# A cohomological study of modified Rota-Baxter algebras

and  Apurba Das  Department of Mathematics, Indian Institute of Technology
Kharagpur, Kharagpur-721302, West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A modified Rota-Baxter algebra is an algebra equipped with an operator that
satisfies the modified Yang-Baxter equation. In this paper, we define the
cohomology of a modified Rota-Baxter algebra with coefficients in a suitable
bimodule. We relate our cohomology of a modified Rota-Baxter algebra with the
known cohomology theory of a Rota-Baxter algebra. As applications of our
cohomology, we study formal one-parameter deformations and abelian extensions
of modified Rota-Baxter algebras.

###### Key words and phrases:

Modified Rota-Baxter operators, Cohomology, Deformations, Abelian extensions

###### 2010 Mathematics Subject Classification:

16E40, 16D20, 16Wxx, 16S80.

###### Contents

  1. 1 Introduction
  2. 2 Modified Rota-Baxter algebras
  3. 3 Cohomology of modified Rota-Baxter algebras
  4. 4 Deformations of modified Rota-Baxter algebras
  5. 5 Abelian extensions of modified Rota-Baxter algebras

##  1\. Introduction

Rota-Baxter operators pay very much attention in the last twenty years as they
have many applications in mathematics and physics. Rota-Baxter operators
originated in a work of Baxter in the fluctuation theory of probability [8].
Subsequently, many well-known mathematicsians, such as Atkinson, Cartier and
Rota studied Rota-Baxter operators from combinatorial points of view [3, 10,
29]. See [1, 2, 5, 11, 18, 19] for the appearence of Rota-Baxter operators in
combinatorics, splitting of algebras, infinitesimal bialgebras and
renormalizations in quantum field theory. Let $A$ be an (associative) algebra
over a field ${\bf k}$ of characteristic $0$. A linear map $P:A\rightarrow A$
is said to be a Rota-Baxter operator of weight $\lambda\in{\bf k}$ if the map
$P$ satisfies

$\displaystyle P(a)\cdot P(b)=P\big{(}P(a)\cdot b+a\cdot
P(b)\big{)}+\lambda~{}\\!P(a\cdot b),\text{ for all }a,b\in A.$

Here $~{}\cdot~{}$ denotes the multiplication map in $A$. A pair $(A,P)$
consisting of an algebra $A$ and a Rota-Baxter operator $P$ of weight
$\lambda$ is said to be a Rota-Baxter algebra of weight $\lambda$. Recall that
[20] a bimodule over a Rota-Baxter algebra $(A,P)$ of weight $\lambda$
consists of a pair $(M,Q)$ in which $M$ is an $A$-bimodule and $Q:M\rightarrow
M$ is a linear map satisfying

$\displaystyle P(a)\cdot Q(u)=~{}$ $\displaystyle Q\big{(}P(a)\cdot u+a\cdot
Q(u)\big{)}+\lambda~{}\\!Q(a\cdot u),$ $\displaystyle Q(u)\cdot P(a)=~{}$
$\displaystyle Q\big{(}Q(u)\cdot a+u\cdot P(a)\big{)}+\lambda~{}\\!Q(u\cdot
a),$

for all $a\in A$ and $u\in M$. Here $~{}\cdot~{}$ also denotes the left and
right $A$-actions on $M$. See also [33] for details.

Rota-Baxter operators can also be defined on other algebraic structures, such
as Lie algebras, Poisson algebras etc. In [30] Semenov-Tyan-Shanskiĭ observed
that under certain conditions, a Rota-Baxter operator of weight $0$ on a Lie
algebra is simply the operator form of the classical Yang-Baxter equation
(CYBE). In the same paper, the author also introduced a closely related notion
of modified classical Yang-Baxter equation (modified CYBE) whose solutions are
called modified $r$-matrices. The modified CYBE has found important
applications in the study of Lax equations, affine geometry on Lie groups,
factorization problems in Lie algebras and compatible Poisson structures [4,
9, 26, 31]. The modified CYBE can be obtained from the CYBE by a suitable
transformation, however they plays independent role in mathematical physics.
This led various authors to consider them independently. Motivated by such
study, the authors in [34, 35, 36] considered the associative analogue of the
modified CYBE, called the modified associative Yang-Baxter equation of weight
$\kappa\in{\bf k}$ (in short modified AYBE of weight $\kappa$). A solution of
the modified AYBE of weight $\kappa$ is called a modified Rota-Baxter operator
of weight $\kappa$. Thus, modified Rota-Baxter operators are the associative
analogue of modified $r$-matrices. Given an associative algebra $A$, a linear
map $R:A\rightarrow A$ is a modified Rota-Baxter operator of weight
$\kappa\in{\bf k}$ if

$\displaystyle R(a)\cdot R(b)=R\big{(}R(a)\cdot b+a\cdot
R(b)\big{)}+\kappa~{}\\!a\cdot b,~{}\text{ for all }a,b\in A.$

An associative algebra equipped with a modified Rota-Baxter operator of weight
$\kappa$ is called a modified Rota-Baxter algebra of weight $\kappa$. It has
been observed [34, 35] that if $(A,P)$ is a Rota-Baxter algebra of weight
$\lambda$, then $(A,R=\lambda~{}\\!\mathrm{id}+2P)$ is a modified Rota-Baxter
algebra of weight $\kappa=-\lambda^{2}$.

Algebraic structures are better understood by their invariants, such as
homology, cohomology and $K$-groups. All these invariants have their own
significance. The cohomology theory of an algebraic structure govern formal
deformations of the structure and classify its abelian extensions. For
example, the classical Hochschild cohomology theory is the cohomology for
associative algebras [22]. To know the better description of the cohomology,
one need to understand representations of the given algebraic structure. In
this paper, we first consider bimodules over modified Rota-Baxter algebras and
construct the corresponding semidirect product. A bimodule over a modified
Rota-Baxter algebra $(A,R)$ consists of a pair $(M,S)$ in which $M$ is an
$A$-bimodule and $S:M\rightarrow M$ is a linear map satisfying some
compatibilities. It follows that any modified Rota-Baxter algebra $(A,R)$ is a
bimodule over itself. Given a modified Rota-Baxter algebra $(A,R)$, we show
that the underlying space $A$ carries a new associative algebra structure with
the product $*_{R}$ given by

$\displaystyle a*_{R}b=R(a)\cdot b+a\cdot R(b),~{}\text{ for }a,b\in A.$

We denote this associative algebra by $A_{R}$. Further, if $(M,S)$ is a
bimodule over the modified Rota-Baxter algebra $(A,R)$, then there is an
$A_{R}$-bimodule structure on $M$ (denoted by $\widetilde{M}$). Thus, we may
define the Hochschild cochain complex of the algebra $A_{R}$ with coefficients
in the $A_{R}$-bimodule $\widetilde{M}$. On the other hand, we can consider
the Hochschild cochain complex of the given algebra $A$ with coefficients in
the $A$-bimodule $M$. As a biproduct of the above two cochain complexes, we
construct a new complex defining the cohomology of the modified Rota-Baxter
algebra $(A,R)$ with coefficients in the bimodule $(M,S)$. We compare our
cohomology of modified Rota-Baxter algebras with the cohomology of Rota-Baxter
algebras recently introduced in [13, 33]. More precisely, we show that the
cohomology of a Rota-Baxter algebra $(A,P)$ is isomorphic to the cohomology of
the modified Rota-Baxter algebra $(A,R=\lambda~{}\\!\mathrm{id}+2P)$.

The formal deformation theory of some algebraic structure began with the
seminal work of Gerstenhaber for associative algebras [16, 15]. Subsequently,
the theory has been generalized by Nijenhuis and Richardson for Lie algebras
[28]. See [7, 14, 6, 17] for more details about deformations of various
algebras and morphisms between them. Recently, various authors have considered
formal deformations of Rota-Baxter operators, Rota-Baxter algebras and
modified $r$-matrices motivated by the deformations of classical $r$-matrices
[12, 13, 23, 25, 32, 33, 24]. In this paper, we consider the formal
deformation theory of a modified Rota-Baxter algebra $(A,R)$. In this theory,
we simultaneously deform the algebra $A$ and the modified Rota-Baxter operator
$R$. We show that the infinitesimal in a formal deformation of $(A,R)$ is a
$2$-cocycle in the cohomology complex of $(A,R)$ with coefficients in itself.
Moreover, the infinitesimals corresponding to equivalent deformations are
cohomologous, hence they corresponds to the same element in the second
cohomology group. We also discuss the rigidity of a modified Rota-Baxter
algebra $(A,R)$ and show that the vanishing of the second cohomology group is
a sufficient condition for the rigidity of $(A,R)$.

In the next, we define and study abelian extensions of a modified Rota-Baxter
algebra $(A,R)$ by a bimodule $(M,S)$ over it. We show that the isomorphism
classes of such abelian extensions can be classified by the second cohomology
of the modified Rota-Baxter algebra $(A,R)$ with coefficients in the bimodule
$(M,S).$

The paper is organized as follows. In Section 2, we recall some basic
definitions about modified Rota-Baxter algebras, and define bimodules over
them. The cohomology of a modified Rota-Baxter algebra with coefficients in a
bimodule is defined in Section 3. We also compare this cohomology with the
cohomology of Rota-Baxter algebras. In Section 4, we study formal deformation
theory and rigidity of modified Rota-Baxter algebras. Finally, abelian
extensions are considered in Section 5.

All vector spaces, linear and multilinear maps, tensor products are over a
field ${\bf k}$ of characteristic $0$.

##  2\. Modified Rota-Baxter algebras

In this section, we recall modified Rota-Baxter algebras and define bimodules
over them. Given a modified Rota-Baxter algebra and a bimodule over it, we
also consider the corresponding semidirect product. We show that a modified
Rota-Baxter algebra induces a new associative structure on the underlying
vector space. We also consider some bimodules over this induced associative
algebra.

######  2.1 Definition.

  1. (1)

Let $A$ be an associative algebra and $\kappa\in{\bf k}$. A linear map
$R:A\rightarrow A$ is said to be a modified Rota-Baxter operator of weight
$\kappa$ if

$\displaystyle R(a)\cdot R(b)=R\big{(}R(a)\cdot b+a\cdot
R(b)\big{)}+\kappa~{}\\!a\cdot b,~{}\text{ for }a,b\in A.$

  2. (2)

A modified Rota-Baxter algebra of weight $\kappa$ is a pair $(A,R)$ consisting
of an associative algebra $A$ and a modified Rota-Baxter operator
$R:A\rightarrow A$ of weight $\kappa$.

