February 09, 2024
The purpose of the present paper is to investigate cohomologies and deformations of weighted Rota-Baxter Lie algebras as well as weighted Rota-Baxter associative algebras with derivations. First we introduce a notion of weighted Rota-Baxter LieDer and weighted Rota-Baxter AssDer pairs. Then we construct cohomologies of weighted Rota-Baxter LieDer pairs, weighted Rota-Baxter AssDer pairs and we discuss the relation between their cohmologies. Finally, as an application, we study deformations of both of them.
Key words:Lie algebras, Lie algebras with derivation, Rota-Baxter operators, cohomology, deformation .
Rota-Baxter operators were first studied in the work of Baxter of the fluctuation theory in probability [1] and further was developed in [2]. These operators can be regarded as an algebraic abstraction representing the integral operator. Many papers have been devoted to various aspects of Rota-Baxter operators in many
mathematical fields, like combinatorics [3], renormalization in quantum field theory [4], multiple zeta values in number theory [5], Yang Baxter equations [6], algebraic operad [7] and other papers. Rota-Baxter operators with arbitrary weight (also called weighted Rota-Baxter operators)
was considered in [8]–[10].
Deformation theory of some algebraic structure goes back to Gerstenhaber [11] for associative algebras and Nijenhuis-Richardson [12] for Lie algebras. There are some advancements in deformation theory and cohomology theory of weighted Rota-Baxter algebras [13], [14]. More precisely, they considered weighted Rota-Baxter Lie algebras and associative algebras and define cohomology of them with coefficients in arbitrary Rota-Baxter
representation.
Derivations have an important role to study many algebraic structures. Homotopy Lie algebras [15] and differential Galois theory [16] can be gained from derivations. Derivations also are important in control theory and gauge theories in quantum field theory [17], [18]. Recently, the cohomologies, extensions and deformations of Lie algebras with derivations (called LieDer pairs) were investigated in [19]. Then had been extended to associative algebras, Leibniz algebras, Lie triple systems, n-Lie algebras and compatible Lie algebras with derivations in [20]–[25]. Also derivations play new role in twisting associative and nonassociative algebras to obtain InvDer algebraic structure,
for more details see [26]
Motivated by these works, we are generalizing the structre of Lie algebras with derivation [19] (respectively associative algebras with derivation [21]) and the structure of weighted Rota-Baxter Lie algebras [13] (respectively
Rota-Baxter associative algebras [14]) to the cohomologies of weighted Rota-Baxter LieDer and AssDer pairs.
The paper is organized as follows. In section 2, we introduce a notion of a weighted Rota-Baxter LieDer pair and its representation. In section 3, we study cohomologies of weighted
Rota-Baxter LieDer pairs and weighted Rota-baxter AssDer pairs. In section 4, we study deformation of two structures.
Let \(\mathfrak{L}=(L,[\cdot,\cdot])\) be a Lie algebra. A linear map \(R:L\rightarrow L\) is said to be a \(\lambda\)-weighted Rota-Baxter operator if
\(R\) satisfies, for \(\lambda \in \mathbb{K}\) \[\label{RBO32eq1} [Rx,Ry]=R([Rx,y]+[x,Ry]+\lambda [x,y]),\quad \forall x,y
\in L.\tag{1}\] According to the previous operator, in this section we introduce the notion of \(\lambda\)-weighted Rota-Baxter LieDer pairs (or simply weighted Rota-Baxter LieDer pairs if there is no
confusion) which is a LieDer pair \((\mathfrak{L},\delta)\) equipped with a \(\lambda\)-weighted Rota-Baxter operator.
This notion is a generalization of weighted Rota-Baxter Lie algebra \((\mathfrak{L},R)\), see [13], [27]–[29] for more details, consisting of a Lie algebra \(\mathfrak{L}\) together with a \(\lambda\)-weighted
Rota-Baxter operator on it.
In this subsection we study the structure of a Lie algebras \(\mathfrak{L}\) equipped with a couple of derivations \(\delta_1,\delta_2\) and we investigate its relation with the LieDer pairs. Recall first the definition of LieDer pair.
Definition 1. [19] A LieDer pair \((\mathfrak{L},\delta)\) is a Lie algebra \(\mathfrak{L}\) equipped with a derivation \(\delta\).
