-associative algebras.
Recall first the notion of an $\mathcal{O}$-operator on a Hom-associative
algebra.

######  Definition 2.3.

Let $(A,\mu,\alpha)$ be a Hom-associative algebra and $(M,l,r,\phi)$ be a
bimodule. A linear map $T:M\to A$ is called an $\mathcal{O}$-operator with
respect to $M$ if it satisfies

$\displaystyle T\phi=\alpha T,\ \quad
T(u)T(v)=T(l(T(u))v+r(T(v))u),~{}~{}~{}\text{ for all }u,v\in M.$

######  Example 2.4.

A Rota-Baxter operator $\mathcal{R}:A\to A$ of weight $0$ is an
$\mathcal{O}$-operator on $A$ with respect to the adjoint representation
$(A,L,R,\alpha)$.

######  Example 2.5.

Let $\\{x_{1},x_{2},x_{3}\\}$ be a basis of a $3$-dimensional vector space $A$
over $\mathbb{K}$. Consider the following multiplication $\mu$ and linear map
$\alpha$ on $A$ that define Hom-associative algebras over $\mathbb{K}^{3}$:

$\begin{array}[]{ll}\begin{array}[]{lll}\mu(x_{1},x_{1})&=&a\ x_{1},\\\
\mu(x_{1},x_{2})&=&\mu(x_{2},x_{1})=a\ x_{2},\\\
\mu(x_{1},x_{3})&=&\mu(x_{3},x_{1})=b\ x_{3},\\\
\end{array}&\quad\begin{array}[]{lll}\mu(x_{2},x_{2})&=&a\ x_{2},\\\
\mu(x_{2},x_{3})&=&b\ x_{3},\\\
\mu(x_{3},x_{2})&=&\mu(x_{3},x_{3})=0,\end{array}\end{array}$

$\alpha(x_{1})=a\ x_{1},\quad\alpha(x_{2})=a\ x_{2},\quad\alpha(x_{3})=b\
x_{3},$

where $a,b$ are parameters in $\mathbb{K}$. The algebras are not associative
when $a\neq b$ and $b\neq 0$, since
$\mu(\mu(x_{1},x_{1}),x_{3})-\mu(x_{1},\mu(x_{1},x_{3}))=(a-b)bx_{3}.$

Let $\mathcal{R}$ be the operator defined with respect to the basis
$\\{x_{1},x_{2},x_{3}\\}$ by

$\mathcal{R}(x_{1})=\rho_{1}x_{3},\ \mathcal{R}(x_{2})=\rho_{2}x_{3},\
\mathcal{R}(x_{3})=0,$

where $\rho_{1},\rho_{2}$ are parameters in $\mathbb{K}$. Then $\mathcal{R}$
is a Rota-Baxter operator of weight $0$ on the Hom- associative algebra
$(A,\mu,\alpha)$.

######  Example 2.6.

Let $(A,\mu,\alpha)$ be a Hom-associative algebra and $(M,l,r,\phi)$ be an
$A$-bimodule. We can easily, verify that $A\oplus M$ is an $A$-bimodule under
the following actions:

$\displaystyle a\bullet(b,m)=(\mu(a,b),a\cdot m),\quad(b,m)\bullet a=(0,m\cdot
a),\forall a,b\in A,\ m\in M.$

Define the linear map $T:A\oplus M\to A,\ (a,m)\mapsto a$. Then $T$ is an
$\mathcal{O}$-operator on $A$ with respect to the bimodule $A\oplus M$.

######  Proposition 2.7.

A linear map $T:M\rightarrow A$ is an $\mathcal{O}$-operator on a Hom-
associative algebra $A$ with respect to the $A$-bimodule $M$ if and only if
the graph of $T$,

$\displaystyle\mathrm{Gr}(T)=\\{(T(u),u)|~{}u\in M\\}$

is a subalgebra of the semi-direct product algebra $A\oplus M$.

In the following proposition, we show that an $\mathcal{O}$-operator can be
lifted up the Rota-Baxter operator.

######  Proposition 2.8.

Let $(A,\mu,\alpha)$ be a Hom-associative algebra, $(M,l,r,\phi)$ be an
$A$-bimodule and $T:M\to A$ be a linear map. Define $\widehat{T}\in
End(A\oplus M)$ by $\widehat{T}(a,m)=(T(m),0)$. Then $T$ is an
$\mathcal{O}$-operator if and only $\widehat{T}$ is a Rota-Baxter operator on
$A\oplus M$.

Another characterization of an $\mathcal{O}$-operator can be given in terms of
Nijenhuis operator on Hom-associative algebras.

######  Definition 2.9.

Let $(A,\mu,\alpha)$ be a Hom-associative algebra. A linear map
$N:A\rightarrow A$ is said to be a Nijenhuis operator if
$N\circ\alpha=\alpha\circ N$ and its Nijenhuis torsion vanishes, i.e.

$\displaystyle N(x)\cdot N(y)=N(N(x)\cdot y+x\cdot N(y)-N(x\cdot
y)),~{}~{}\text{ for all }x,y\in A.$

Note that the deformed multiplication $\mu_{N}:A\otimes A\rightarrow A$ given
by

$\mu_{N}(x,y)=N(x)\cdot y+x\cdot N(y)-N(x\cdot y),$

defines a new Hom-associative multiplication on $A$, and $N$ becomes an
algebra morphism from $(A,\mu_{N},\alpha)$ to $(A,\mu,\alpha).$ In addition
$\mu$ and $\mu_{N}$ are compatible, i.e. $\mu+\lambda\mu_{N}$ still a Hom-
associative structure for each $\lambda\in\mathbb{K}$. Then the pair
$(\mu,\mu_{N})$ becomes a Hom-quantum bi-Hamiltonian system.

The following result is straightforward, so we omit details.

######  Proposition 2.10.

A linear map $T:M\rightarrow A$ is an $\mathcal{O}$-operator on
$(A,\cdot,\alpha)$ with respect to the bimodule $(M,l,r,\phi)$ if and only if
$N_{T}=\begin{pmatrix}0&T\\\ 0&0\end{pmatrix}:A\oplus M\rightarrow A\oplus M$
is a Nijenhuis operator on the Hom-associative algebra $A\oplus M$.

Dendriform structures were first introduced by Loday in his study of the
periodicity phenomenons in algebraic $K$-theory [36]. They become rather
popular in the last 20 years due to their connection with Rota-Baxter
algebras, shuffle algebras and combinatorics [6, 48, 27]. In [43], Makhlouf
introduced the notion of Hom-dendriform algebra.

######  Definition 2.11.

A Hom-dendriform algebra is a vector space $D$ together with three linear maps
$\prec,\succ:D\otimes D\rightarrow D$ and $\alpha:D\to D$ satisfying the
following three identities

$\displaystyle(x\prec y)\prec\alpha(z)=~{}\alpha(x)\prec(y\prec z+y\succ z),$
(2.2) $\displaystyle(x\succ y)\prec\alpha(z)=~{}\alpha(x)\succ(y\prec z),$
(2.3) $\displaystyle(x\prec y+x\succ y)\succ\alpha(z)=~{}\alpha(x)\succ(y\succ
z),$ (2.4)

for all $x,y,z\in D$.

It follows from (2.2)-(2.4) that the new product $\star:D\otimes D\rightarrow
D$ defined by $x\star y=x\prec y~{}+~{}y\succ x$ turns out to be Hom-
associative. Thus, a Hom-dendriform algebra can be seen as a splitting of a
Hom-associative algebra.

An $\mathcal{O}$-operator has an underlying Hom-dendriform structure on the
bimodule.

######  Proposition 2.12.

Let $T:M\rightarrow A$ be an $\mathcal{O}$-operator on a Hom-associative
algebra $A$ with respect to the bimodule $M$. Then the vector space $M$
carries a Hom-dendriform structure with

$\displaystyle u\prec v=r(T(v))u~{}~{}~{}\text{ and }~{}~{}~{}u\succ
v=l(T(u))v,~{}~{}~{}\text{ for all }u,v\in M.$

Moreover, $(M,\star_{T}=\prec+\succ,\phi)$ is a Hom-associative algebra.

###  2.2 Graded Lie algebras, Lift and Bidegree

In this section, we discuss a graded Lie algebra related to Hom-associative
algebras and its corresponding Maurer-Cartan equation for which a solution
characterize an $\mathcal{O}$-operator. It is shown in [4] that a Gerstenhaber
algebra structure is constructed on the cohomology of Hom-associative algebras
using a cup product together with the degree $-1$ graded Lie bracket. We aim
to show that an $\mathcal{O}$-operator can be characterized by certain
solutions of the corresponding Maurer-Cartan equation (see Theorem 3.2).

######  Definition 2.13.

Let $(G=\oplus_{k=1}^{\infty}G_{i},[\cdot,\cdot],d)$ be a differential graded
Lie algebra. A degree $1$ element $\theta\in G_{2}$ is called a Maurer-Cartan
element of $G$ if it satisfies the following Maurer-Cartan equation:

$d\theta+\frac{1}{2}[\theta,\theta]=0.$ (2.5)

A graded Lie algebra is a differential graded Lie algebra with $d=0$. Then we
have

######  Proposition 2.14.

Let $(G=\oplus_{k=1}^{\infty}G_{i},[\cdot,\cdot])$ be a graded Lie algebra and
let $\mu\in G_{2}$ be a Maurer-Cartan element. Then the map

$d_{\mu}:G\longrightarrow G,\ d_{\mu}(\nu):=[\mu,\nu],\quad\forall\nu\in G,$

is a differential on $G$. For any $\nu\in G$, the sum $\mu+\nu$ is a Maurer-
Cartan element of the graded Lie algebra $(G,[\cdot,\cdot])$ if and only if
$\nu$ is a Maurer-Cartan element of the differential graded Lie algebra
$(G,[\cdot,\cdot],d_{\mu})$.