######  2.2 Definition.

Let $(A,R)$ and $(A^{\prime},R^{\prime})$ be two modified Rota-Baxter algebras
of weight $\kappa$. A morphism of modified Rota-Baxter algebras from $(A,R)$
to $(A^{\prime},R^{\prime})$ is an algebra homomorphism $\varphi:A\rightarrow
A^{\prime}$ that satisfies $R^{\prime}\circ\varphi=\varphi\circ R$.

There is a close relationship between Rota-Baxter algebras and modified Rota-
Baxter algebras.

######  2.3 Lemma.

Let $A$ be an associative algebra. Then a linear map $P:A\rightarrow A$ is a
Rota-Baxter operator of weight $\lambda\in{\bf k}$ if and only if the map
$R=\lambda~{}\\!\mathrm{id}+2P$ is a modified Rota-Baxter operator of weight
$\kappa=-\lambda^{2}.$ Equivalently, $(A,P)$ is a Rota-Baxter algebra of
weight $\lambda$ if and only if $(A,R=\lambda~{}\\!\mathrm{id}+2P)$ is a
modified Rota-Baxter algebra of weight $\kappa=-\lambda^{2}.$

Let $A$ be an associative algebra. Consider the direct sum $A\oplus A$ with
the direct product associative structure

$\displaystyle(a,b)\bullet(a^{\prime},b^{\prime})=(a\cdot a^{\prime},b\cdot
b^{\prime}),\text{ for }(a,b),(a^{\prime},b^{\prime})\in A\oplus A.$

Note that the subspace $A_{\triangle}=\\{(a,a)|~{}a\in A\\}\subset A\oplus A$
is an associative subalgebra. Modified Rota-Baxter operators of weight $-1$
are useful to find compliments of $A_{\triangle}$ which is also a subalgebra.
To see this, let $A_{-\triangle}\subset A\oplus A$ be the subspace given by
$A_{-\triangle}=\\{(a,-a)|~{}a\in A\\}$. For any linear map $R:A\rightarrow
A$, we define a map $\widehat{R}:A_{-\triangle}\rightarrow A_{\triangle}$ by
$\widehat{R}(a,-a)=(-R(a),-R(a))$, for $(a,-a)\in A_{-\triangle}$. Consider
the graph

$\displaystyle\mathrm{Gr}(\widehat{R})=~{}$
$\displaystyle\\{\widehat{R}(a,-a)+(a,-a)|~{}a\in A\\}$ $\displaystyle=~{}$
$\displaystyle\\{(-R(a)+a,-R(a)-a)|~{}a\in A\\}\subset A\oplus A$

of the map $\widehat{R}$. Note that $\mathrm{Gr}(\widehat{R})$ is a compliment
(as a subspace) of $A_{\triangle}$ in the space $A\oplus A$. However,
$\mathrm{Gr}(\widehat{R})$ is in general not a subalgebra. For any $a,b\in A$,
we observe that

$\displaystyle\big{(}-R(a)+a,-R(a)-a\big{)}\bullet\big{(}-R(b)+b,-R(b)-b\big{)}$
$\displaystyle=\big{(}R(a)\cdot R(b)-R(a)\cdot b-a\cdot R(b)+a\cdot
b,~{}R(a)\cdot R(b)+R(a)\cdot b+a\cdot R(b)+a\cdot b\big{)}$

This is in $\mathrm{Gr}(\widehat{R})$ if and only if

$\displaystyle R(a)\cdot R(b)+a\cdot b=R\big{(}R(a)\cdot b+a\cdot
R(b)\big{)}.$

In other words, $\mathrm{Gr}(\widehat{R})$ is a subalgebra of $A\oplus A$ if
and only if $R$ is a modified Rota-Baxter operator of weight $-1$.

Next we define bimodules over a modified Rota-Baxter algebra.

######  2.4 Definition.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$. A bimodule
over $(A,R)$ consists of a pair $(M,S)$ in which $M$ is an $A$-bimodule and
$S:M\rightarrow M$ is a linear map satisfying for $a\in A,$ $u\in M$,

$\displaystyle R(a)\cdot S(u)=S\big{(}R(a)\cdot u+a\cdot
S(u)\big{)}+\kappa~{}\\!a\cdot u,$ $\displaystyle S(u)\cdot
R(a)=S\big{(}S(u)\cdot a+u\cdot R(a)\big{)}+\kappa~{}\\!u\cdot a.$

If $(A,R)$ is a modified Rota-Baxter algebra of weight $\kappa$, then the pair
$(A,R)$ itself is a bimodule over it, where the $A$-bimodule structure on $A$
is given by the algebra multiplication. This is called the adjoint bimodule.

The following result finds the relation between bimodules over Rota-Baxter
algebras and over modified Rota-Baxter algebras.

######  2.5 Proposition.

Let $(A,P)$ be a Rota-Baxter algebra of weight $\lambda$, and $(M,Q)$ be a
bimodule over it. Then $(M,S=\lambda~{}\\!\mathrm{id}_{M}+2Q)$ is a bimodule
over the modified Rota-Baxter algebra $(A,R=\lambda~{}\\!\mathrm{id}_{A}+2P)$
of weight $\kappa=-\lambda^{2}$.

###### Proof.

For any $a\in A$ and $u\in M$, we have

$\displaystyle R(a)\cdot S(u)=~{}$ $\displaystyle\big{(}\lambda
a+2P(a)\big{)}\cdot\big{(}\lambda u+2Q(u)\big{)}$ $\displaystyle=~{}$
$\displaystyle\lambda^{2}~{}\\!a\cdot u+2\lambda\big{(}P(a)\cdot u+a\cdot
Q(u)\big{)}+4~{}\\!P(a)\cdot Q(u)$ $\displaystyle=~{}$
$\displaystyle\lambda^{2}~{}\\!a\cdot u+2\lambda\big{(}P(a)\cdot u+a\cdot
Q(u)\big{)}+4~{}\\!Q\big{(}P(a)\cdot u+a\cdot Q(u)+\lambda~{}\\!a\cdot
u\big{)}$ $\displaystyle=~{}$
$\displaystyle(\lambda~{}\\!\mathrm{id}_{M}+2Q)\big{(}(\lambda~{}\\!\mathrm{id}_{A}+2P)(a)\cdot
u~{}+~{}a\cdot(\lambda\mathrm{id}_{M}+2Q)(u)\big{)}~{}-~{}\lambda^{2}~{}\\!a\cdot
u$ $\displaystyle=~{}$ $\displaystyle S\big{(}R(a)\cdot u+a\cdot
S(u)\big{)}+\kappa~{}\\!a\cdot u.$

Similarly, we can show that

$\displaystyle S(u)\cdot R(a)=S\big{(}S(u)\cdot a+u\cdot
R(a)\big{)}+\kappa~{}\\!u\cdot a.$

This completes the proof. ∎

######  2.6 Proposition.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$, and
$\\{(M_{i},S_{i})\\}_{i\in I}$be a family of bimodules over it. Then
$(\oplus_{i\in I}M_{i},\oplus_{i\in I}S_{i})$ is also a bimodule over $(A,R)$.

###### Proof.

The $A$-bimodule structure on $\oplus_{i\in I}M_{i}$ is given by the
componentwise $A$-bimodule structure on each $M_{i}$. For any $a\in A$ and
$(u_{i})_{i\in I}\in\oplus_{i\in I}M_{i}$, we have

$\displaystyle R(a)\cdot(\oplus_{i\in I}S_{i})(u_{i})_{i\in I}=~{}$
$\displaystyle R(a)\cdot(S_{i}(u_{i}))_{i\in I}$ $\displaystyle=~{}$
$\displaystyle\big{(}R(a)\cdot S_{i}(u_{i})\big{)}_{i\in I}$
$\displaystyle=~{}$ $\displaystyle\big{(}S_{i}(R(a)\cdot u_{i}+a\cdot
S_{i}(u_{i}))+\kappa~{}\\!a\cdot u_{i}\big{)}_{i\in I}$ $\displaystyle=~{}$
$\displaystyle(\oplus_{i\in I}S_{i})\bigg{(}R(a)\cdot(u_{i})_{i\in
I}+a\cdot\big{(}(\oplus_{i\in I}S_{i})(u_{i})_{i\in
I}\big{)}\bigg{)}+\kappa~{}\\!a\cdot(u_{i})_{i\in I}.$

This shows that $(\oplus_{i\in I}M_{i},\oplus_{i\in I}S_{i})$ is a bimodule
over the modified Rota-Baxter algebra $(A,R)$ of weight $\kappa$. ∎

Recall that, if $M$ is an $A$-bimodule, then $\mathrm{End}(M)$ can be equipped
with an $A$-bimodule structure with left and right $A$-actions

$\displaystyle(a\odot f)(u)=f(u\cdot a)~{}~{}~{}\text{ and }~{}~{}~{}(f\odot
a)(u)=f(a\cdot u),\text{ for }a\in A,f\in\mathrm{End}(M),u\in M.$

######  2.7 Proposition.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$, and $(M,S)$
be a bimodule over it. Then $(\mathrm{End}(M),\widetilde{S})$ is also a
bimodule over $(A,R)$, where
$\widetilde{S}:\mathrm{End}(M)\rightarrow\mathrm{End}(M)$ is given by
$\widetilde{S}(f)=-f\circ S$, for $f\in\mathrm{End}(M)$.