So a Lie algebra equipped with a derivation leads to construct a LieDer pair. Next we combine a Lie algebra \(\mathfrak{L}\) with two derivations \(\delta_1\) and \(\delta_2\) to construct a LieDer pair structure with an additionally condition.
Remark 1. Let \(\mathfrak{L}\) be a Lie algebra and \(\delta_1,\delta_2:L\rightarrow L\) be two derivations on \(\mathfrak{L}\). Then \(\delta_1\circ\delta_2\) is a derivation on \(\mathfrak{L}\) if and only if for all \(x,y\in L\) the following condition holds \[\label{condition32biderivation} [\delta_1x,\delta_2y]=[\delta_1y,\delta_2x]\tag{2}\]
Definition 2. Let \(\mathfrak{L}\) be a Lie algebra, \(\delta_1\) and \(\delta_2\) be two derivations on it. Then \((\mathfrak{L},\delta_1,\delta_2)\) is called a Lie-BiDer pair if \((\mathfrak{L},\delta_1\circ\delta_2)\) is a LieDer pair.
The previous definition means that \((\mathfrak{L},\delta_1,\delta_2)\) is a Lie-BiDer pair if and only if the condition 2 is satisfied.
Example 1. Let \(L\) be two dimentional Lie algebra such that \([e_1,e_2]=e_2\). Then \((\mathfrak{L},\delta_1,\delta_2)\) is a Lie-BiDer pair where, for all \(a,b,c,d\in \mathbb{K}\) \[\delta_1=\begin{pmatrix} 0 & 0 \\ a & b \end{pmatrix}\quad \text{ and }\quad \delta_2=\begin{pmatrix} 0 & 0 \\ c & d \end{pmatrix} \quad \text{ with }\quad ad=bc\]
Definition 3. Let \((\mathfrak{L},\delta_1,\delta_2)\) and \((\mathfrak{K},\partial_1,\partial_2)\) be two Lie-BiDer pairs. A Lie-BiDer homomorphism \(f:\mathfrak{L} \rightarrow \mathfrak{K}\) is a Lie algebra homomorphism from \(\mathfrak{L}\) to \(\mathfrak{K}\) such that \[\label{morphism32Lie-BiDer32pair} f\circ\delta_1=\partial_1\circ f,\quad f\circ\delta_2=\partial_2\circ f\tag{3}\]
Definition 4. Let \((\mathfrak{L},\delta_1,\delta_2)\) be a Lie-BiDer pair. A representation of it on a vector space \(V\) with respect to \((\varphi_1,\varphi_2)\in \mathrm{gl}(V)\) is a Lie algebra morphism \(\rho:L\rightarrow \mathrm{gl}(V)\) such that, for \(x\in L\), we have \[\label{representation32Lie-BiDer32pair} \rho(\delta_1x)\circ\varphi_2=-\rho(\delta_2x)\circ\varphi_1.\tag{4}\] Such a representation is denoted by \((\mathcal{V}=(V,\rho),\varphi_1,\varphi_2)\).
Example 2. Let \(\mathfrak{L}\) be a Lie algebra and a map \(\mathrm{ad}_x:L \rightarrow L\) defined by \[\mathrm{ad}_x(y)=[x,y],\quad \forall y\in L.\] Then \((L,\mathrm{ad},\delta_1,\delta_2)\) is a representation of the Lie-BiDer pair \((L,\delta_1,\delta_2)\) on \(L\) with respect to \(\delta_1\) and \(\delta_2\), it is called the adjoint representation.
Proposition 3. Let \((\mathfrak{L},\delta_1,\delta_2)\) be a Lie-BiDer pair and \((\mathcal{V},\varphi_1,\varphi_2)\) be a representation of it. Then \((L\oplus V,\delta_1\oplus\varphi_1,\delta_2\oplus\varphi_2)\) is a Lie-BiDer pair with the following Lie structure \[_{\ltimes}=[x,y]+\rho(x)b-\rho(y)a;\quad x,y \in L \text{ and }a,b \in V.\] and the maps \(\delta_\mathrm{i}\oplus\varphi_\mathrm{i}\) are given by \[\delta_\mathrm{i}\oplus\varphi_\mathrm{i}(x+a):=\delta_\mathrm{i}x+\varphi_\mathrm{i}(a),\quad \forall i\in \{1,2\}.\]
Such a Lie-BiDer pair is called the semi-direct product of \((\mathfrak{L},\delta_1,\delta_2)\) by a representation representation \((\mathcal{V},\varphi_1,\varphi_2)\) and it is denoted by \(L\ltimes_{\boldsymbol{BiDer}} V\).