Let $A$ be a vector space and $\alpha:A\to A$ be a linear map. Denote by
$V_{n}^{\alpha}$ the set of all linear maps $f:A^{\otimes n}\to A$ satisfying

$\alpha(f(a_{1},\cdots,a_{n}))=f(\alpha(a_{1}),\cdots,\alpha(a_{n})),~{}~{}~{}~{}\textrm{for
all}\ a_{i}\in A.$

The graded space $G=\oplus_{n\geq 1}V_{n}^{\alpha}$ carries a graded Lie
algebra structure $[-,-]_{\alpha}:V_{m}^{\alpha}\times V_{n}^{\alpha}\to
V_{m+n-1}^{\alpha}$ given by

$[f,g]_{\alpha}=f\circ g-(-1)^{(m-1)(n-1)}g\circ f,~{}~{}~{}~{}\textrm{for all
}\ f\in V_{m}^{\alpha},g\in V_{n}^{\alpha},$

where

$\displaystyle(f\circ g)(a_{1},\ldots,a_{m+n-1})=$
$\displaystyle\sum_{i=1}^{m}(-1)^{(i-1)(n-1)}~{}f(\alpha^{n-1}(a_{1}),\ldots,\alpha^{n-1}(a_{i-1}),g(a_{i},\ldots,a_{i+n-1}),\ldots,\alpha^{n-1}(a_{m+n-1})).$

Let $\mu\in V_{2}^{\alpha}$. By a direct computation, we have

$\displaystyle\frac{1}{2}[\mu,\mu]_{\alpha}(x,y,z)=\mu\circ\mu(x,y,z)=\mu(\mu(x,y),\alpha(z))-\mu(\alpha(x),\mu(y,z)),$

which implies that $\mu$ defines a Hom-associative structure on $A$ if an only
if $[\mu,\mu]_{\alpha}=0$, i.e. $\mu$ is a Maurer-Cartan element of the graded
Lie algebra $G$.

In the following, we introduce the notion of bidegree and discuss it with
respect to Gerstenhaber bracket. Let $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$
be two vector spaces and $c:\mathcal{A}^{\otimes n}_{2}\to\mathcal{A}_{1}$ be
a linear map, or a cochain in $C^{n}(\mathcal{A}_{2},\mathcal{A}_{1})$. We
construct a cochain $\hat{c}\in C^{n}(\mathcal{A}_{1}\oplus\mathcal{A}_{2})$
by

$\hat{c}\Big{(}(a_{1},x_{1})\otimes...\otimes(a_{n},x_{n})\Big{)}:=(c(x_{1},...,x_{n}),0).$

In general, for a given multilinear map
$f:\mathcal{A}_{i_{1}}\otimes\mathcal{A}_{i_{2}}\otimes...\otimes\mathcal{A}_{i_{n}}\to\mathcal{A}_{j}$,
$i_{1},...,i_{n},j\in\\{1,2\\}$, we define a cochain $\hat{f}\in
C^{n}(\mathcal{A}_{1}\oplus\mathcal{A}_{2})$ by

$\displaystyle\hat{f}:=\left\\{\begin{array}[]{ll}f&\text{on
$\mathcal{A}_{i_{1}}\otimes\mathcal{A}_{i_{2}}\otimes...\otimes\mathcal{A}_{i_{n}}$,}\\\
0&\text{all other cases.}\end{array}\right.$

We call the cochain $\hat{f}$ a horizontal lift of $f$, or simply a lift. For
instance, the lifts of
$\alpha:\mathcal{A}_{1}\otimes\mathcal{A}_{1}\to\mathcal{A}_{1}$,
$\beta:\mathcal{A}_{1}\otimes\mathcal{A}_{2}\to\mathcal{A}_{2}$ and
$\gamma:\mathcal{A}_{2}\otimes\mathcal{A}_{1}\to\mathcal{A}_{2}$ are defined
respectively by

$\displaystyle\hat{\alpha}((a,x),(b,y))$ $\displaystyle=$
$\displaystyle(\alpha(a,b),0),$ (2.7) $\displaystyle\hat{\beta}((a,x),(b,y))$
$\displaystyle=$ $\displaystyle(0,\beta(a,y)),$ (2.8)
$\displaystyle\hat{\gamma}((a,x),(b,y))$ $\displaystyle=$
$\displaystyle(0,\gamma(x,b)).$ (2.9)

Let $H:\mathcal{A}_{2}\to\mathcal{A}_{1}$ (resp.
$H:\mathcal{A}_{1}\to\mathcal{A}_{2}$) be a 1 cochain. The lift is defined by

$\widehat{H}(a,x)=(H(x),0)\ \ \ (\text{resp. $\widehat{H}(a,x)=(0,H(a))$}).$

For any $(a,x)\in\mathcal{A}_{1}\oplus\mathcal{A}_{2}$,

We denote by $\mathcal{A}^{l,k}$ the direct sum of all $(l+k)$-tensor powers
of $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$, where $l$ (resp. $k$) is the
number of $\mathcal{A}_{1}$ (resp. $\mathcal{A}_{2}$). For instance,

$\mathcal{A}^{1,2}:=(\mathcal{A}_{1}\otimes\mathcal{A}_{2}\otimes\mathcal{A}_{2})\oplus(\mathcal{A}_{2}\otimes\mathcal{A}_{1}\otimes\mathcal{A}_{2})\oplus(\mathcal{A}_{2}\otimes\mathcal{A}_{2}\otimes\mathcal{A}_{1}).$

The tensor space $(\mathcal{A}_{1}\oplus\mathcal{A}_{2})^{\otimes n}$ is
expanded into the direct sum of $\mathcal{A}^{l,k}$, $l+k=n$. For instance,

$(\mathcal{A}_{1}\oplus\mathcal{A}_{2})^{\otimes
2}=\mathcal{A}^{2,0}\oplus\mathcal{A}^{1,1}\oplus\mathcal{A}^{0,2}.$

Let $f$ be a $n$-cochain in $C^{n}(\mathcal{A}_{1}\oplus\mathcal{A}_{2})$. We
say that the bidegree of $f$ is $k|l$ if $f$ is an element in
$C^{n}(\mathcal{A}^{l,k-1},\mathcal{A}_{1})$ or in
$C^{n}(\mathcal{A}^{l-1,k},\mathcal{A}_{2})$, where $n=l+k-1$. We denote the
bidegree of $f$ by $||f||=k|l$. In general, cochains do not have bidegree. We
call a cochain $f$ a homogeneous cochain, if $f$ has a bidegree.
We have $k+l\geq 2$, because $n\geq 1$. Thus there are no cochains of bidegree
$0|0$ or $1|0$ or $0|1$. We recall $\hat{\alpha},\hat{\beta},\hat{\gamma}\in
C^{2}(\mathcal{A}_{1}\oplus\mathcal{A}_{2})$ in (2.7), (2.8) and (2.9). One
can easily see $||\hat{\alpha}||=||\hat{\beta}||=||\hat{\gamma}||=1|2$.

######  Lemma 2.15.

If $||f||=k|0$ (resp. $0|k$) and $||g||=l|0$ (resp. $0|l$), then
$[f,g]_{\alpha}=0$.

######  Proposition 2.16.

Let $f,g\in C^{\bullet}(\mathcal{A}_{1}\oplus\mathcal{A}_{2})$. If
$||f||=k_{f}|l_{f}$ and $||g||=k_{g}|l_{g}$, then the Gerstenhaber bracket
$[f,g]_{\alpha}$ has the bidegree $k_{f}+k_{g}-1|l_{f}+l_{g}-1$.

##  3 Cohomologies of $\mathcal{O}$-operators on Hom-associative algebras

###  3.1 Cohomology of $\mathcal{O}$-operator

Let $A$ and $M$ be two vector spaces equipped with maps $\mu:A^{\otimes
2}\rightarrow A$, $l:A\otimes M\rightarrow M$, $r:M\otimes A\rightarrow M$,
$\alpha:A\to A$ and $\phi:M\to M$. Set $V=A\oplus M$. It is, naturally
equipped with the linear map $\alpha+\phi:(x,u)\mapsto(\alpha(x),\phi(u))$.
Now, consider the graded Lie algebra structure on $\mathfrak{g}=\oplus_{n\geq
1}V_{n}^{\alpha+\phi}$ Observe that the elements $\mu,l,r\in
V_{2}^{\alpha+\phi}$. Therefore, $\mu+l+r\in V_{2}^{\alpha+\phi}$

######  Proposition 3.1.

The product $\mu$ defines a Hom-associative structure on $A$ and $l,r$ define
an $A$-bimodule structure on $M$ with respect to $\phi$ if and only if
$\mu+l+r\in V_{2}^{\alpha+\phi}$ is a Maurer-Cartan element in $\mathfrak{g}.$