###### Proof.

For any $a\in A$, $f\in\mathrm{End}(M)$ and $u\in M$, we have

$\displaystyle\big{(}R(a)\odot\widetilde{S}(f)\big{)}(u)=\widetilde{S}(f)\big{(}u\cdot
R(a)\big{)}=~{}$ $\displaystyle-f\big{(}S(u\cdot R(a))\big{)}$
$\displaystyle=~{}$ $\displaystyle-f\big{(}S(u)\cdot R(a)-S(S(u)\cdot
a)-\kappa~{}\\!u\cdot a\big{)}$ $\displaystyle=~{}$
$\displaystyle-f\big{(}S(u)\cdot R(a)\big{)}-\widetilde{S}(f)(S(u)\cdot
a)+\kappa~{}\\!f(u\cdot a)$ $\displaystyle=~{}$
$\displaystyle-\big{(}R(a)\odot
f+a\odot\widetilde{S}(f)\big{)}(S(u))+\kappa~{}\\!f(u\cdot a)$
$\displaystyle=~{}$ $\displaystyle\big{(}\widetilde{S}\big{(}R(a)\odot
f+a\odot\widetilde{S}(f)\big{)}+\kappa~{}\\!a\odot f\big{)}(u)$

and

$\displaystyle\big{(}\widetilde{S}(f)\odot
R(a)\big{)}(u)=\widetilde{S}(f)\big{(}R(a)\cdot u\big{)}=~{}$
$\displaystyle-f\big{(}S(R(a)\cdot u)\big{)}$ $\displaystyle=~{}$
$\displaystyle-f\big{(}R(a)\cdot S(u)-S(a\cdot S(u))-\kappa~{}\\!a\cdot
u\big{)}$ $\displaystyle=~{}$ $\displaystyle-f\big{(}R(a)\cdot
S(u)\big{)}-\widetilde{S}(f)(a\cdot S(u))+\kappa~{}\\!f(a\cdot u)$
$\displaystyle=~{}$ $\displaystyle-\big{(}f\odot R(a)+\widetilde{S}(f)\odot
a\big{)}(S(u))+\kappa~{}\\!f(a\cdot u)$ $\displaystyle=~{}$
$\displaystyle\big{(}\widetilde{S}\big{(}\widetilde{S}(f)\odot a+f\odot
R(a)\big{)}+\kappa~{}\\!f\odot a\big{)}(u).$

This shows that $(\mathrm{End}(M),\widetilde{S})$ is a bimodule over the
modified Rota-Baxter algebra $(A,R)$ of weight $\kappa$. ∎

######  2.8 Proposition.

(Semidirect product) Let $(A,R)$ be a modified Rota-Baxter algebra of weight
$\kappa$, and $(M,S)$ be a bimodule over it. Then the pair $(A\oplus M,R\oplus
S)$ is a modified Rota-Baxter algebra of weight $\kappa$, where the
associative algebra structure on $A\oplus M$ is given by the semidirect
product

$\displaystyle(a,u)\cdot_{\ltimes}(b,v)=(a\cdot b,a\cdot v+u\cdot b),\text{
for }(a,u),(b,v)\in A\oplus M.$

###### Proof.

For $(a,u),(b,v)\in A\oplus M$, we have

$\displaystyle(R\oplus S)(a,u)\cdot_{\ltimes}(R\oplus S)(b,v)$
$\displaystyle=(R(a),S(u))\cdot_{\ltimes}(R(b),S(v))$
$\displaystyle=\big{(}R(a)\cdot R(b),~{}R(a)\cdot S(v)+S(u)\cdot R(b)\big{)}$
$\displaystyle=\bigg{(}R(R(a)\cdot b+a\cdot R(b))+\kappa~{}\\!a\cdot
b,~{}S(R(a)\cdot v+a\cdot S(v))+\kappa~{}\\!a\cdot v+S(S(u)\cdot b+u\cdot
R(b))+\kappa~{}\\!u\cdot b\bigg{)}$ $\displaystyle=(R\oplus
S)\bigg{(}\big{(}R(a)\cdot b,~{}R(a)\cdot v+S(u)\cdot b\big{)}+\big{(}a\cdot
R(b),~{}a\cdot S(v)+u\cdot R(b)\big{)}\bigg{)}+\kappa~{}\\!\big{(}a\cdot
b,~{}a\cdot v+u\cdot b\big{)}$ $\displaystyle=(R\oplus S)\bigg{(}(R\oplus
S)(a,u)\cdot_{\ltimes}(b,v)~{}+~{}(a,u)\cdot_{\ltimes}(R\oplus
S)(b,v)\bigg{)}+\kappa~{}\\!(a,u)\cdot_{\ltimes}(b,v).$

This shows that $R\oplus S$ is a modified Rota-Baxter operator of weight
$\kappa$ on the semidirect product associative algebra $A\oplus M$. In other
words, $(A\oplus M,R\oplus S)$ is a modified Rota-Baxter algebra of weight
$\kappa$. ∎

In the following, we show that a modified Rota-Baxter algebra of weight
$\kappa$ induces a new associative algebra structure. This associative algebra
will be useful in the construction of the cohomology of the modified Rota-
Baxter algebra.

######  2.9 Proposition.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$. Then we have
the followings.

(i) The underlying vector space $A$ inherits a new associative algebra
structure with the product

$\displaystyle a\ast_{R}b=R(a)\cdot b+a\cdot R(b),~{}\text{ for }a,b\in A.$

(We denote this associative algebra simply by $A_{R}$).

(ii) The pair $(A_{R},R)$ is a modified Rota-Baxter algebra of weight
$\kappa$.

###### Proof.

For any $a,b,c\in A$, we have

$\displaystyle(a\ast_{R}b)\ast_{R}c=~{}$ $\displaystyle R(a\ast_{R}b)\cdot
c+(a\ast_{R}b)\cdot R(c)$ $\displaystyle=~{}$ $\displaystyle\big{(}R(a)\cdot
R(b)-\kappa~{}\\!a\cdot b\big{)}\cdot c+\big{(}R(a)\cdot b+a\cdot
R(b)\big{)}\cdot R(c)$ $\displaystyle=~{}$ $\displaystyle
R(a)\cdot\big{(}R(b)\cdot c+b\cdot R(c)\big{)}+a\cdot\big{(}R(b)\cdot
R(c)-\kappa~{}\\!b\cdot c\big{)}$ $\displaystyle=~{}$ $\displaystyle
R(a)\cdot(b\ast_{R}c)+a\cdot R(b\ast_{R}c)$ $\displaystyle=~{}$ $\displaystyle
a\ast_{R}(b\ast_{R}c).$

This shows that $A_{R}$ is an associative algebra.

(ii) For any $a,b\in A$,

$\displaystyle R(a)\ast_{R}R(b)=~{}$ $\displaystyle R^{2}(a)\cdot
R(b)+R(a)\cdot R^{2}(b)$ $\displaystyle=~{}$ $\displaystyle
R(R(a)\ast_{R}b)+\kappa~{}\\!R(a)\cdot b+R(a\ast_{R}R(b))+\kappa~{}\\!a\cdot
R(b)$ $\displaystyle=~{}$ $\displaystyle
R\big{(}R(a)\ast_{R}b+a\ast_{R}R(b)\big{)}+\kappa~{}\\!a\ast_{R}b.$

This shows that $R$ is a modified Rota-Baxter operator of weight $\kappa$ on
the associative algebra $A_{R}$. In other words, $(A_{R},R)$ is a modified
Rota-Baxter algebra of weight $\kappa$. ∎

######  2.10 Proposition.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$, and $(M,S)$
be a bimodule over it.

(i) Then there is an $A_{R}$-bimodule structure on $M$ with left and right
actions (denoted by the same notation) given by

$\displaystyle a~{}{\cdot}_{S}~{}u:=R(a)\cdot u+a\cdot S(u)\quad\text{ and
}\quad u~{}{\cdot}_{S}~{}a:=S(u)\cdot a+u\cdot R(a),$

for $a\in A_{R}$, $u\in M$. We denote this $A_{R}$-bimodule structure simply
by ${M}_{S}$.

(ii) The pair $({M}_{S},S)$ is bimodule over the modified Rota-Baxter algebra
$(A_{R},R)$ of weight $\kappa$.

###### Proof.

(i) Consider the (semidirect product) modified Rota-Baxter algebra $(A\oplus
M,R\oplus S)$ of weight $\kappa$ given in Proposition 2.8. This induces an
associative algebra structure on the direct sum $A\oplus B$ (which we denote
by $(A\oplus B)_{R\oplus S}$) with the product given by

$\displaystyle(a,u)\ltimes_{R\oplus S}(b,v)=~{}$
$\displaystyle(R(a),S(u))\ltimes_{R\oplus S}(b,v)~{}+~{}(a,u)\ltimes_{R\oplus
S}(R(b),S(v))$ $\displaystyle=~{}$ $\displaystyle\big{(}R(a)\cdot b+a\cdot
R(b),~{}R(a)\cdot v+S(u)\cdot b+a\cdot S(v)+u\cdot R(b)\big{)}$
$\displaystyle=~{}$
$\displaystyle\big{(}a\ast_{R}b,~{}a~{}{\cdot}_{S}~{}v+u~{}{\cdot}_{S}~{}b\big{)}.$

This expression of the product says that ${M}_{S}$ is an $A_{R}$-bimodule.

(ii) For any $a\in A$ and $u\in M$, we have

$\displaystyle R(a)~{}{\cdot}_{S}~{}S(u)=~{}$ $\displaystyle R^{2}(a)\cdot
S(u)+R(a)\cdot S^{2}(u)$ $\displaystyle=~{}$ $\displaystyle
S\big{(}R^{2}(a)\cdot u+R(a)\cdot S(u)\big{)}+\kappa~{}\\!R(a)\cdot
u+S\big{(}R(a)\cdot S(u)+a\cdot S^{2}(u)\big{)}+\kappa~{}\\!a\cdot S(u)$
$\displaystyle=~{}$ $\displaystyle
S\big{(}R(a)~{}{\cdot}_{S}~{}u\big{)}+\kappa~{}\\!R(a)\cdot
u+S\big{(}a~{}{\cdot}_{S}~{}S(u)\big{)}+\kappa~{}\\!a\cdot S(u)$
$\displaystyle=~{}$ $\displaystyle
S\big{(}R(a)~{}{\cdot}_{S}~{}u+a~{}{\cdot}_{S}~{}S(u)\big{)}+\kappa~{}\\!a~{}{\cdot}_{S}~{}u$

and

$\displaystyle S(u)~{}{\cdot}_{S}~{}R(a)=~{}$ $\displaystyle S^{2}(u)\cdot
R(a)+S(u)\cdot R^{2}(a)$ $\displaystyle=~{}$ $\displaystyle
S\big{(}S^{2}(u)\cdot a+S(u)\cdot R(a)\big{)}+\kappa~{}\\!S(u)\cdot
a+S\big{(}S(u)\cdot R(a)+u\cdot R^{2}(a)\big{)}+\kappa~{}\\!u\cdot R(a)$
$\displaystyle=~{}$ $\displaystyle
S\big{(}S(u)~{}{\cdot}_{S}~{}a\big{)}+\kappa~{}\\!S(u)\cdot
a+S\big{(}u~{}{\cdot}_{S}~{}R(a)\big{)}+\kappa~{}\\!u\cdot R(a)$
$\displaystyle=~{}$ $\displaystyle
S\big{(}S(u)~{}{\cdot}_{S}~{}a+u~{}{\cdot}_{S}~{}R(a)\big{)}+\kappa~{}\\!u~{}{\cdot}_{S}~{}a.$

This shows that $({M}_{S},S)$ is a bimodule over the modified Rota-Baxter
algebra $(A_{R},R)$. ∎

######  2.11 Remark.

When $(M,S)=(A,R)$ is the adjoint bimodule over the modified Rota-Baxter
algebra $(A,R)$ of weight $\kappa$, then the $A_{R}$-bimodule ${M}_{S}$
coincides with the adjoint $A_{R}$-bimodule. Hence we have
$({M}_{S},S)=(A_{R},R)$ the adjoint bimodule over the modified Rota-Baxter
algebra $(A_{R},R)$ of weight $\kappa$.