Proof. We need just to prove that the condition 2 is satisfied for semi-direct product. \[\begin{align} _\ltimes&=[\delta_1x+\varphi_1(a),\delta_2y+\varphi_2(b)]_\ltimes\\ &=[\delta_1x,\delta_2y]+\rho(\delta_1x)\varphi_2(b)-\rho(\delta_2y)\varphi_1(a)\\ &\overset{\ref{condition32biderivation}}{=}[\delta_1y,\delta_2x]+\rho(\delta_1x)\varphi_2(b)-\rho(\delta_2y)\varphi_1(a)\\ &\overset{\ref{representation32Lie-BiDer32pair}}{=}[\delta_1y,\delta_2x]-\rho(\delta_2x)\varphi_1(b)+\rho(\delta_1y)\varphi_2(a)\\ &=[(\delta_1\oplus\varphi_1)(y+b),(\delta_2\oplus\varphi_2)(x+a)]_\ltimes \end{align}\] This complete the proof. ◻
In this subsection, we introduce a notion of \(\lambda\)-weighted Rota-Baxter LieDer pair (or simply weighted Rota-Baxter LieDer pair) and some basic definitions. Denote the Lie algebra \((L,[\cdot,\cdot])\) by \(\mathfrak{L}\) and its representation on a vector space \(V\) by \(\mathcal{V}=(V;\rho)\).
A weighted Rota-Baxter Lie algebra consists of a Lie algebra \(\mathfrak{L}\) equipped with a Rota-Baxter operator of weight \(\lambda\) denoted by \(R\).
Inspired by the notion of LieDer pair [19] and the definition of weighted Rota-Baxter Lie algebra [28] we introduce the following.
Definition 5. A weighted Rota-Baxter LieDer pair consists of a LieDer pair \((\mathfrak{L},\delta)\) together with a \(\lambda\)-weighted Rota-Baxter operator \(R\) such that \[\label{condition132RBLieDer32pair} R\circ \delta=\delta\circ R.\tag{5}\]
Example 4. Let \(\{e_1,e_2\}\) be a basis of a \(2\)-dimensional vector space \(L\) over \(\mathbb{K}\). Given a Lie structure \([e_1,e_2]=e_2\), then the triple \((\mathfrak{L},\delta,R)\) is a \(\lambda\)-weighted Rota-Baxter LieDer pair with \[\begin{align} \delta=\begin{pmatrix} 0 & 0 \\ 0 & a \end{pmatrix}\quad \text{ and } \quad R=\begin{pmatrix} 0 & 0 \\ 0 & b \end{pmatrix},\quad \forall a,b \in \mathbb{K}. \end{align}\]
Definition 6. A \(\lambda\)-weighted Rota-Baxter LieDer pair morphism from \((L_1,\delta_1,R_1)\) to \((L_2,\delta_2,R_2)\) is a Lie algebra morphism \(\varphi:L_1\rightarrow L_2\) such that the following identities holds, for all \(x,y\in L_1\) \[\begin{align} \varphi\circ \delta_1&=&\delta_2\circ\varphi,\tag{6}\\ \varphi\circ R_1&=&R_2\circ\varphi.\tag{7} \end{align}\]
Let \((\mathfrak{L},\delta)\) be a LieDer pair. Recall that a representation of it is a vector space \(V\) with two linear maps \(\rho:L\rightarrow gl(V)\) and \(\delta_\mathrm{V}:L\rightarrow L\) such that, forall \(x,y\in L\) \[\begin{align} \rho([x,y])&=&\rho(x)\circ \rho(y)-\rho(y)\circ \rho(x),\\ \delta_\mathrm{V}\circ \rho(x)&=&\rho(\delta x)+\rho(x)\circ \delta_\mathrm{V}. \end{align}\]
Definition 7. Let \((\mathfrak{L},\delta,R)\) be a weighted Rota-Baxter LieDer pair. A representation of it is a triple \((\mathcal{V},\delta_\mathrm{V},T)\) where \(T:V\rightarrow V\) is a linear map such that for all \(x\in L\) and \(u\in V\) \[\begin{align} \rho([x,y])&=&\rho(x)\circ \rho(y)-\rho(y)\circ \rho(x),\tag{8} \\ \delta_V\circ \rho(x)&=&\rho(\delta x)+\rho(x)\circ \delta_V,\tag{9}\\ \rho(Rx)(Tu)&=&T(\rho(Rx)(u)+\rho(x)(Tu)+\lambda \rho(x)u),\tag{10}\\ T\circ\delta_V&=&\delta_V\circ T. \tag{11} \end{align}\]
Example 5. Let \((L,\delta,R)\) be a \(\lambda\)-weighted Rota-Baxter LieDer pair and \((\mathcal{V},\delta_\mathrm{V},T)\) be a representation of it. Then for any scalar \(\mu\in \mathbb{K}\), the triple \((\mathcal{V},\delta_\mathrm{V},\mu T)\) is a representation of the \((\mu\lambda)\)-weighted Rota-Baxter LieDer pair \((L,\delta_\mathrm{V},\mu R)\).