###### Proof.

By a direct computation, we have

$\displaystyle\frac{1}{2}[\mu+l+r,\mu+l+r]_{\alpha}((a_{1},m_{1}),(a_{2},m_{2}),(a_{3},m_{3}))$
$\displaystyle=$
$\displaystyle(\mu+l+r)\circ(\mu+l+r)((a_{1},m_{1}),(a_{2},m_{2}),(a_{3},m_{3})$
$\displaystyle=$
$\displaystyle(\mu+l+r)\big{(}(\mu+l+r)((a_{1},m_{1}),(a_{2},m_{2})),(\alpha(a_{3}),\phi(m_{3}))\big{)}$
$\displaystyle-(\mu+l+r)\big{(}(\alpha(a_{1}),\phi(m_{1})),(\mu+l+r)((a_{2},m_{2}),(a_{3},m_{3}))\big{)}$
$\displaystyle=$
$\displaystyle\big{(}\mu(\mu(a_{1},a_{2}),\alpha(a_{3})),~{}l(\mu(a_{1},a_{2}),\phi(m_{3}))+r(l(a_{1},m_{2}),\alpha(a_{3}))+r(r(m_{1},a_{2}),\alpha(a_{3}))\big{)}$
$\displaystyle-\big{(}\mu(\alpha(a_{1}),\mu(a_{2},a_{3})),~{}l(\alpha(a_{1}),l(a_{2},m_{3}))+l(\alpha(a_{1}),r(m_{2},a_{3}))+r(r(m_{1},a_{2}),\alpha(a_{3}))\big{)}.$

Then $\mu$ defines a Hom-associative structure on $A$ and $l,r$ define an
$A$-bimodule structure on $M$ with respect to $\phi$ if and only if $\mu+l+r$
is a Maurer-Cartan element in $\mathfrak{g}.$ ∎

We aim in the following the Voronov’s derived bracket construction [50] (see
also [49]) to get a graded Lie algebra structure on $\oplus_{n\geq
1}\text{Hom}(M^{\otimes n},A)$, where $A$ is a regular Hom-associative algebra
and $(l,r)$ defines an $A$-bimodule structure on $M$ with respect to $\phi$
supposed to be invertible.

According to the above proposition, the graded Lie algebra
$(\mathfrak{g},[~{},~{}])$ together with the differential
$d_{\mu+l+r}=[\mu+l+r,~{}]_{\alpha}$ becomes a differential graded Lie
algebra. Moreover, it is easy to see that the graded subspace $\oplus_{n\geq
1}\text{Hom}(M^{\otimes n},A)$ is an abelian subalgebra.

Define, for $P\in\text{Hom}(M^{\otimes m},A),~{}Q\in\text{Hom}(M^{\otimes
n},A)$, $m,n\geq 1$, the bracket

$\displaystyle\llbracket
P,Q\rrbracket_{\alpha}:=(-1)^{m}[[\mu+l+r,P]_{\alpha},Q]_{\alpha},$ (3.1)

It is well defined according to Proposition 2.16,
$\llbracket-,-\rrbracket_{\alpha}$ and may be explicitly given by

$\displaystyle\llbracket P,Q\rrbracket_{\alpha}(u_{1},\ldots,u_{m+n})$ (3.2)
$\displaystyle=\sum_{i=1}^{m}(-1)^{(i-1)n}P(\phi^{m}(u_{1}),\ldots,\phi^{m}(u_{i-1}),l(Q(u_{i},\ldots,u_{i+n-1}),\phi^{m-1}(u_{i+n})),\ldots,\phi^{m}(u_{m+n}))$
$\displaystyle-\sum_{i=1}^{m}(-1)^{in}P(\phi^{m}(u_{1}),\ldots,\phi^{m}(u_{i-1}),r(\phi^{m-1}(u_{i}),Q(u_{i+1},\ldots,u_{i+n})),\phi^{m}(u_{i+n+1}),\ldots,\phi^{m}(u_{m+n}))$
$\displaystyle-(-1)^{mn}\bigg{\\{}\sum_{i=1}^{n}(-1)^{(i-1)m}~{}Q(\phi^{n}(u_{1}),\ldots,\phi^{n}(u_{i-1}),l(P(u_{i},\ldots,u_{i+m-1}),\phi^{n-1}(u_{i+m})),\ldots,\phi^{n}(u_{m+n}))$
$\displaystyle-\sum_{i=1}^{n}(-1)^{im}~{}Q(\phi^{n}(u_{1}),\ldots,\phi^{n}(u_{i-1}),r(\phi^{n-1}(u_{i}),P(u_{i+1},\ldots,u_{i+m})),\phi^{n}(u_{i+m+1}),\ldots,\phi^{n}(u_{m+n}))\bigg{\\}}$
$\displaystyle+(-1)^{mn}\bigg{\\{}\mu(P(\phi^{n-1}(u_{1}),\ldots,\phi^{n-1}(u_{m})),Q(\phi^{m-1}(u_{m+1}),\ldots,\phi^{m-1}(u_{m+n})))$
$\displaystyle-(-1)^{mn}~{}\mu(Q(\phi^{m-1}(u_{1}),\ldots,\phi^{m-1}(u_{n})),P(\phi^{n-1}(u_{n+1}),\ldots,\phi^{n-1}(u_{m+n})))\bigg{\\}},$

for $u_{1},\ldots,u_{m+n}\in M.$ We may extend the graded Lie bracket to
$\oplus_{n\geq 0}\text{Hom}(M^{\otimes n},A)$ as follows. For
$P\in\text{Hom}(M^{\otimes m},A)$ and $a\in A$, the bracket is

$\displaystyle\llbracket P,a\rrbracket_{\alpha}(u_{1},\ldots,u_{m})$ (3.3)
$\displaystyle=$
$\displaystyle\sum_{i=1}^{m}P(\phi^{m}(u_{1}),\ldots,\phi^{m}(u_{i-1}),l(a,\phi^{m-1}(u_{i}))-r(\phi^{m-1}(u_{i}),a),\phi^{m}(u_{i+1}),\ldots,\phi^{m}(u_{m}))$
$\displaystyle+\mu(P(\phi^{-1}(u_{1}),\ldots,\phi^{-1}(u_{m})),a)-\mu(a,\cdot
P(\phi^{-1}(u_{1}),\ldots,\phi^{-1}(u_{m}))$

and for $a,b\in A$, we define $\llbracket a,b\rrbracket_{\alpha}$ to be the
Lie bracket $[a,b]_{C}=\mu(a,b)-\mu(b,a).$

Thus, for any $T,T^{\prime}\in\text{Hom}(M,A)$, we have from (3.2) that

$\displaystyle\llbracket
T,T^{\prime}\rrbracket_{\alpha}(u,v)=T(T^{\prime}(u)v)+T(uT^{\prime}(v))+T^{\prime}(T(u)v)+T^{\prime}(uT(v))-T(u)\cdot
T^{\prime}(v)-T^{\prime}(u)\cdot T(v).$ (3.4)

For any $n\geq 0$, we define $C^{n}_{\alpha}(M,A):=\mathrm{Hom}(M^{\otimes
n},A)$ and consider the graded vector space $C^{\bullet}(M,A)=\oplus_{n\geq
0}C^{n}_{\alpha}(M,A)=\oplus_{n\geq 0}\mathrm{Hom}(M^{\otimes n},A)$. Thus, we
obtain the following result.

######  Theorem 3.2.

The graded vector space $C^{\bullet}_{\alpha}(M,A)=\oplus_{n\geq
0}\mathrm{Hom}(M^{\otimes n},A)$ together with the above defined bracket
$\llbracket~{},~{}\rrbracket_{\alpha}$ form a graded Lie algebra. A linear map
$T:M\rightarrow A$ is an $\mathcal{O}$-operator on $A$ with respect to the
bimodule $M$ if and only if $T\in C^{1}_{\alpha}(M,A)$ is a Maurer-Cartan
element in $(C^{\bullet}_{\alpha}(M,A),\llbracket~{},~{}\rrbracket_{\alpha})$,
i.e. $T$ satisfies $\llbracket T,T\rrbracket_{\alpha}=0$.

Therefore, from the general principle of Maurer-Cartan elements, we obtain.

######  Theorem 3.3.

Let $T:M\rightarrow A$ be an $\mathcal{O}$-operator on $A$ with respect to the
bimodule $M$. Then $T$ induces a differential $d_{T}=\llbracket
T,~{}\rrbracket_{\alpha}$ which makes the graded Lie algebra
$(C^{\bullet}_{\alpha}(M,A),\llbracket~{},~{}\rrbracket_{\alpha})$ into a
differential graded Lie algebra (dgLa).

Moreover, for any linear map $T^{\prime}:M\rightarrow A$, the sum
$T+T^{\prime}$ is still an $\mathcal{O}$-operator if and only if $T^{\prime}$
is a Maurer-Cartan element in the dgLa
$(C^{\bullet}_{\alpha}(M,A),\llbracket~{},~{}\rrbracket_{\alpha},d_{T})$, that
is

$\displaystyle\llbracket
T+T^{\prime},T+T^{\prime}\rrbracket_{\alpha}=0~{}\Longleftrightarrow~{}d_{T}(T^{\prime})+\frac{1}{2}\llbracket
T^{\prime},T^{\prime}\rrbracket_{\alpha}=0.$

It follows from the above theorem that if $T$ is an $\mathcal{O}$-operator,
then $(C^{\bullet}_{\alpha}(M,A),d_{T}=\llbracket T,~{}\rrbracket_{\alpha})$
is a cochain complex. Its cohomology is called the cohomology of the
$\mathcal{O}$-operator $T$.

In the next paragraph, we interpret this cohomology as the Hochschild
cohomology of a certain algebra with coefficients in a suitable bimodule. Then
we will use the usual notation for Hochschild cohomology to denote the
cohomology of an $\mathcal{O}$-operator.