Given a bimodule over a modified Rota-Baxter algebra $(A,R)$, in Proposition
2.10, we constructed a bimodule over the induced modified Rota-Baxter algebra
$(A_{R},R)$. In the following, we will construct another bimodule over
$(A_{R},R)$. This new construction will be helpful to construct the cohomology
of the modified Rota-Baxter algebra.

######  2.12 Proposition.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$, and $(M,S)$
be a bimodule over it. Then $M$ can be equipped with an $A_{R}$-bimodule
structure with left and right $A_{R}$-actions

$\displaystyle a~{}\widetilde{\cdot}~{}u:=R(a)\cdot u-S(a\cdot u)\quad\text{
and }\quad u~{}\widetilde{\cdot}~{}a:=u\cdot R(a)-S(u\cdot a),$

for $a\in A_{R}$, $u\in M$. (We denote this $A_{R}$-bimodule structure simply
by $\widetilde{M}$). Moreover, the pair $(\widetilde{M},S)$ is a bimodule over
the modified Rota-Baxter algebra $(A_{R},R)$ of weight $\kappa$.

###### Proof.

For $a,b\in A_{R}$ and $u\in M$, we have

$\displaystyle(a\ast_{R}b)~{}\widetilde{\cdot}~{}u=~{}$ $\displaystyle
R(a\ast_{R}b)\cdot u-S\big{(}(a\ast_{R}b)\cdot u\big{)}$ $\displaystyle=~{}$
$\displaystyle\big{(}R(a)\cdot R(b)-\kappa~{}\\!a\cdot b\big{)}\cdot
u-S\big{(}(R(a)\cdot b+a\cdot R(b))\cdot u\big{)}$ $\displaystyle=~{}$
$\displaystyle R(a)\cdot R(b)\cdot u-\kappa~{}\\!a\cdot b\cdot
u-S\big{(}R(a)\cdot b\cdot u+a\cdot S(b\cdot u)\big{)}-S\big{(}a\cdot
R(b)\cdot u-a\cdot S(b\cdot u)\big{)}$ $\displaystyle=~{}$ $\displaystyle
R(a)\cdot(R(b)\cdot u)-R(a)\cdot S(b\cdot u)-S\big{(}a\cdot R(b)\cdot u-a\cdot
S(b\cdot u)\big{)}$ $\displaystyle=~{}$ $\displaystyle
R(a)\cdot\big{(}R(b)\cdot u-S(b\cdot u)\big{)}-S\big{(}a\cdot(R(b)\cdot
u-S(b\cdot u))\big{)}$ $\displaystyle=~{}$ $\displaystyle
a~{}\widetilde{\cdot}~{}\big{(}R(b)\cdot u-S(b\cdot u)\big{)}$
$\displaystyle=~{}$ $\displaystyle
a~{}\widetilde{\cdot}~{}(b~{}\widetilde{\cdot}~{}u).$

Similarly,

$\displaystyle(a~{}\widetilde{\cdot}~{}u)~{}\widetilde{\cdot}~{}b=~{}$
$\displaystyle(a~{}\widetilde{\cdot}~{}u)\cdot
R(b)-S\big{(}(a~{}\widetilde{\cdot}~{}u)\cdot b\big{)}$ $\displaystyle=~{}$
$\displaystyle\big{(}R(a)\cdot u-S(a\cdot u)\big{)}\cdot
R(b)-S\big{(}(R(a)\cdot u-S(a\cdot u))\cdot b\big{)}$ $\displaystyle=~{}$
$\displaystyle R(a)\cdot u\cdot R(b)-S\big{(}S(a\cdot u)\cdot b+a\cdot u\cdot
R(b)\big{)}-\kappa~{}\\!a\cdot u\cdot b-S\big{(}R(a)\cdot u\cdot
b\big{)}+S\big{(}S(a\cdot u)\cdot b\big{)}$ $\displaystyle=~{}$ $\displaystyle
R(a)\cdot u\cdot R(b)-S\big{(}R(a)\cdot u\cdot b+a\cdot S(u\cdot
b)\big{)}-\kappa~{}\\!a\cdot u\cdot b-S\big{(}a\cdot u\cdot
R(b)\big{)}+S\big{(}a\cdot S(u\cdot b)\big{)}$ $\displaystyle=~{}$
$\displaystyle R(a)\cdot\big{(}u\cdot R(b)-S(u\cdot
b)\big{)}-S\big{(}a\cdot(u\cdot R(b)-S(u\cdot b))\big{)}$ $\displaystyle=~{}$
$\displaystyle
R(a)\cdot(u~{}\widetilde{\cdot}~{}b)-S\big{(}a\cdot(u~{}\widetilde{\cdot}~{}b)\big{)}$
$\displaystyle=~{}$ $\displaystyle
a~{}\widetilde{\cdot}~{}(u~{}\widetilde{\cdot}~{}b)$

and

$\displaystyle(u~{}\widetilde{\cdot}~{}a)~{}\widetilde{\cdot}~{}b=~{}$
$\displaystyle(u~{}\widetilde{\cdot}~{}a)\cdot
R(b)-S\big{(}(u~{}\widetilde{\cdot}~{}a)\cdot b\big{)}$ $\displaystyle=~{}$
$\displaystyle\big{(}u\cdot R(a)-S(u\cdot a)\big{)}\cdot R(b)-S\big{(}(u\cdot
R(a)-S(u\cdot a))\cdot b\big{)}$ $\displaystyle=~{}$ $\displaystyle u\cdot
R(a)\cdot R(b)-S(u\cdot a)\cdot R(b)-S\big{(}u\cdot R(a)\cdot
b\big{)}+S\big{(}S(u\cdot a)\cdot b\big{)}$ $\displaystyle=~{}$ $\displaystyle
u\cdot R(a)\cdot R(b)-S\big{(}S(u\cdot a)\cdot b+u\cdot a\cdot
R(b)\big{)}-\kappa~{}\\!u\cdot a\cdot b-S\big{(}u\cdot R(a)\cdot
b\big{)}+S\big{(}S(u\cdot a)\cdot b\big{)}$ $\displaystyle=~{}$ $\displaystyle
u\cdot\big{(}R(a)\cdot R(b)-\kappa~{}\\!a\cdot b\big{)}-S\big{(}u\cdot
R(a)\cdot b+u\cdot a\cdot R(b)\big{)}$ $\displaystyle=~{}$ $\displaystyle
u\cdot R(a\ast_{R}b)-S\big{(}u\cdot(a\ast_{R}b)\big{)}$ $\displaystyle=~{}$
$\displaystyle u~{}\widetilde{\cdot}~{}(a\ast_{R}b).$

This shows that $\widetilde{M}$ is an $A_{R}$-bimodule. For the second part,
we observe that

$\displaystyle R(a)~{}\widetilde{\cdot}~{}S(u)=~{}$ $\displaystyle
R^{2}(a)\cdot S(u)-S\big{(}R(a)\cdot S(u)\big{)}$ $\displaystyle=~{}$
$\displaystyle S\big{(}R^{2}(a)\cdot u+R(a)\cdot
S(u)\big{)}+\kappa~{}\\!R(a)\cdot u-S^{2}\big{(}R(a)\cdot u+a\cdot
S(u)\big{)}-\kappa~{}\\!S(a\cdot u)$ $\displaystyle=~{}$ $\displaystyle
S\big{(}R^{2}(a)\cdot u-S(R(a)\cdot u)+R(a)\cdot S(u)-S(a\cdot
S(u))\big{)}+\kappa~{}\\!\big{(}R(a)\cdot u-S(a\cdot u)\big{)}$
$\displaystyle=~{}$ $\displaystyle
S\big{(}R(a)~{}\widetilde{\cdot}~{}u+a~{}\widetilde{\cdot}~{}S(u)\big{)}+\kappa~{}\\!a~{}\widetilde{\cdot}~{}u$

and

$\displaystyle S(u)~{}\widetilde{\cdot}~{}R(a)=~{}$ $\displaystyle S(u)\cdot
R^{2}(a)-S\big{(}S(u)\cdot R(a)\big{)}$ $\displaystyle=~{}$ $\displaystyle
S\big{(}S(u)\cdot R(a)+u\cdot R^{2}(a)\big{)}+\kappa~{}\\!u\cdot
R(a)-S^{2}\big{(}S(u)\cdot a+u\cdot R(a)\big{)}-\kappa~{}\\!S(u\cdot a)$
$\displaystyle=~{}$ $\displaystyle S\big{(}S(u)\cdot R(a)-S(S(u)\cdot
a)+u\cdot R^{2}(a)-S(u\cdot R(a))\big{)}+\kappa~{}\\!\big{(}u\cdot
R(a)-S(u\cdot a)\big{)}$ $\displaystyle=~{}$ $\displaystyle
S\big{(}S(u)~{}\widetilde{\cdot}~{}a+u~{}\widetilde{\cdot}~{}R(a)\big{)}+\kappa~{}\\!u~{}\widetilde{\cdot}~{}a,$

which shows that $(\widetilde{M},S)$ is a bimodule over the modified Rota-
Baxter algebra $(A_{R},R)$ of weight $\kappa$. ∎

##  3\. Cohomology of modified Rota-Baxter algebras

In this section, we introduce the cohomology of a modified Rota-Baxter algebra
$(A,R)$ with coefficients in a bimodule $(M,S)$. This cohomology is obtained
as a biproduct of the Hochschild cohomology of the underlying associative
algebra $A$ and the Hochschild cohomology of the induced associative algebra
$A_{R}$. We also compare our cohomology of a modified Rota-Baxter algebra with
the cohomology of a Rota-Baxter algebra defined in [13, 33].

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$, and $(M,S)$
be a bimodule over it. First, consider the Hochschild cochain complex of $A$
with coefficients in the $A$-bimodule $M$. The complex is given by
$\\{C^{\bullet}(A,M),\delta_{\mathrm{Hoch}}\\}$, where
$C^{n}(A,M)=\mathrm{Hom}(A^{\otimes n},M)$ for $n\geq 0$, and the coboundary
map $\delta_{\mathrm{Hoch}}:C^{n}(A,M)\rightarrow C^{n+1}(A,M)$ is given by

$\displaystyle(\delta_{\mathrm{Hoch}}f)(a_{1},\ldots,a_{n+1})=~{}$
$\displaystyle(-1)^{n+1}~{}a_{1}\cdot
f(a_{2},\ldots,a_{n+1})+f(a_{1},\ldots,a_{n})\cdot a_{n+1}$
$\displaystyle+\sum_{i=1}^{n}(-1)^{i+n+1}~{}f(a_{1},\ldots,a_{i}\cdot
a_{i+1},\ldots,a_{n+1}),$

for $f\in C^{n}(A,M)$ and $a_{1},\ldots,a_{n+1}\in A$. We denote the
corresponding cohomology groups by $H^{\bullet}(A,M)$. Second, we consider the
Hochschild cochain complex
$\\{C^{\bullet}(A_{R},\widetilde{M}),\widetilde{\delta}_{\mathrm{Hoch}}\\}$ of
the induced associative algebra $A_{R}$ with coefficients in the
$A_{R}$-bimodule $\widetilde{M}$, where
$C^{n}(A_{R},\widetilde{M})=\mathrm{Hom}(A^{\otimes n},M)$ for $n\geq 0$, and
the coboundary map
$\widetilde{\delta}_{\mathrm{Hoch}}:C^{n}(A_{R},\widetilde{M})\rightarrow
C^{n+1}(A_{R},\widetilde{M})$ is given by

$\displaystyle(\widetilde{\delta}_{\mathrm{Hoch}}f)(a_{1},\ldots,a_{n+1})=~{}$
$\displaystyle(-1)^{n+1}R(a_{1})\cdot
f(a_{2},\ldots,a_{n+1})-(-1)^{n+1}~{}S\big{(}a_{1}\cdot
f(a_{2},\ldots,a_{n+1})\big{)}$ $\displaystyle+f(a_{1},\ldots,a_{n})\cdot
R(a_{n+1})-S\big{(}f(a_{1},\ldots,a_{n})\cdot a_{n+1}\big{)}$
$\displaystyle+\sum_{i=1}^{n}(-1)^{i+n+1}~{}f\big{(}a_{1},\ldots,R(a_{i})\cdot
a_{i+1}+a_{i}\cdot R(a_{i+1}),\ldots,a_{n+1}\big{)},$

for $f\in C^{n}(A_{R},\widetilde{M})$ and $a_{1},\ldots,a_{n+1}\in A$. The
cohomology groups of this cochain complex are denoted by
$H^{\bullet}_{\mathrm{Hoch}}(A_{R},\widetilde{M})$.

######  3.1 Remark.

Let $(M,S)=(A,R)$ be the adjoint bimodule over the modified Rota-Baxter
algebra $(A,R)$ of weight $\kappa$. Then the cochain complex constructed above
is given by
$\\{C^{\bullet}(A_{R},\widetilde{A}),\widetilde{\delta}_{\mathrm{Hoch}}\\}$,
which is the Hochschild cochain complex of the induced algebra $A_{R}$ with
coefficients in the $A_{R}$-bimodule $\widetilde{A}$. It can be easily check
that the corresponding cohomology governs the formal deformation theory of the
modified Rota-Baxter operator $R$ keeping the underlying associative structure
on $A$ intact. In the following, we will be interested in a more general
cohomology that simultaneously controls the formal deformations of the
underlying algebra and the modified Rota-Baxter operator.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$, and $(M,S)$
be a bimodule over it. Our construction of the general cohomology is based on
the following result. This is highly motivated by the similar result in the
context of Rota-Baxter algebras [33, Proposition 5.2]. The proof can be done
by considering the correspondence between Rota-Baxter algebras and modified
Rota-Baxter algebras, and using the above mentioned reference.

######  3.2 Proposition.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$, and $(M,S)$
be a bimodule. Then the collection of maps $\\{\Psi^{n}:C^{n}(A,M)\rightarrow
C^{n}(A_{R},\widetilde{M})\\}_{n\geq 0}$ given by

$\displaystyle\Psi^{0}=~{}$ $\displaystyle\mathrm{id}_{M},~{}\text{ and }$
$\displaystyle\Psi^{n}(f)(a_{1},\ldots,a_{n})=~{}$ $\displaystyle
f\big{(}R(a_{1}),\ldots,R(a_{n})\big{)}$ $\displaystyle-\sum_{1\leq
i_{1}<\cdots<i_{r}\leq n,,r\text{ odd }}(-\kappa)^{\frac{r-1}{2}}S\circ
f\big{(}R(a_{1}),\ldots,a_{i_{1}},\ldots,a_{i_{2}},\ldots,a_{i_{r}},\ldots,R(a_{n})\big{)}$
$\displaystyle-\sum_{1\leq i_{1}<\cdots<i_{r}\leq n,,r\text{ even
}}(-\kappa)^{\frac{r}{2}+1}S\circ
f\big{(}R(a_{1}),\ldots,a_{i_{1}},\ldots,a_{i_{2}},\ldots,a_{i_{r}},\ldots,R(a_{n})\big{)}$

defines a morphism of cochain complexes from
$\\{C^{\bullet}(A,M),\delta_{\mathrm{Hoch}}\\}$ to
$\\{C^{\bullet}(A_{R},\widetilde{M}),\widetilde{\delta}_{\mathrm{Hoch}}\\}$.

The above proposition suggests to consider a new cochain complex
$\\{C^{\bullet}_{\mathrm{mRBA}}((A,R),(M,S)),\delta_{\mathrm{mRBA}}\\}$, where

$\displaystyle C^{n}_{\mathrm{mRBA}}((A,R),(M,S))=\begin{cases}M&\text{ if
}n=0,\\\ C^{n}(A,M)\oplus C^{n-1}(A_{R},\widetilde{M})=\mathrm{Hom}(A^{\otimes
n},M)\oplus\mathrm{Hom}(A^{\otimes n-1},{M})&\text{ if }n\geq 1.\end{cases}$

The coboundary operator
$\delta_{\mathrm{mRBA}}:C^{n}_{\mathrm{mRBA}}((A,R),(M,S))\rightarrow
C^{n+1}_{\mathrm{mRBA}}((A,R),(M,S))$ is given by

$\displaystyle\delta_{\mathrm{mRBA}}(u)=~{}$
$\displaystyle(\delta_{\mathrm{Hoch}}(u),-u),~{}\text{ for }u\in M,$
$\displaystyle\delta_{\mathrm{mRBA}}(\chi,\Phi)=~{}$
$\displaystyle\big{(}\delta_{\mathrm{Hoch}}(\chi),~{}-\widetilde{\delta}_{\mathrm{Hoch}}(\Phi)-\Psi^{n}(\chi)\big{)},~{}\text{
for }(\chi,\Phi)\in C^{n}_{\mathrm{mRBA}}((A,R),(M,S)).$

Note that $(\delta_{\mathrm{mRBA}})^{2}=0$ as we have

$\displaystyle(\delta_{\mathrm{mRBA}})^{2}(u)=~{}$
$\displaystyle\delta_{\mathrm{mRBA}}(\delta_{\mathrm{Hoch}}(u),-u)$
$\displaystyle=~{}$
$\displaystyle\big{(}(\delta_{\mathrm{Hoch}})^{2}(u),~{}\widetilde{\delta}_{\mathrm{Hoch}}(u)-\Psi^{1}\circ\delta_{\mathrm{Hoch}}(u)\big{)}=0$

and

$\displaystyle(\delta_{\mathrm{mRBA}})^{2}(\chi,\Phi)=~{}$
$\displaystyle\delta_{\mathrm{mRBA}}\big{(}\delta_{\mathrm{Hoch}}(\chi),~{}-\widetilde{\delta}_{\mathrm{Hoch}}(\Phi)-\Psi^{n}(\chi)\big{)}$
$\displaystyle=~{}$
$\displaystyle\big{(}(\delta_{\mathrm{Hoch}})^{2}(\chi),~{}(\widetilde{\delta}_{\mathrm{Hoch}})^{2}(\Phi)+\widetilde{\delta}_{\mathrm{Hoch}}\circ\Psi^{n}(\chi)-\Psi^{n+1}\circ\delta_{\mathrm{Hoch}}(\chi)\big{)}=0.$

It is clear from the above description that a pair
$(\chi,\Phi)\in\mathrm{Hom}(A^{\otimes 2},M)\oplus\mathrm{Hom}(A,M)$ is a
$2$-cocycle if they satisfy

$\displaystyle\begin{cases}a\cdot\chi(b,c)-\chi(a\cdot b,c)+\chi(a,b\cdot
c)-\chi(a,b)\cdot c=0,\\\ \\\
R(a)\cdot\Phi(b)-S(a\cdot\Phi(b))-\Phi\big{(}R(a)\cdot b+a\cdot
R(b)\big{)}+\Phi(a)\cdot R(b)-S(\Phi(a)\cdot b)\\\
\quad+\chi(R(a),R(b))-\kappa~{}\\!\chi(a,b)-R\big{(}\chi(R(a),b)+\chi(a,R(b))\big{)}=0.\end{cases}$

Further, it is a $2$-coboundary if there exists a pair
$(\varphi_{1},u)\in\mathrm{Hom}(A,M)\oplus M$ such that

$\displaystyle\chi=\delta_{\mathrm{Hoch}}(\varphi_{1})~{}~{}~{}\text{ and
}~{}~{}~{}\Phi=-\widetilde{\delta}_{\mathrm{Hoch}}(u)-\Psi^{1}(\varphi_{1}).$

Let $Z^{n}_{\mathrm{mRBA}}((A,R),(M,S))$ denote the space of $n$-cocycles, and
$B^{n}_{\mathrm{mRBA}}((A,R),(M,S))$ denote the space of $n$-coboundaries.
Then we have $B^{n}_{\mathrm{mRBA}}((A,R),(M,S))\subset
Z^{n}_{\mathrm{mRBA}}((A,R),(M,S))$, for $n\geq 0$. The quotient groups

$\displaystyle
H^{n}_{\mathrm{mRBA}}((A,R),(M,S)):=\frac{Z^{n}_{\mathrm{mRBA}}((A,R),(M,S))}{B^{n}_{\mathrm{mRBA}}((A,R),(M,S))},\text{
for }n\geq 0$

are called the cohomology of the modified Rota-Baxter algebra $(A,R)$ of
weight $\kappa$ with coefficients in the bimodule $(M,S)$.

It is easy to see that there is a short exact sequence

$\displaystyle
0\rightarrow\big{\\{}C^{\bullet-1}(A_{R},\widetilde{M}),\widetilde{\delta}_{\mathrm{Hoch}}\big{\\}}\xrightarrow{\iota}\big{\\{}C^{\bullet}_{\mathrm{mRBA}}((A,R),(M,S)),\delta_{\mathrm{mRBA}}\big{\\}}\xrightarrow{p}\big{\\{}C^{\bullet}(A,M),\delta_{\mathrm{Hoch}}\big{\\}}\rightarrow
0$

of cochain complexes, where the maps $\iota$ and $p$ are given by
$\iota(\Phi)=(0,(-1)^{n-1}\Phi)$, for $\Phi\in C^{n-1}(A_{R},\widetilde{M})$
and $p(\chi,\Phi)=\chi$, for $(\chi,\Phi)\in
C^{n}_{\mathrm{mRBA}}((A,R),(M,S))$. As a consequence, we obtain the
following.