###  3.2 Cohomology of $\mathcal{O}$-operators as Hochschild cohomology

The aim of this section is to show that the cohomology of an
$\mathcal{O}$-operator can also be described as the Hochschild cohomology of a
certain Hom-associative algebra with coefficients in a suitable bimodule.

Let $T:M\rightarrow A$ be an $\mathcal{O}$-operator on $A$ with respect to the
bimodule $(M,l,r,\phi)$. Then by Proposition 2.12 the vector space $M$ carries
a Hom-associative algebra structure with the product

$\displaystyle u\star_{T}v=r(u,T(v))+l(T(u),v),~{}~{}~{}\text{ for }u,v\in M.$
(3.5)

According to (2.10), there is a Hom-associative algebra structure on $A\oplus
M\backsimeq M\oplus A$ defined as

$\displaystyle(x,u)\cdot_{\overline{T}}(y,v)=$
$\displaystyle(Tu,0)\cdot_{(l,r)}(y,v)+(x,u)\cdot_{(l,r)}(Tv,0)-(T(l(x,v)+r(u,y),0)$
$\displaystyle=$ $\displaystyle(Tu\cdot y,l(Tu,v))+(x\cdot
Tv,-r(u,Tv))-(T(l(x,v)+T(r(u,y)),0)$ $\displaystyle=$
$\displaystyle(r_{T}(u,y)+l_{T}(v,x),u\star v),$

where

$l_{T}(u,x)=Tu\cdot x-T(r(u,x)),\ r_{T}(x,u)=x\cdot Tu-T(l(x,u),\ \forall x\in
A,u\in M.$ (3.6)

Therefore we have the following result.

######  Lemma 3.4.

Let $T:M\rightarrow A$ be an $\mathcal{O}$-operator on $A$ with respect to the
bimodule $(M,l,r,\phi)$. Then $(l_{T},~{}r_{T})$ defines an $M$-bimodule
structure on $A$ with respect to $\alpha$.

By Lemma 3.4 we obtain an $M$-bimodule structure on the vector space $A$.
Therefore, we may consider the corresponding Hochschild cohomology of $M$ with
coefficients in $A$. We define

$\displaystyle C^{n}_{\alpha}(M,A):=\text{Hom}(M^{\otimes n},A),~{}~{}\text{
for }n\geq 0$

and the differential by

$\displaystyle d_{H}(a)(m)=l_{T}(m,a)-r_{T}(a,m)=T(m)\cdot a-T(ma)-a\cdot
T(m)+T(am),~{}~{}~{}\text{ for }a\in A=C^{0}_{\alpha}(M,A)$ (3.7)

and

$\displaystyle(d_{H}f)(u_{1},\ldots,u_{n+1})=~{}T(\phi^{n-1}(u_{1}))\cdot
f(u_{2},\ldots,u_{n+1})-T(r(\phi^{n-1}(u_{1}),f(u_{2},\ldots,u_{n+1})))$
$\displaystyle+\sum_{i=1}^{n}(-1)^{i}~{}f(\phi(u_{1}),\ldots,\phi(u_{i-1}),r(u_{i},T(u_{i+1}))+l(T(u_{i}),u_{i+1}),\ldots,\phi(u_{n+1}))$
$\displaystyle+(-1)^{n+1}~{}f(u_{1},\ldots,u_{n})\cdot
T(\phi^{n-1}(u_{n+1}))-(-1)^{n+1}T(l(f(u_{1},\ldots,u_{n}),\phi^{n-1}(u_{n+1}))).$

We denote the group of $n$-cocycles by $Z^{n}_{\alpha}(M,A)$ and the group of
$n$-coboundaries by $B^{n}_{\alpha}(M,A)$. The corresponding cohomology groups
are defined by

$H^{n}_{\alpha}(M,A)=Z^{n}_{\alpha}(M,A)/B^{n}_{\alpha}(M,A),\quad n\geq 0.$

In particular, we have

$\displaystyle H^{0}(M,A)=~{}$ $\displaystyle\\{a\in A|~{}d_{H}(a)=0\\}$
$\displaystyle=~{}$ $\displaystyle\\{a\in A|~{}a\cdot T(m)-T(m)\cdot
a=T(l(a,m)-r(m,a)),~{}\forall m\in M\\}.$

From this definition, it is easy to see that if $a,b\in H^{0}(M,A)$, then
their commutator $[a,b]:=a\cdot b-b\cdot a$ is also in $H^{0}(M,A)$. This
shows that $H^{0}(M,A)$ has a Hom-Lie algebra structure induced from that of
$A$.

Note that a linear map $f:M\rightarrow A$ (i.e. $f\in C^{1}(M,A)$) is closed
if it satisfies

$\displaystyle T(u)\cdot f(v)+f(u)\cdot T(v)-T(uf(v)+f(u)v)-f(uT(v)+T(u)v)=0,$
(3.8)

for all $u,v\in M.$

For an $\mathcal{O}$-operator $T$ on $A$ with respect to the bimodule $M$, we
get two coboundary operators $d_{T}=\llbracket T,~{}\rrbracket_{\alpha}$ and
$d_{H}$ on the same graded vector space
$C^{\bullet}_{\alpha}(M,A)=\oplus_{n\geq 0}C^{n}_{\alpha}(M,A)$. The following
proposition relates the above two coboundary operators.

######  Proposition 3.5.

Let $T:M\rightarrow A$ be an $\mathcal{O}$-operator on $A$ with respect to the
$A$-bimodule $M$. Then the two coboundary operators are related by

$\displaystyle d_{T}f=(-1)^{n}~{}d_{H}f,~{}~{}~{}~{}\text{ for }f\in
C^{n}_{\alpha}(M,A).$

###### Proof.

For any $f\in C^{n}_{\alpha}(M,A)=\text{Hom}(M^{\otimes n},A)$ and
$u_{1},\ldots,u_{n+1}\in M$, we have from (3.2) that

$\displaystyle(d_{T}f)(u_{1},\ldots,u_{n+1})=~{}$ $\displaystyle\llbracket
T,f\rrbracket_{\alpha}(u_{1},\ldots,u_{n+1})$ $\displaystyle=~{}$
$\displaystyle
T(l(f(u_{1},\ldots,u_{n})\phi^{n-1}(u_{n+1})))-(-1)^{n}~{}T(r(\phi^{n-1}(u_{1}),f(u_{2},\ldots,u_{n+1}))$
$\displaystyle-~{}$
$\displaystyle(-1)^{n}\bigg{\\{}\sum_{i=1}^{n}(-1)^{i-1}~{}f(\phi(u_{1}),\ldots,\phi(u_{i-1}),T(u_{i})u_{i+1},u_{i+2},\ldots,\phi(u_{n+1}))$
$\displaystyle-~{}$
$\displaystyle\sum_{i=1}^{n}(-1)^{i}f(\phi(u_{1}),\ldots,\phi(u_{i-1}),u_{i}T(u_{i+1}),u_{i+2},\ldots,\phi(u_{n+1}))\bigg{\\}}$
$\displaystyle+~{}$ $\displaystyle(-1)^{n}~{}T(\phi^{n-1}(u_{1}))\cdot
f(u_{2},\ldots,u_{n+1})-f(u_{1},\ldots,u_{n})\cdot T(\phi^{n-1}(u_{n+1}))$
$\displaystyle=~{}$ $\displaystyle(-1)^{n}~{}(d_{H}f)(u_{1},\ldots,u_{n+1}).$

The same holds true when $f=a\in A$. Compare (3.3) with (3.7). Hence the
proof. ∎

This shows that the cohomology of the either complex
$(C^{\bullet}_{\alpha}(M,A),d_{T})$ or $(C^{\bullet}(M,A),d_{H})$ are
isomorphic. Thus, we may use the same notation $H^{\bullet}_{\alpha}(M,A)$ to
denote the cohomology of an $\mathcal{O}$-operator.

##  4 Deformations of $\mathcal{O}$-operators

In this section, we study the Infinitesimal and formal deformation theory of
$\mathcal{O}$-operators following the classical approaches initiated by
Gerstenhaber [24, 25].

###  4.1 Infinitesimal deformations

Let $(A,\cdot_{A},\alpha_{A})$ be a Hom-associative algebra and
$(V,l_{A},r_{A},\phi)$ an $A$-bimodule, and $(B,\cdot_{B},\alpha_{B})$ be a
Hom-associative algebra with $(W,l_{B},l_{B},\psi)$ a $B$-bimodule. Suppose
$T:V\rightarrow A$ is an $\mathcal{O}$-operator on $A$ with respect to the
$A$-bimodule $(V,l_{A},r_{A},\phi)$ and $T^{\prime}:W\rightarrow B$ is an
$\mathcal{O}$-operator on $B$ with respect to the bimodule
$(B,\cdot_{B},\alpha_{B})$.

######  Definition 4.1.

A morphism of $\mathcal{O}$-operators from $T$ to $T^{\prime}$ consists of a
pair $(f,g)$ of an algebra morphism $f:A\rightarrow B$ and a linear map
$g:V\rightarrow W$ satisfying

$\displaystyle\psi g=~{}$ $\displaystyle g\phi,$ (4.1) $\displaystyle
T^{\prime}g=~{}$ $\displaystyle fT,$ (4.2) $\displaystyle
l_{B}(f(x),g(u))=~{}$ $\displaystyle g(l_{A}(x,u)),$ (4.3) $\displaystyle
r_{B}(g(u),f(x))=~{}$ $\displaystyle g(r_{A}(u,x),$ (4.4)

for all $x\in A$ and $u\in V$.