######  3.3 Theorem.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$, and $(M,S)$
be a bimodule over it. Then there is a long exact sequence connecting various
cohomology groups

$\displaystyle 0\rightarrow H^{0}_{\mathrm{mRBA}}((A,R),(M,S))\rightarrow
H^{0}_{\mathrm{Hoch}}(A,M)\rightarrow
H^{0}_{\mathrm{Hoch}}(A_{R},\widetilde{M})\rightarrow
H^{1}_{\mathrm{mRBA}}((A,R),(M,S))\rightarrow\cdots.$

A relation with the cohomology of Rota-Baxter algebras. In [13, 33] the
authors introduced the cohomology of Rota-Baxter algebras. Here we will show
that the cohomology of a Rota-Baxter algebra $(A,P)$ of weight $\lambda$ is
isomorphic to the cohomology of the modified Rota-Baxter algebra
$(A,R=\lambda~{}\\!\mathrm{id}+2P)$ of weight $\kappa=-\lambda^{2}$.

We first recall the cohomology of Rota-Baxter algebras from [33]. Let $(A,P)$
be a Rota-Baxter algebra of weight $\lambda$. Then the vector space $A$
carries an associative algebra structure with the multiplication given by

$\displaystyle a\ast_{P}b:=P(a)\cdot b~{}+~{}a\cdot P(b)+\lambda~{}\\!a\cdot
b,\text{ for }a,b\in A.$

We denote this associative algebra simply by $A_{P}$. Moreover, if $(M,Q)$ is
a bimodule over the Rota-Baxter algebra $(A,P)$ of weight $\lambda$, then the
maps

$\displaystyle A\otimes M\rightarrow M,~{}(a,u)\mapsto P(a)\cdot u-Q(a\cdot
u),$ $\displaystyle M\otimes A\rightarrow M,~{}(u,a)\mapsto u\cdot
P(a)-Q(u\cdot a)$

defines an $A_{P}$-bimodule structure on $M$ (denoted by $\overline{M}$). Then
the authors considered the cochain complex
$\\{C^{\bullet}_{\mathrm{RBA}}((A,P),(M,Q)),\delta_{\mathrm{RBA}}\\}$, where

$\displaystyle C^{n}_{\mathrm{RBA}}((A,P),(M,Q))=\begin{cases}M&\text{ if
}n\geq 0,\\\ C^{n}(A,M)\oplus
C^{n-1}(A_{P},\overline{M})=\mathrm{Hom}(A^{\otimes
n},M)\oplus\mathrm{Hom}(A^{\otimes n-1},M)&\text{ if }n\geq 1.\end{cases}$

The coboundary map is given by

$\displaystyle\delta_{\mathrm{RBA}}(u)=~{}$
$\displaystyle\big{(}\delta_{\mathrm{Hoch}}(u),-u\big{)},$
$\displaystyle\delta_{\mathrm{RBA}}(f,g)=~{}$
$\displaystyle\big{(}\delta_{\mathrm{Hoch}}(f),~{}-\overline{\delta}_{\mathrm{Hoch}}(g)-\Phi^{n}(f)\big{)},$

where $\overline{\delta}_{\mathrm{Hoch}}$ is the Hochschild coboundary
operator of the associative algebra $A_{P}$ with coefficients in the
$A_{P}$-bimodule $\overline{M}$, and $\\{\Phi^{n}:C^{n}(A,M)\rightarrow
C^{n}(A_{P},\overline{M})\\}_{n\geq 0}$ is the sequence of maps given by
$\Phi^{0}=\mathrm{id}_{M}$ and

$\displaystyle\Phi^{n}(f)(a_{1},\ldots,a_{n})=f\big{(}P(a_{1}),\ldots,P(a_{n})\big{)}-\sum_{k=0}^{n-1}\lambda^{n-k-1}\sum_{1\leq
i_{1}<\cdots<i_{k}\leq n}~{}Q\circ
f\big{(}a_{1},\ldots,P(a_{i_{1}}),\ldots,P(a_{i_{k}}),\ldots,a_{n}\big{)}.$

The cohomology groups of the cochain complex
$\\{C^{\bullet}_{\mathrm{RBA}}((A,P),(M,Q)),\delta_{\mathrm{RBA}}\\}$ are
called the cohomology of the Rota-Baxter algebra $(A,P)$ with coefficients in
the bimodule $(M,Q).$

######  3.4 Theorem.

Let $(A,P)$ be a Rota-Baxter algebra of weight $\lambda$, and $(M,Q)$ be a
bimodule over it. Then the cohomology of the Rota-Baxter algebra $(A,P)$ with
coefficients in $(M,Q)$ is isomorphic to the cohomology of the modified Rota-
Baxter algebra $(A,R=\lambda~{}\\!\mathrm{id}_{A}+2P)$ with coefficients in
the bimodule $(M,S=\lambda~{}\\!\mathrm{id}_{M}+2Q)$.

###### Proof.

For each $n\geq 0$, we define an isomorphism of vector spaces

$\displaystyle\Theta_{n}:C^{n}_{\mathrm{RBA}}((A,P),(M,Q))\rightarrow
C^{n}_{\mathrm{mRBA}}((A,R),(M,S))$

by

$\displaystyle\Theta_{0}(u)=~{}$ $\displaystyle\frac{u}{2},~{}\text{ for }u\in
C^{0}_{\mathrm{RBA}}((A,P),(M,Q))=M,$ $\displaystyle\Theta_{n}(f,g)=~{}$
$\displaystyle(f,~{}2^{n-2}g),~{}\text{ for }(f,g)\in C^{n\geq
1}_{\mathrm{RBA}}((A,P),(M,Q)).$

Then it is straightforward to verify that
$\delta_{\mathrm{mRBA}}\circ\Theta_{n}=\Theta_{n+1}\circ\delta_{\mathrm{RBA}}$,
for all $n\geq 0$. In other words, the collections $\\{\Theta_{n}\\}_{n\geq
0}$ defines an isomorphism of cochain complexes from
$\\{C^{\bullet}_{\mathrm{RBA}}((A,P),(M,Q)),\delta_{\mathrm{RBA}}\\}$ to
$\\{C^{\bullet}_{\mathrm{RBA}}((A,R),(M,S)),\delta_{\mathrm{mRBA}}\\}$. Hence
the result follows. ∎

##  4\. Deformations of modified Rota-Baxter algebras

In this section, we study formal one-parameter deformations of modified Rota-
Baxter algebras. In particular, we show that the infinitesimal in a formal
deformation of a modified Rota-Baxter algebra $(A,R)$ is a $2$-cocycle in the
cohomology complex of $(A,R)$ with coefficients in itself. Further, the
vanishing of the second cohomology implies the rigidity of the modified Rota-
Baxter algebra.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$. Let $\mu$
denotes the associative multiplication on $A$, i.e. $\mu(a,b)=a\cdot b$, for
$a,b\in A$. Consider the space $A[[t]]$ of formal power series in $t$ with
coefficients from $A$. Then $A[[t]]$ is a $\mathbf{k}[[t]]$-module.

######  4.1 Definition.

A formal one-parameter deformation of the modified Rota-Baxter algebra $(A,R)$
of weight $\kappa$ consists of two formal power series of the form

$\displaystyle\mu_{t}=\sum_{i=0}^{\infty}\mu_{i}t^{i},\text{ where
}\mu_{i}\in\mathrm{Hom}(A^{\otimes 2},A)\text{ with }\mu_{0}=\mu,$
$\displaystyle R_{t}=\sum_{i=0}^{\infty}R_{i}t^{i},\text{ where
}R_{i}\in\mathrm{Hom}(A,A)\text{ with }R_{0}=R,$

such that the $\mathbf{k}[[t]]$-module $A[[t]]$ is an associative algebra with
the $\mathbf{k}[[t]]$-bilinear multiplication $\mu_{t}$ and the
$\mathbf{k}[[t]]$-linear map $R_{t}:A[[t]]\rightarrow A[[t]]$ is a modified
Rota-Baxter operator of weight $\kappa$. In other words,
$\big{(}A[[t]]=(A[[t]],\mu_{t}),R_{t}\big{)}$ is a modified Rota-Baxter
algebra of weight $\kappa$.

We often denote a formal one-parameter deformation as above by the pair
$(\mu_{t},R_{t})$.

It follows that $(\mu_{t},R_{t})$ is a formal one-parameter deformation of the
modified Rota-Baxter algebra $(A,R)$ of weight $\kappa$ if and only if the
followings are hold:

$\displaystyle\mu_{t}\big{(}\mu_{t}(a,b),c\big{)}=~{}$
$\displaystyle\mu_{t}\big{(}a,\mu_{t}(b,c)\big{)},$
$\displaystyle\mu_{t}\big{(}R_{t}(a),R_{t}(b)\big{)}=~{}$ $\displaystyle
R_{t}\big{(}\mu_{t}(R_{t}(a),b)+\mu_{t}(a,R_{t}(b))\big{)}+\kappa~{}\mu_{t}(a,b),$

for all $a,b,c\in A$. By expanding these equations and comparing the
coefficients of $t^{n}$ (for $n\geq 0$) in both sides, we obtain

$\displaystyle\sum_{i+j=n}\mu_{i}\big{(}\mu_{j}(a,b),c\big{)}=~{}$
$\displaystyle\sum_{i+j=n}\mu_{i}\big{(}a,\mu_{j}(b,c)\big{)},$
$\displaystyle\sum_{i+j+k=n}\mu_{i}\big{(}R_{j}(a),R_{j}(b)\big{)}=~{}$
$\displaystyle\sum_{i+j+k=n}R_{i}\big{(}\mu_{j}(R_{k}(a),b)~{}+~{}\mu_{j}(a,R_{k}(b))\big{)}+\kappa~{}\mu_{n}(a,b),$

for $n\geq 0$. The equations are obviously hold for $n=0$ as $\mu$ is the
associative multiplication on $A$ and the linear map $R:A\rightarrow A$ is a
modified Rota-Baxter algebra of weight $\kappa$. However, for $n=1$, we get

(1) $\displaystyle\mu_{1}(a\cdot b,c)+\mu_{1}(a,b)\cdot c=~{}$
$\displaystyle\mu_{1}(a,b\cdot c)+a\cdot\mu_{1}(b,c),$ (2) $\displaystyle
R_{1}(a)\cdot R(b)+R(a)\cdot R_{1}(b)+\mu_{1}(R(a),R(b))=~{}$ $\displaystyle
R_{1}\big{(}R(a)\cdot b+a\cdot
R(b)\big{)}+R\big{(}\mu_{1}(R(a),b)~{}+~{}\mu_{1}(a,R(b))\big{)}$
$\displaystyle+R\big{(}R_{1}(a)\cdot b+a\cdot
R_{1}(b)\big{)}+\kappa~{}\mu_{n}(a,b),$

for all $a,b,c\in A$. Note the the equation (1) is equivalent to
$(\delta_{\mathrm{Hoch}}\mu_{1})(a,b,c)=0$ while the equation (2) is
equivalent to

$\displaystyle\widetilde{\delta}_{\mathrm{Hoch}}(R_{1})+\Psi^{2}(\mu_{1})=0.$

Thus, it follows that

(3)
$\displaystyle\delta_{\mathrm{mRBA}}(\mu_{1},R_{1})=(\delta_{\mathrm{Hoch}}(\mu_{1}),~{}-\widetilde{\delta}_{\mathrm{Hoch}}(R_{1})-\Psi^{2}(\mu_{1}))=0.$

This shows that $(\mu_{1},R_{1})$ is a $2$-cocycle in the cohomology complex
of $(A,R)$ with coefficients in itself. This is called the infinitesimal of
the formal one-parameter deformation $(\mu_{t},R_{t})$.