It is called an isomorphism if both $f$ and $g$ are bijective. According to
Lemma 2.12, the vector spaces $V$ and $W$ can be endowed with a Hom-dendriform
algebra structures and $g$ becomes then a morphism between Hom-dendriform
algebras $V$ and $W$ and also a morphism between the sub-adjacent Hom-
associative algebras .

The proof of the following result is straightforward so we omit the details.

######  Proposition 4.2.

A pair of linear maps $(f:A\rightarrow B,~{}g:V\rightarrow W)$ is a morphism
of $\mathcal{O}$-operators from $T$ to $T^{\prime}$ if and only if

$\displaystyle\mathrm{Gr}((f,g)):=\big{\\{}\big{(}(a,u),(f(a),g(u))\big{)}|~{}a\in
A,u\in V\big{\\}}\subset(A\oplus V)\oplus(B\oplus W)$

is a Hom-subalgebra, where $A\oplus V$ and $B\oplus W$ are equipped with semi-
direct product Hom-algebra structures.

In the rest of the paper, we will be most interested in morphisms between
$\mathcal{O}$-operators on the same Hom-algebra with respect to the same
bimodule.

Let $T:M\rightarrow A$ be an $\mathcal{O}$-operator on a Hom-associative
algebra $A$ with respect to the $A$-bimodule $M$.

######  Definition 4.3.

An infinitesimal (one-parameter) deformation of $T$ consists of a sum
$T_{t}=T+t\mathcal{T}$, for some $\mathcal{T}\in\text{Hom}(M,A)$, such that
$T_{t}$ is an $\mathcal{O}$-operator on $A$ with respect to the bimodule $M$,
for all $t$. In such a case, we say that $\mathcal{T}$ generates a
infinitesimal or a linear deformation of $T$.

It is easy to check that $T_{t}=T+t\mathcal{T}$ is an infinitesimal
deformation of an $\mathcal{O}$-operator $T$ if and only if for any $u,v\in
M$,

$\displaystyle T_{t}\phi=$ $\displaystyle\alpha T_{t},~{}~{}~{}~{}\text{ for
all }t,$ $\displaystyle T_{t}(u)\cdot T_{t}(v)=$ $\displaystyle
T_{t}(l(T_{t}(u))v+r(T_{t}(v))u),~{}~{}~{}~{}\text{ for all }t\text{ and
}u,v\in M.$

By equating coefficients of $t$ and $t^{2}$ from both side, we obtain

$\displaystyle\mathcal{T}\phi=~{}$ $\displaystyle\alpha\mathcal{T},,$ (4.5)
$\displaystyle T(u)\cdot\mathcal{T}(v)+\mathcal{T}(u)\cdot T(v)=~{}$
$\displaystyle
T(r(\mathcal{T}(v))u+l(\mathcal{T}(u))v)+\mathcal{T}(r(T(v))u+l(T(u))v),$
(4.6) $\displaystyle\mathcal{T}(u)\cdot\mathcal{T}(v)=~{}$
$\displaystyle\mathcal{T}(r(\mathcal{T}(v))u+l(\mathcal{T}(u))v).$ (4.7)

The identity (4.6) means that $d_{T}(\mathcal{T})=0$, that is $\mathcal{T}$ is
a $1$-cocycle with respect to the cohomology of the $\mathcal{O}$-operator
$T$. Eqs. (4.5) and (4.7) mean that $\mathcal{T}$ is an $\mathcal{O}$-operator
on $A$ associated to the bimodule $M$.

Given a Hom-dendriform algebra $(D,\prec,\succ,\phi)$. Let
$\omega_{\prec},\omega_{\succ}:D\otimes D\to D$ be two linear maps. If for any
$t\in\mathbb{K}$, the multiplications $\prec_{t}$ and $\succ_{t}$ defined by

$\displaystyle u\prec_{t}v=u\prec v+t\omega_{\prec}(u,v),\quad
u\succ_{t}v=u\succ v+t\omega_{\succ}(u,v),\ \forall u,v\in D$

give a Hom-dendriform algebra structure on $D$, we say that the pair
$(\omega_{\prec},\omega_{\succ})$ generates a (one parameter) infinitesimal
deformation of the Hom-dendriform algebra $(D,\prec,\succ,\phi)$.

In the following, we show that an infinitesimal deformation of an
$\mathcal{O}$-operator induces an infinitesimal deformation of the
corresponding Hom-dendriform structure on the module.

######  Proposition 4.4.

If $\mathcal{T}$ generates an infinitesimal deformation of an
$\mathcal{O}$-operator $T$ on a Hom-associative algebra $A$ with respect to
the bimodule $M$, then the multiplications

$\displaystyle
u\prec_{t}v=uT(v)+t~{}u\mathcal{T}(v),~{}u\succ_{t}v=T(u)v+t~{}\mathcal{T}(u)v$

define an infinitesimal deformation of the corresponding Hom-dendriform
structure on $M$.

######  Corollary 4.5.

If $\mathcal{T}$ generates an infinitesimal deformation of an
$\mathcal{O}$-operator $T$ on a Hom-associative algebra $A$ with respect to
the bimodule $M$, then the multiplication

$\displaystyle u\star_{t}v=u\star v+t(u\mathcal{T}(v)+\mathcal{T}(u)v)$

defines an infinitesimal deformation of the corresponding Hom-associative
structure on $M$.

######  Definition 4.6.

Let $T$ be an $\mathcal{O}$-operator on a Hom-associative algebra $A$ with
respect to a bimodule $(M,l,r,\phi)$. Two infinitesimal deformations
$T^{1}_{t}=T+t\mathcal{T}_{1}$ and $T^{2}_{t}=T+t\mathcal{T}_{2}$ of $T$ are
equivalent if there exists an $x\in A$ such that $\alpha(x)=x$ and the pair
$\big{(}\text{id}_{A}+t(L_{x}-R_{x}),\text{id}_{M}+t(l_{x}-r_{x})\big{)}$ is a
morphism of $\mathcal{O}$-operators from $T^{1}_{t}$ to $T^{2}_{t}$.

Let us recall from Definition 4.1 that the pair
$\big{(}\text{id}_{A}+t(L_{x}-R_{x}),\text{id}_{M}+t(l_{x}-r_{x})\big{)}$ is a
morphism of $\mathcal{O}$-operators from $T^{1}_{t}$ to $T^{2}_{t}$ if the
following conditions are satisfied

  * (i)

The map $\text{id}_{A}+t(L_{x}-R_{x})$ is a Hom-associative algebra
homomorphism.

  * (ii)

$\phi\circ(\text{id}_{M}+t(l_{x}-r_{x}))=(\text{id}_{M}+t(l_{x}-r_{x}))\circ\phi$.

  * (iii)

$(T+t\mathcal{T}_{2})\circ(\text{id}_{M}+t(l_{x}-r_{x}))=(\text{id}_{A}+t(L_{x}-R_{x}))\circ(T+t\mathcal{T}_{1})$.

  * (iv)

$l(y+t[x,y],u+t(l(x,u)-r(u,x)))=l(y,u)+t(l(x,l(y,u))-r(l(y,u),x))$ .

  * (v)

$r(u+t(l(x,u)-r(u,x)),y+t[x,y])=r(u,y)+tl(x,r(u,y))-tr(r(u,y),x)$.

From (i), we obtain, for all $y,z\in A$

$\displaystyle(\text{id}_{A}+t(L_{x}-R_{x}))(y\cdot
z)=(\text{id}_{A}+t(L_{x}-R_{x}))(y)\cdot(\text{id}_{A}+t(L_{x}-R_{x}))(z),$

or equivalently

$\displaystyle(x\cdot y-y\cdot x)\cdot(x\cdot z-z\cdot x)=0,~{}~{}\forall
y,z\in A.$ (4.8)

Since $\alpha(x)=x$, then condition (ii) holds. Moreover, from (iii) we get,
for any $u\in M$,

$\displaystyle(T+t\mathcal{T}_{2})\circ(\text{id}_{M}+t(l_{x}-r_{x}))(u)=(\text{id}_{A}+t(L_{x}-R_{x}))\circ(T+t\mathcal{T}_{1})(u),$

On comparing the coefficients of $t$ and $t^{2}$, respectively, from both
sides of the above identity, we obtain

$\displaystyle\mathcal{T}_{1}(u)-\mathcal{T}_{2}(u)=~{}$ $\displaystyle
T(l(x,u)-r(u,x))-(x\cdot T(u)-T(u)\cdot x)$ $\displaystyle=~{}$ $\displaystyle
l_{T}(u,x)-r_{T}(x,u),$ (4.9) $\displaystyle
x\cdot\mathcal{T}_{1}(u)-\mathcal{T}_{1}(u)\cdot x=~{}$
$\displaystyle\mathcal{T}_{2}(l(x,u)-r(u,x)).$ (4.10)

From (iv), on comparing just the coefficients of $t^{2}$ (since the
coefficients of $t$ vanish) from both sides of the identity, we obtain

$l(x\cdot y-y\cdot x,l(x,u)-r(u,x))=0$

or equivalently,

$\displaystyle l_{x\cdot y-y\cdot x}\circ l_{x}=l_{x\cdot y-y\cdot x}\circ
r_{x},~{}~{}~{}~{}\text{ for }y\in A.$ (4.11)