######  4.2 Definition.

Let $(\mu_{t},R_{t})$ and $(\mu_{t}^{\prime},R_{t}^{\prime})$ be two formal
one-parameter deformations of the modified Rota-Baxter algebra $(A,R)$ of
weight $\kappa$. These two deformations are said to be equivalent if there
exists a formal isomorphism

$\displaystyle\varphi_{t}=\sum_{i=0}^{\infty}\varphi_{i}t^{i}:\big{(}A[[t]]=(A[[t]],\mu_{t}),R_{t}\big{)}\rightarrow\big{(}A[[t]]^{\prime}=(A[[t]],\mu_{t}^{\prime}),R_{t}^{\prime}\big{)}~{}~{}\text{
with }\varphi_{0}=\mathrm{id}_{A}$

between modified Rota-Baxter algebras of weight $\kappa$, i.e.

(4)
$\displaystyle\varphi_{t}\big{(}\mu_{t}(a,b)\big{)}=\mu_{t}^{\prime}\big{(}\varphi_{t}(a),\varphi_{t}(b)\big{)}~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}\varphi_{t}\circ
R_{t}=R_{t}^{\prime}\circ\varphi_{t},~{}\text{ for }a,b\in A.$

By expanding the equations in (4) and comparing the coefficients of $t^{n}$
(for $n\geq 0$) in both sides, we obtain

$\displaystyle\sum_{i+j=n}\varphi_{i}\big{(}\mu_{j}(a,b)\big{)}=~{}$
$\displaystyle\sum_{i+j+k=n}\mu_{i}^{\prime}\big{(}\varphi_{j}(a),\varphi_{k}(b)\big{)},$
$\displaystyle\sum_{i+j=n}\varphi_{i}\circ R_{j}=~{}$
$\displaystyle\sum_{i+j=n}R_{i}^{\prime}\circ\varphi_{j},$

for $n\geq 0$. The equations are hold for $n=0$ as
$\varphi_{0}=\mathrm{id}_{A}$. However, for $n=1$, we get

(5) $\displaystyle\mu_{1}(a,b)+\varphi_{1}(a\cdot b)=~{}$
$\displaystyle\mu_{1}^{\prime}(a,b)+\varphi_{1}(a)\cdot
b+a\cdot\varphi_{1}(b),$ (6) $\displaystyle R_{1}+\varphi_{1}\circ R=~{}$
$\displaystyle R_{1}^{\prime}+R\circ\varphi_{1},.$

Note that the equation (5) is same as
$\mu_{1}-\mu_{1}^{\prime}=\delta_{\mathrm{Hoch}}(\varphi_{1})$, while the
equation (6) is same as

$\displaystyle R_{1}-R_{1}^{\prime}=-\Psi^{1}(\varphi_{1}).$

Thus, we have

$\displaystyle(\mu_{1},R_{1})-(\mu_{1}^{\prime},R_{1}^{\prime})=\big{(}\delta_{\mathrm{Hoch}}(\varphi_{1}),~{}-\Psi^{1}(\varphi_{1})\big{)}=\delta_{\mathrm{mRBA}}(\varphi_{1},0).$

As a consequence of the above discussions, we obtain the following.

######  4.3 Theorem.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$. Suppose
$(\mu_{t},R_{t})$ is a formal one-parameter deformation of $(A,R).$ Then the
infinitesimal is a $2$-cocycle in the cohomology complex of $(A,R)$ with
coefficients in itself. Moreover, the corresponding cohomology class depends
only on the equivalence class of the deformation.

We end this section by considering the rigidity of a modified Rota-Baxter
algebra of weight $\kappa$. We also find a sufficient condition for rigidity.

######  4.4 Definition.

A modified Rota-Baxter algebra $(A,R)$ is said to be rigid if any formal one-
parameter deformation $(\mu_{t},R_{t})$ is equivalent to the undeformed one
$(\mu,R)$.

######  4.5 Theorem.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$. If
$H^{2}_{\mathrm{mRBA}}((A,R),(A,R))=0$ then $(A,R)$ is rigid.

###### Proof.

Let $(\mu_{t},R_{t})$ be any formal one-parameter deformation of the modified
Rota-Baxter algebra $(A,R)$ of weight $\kappa$. Then the infinitesimal
$(\mu_{1},R_{1})$ is a $2$-cocycle. Thus, it follows from the hypothesis that

(7) $\displaystyle(\mu_{1},R_{1})=\delta_{\mathrm{mRBA}}(\varphi_{1},a),\text{
for some }(\varphi_{1},a)\in C^{1}_{\mathrm{mRBA}}((A,R),(A,R)).$

Note that

$\displaystyle\delta_{\mathrm{mRBA}}(\varphi_{1}+\delta_{\mathrm{Hoch}}(a),0)=~{}$
$\displaystyle\big{(}\delta_{\mathrm{Hoch}}(\varphi_{1}),~{}-\Psi^{1}(\varphi_{1})-\Psi^{1}\circ\delta_{\mathrm{Hoch}}(a)\big{)}$
$\displaystyle=~{}$
$\displaystyle\big{(}\delta_{\mathrm{Hoch}}(\varphi_{1}),~{}-\Psi^{1}(\varphi_{1})-\widetilde{\delta}_{\mathrm{Hoch}}(a)\big{)}=\delta_{\mathrm{mRBA}}(\varphi_{1},a).$

Thus, without loss of any generality, we may assume that
$(\mu_{1},R_{1})=\delta_{\mathrm{mRBA}}(\varphi_{1},0)$, for some
$(\varphi_{1},0)\in C^{1}_{\mathrm{mRBA}}((A,R),(A,R))$. Then we define a map
$\varphi_{t}=\mathrm{id}_{A}+t\varphi_{1}:A[[t]]\rightarrow A[[t]]$ and
consider the pair

$\displaystyle\big{(}\overline{\mu}_{t}=\varphi_{t}\circ\mu_{t}\circ(\varphi_{t}^{-1}\otimes\varphi_{t}^{-1}),~{}\overline{R}_{t}=\varphi_{t}\circ
R_{t}\circ\varphi_{t}^{-1}\big{)}.$

Then $(\overline{\mu}_{t},\overline{R}_{t})$ is a formal one-parameter
deformation equivalent to $(\mu_{t},R_{t})$. Moreover, it follows that
$\overline{\mu}_{t}$ and $\overline{R}_{t}$ are of the following forms

$\displaystyle\overline{\mu}_{t}=\mu+\overline{\mu}_{2}t^{2}+\cdots~{}~{}~{}~{}\text{
and }~{}~{}~{}~{}\overline{R}_{t}=R+\overline{R}_{2}t^{2}+\cdots.$

By repeating this argument, one can conclude that $(\mu_{t},R_{t})$ is
equivalent to $(\mu,R)$. This completes the proof. ∎

##  5\. Abelian extensions of modified Rota-Baxter algebras

In this section, we define and study abelian extensions of a modified Rota-
Baxter algebra $(A,R)$ by a bimodule $(M,S)$. We show that the equivalence
classes of such abelian extensions are classified by the second cohomology
group $H^{2}_{\mathrm{mRBA}}((A,R),(M,S))$.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$. Let $(M,S)$
be a pair consisting of a vector space $M$ and a linear map $S:M\rightarrow
M$. Note that the pair $(M,S)$ can be regarded as a modified Rota-Baxter
algebra of weight $\kappa$, where we equip $M$ with the trivial associative
multiplication.

######  5.1 Definition.

(i) An abelian extension of the modified Rota-Baxter algebra $(A,R)$ of weight
$\kappa$ by a pair $(M,S)$ is a short exact sequence of modified Rota-Baxter
algebras

(10)

We often denote an abelian extension as above simply by $(E,U)$ when the
structure maps $i$ and $\pi$ are clear from the context.

(ii) Two abelian extensions $(E,U)$ and $(E^{\prime},{U^{\prime}})$ of the
modified Rota-Baxter algebra $(A,R)$ by a pair $(M,S)$ are said to be
isomorphic if there exists an isomorphism
$\varphi:(E,U)\rightarrow(E^{\prime},U^{\prime})$ of modified Rota-Baxter
algebras making the following diagram commutative

(15)

Let $(E,U)$ be an abelian extension as of (10). A section of the map $\pi$ is
a linear map $s:A\rightarrow E$ that satisfies $\pi s=\mathrm{id}_{A}$. There
is always a section of the map $\pi$. For any section $s$, we define two
bilinear maps $A\times M\rightarrow M$, $(a,u)\mapsto s(a)\cdot_{E}i(u)$ and
$M\times A\rightarrow M$, $(u,a)\mapsto i(u)\cdot_{E}s(a)$, for $a\in A$,
$u\in M$. Here $~{}\cdot_{E}~{}$ denotes the associative multiplication on
$E$. With these bilinear maps, it is easy to see that $M$ is an $A$-bimodule
(see [27]). More generally, the pair $(M,S)$ is a bimodule over the modified
Rota-Baxter algebra $(A,R)$ of weight $\kappa$. This bimodule is called the
induced bimodule over $(A,R)$.

Let $(M,S)$ be a given bimodule over a modified Rota-Baxter algebra $(A,R)$ of
weight $\kappa$. We denote $\mathrm{Ext}((A,R),(M,S))$ by the set of all
isomorphism classes of abelian extensions of $(A,R)$ by the pair $(M,S)$ for
which the induced bimodule on $(M,S)$ is the given one.

######  5.2 Theorem.

Let $(A,R)$ be a modified Rota-Baxter algebra of weight $\kappa$, and $(M,S)$
be a bimodule over it. Then there is a one-to-one correspondence between
$\mathrm{Ext}((A,R),(M,S))$ and the second cohomology group
$H^{2}_{\mathrm{mRBA}}((A,R),(M,S))$.