Similarly, (v) gives rise to

$\displaystyle r_{x\cdot y-y\cdot x}\circ l_{x}=r_{x\cdot y-y\cdot x}\circ
r_{x},~{}~{}~{}~{}\text{ for }y\in A.$ (4.12)

######  Theorem 4.7.

Let $T^{1}_{t}=T+t\mathcal{T}_{1}$ and $T^{2}_{t}=T+t\mathcal{T}_{2}$ be two
equivalent infinitesimal deformations of an $\mathcal{O}$-operator $T$. Then
$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ defines the same cohomology class in
$H^{1}_{\alpha}(M,A).$

###### Proof.

The proof follows from equation (4.9). ∎

######  Definition 4.8.

Let $T:M\to A$ be an $\mathcal{O}$-operator on a Hom-associative algebra $A$
with respect to a representation $(M,l,r,\phi)$. An infinitesimal deformation
$T_{t}=T+t\mathcal{T}$ is said to be trivial if it is equivalent to the
deformation $T_{0}=T.$

According to the above computation, $T_{t}=T+t\mathcal{T}$ is trivial if there
exists an element $x\in A$ such that $\alpha(x)=x$ and satisfying

$\displaystyle(x\cdot y-y\cdot x)\cdot(x\cdot z-z\cdot x)=0,$ (4.13)
$\displaystyle x\cdot(l_{T}(u,x)-r_{T}(x,u))-(l_{T}(u,x)-r_{T}(x,u))\cdot
x=0,$ (4.14) $\displaystyle l_{x\cdot y-y\cdot x}\circ l_{x}=l_{x\cdot
y-y\cdot x}\circ r_{x},$ (4.15) $\displaystyle r_{x\cdot y-y\cdot x}\circ
l_{x}=r_{x\cdot y-y\cdot x}\circ r_{x},$ (4.16)

for any $y,z\in A$ and $u\in M$. Then we introduce the notion of a Nijenhuis
element associated to an $\mathcal{O}$-operator on a Hom-associative algebra.

######  Definition 4.9.

Let $T$ be an $\mathcal{O}$-operator on a Hom-associative algebra $(A,\alpha)$
with respect to a representation $(M,l,r,\phi)$. An element $x\in A$ is called
a Nijenhuis element associated to $T$ if $x$ satisfies $\alpha(x)=x$ and the
conditions (4.13)–(4.16) hold.

Let us denote the set of Nijenhuis elements associated to the
$\mathcal{O}$-operator $T$ by$\textrm{Nij}(T)$.

In [3], the authors introduce Nijenhuis elements associated to an
$\mathcal{O}$\- operator on a Hom-Lie algebra to study their trivial
deformations. If $T$ is an $\mathcal{O}$-operator on a Hom-Lie algebra
$(A,[\cdot,\cdot],\alpha)$ with respect to a representation $(M,\rho,\beta)$.
Then an element $x\in A$ is called a Nijenhuis element if $\alpha(x)=x$ and

$\displaystyle[[x,y],[x,z]]=0,\ \rho([x,y])\rho(x)=0,\
[x,T\rho(x)v+[Tv,x]]=0,$

for all $y,z\in A$ and $v\in M$.

######  Proposition 4.10.

Let $x\in A$ be a Nijenhuis element with respect to an $\mathcal{O}$-operator
$T$ on a Hom-associative $A$. Then $x$ is also a Nijenhuis element for the
$O$-operator $T$ on the commutator Hom-Lie algebra
$(A,[\cdot,\cdot]_{C},\alpha)$.

###### Proof.

Straightforward. ∎

By Eqs. (4.8)–(4.12), it is obvious that a trivial infinitesimal deformation
gives rise to a Nijenhuis element. Conversely, a Nijenhuis element can also
generate a trivial infinitesimal deformation as the following theorem shows.

######  Theorem 4.11.

Let $T$ be an $\mathcal{O}$-operator on a Hom-associative algebra $(A,\alpha)$
with respect to a representation $(M,l,r,\phi)$. For any element
$x\in\textrm{Nij}(T)$, $T_{t}:=T+t\mathcal{T}$, with $\mathcal{T}=d_{H}(x)$,
is a trivial infinitesimal deformation of $T$.

###### Proof.

First, we need to prove that $T_{t}:=T+t\mathcal{T}$ is a infinitesimal
deformation generated by $\mathcal{T}=d_{H}(x)$, where $x\in\textrm{Nij}(T)$.
That is $\mathcal{T}$ must satisfy conditions (4.5), (4.6) and (4.7). First,

$\displaystyle\alpha\mathcal{T}(u)=\alpha
d_{H}(x)(u)=d_{H}(x)(\phi(u))=\mathcal{T}(\phi(u)),\forall u\in M.$

Then $\mathcal{T}$ satisfies (4.5). The identity (4.6) holds immediately since
$\mathcal{T}=d_{H}(x)$. The identity (4.7) is also straightforward. We omit
details since it follows from a long computation. We refer to [46] for a
similar computation. Finally, since $x$ is a Nijenhuis element, it follows
that the pair $(id_{A}+t(L_{x}-R_{x},id_{M}+t(l_{x}-r_{x}))$ is a morphism of
$\mathcal{O}$-operators from $T_{t}$ to $T$. Hence the result follows. ∎

###  4.2 Formal Deformations

In this section, we study one-parameter formal deformations of O-operators.
Let $(A,\mu,\alpha)$ be a Hom-associative algebra with a representation
$(M,l,r,\phi)$. Let $\mathbb{K}[[t]]$ be the formal power series ring in one
variable $t$ and $A[[t]]$ be the formal power series in $t$ with coefficients
in $A$. Then the tuple $(A[[t]],\mu_{t},\alpha_{t})$ is a Hom-associative
algebra, where the product $\mu_{t}$ and the structure map $\alpha_{t}$ are
obtained by extending $\mu$ and $\alpha$ linearly over the ring
$\mathbb{K}[[t]]$. Moreover the maps $l$, $r$ and $\phi$ can be extended
linearly over $\mathbb{K}[[t]]$ to obtain $\mathbb{K}[[t]]$-linear maps
$l_{t}:A[[t]]\otimes M[[t]]\to M[[t]],\ r_{t}:M[[t]]\otimes A[[t]]\to M[[t]]$
and $\phi_{t}:M[[t]]\to M[[t]]$. Then $(M[[t]],l_{t},r_{t},\phi_{t})$ is a
representation of the Hom-associative algebra $(A[[t]],\mu_{t},\alpha_{t})$.

######  Definition 4.12.

Let $T$ be an $\mathcal{O}$-operator on a Home associative algebra
$(A,\mu,\alpha)$ with respect to a representation $(M,l,r,\phi)$. A formal
one-parameter deformation of $T$ is given by

$\displaystyle T_{t}=T_{0}+\sum_{i\geq 1}t^{i}T_{i},\quad\textrm{with}\
T_{0}=T,\ T_{i}\in C^{1}_{\phi,\alpha}(M,A)$

such that $T_{t}:M[[t]]\to A[[t]]$ is an $\mathcal{O}$-operator on
$(A[[t]],\mu_{t},\alpha_{t})$ with respect to $(M[[t]],l_{t},r_{t},\phi_{t})$.

That is

$\displaystyle T_{t}\phi_{t}=\alpha_{t}T_{t},$ (4.17) $\displaystyle
T_{t}(u)\cdot T_{t}(v)=T_{t}(T_{t}(u)v+uT_{t}(v)),\ \forall u,v\in M.$ (4.18)

Condition (4.17) is immediate since $T_{i}\in C^{1}_{\phi,\alpha}(M,A),\
\forall i$. For any $k\geq 0$, by equating the coefficients of $t^{k}$ from
both sides of equation (4.18), we obtain the following system of equations

$\displaystyle\sum_{i+j=k}T_{i}(u)\cdot
T_{j}(v)=\sum_{i+j=k}T_{i}(T_{j}(u)v+uT_{j}(v)).$ (4.19)

The above identity holds for $k=0$, since $T$ is an $\mathcal{O}$-operator.
For $k=1$, we get

$\displaystyle T(u)\cdot T_{1}(v)+T_{1}(u)\cdot
T(v)=T(uT_{1}(v)+T_{1}(u)v)+T_{1}(uT(v)+T(u)v),\forall u,v\in M,$ (4.20)

which is equivalent to

$\displaystyle\llbracket T,T_{1}\rrbracket_{\alpha}=0.$ (4.21)

That is $d_{T}(T_{1})=0$. Therefore, we get the following proposition.

######  Proposition 4.13.

Let $T_{t}=\sum_{i\geq 0}t^{i}T_{i}$ be a formal deformation of an
$\mathcal{O}$-operator $T$ on $A$ with respect to the bimodule $M$. Then the
$1$-cochain $T_{1}\in C^{1}_{\phi,\alpha}(M,A)$ is a $1$-cocycle with respect
to the cohomology of the $\mathcal{O}$-operator $T$.

The $1$-cochain $T_{1}$ is called the infinitesimal of the deformation
$T_{t}$. More generally, if $T_{i}=0$ for $1\leq i\leq(n-1)$ and $T_{n}$ is a
non-zero cochain, then $T_{n}$ is called the $n$-infinitesimal of the
deformation $T_{t}$.

In [19], the author study formal deformations of Hom-dendriform algebras and
their relation with the cohomology. Here we provide a connection between
deformations of an $\mathcal{O}$-operator with deformations of the
corresponding Hom-dendriform algebra. We mean by a deformation of a Hom-
dendriform algebra, a formal deformation of its defining two multiplications.