###### Proof.

Let $(\chi,\Phi)\in Z^{2}_{\mathrm{mRBA}}((A,R),(M,S))$ be a $2$-cocycle that
represents an element in $H^{2}_{\mathrm{mRBA}}((A,R),(M,S))$. Since
$(\chi,\Phi)$ is a $2$-cocycle, we have $\delta_{\mathrm{Hoch}}(\chi)=0$ and
$\widetilde{\delta}_{\mathrm{Hoch}}(\Phi)+\Psi^{2}(\chi)=0$. Consider the
direct sum $A\oplus M$ with the multiplication

$\displaystyle(a,u)\cdot_{\ltimes}(b,v)=(a\cdot b,~{}a\cdot v+u\cdot
a+\chi(a,b)),\text{ for }(a,u),(b,v)\in A\oplus M.$

Since $\chi$ is a Hochschild $2$-cocycle, it follows that the above
multiplication is associative. Thus it makes the space $A\oplus M$ into an
associative algebra, denoted by $A\oplus_{\chi}M$. Further, we define a map
$R_{\Phi}:A\oplus_{\chi}M\rightarrow A\oplus_{\chi}M$ by

$\displaystyle R_{\Phi}(a,u)=\big{(}R(a),S(u)+\Phi(a)\big{)},\text{ for
}(a,u)\in A\oplus M.$

Since $\widetilde{\delta}_{\mathrm{Hoch}}(\Phi)+\Psi^{2}(\chi)=0$, it follows
that $R_{\Phi}$ is a modified Rota-Baxter operator of weight $\kappa$ on the
algebra $A\oplus_{\chi}M$. In other words, $(A\oplus_{\chi}M,R_{\Phi})$ is a
modified Rota-Baxter algebra of weight $\kappa$. Then it is easy to see that

is an abelian extension of $(A,R)$ by $(M,S)$, where the structure maps $i$
and $\pi$ are respectively given by $i(u)=(0,u)$ and $\pi(a,u)=a$, for $u\in
M$, $(a,u)\in A\oplus M$. Note that the map $s:A\rightarrow A\oplus M$,
$s(a)=(a,0)$ is a section of the map $\pi$, and the section $s$ induces the
given bimodule structure on $(M,S).$

Let $(\chi^{\prime},\Phi^{\prime})\in Z^{2}_{\mathrm{mRBA}}((A,R),(M,S))$ be
another $2$-cocycle cohomologous to $(\chi,\Phi)$. In other words, they
corresponds to the same element in $H^{2}_{\mathrm{mRBA}}((A,R),(M,S))$. Then
there exists a map $\theta\in\mathrm{Hom}(A,M)$ such that

$\displaystyle(\chi,\Phi)-(\chi^{\prime},\Phi^{\prime})=\delta_{\mathrm{mRBA}}(\theta,0).$

We now define a map $\varphi:A\oplus M\rightarrow A\oplus M$ by

$\displaystyle\varphi(a,u)=(a,u-\theta(a)).$

It is then easy to verify that
$\varphi:(A\oplus_{\chi}M,R_{\Phi})\rightarrow(A\oplus_{\chi^{\prime}}M,R_{\Phi^{\prime}})$
is an isomorphism of modified Rota-Baxter algebras, which makes the abelian
extensions $(A\oplus_{\chi}M,R_{\Phi})$ and
$(A\oplus_{\chi^{\prime}}M,R_{\Phi^{\prime}})$ isomorphic. Therefore, there
exists a well-defined map
$\Upsilon:H^{2}_{\mathrm{mRBA}}((A,R),(M,S))\rightarrow\mathrm{Ext}((A,R),(M,S))$.

Conversely, let $(E,U)$ be an abelian extension of $(A,R)$ by $(M,S)$ so that
the induced bimodule structure on $(M,S)$ coincides with the given one. Let
$s:A\rightarrow E$ be any section of the map $\pi$. We define maps
$\chi^{s}\in\mathrm{Hom}(A^{\otimes 2},M)$ and $\Phi^{s}\in\mathrm{Hom}(A,M)$
by

$\displaystyle\chi^{s}(a,b)=s(a)\cdot_{E}s(b)-s(a\cdot b)~{}~{}~{}\text{ and
}~{}~{}~{}\Phi^{s}(a)=Us(a)-sR(a),\text{ for }a,b\in A.$

Then straightforward calculations show that $\chi^{s}$ is a Hochschild
$2$-cocycle (i.e. $\delta_{\mathrm{Hoch}}(\psi^{s})=0$) and

$\displaystyle\widetilde{\delta}_{\mathrm{Hoch}}(\Phi^{s})+\Psi^{2}(\chi^{s})=0.$

Thus, we have
$\delta_{\mathrm{mRBA}}(\chi^{s},\Phi^{s})=\big{(}\delta_{\mathrm{Hoch}}(\chi^{s}),~{}-\widetilde{\delta}_{\mathrm{Hoch}}(\Phi^{s})-\Psi^{2}(\chi^{s})\big{)}=0$,
which shows that $(\chi^{s},\Phi^{s})$ is a $2$-cocycle. It is easy to see
that the corresponding class in $H^{2}_{\mathrm{mRBA}}((A,R),(M,S))$ doesn’t
depend on the choice of the section $s$.

Let $(E,U)$ and $(E^{\prime},U^{\prime})$ be two isomorphic abelian extensions
of as (15). If $s:A\rightarrow E$ is a section of the map $\pi$, then the map
$s^{\prime}=\varphi s:A\rightarrow E^{\prime}$ is a section of the map
$\pi^{\prime}$. Let $(\chi^{s^{\prime}},\Phi^{s^{\prime}})$ be the $2$-cocycle
corresponding to the abelian extension $(E^{\prime},{U^{\prime}})$ and the
section $s^{\prime}$ of the map $\pi^{\prime}$. Then we have

$\displaystyle\chi^{s^{\prime}}(a,b)=~{}$ $\displaystyle
s^{\prime}(a)\cdot_{E^{\prime}}s^{\prime}(b)-s^{\prime}(a\cdot b)$
$\displaystyle=~{}$ $\displaystyle(\varphi s)(a)\cdot_{E^{\prime}}(\varphi
s)(b)-(\varphi s)(a\cdot b)$ $\displaystyle=~{}$
$\displaystyle\varphi\big{(}s(a)\cdot_{E}s(b)-s(a\cdot
b)\big{)}=\varphi(\chi^{s}(a,b))=\chi^{s}(a,b)$

and

$\displaystyle\Phi^{s^{\prime}}(a)=~{}$ $\displaystyle
Us^{\prime}(a)-s^{\prime}R(a)$ $\displaystyle=~{}$ $\displaystyle U\varphi
s(a)-\varphi sR(a)$ $\displaystyle=~{}$ $\displaystyle\varphi Us(a)-\varphi
sR(a)=\varphi(\Phi^{s}(a))=\Phi^{s}(a).$

Hence $(\chi^{s^{\prime}},\Phi^{s^{\prime}})=(\chi^{s},\Phi^{s})$. Therefore,
they corresponds to the same element in $H^{2}_{\mathrm{mRBA}}((A,R),(M,S))$.
As a summary, we obtain a well-defined map
$\Omega:\mathrm{Ext}((A,R),(M,S))\rightarrow
H^{2}_{\mathrm{mRBA}}((A,R),(M,S))$. Finally, the maps $\Upsilon$ and $\Omega$
are inverses to each other. This completes the proof. ∎

We end this paper by suggesting some questions of further interest. We will
address some of these questions in future.

We have seen that Rota-Baxter algebras are closely related to modified Rota-
Baxter algebras of negative weights. In [13, 33] the authors constructs an
$L_{\infty}$-algebra whose Maurer-Cartan elements are precisely Rota-Baxter
algebra structures on a vector space $A$. One requires a suitable linear
transformation of this $L_{\infty}$-algebra to obtain a new one whose Maurer-
Cartan elements are precisely modified Rota-Baxter algebras of fixed negative
weight. In a subsequent work, we aim to provide a systematic construction of
the $L_{\infty}$-algebra that characterize modified Rota-Baxter algebras of
any weight as Maurer-Cartan elements.

Given an algebraic structure, it is an interesting question to find the
corresponding homotopy version. Such homotopy version is better understood in
terms of the minimal model of the operad governing such structure. The minimal
model of the operad for Rota-Baxter algebras was considered in [33]. It could
be interesting to find the similar theory for modified Rota-Baxter algebras.

In [24] the authors considered modified Rota-Baxter operators of weight $-1$
on Lie algebras and studied their cohomology and deformation theory. One can
easily generalize this to modified Rota-Baxter operators of arbitrary weight.
Let $\mathfrak{g}=(\mathfrak{g},[~{},~{}])$ be a Lie algebra. A linear map
$R:\mathfrak{g}\rightarrow\mathfrak{g}$ is a modified Rota-Baxter operator of
weight $\kappa$ if

$\displaystyle[R(x),R(y)]=R\big{(}[R(x),y]+[x,R(y)]\big{)}+\kappa~{}\\![x,y],\text{
for }x,y\in\mathfrak{g}.$

It follows that if $(A,R)$ is a modified Rota-Baxter algebra of weight
$\kappa$ then $R$ is a modified Rota-Baxter operator of the same weight on the
commutator Lie algebra structure on $A$. We define a modified Rota-Baxter Lie
algebra of weight $\kappa$ as a pair $(\mathfrak{g},R)$ in which
$\mathfrak{g}$ is a Lie algebra and $R:\mathfrak{g}\rightarrow\mathfrak{g}$ is
a modified Rota-Baxter operator of weight $\kappa$. By following the results
of the present paper, one could also develop the cohomology and deformation
theory of modified Rota-Baxter Lie algebras of any weight.

In the study of integrations of Rota-Baxter Lie algebras, the authors in [21]
introduced Rota-Baxter operators of weight $1$ on abstract groups. Let
$(G,\diamond)$ be a group. A set map $B:G\rightarrow G$ is a Rota-Baxter
operator of weight $1$ if

$\displaystyle B(x)\diamond B(y)=B\big{(}x\diamond B(x)\diamond y\diamond
B(x)^{-1}\big{)},\text{ for }x,y\in\mathfrak{g}.$

An interesting problem is to find the suitable notion of modified Rota-Baxter
operator of weight $-1$ on abstract groups. With this notion, one could expect
integrations of modified Rota-Baxter Lie algebras of weight $-1$ by modified
Rota-Baxter Lie groups of weight $-1$.

Acknowledgements. The author would like to thank IIT Kharagpur (India) for
providing a beautiful academic atmosphere where his research has been carried
out.

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