######  Proposition 4.14.

Let $T_{t}=\sum_{i\geq 0}t^{i}T_{i}$ be a formal deformation of an
$\mathcal{O}$-operator $T$. Then the formal sums

$\displaystyle u\prec_{t}v=\sum_{i\geq 0}t^{i}uT_{i}(v)\quad\text{ and }\quad
u\succ_{t}v=\sum_{i\geq 0}t^{i}T_{i}(u)v$

define a formal deformation of the Hom-dendriform structure on $M$.

######  Corollary 4.15.

It follows that the formal sum $u\star_{t}v=\sum_{i\geq
0}t^{i}(uT_{i}(v)+T_{i}(u)v)$ defines a formal deformation of the Hom-
associative structure on $M$.

Now, we consider the equivalence of two formal deformations of an
$\mathcal{O}$-operator. The definition is motivated by the associative algebra
case [20].

######  Definition 4.16.

Two formal deformations $T_{t}=\sum_{i\geq 0}t^{i}T_{i}$ and
$\overline{T}_{t}=\sum_{i\geq 0}t^{i}\overline{T}_{i}$ of an
$\mathcal{O}$-operator $T$ are said to be equivalent if there exists an
element $x\in A$ satisfying $\alpha(x)=x$, linear maps $f_{i}:A\to A$ and
$g_{i}:M\to M$, for $i\geq 2$, such that the pair $(f_{t},g_{t})$ given by

$\displaystyle f_{t}=id_{A}+t(L_{x}-R_{x})+\sum_{i\geq 2}t^{i}f_{i}\quad\text{
and }\quad g_{t}=id_{M}+t(l_{x}-r_{x})+\sum_{i\geq 2}t^{i}g_{i}$

defines a formal morphism of $\mathcal{O}$-operators from $T_{t}$ to
$\overline{T}_{t}$.

With the above notations, for all $y,z\in A$ and $u\in M$, the equivalence of
two deformations $T_{t}$ and $\overline{T}_{t}$ corresponds to the following
conditions

$\displaystyle f_{t}T_{t}=$ $\displaystyle T_{t}g_{t},$ (4.22) $\displaystyle
f_{t}(y\cdot z)=$ $\displaystyle f_{t}(y)\cdot f_{t}(z),$ (4.23)
$\displaystyle g_{t}(yu)=$ $\displaystyle f_{t}(y)g_{t}(u),$ (4.24)
$\displaystyle g_{t}(uy)=$ $\displaystyle g_{t}(u)f_{t}(y),$ (4.25)
$\displaystyle f_{t}\alpha=\alpha f_{t},\ $ $\displaystyle g_{t}\phi=\phi
g_{t}.$ (4.26)

In particular, a formal deformation $T_{t}$ of an $\mathcal{O}$-operator $T$
is said to be trivial if it is equivalent to $T$ (here $T$ is regarded as a
deformation of itself).

Now, on comparing the coefficients of $t$ from both sides of the condition
(4.22), we get

$\displaystyle T_{1}(u)-\overline{T}_{1}(u)=T(u)\cdot x-T(ux)-x\cdot
T(u)+T(xu)=d_{H}(x)(u).$

Consequently, we obtain the following result.

######  Proposition 4.17.

The infinitesimals of equivalent deformations are cohomologous, i.e. they lie
on the same cohomology class .

######  Definition 4.18.

An $\mathcal{O}$-operator $T$ is said to be rigid if every deformation $T_{t}$
of $T$ is equivalent to the deformation $\overline{T}_{t}=T$.

As a cohomological condition of the rigidity, we have the following result
which suggests that the rigidity of an $\mathcal{O}$-operator is a very strong
condition.

######  Proposition 4.19.

Let $T$ be an $\mathcal{O}$-operator on a Hom-associative algebra $A$ with
respect to a bimodule $M$. If $Z^{1}(M,A)=d_{H}(\mathrm{Nij}(T))$ then $T$ is
rigid.

###### Proof.

Let $T_{t}=\sum_{i\geq 0}t^{i}T_{i}$ be any formal deformation of the
$\mathcal{O}$-operator $T$. Then by Proposition 4.13, the linear term $T_{1}$
is in $Z^{1}(M,A)$. Therefore, by assumption, we have $T_{1}=d_{H}(x)$, for
some $x\in\text{Nij}(T)$. Set

$\displaystyle f_{t}:=id_{A}+t(L_{x}-R_{x})\quad\text{ and }\quad
g_{t}:=id_{M}+t(l_{x}-r_{x})$

and define $\overline{T}_{t}=f_{t}\circ T_{t}\circ g_{t}^{-1}$. Then $T_{t}$
is equivalent to $\overline{T}_{t}$. Moreover, from the definition of
$\overline{T}_{t}$, we have

$\displaystyle\overline{T}_{t}(u)=(id_{A}+t(L_{x}-R_{x}))\circ(\sum_{i\geq
0}t^{i}T_{i})\circ(id_{M}-t(l_{x}-r_{x})+t^{2}(l_{x}-r_{x})^{2}-\cdots)(u),~{}~{}~{}\text{
for }u\in M.$

Hence

$\displaystyle\overline{T}_{t}(u)\quad(\text{mod }t^{2})=~{}$
$\displaystyle(id_{A}+t(L_{x}-R_{x}))\circ(T+tT_{1})(u-t(xu-
ux))\quad(\text{mod }t^{2})$ $\displaystyle=~{}$
$\displaystyle(id_{A}+t(L_{x}-R_{x}))(T(u)+tT_{1}(u)-tT(xu-ux))\quad(\text{mod
}t^{2})$ $\displaystyle=~{}$ $\displaystyle T(u)+t\big{(}x\cdot T(u)-T(u)\cdot
x-T(xu-ux)+T_{1}(u)\big{)}.$

The coefficient of $t$ is zero as $T_{1}=d_{H}(x)$. See (3.7) for instance.
Therefore, $\overline{T}_{t}$ is of the form $\overline{T}_{t}=T+\sum_{i\geq
2}t^{i}\overline{T}_{i}$. By repeating this argument, one get the equivalence
between $T_{t}$ and $T$. Hence the proof. ∎

###  4.3 Order $n$ deformation of an $\mathcal{O}$-operator

In this section, We introduce a cohomology class associated to any order $n$
deformation of an $\mathcal{O}$-operator. We prove that an order $n$
deformation of an $\mathcal{O}$-operator is extensible if and only if this
cohomology class is trivial. This cohomology class is called the obstruction
class of an order $n$ deformation being extensible. Let $(A,\mu,\alpha)$ be a
Hom-associative algebra with a representation $(M,l,r,\phi)$. Let $T:M\to A$
be an $\mathcal{O}$-operator on $A$ with respect to $M$. An order $n$
deformation of the $\mathcal{O}$-operator $T$ is given by a
$\mathbb{K}[[t]]/(t^{n+1})$-linear map

$T_{t}=T+\sum_{i=1}^{n}t^{i}T_{i},\quad T_{i}\in C^{1}_{\phi,\alpha}(M,A)$

such that

$\displaystyle T_{t}(u)\cdot
T_{t}(v)=T_{t}(T_{t}(u)v+uT_{t}(v))\quad\textrm{mod}(t^{n+1}),\ \forall u,v\in
M.$

Which is equivalent to the following system of equations

$\displaystyle\sum_{i+j=k}T_{i}(u)\cdot
T_{j}(v)=\sum_{i+j=k}T_{i}(T_{j}(u)v+uT_{j}(v)),\quad\textrm{for any}\
k=0,1,\cdots n.$

or equivalently,

$\displaystyle\llbracket
T,T_{k}\rrbracket_{\alpha}=-\frac{1}{2}\sum_{i+j=k,i,j\geqslant 1}\llbracket
T_{i},T_{j}\rrbracket_{\alpha},\quad\textrm{for any}\ k=0,1,\cdots n.$ (4.27)

######  Definition 4.20.

Let $T_{t}=T+\sum_{i=1}^{n}t^{i}T_{i}$ be an order $n$ deformation of an
$\mathcal{O}$-operator $T$ on $A$ with respect to $M$. If there exists a
$1$-cochain $T_{n+1}\in C^{1}_{\phi,\alpha}(M,A)$ such that
$\widetilde{T_{t}}=T_{t}+t^{n+1}T_{n+1}$ is an order $(n+1)$ deformation of
$T$, then we say that $T_{t}$ extends to a deformation of order $(n+1)$. In
this case, we say that $T_{t}$ is extendable.

Let us observe that the map $\widetilde{T}_{t}:=T_{t}+t^{n+1}T_{n+1}$ is an
extension of the order $n$ deformation $T_{t}$ if and only if

$\sum_{\begin{subarray}{c}i+j=n+1\\\ i,j\geq 0\end{subarray}}\llbracket
T_{i},T_{j}\rrbracket_{\alpha}=0.$

That is

$\displaystyle\llbracket
T,T_{n+1}\rrbracket_{\alpha}=-\frac{1}{2}\sum_{\begin{subarray}{c}i+j=n+1\\\
i,j>0\end{subarray}}\llbracket T_{i},T_{j}\rrbracket_{\alpha}.$ (4.28)

######  Definition 4.21.

Let $T_{t}$ be an order $n$ deformation of the $\mathcal{O}$-operator $T$. Let
us consider a $2$-cochain $\Theta_{T}\in C^{2}_{\phi,\alpha}(M,A)$ defined as
follows

$\Theta_{T}=-\frac{1}{2}\sum_{i+j=n+1;~{}i,j>0}\llbracket
T_{i},T_{j}\rrbracket_{\alpha}.$ (4.29)

The $2$-cochain $\Theta_{T}$ is called the obstruction cochain for extending
the deformation $T_{t}$ of order $n$ to a deformation of order $n+1$. From
equation (4.29) and using graded Jacobi identity of the bracket
$\llbracket\cdot,\cdot\rrbracket_{\alpha}$, it follows that $\Theta_{T}$ is a
$2$-cocycle.

The cohomology class $|\Theta_{T}|\in H^{2}(M,A)$ is called the obstruction
class of $T_{t}$ being extendable.

######  Theorem 4.22.

An order $n$ deformation $T_{t}$ extends to a deformation of next order if and
only if the obstruction class $[\Theta_{T}]$ is trivial.

###### Proof.

Suppose that a deformation of $\mathcal{O}$-operator $T_{t}$ of order $n$
extends to a deformation of order $n+1$. Then

$\sum_{i+j=n+1}(T_{i}(u)\cdot T_{j}(v)-T_{i}(T_{j}(u)v+uT_{j}(v)))=0.$

As a result, we get $\Theta_{T}=-\frac{1}{2}\sum_{i+j=n+1;~{}i,j>0}\llbracket
T_{i},T_{j}\rrbracket_{\alpha}.$ So, the obstruction class vanishes.
Conversely, if the obstruction class $|\Theta_{T}|$ is trivial, suppose that
$\Theta_{T}=-d_{T}\mathcal{T}_{n+1}$ for some $1$-cochain $\mathcal{T}_{n+1}$.
Put $\tilde{T}_{t}=T_{t}+\mathcal{T}t^{n+1}$. Then $\tilde{T}$ is a
deformation of order $n+1$, which means that $T_{t}$ is extendable. ∎

######  Corollary 4.23.

If $H^{2}(M,A)=0$, then every finite order deformation of $T$ extends to a
deformation of next order.

######  Corollary 4.24.

If $H^{2}(M,A)=0$, then every $1$-cocycle in $Z^{1}(M,A)$ is the infinitesimal
of some formal deformation of the $\mathcal{O}$-operator $T$.

## References

  * [1] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960) 731-742.
  * [2] G.-C. Rota, Baxter algebras and combinatorial identities I, II, Bull. Amer. Math. Soc. 75 (1969) 325–329, 330–334.
  * [3] S. K. Mishra, and A. Naolekar, $\mathcal{O}$-Operators on Hom-Lie algebras. Preprint arXiv:2007.09440 (2020).
  * [4] F. Ammar, Z. Ejbehi and A. Makhlouf, Cohomology and deformations of Hom-algebras, J. Lie Theory 21 (2011), no. 4, 813-836.
  * [5] M. Aguiar, Infinitesimal Hopf algebras, New trends in Hopf algebra theory (La falda, 1999), 1-29, Contemp. Math., 267, Amer. Math. Soc., Providence, RI, 2000.
  * [6] M. Aguiar, Pre-Poisson algebras, Lett. Math. Phys. 54 (2000), no. 4, 263-277.
  * [7] M. Aguiar, On the associative analog of Lie bialgebras, J. Algebra 244 (2001), no. 2, 492-532.
  * [8] C. Bai, Double constructions of Frobenius algebras, Connes cocycles and their duality, J. Noncommut. Geom., 4 (2010), pp. 475-530
  * [9] D. Balavoine, Deformations of algebras over a quadratic operad, Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), 207-234, Contemp. Math., 202, Amer. Math. Soc., Providence, RI, 1997\.
  * [10] D. Balavoine, Déformations des algébres de Leibniz (in French), C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 8, 783-788.
  * [11] K. H. Bhaskara and K. Viswanath, Calculus on Poisson manifolds, Bull. London Math. Soc. 20 (1988), no. 1, 68-72.
  * [12] J. F. Cariñena, J. Grabowski and G. Marmo, Quantum Bi-Hamiltonian Systems, Internat. J. Modern. Phys. A 15 (2000) 4797-4810.
  * [13] L. Cai and Y. Sheng, Hom-big brackets: theory and applications, _SIGMA_ 12 (2016), 014, 18 pages.
  * [14] P. Cartier, On the structure of free Baxter algebras, Advances in Math. 9 (1972), 253-265.
  * [15] A. Connes and D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), no. 1, 249-273.
  * [16] T. J. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631-661.
  * [17] A. Das, Cohomology and deformations of dendriform algebras, and $\text{Dend}_{\infty}$-algebras, preprint, arXiv:1903.11802
  * [18] A. Das, Gerstenhaber algebra structure on the cohomology of a hom-associative algebra. Proc. Indian Acad. Sci. Math. Sci. 130 (2020), no. 1, Paper No. 20, 20 pp.
  * [19] A. Das, Cohomology and deformations of Hom-dendriform algebras and coalgebras. Colloq. Math. 163 (2021), no. 1, 37–52.
  * [20] A. Das, Deformations of associative Rota-Baxter operators. J. Algebra  560 (2020), 144–180.
  * [21] A. Dzhumadil’daev, Cohomologies and deformations of right-symmetric algebras, J. Math. Sci. (New York) 93 (1999), no. 6, 836-876.
  * [22] K. Ebrahimi-Fard and L. Guo, Rota-Baxter algebras and dendriform algebras, J. Pure Appl. Algebra 212 (2008), no. 2, 320-339.
  * [23] A. Frölicher and A. Nijenhuis, A theorem on stability of complex structures, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 239-241.
  * [24] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59-103.
  * [25] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267-288.
  * [26] L. Guo, An introduction to Rota-Baxter algebra, Surveys of Modern Mathematics, 4. International Press, Somerville, MA; Higher Education Press, Beijing, 2012\.
  * [27] L. Guo and W. Keigher, Baxter algebras and shuffle products, Adv. Math. 150 (2000), no. 1, 117-149.
  * [28] J.T. Hartwig, D. Larsson, and S.D. Silvestrov, Deformations of Lie algebras using $\sigma$-derivations, J. Algebra 295 (2006) 314–361.
  * [29] G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58-67.
  * [30] B. Hurle and A. Makhlouf, $\alpha$-type Hochschild cohomology of Hom-associative algebras and bialgebras. J. Korean Math. Soc. 56 (2019), no. 6, 1655–1687
  * [31] K. Kodaira, L. Nirenberg and D. C. Spencer, On the existence of deformations of complex analytic structures, Ann. of Math. (2) 68 (1958), 450â€“459.
  * [32] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I, II, Ann. of Math. (2) 67 (1958), 328-466.
  * [33] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157-216.
  * [34] J. L. Kelley, Averging operators on $C_{\infty}(X)$, Illinois J. Math. 2 (1958), 214-223
  * [35] A. Lazarev, Y. Sheng and R. Tang, Deformations and homotopy theory of relative Rota-Baxter Lie algebras, Comm. Math. Phys. 383 (2021), 595–631.
  * [36] J.-L. Loday, Dialgebras, Dialgebras and related operads, 7-66, Lecture Notes in Math., 1763, Springer, Berlin, 2001\.
  * [37] A. Nijenhuis and R. W. Richardson, Cohomology and deformations in graded Lie algebras, Bull. Amer. Math. Soc. 72 (1966), 1-29.
  * [38] J. Pei and L. Guo, Averaging algebras, Schröder numbers, rooted trees and operads, J. Algebraic Combin. 42 (2015), no. 1, 73-109.
  * [39] G.-C. Rota, On the representation of averaging operator, Rendiconti del Seminario Matematico della Universita di Padova 30 (1960), 52-64.
  * [40] O. Reynolds, On the dynamic theory of incompressible viscous fluids, Phil. Trans. Roy. Soc. A 136 (1895), 123-164
  * [41] J. Liu, C. Bai and Y. Sheng, Compatible $\mathcal{O}$-operators on bimodules over associative algebras, J. Algebra 532 (2019), 80-118.
  * [42] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2 (2008) 51–64.
  * [43] A. Makhlouf. Hom-dendriform algebras and Rota-Baxter Hom-algebras. In : Operads And Universal Algebra. p. 147–171. Nankai Ser. Pure Appl. Math. Theoret. Phys., 9, World Sci. Publ., Hackensack, NJ, 2012.
  * [44] A. Makhlouf and S. Silvestrov, Notes on $1$-parameter formal deformations of Hom-associative and Hom-Lie algebras, Forum Math. 22 (2010), no. 4, 715-739.
  * [45] Y. Sheng, Representations of hom-Lie algebras. Algebr. Represent. Theory 15 (2012), no. 6, 1081–1098.
  * [46] R. Tang, C. Bai, L. Guo and Y. Sheng, Deformations and their controlling cohomologies of $\mathcal{O}$-operators, Comm. Math. Phys. 368 (2019), no. 2, 665-700.
  * [47] K. Uchino, Quantum analogy of Poisson geometry, related dendriform algebras and O-operators, Lett. Math. Phys. 85 (2008) 91-109.
  * [48] K. Uchino, Twisting on associative algebras and Rota-Baxter type operators, J. Noncommut. Geom.  4(3) (2010) 349-379.
  * [49] K. Uchino, Derived bracket construction and Manin products, Lett. Math. Phys. 93 (2010), no. 1, 37-53.
  * [50] Th. Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (2005), no. 1-3, 133-153.
  * [51] Q. Wang, Y. Sheng, C. Bai and J. Liu, Nijenhuis operators on pre-Lie algebras, Commun. Contemp. Math. 21 (2019), no. 7, 1850050, 37 pp.