Example 6. Let \((L,\delta,R)\) be a \(\lambda\)-weighted Rota-Baxter LieDer pair and \((\mathcal{V},\delta_\mathrm{V},T)\) be a representation of it. Then the quadruple \((\mathcal{V},\delta_\mathrm{V},-\lambda \mathrm{Id}_\mathrm{V}-T)\) is a representation of the \(\lambda\)-weighted Rota-Baxter LieDer pair \((L,\delta,-\lambda\mathrm{Id}_\mathrm{L}-R)\).
Example 7. Any \(\lambda\)-weighted Rota-Baxter LieDer pair \((L,\delta,R)\) is a representation of itself. Such a representation is called the adjoint representation.
Next, we construct the semi-direct product in the context of \(\lambda\)-weighted Rota-Baxter LieDer pair.
Proposition 8. Let \((\mathfrak{L},\delta,R)\) be a weighted Rota-Baxter LieDer pair and \((\mathcal{V},\delta_\mathrm{V},T)\) be a representation of it. Then \((L\oplus V,\delta\oplus \delta_{\mathrm{V}},R\oplus T)\) is a weighted Rota-Baxter LieDer pair where the Lie bracket on \(L\oplus V\) is given by \[_\ltimes:=[x,y]+\rho(x)b-\rho(y)a,\] and the derivation is given by \[\begin{align} \delta\oplus\delta_\mathrm{V}(x+a)=\delta x+\delta_\mathrm{V}a \end{align}\] and The \(\lambda\)-weighted Rota-Baxter LieDer pair is given by \[\begin{align} R\oplus T(x+a)=R x+Ta,\quad \forall x,y\in L \quad \forall a,b\in V. \end{align}\] We call such structure by the semi-direct product of the \(\lambda\)-weighted Rota-Baxter LieDer pair \((L,\delta,R)\) by a representation of it \((\mathcal{V},\delta_\mathrm{V},T)\) and denoted by \(L\ltimes_{\mathrm{R.B.LieDer}}V\).
Proof. According to the [28] and [19] the proof is straightforward. The idea is to show that \(\delta\oplus \delta_{\mathrm{V}}\) is a derivation on \(L\oplus V\) and to show that \(R\oplus T\) is a weighted Rota-Baxter operator. ◻
Proposition 9. Let \((\mathfrak{L},\delta,R)\) be a weighted Rota-Baxter LieDer pair. Then we have the triple \((L,[\cdot,\cdot]_\mathrm{R},\delta)\) is a LieDer pair with \[\label{induced32RBLieDer} [x,y]_R:=[Rx,y]+[x,Ry]+\lambda [x,y]\qquad{(1)}\] and it is denoted simply by \((\mathfrak{L}_\mathrm{R},\delta)\).
Proof. Let \(x,y\in L\) \[\begin{align} \delta([x,y]_R)&=\delta([Rx,y]+[x,Ry]+\lambda [x,y])\\ &=\delta([Rx,y])+\delta ([x,Ry]) +\lambda \delta([x,y])\\ &=[\delta\circ Rx,y]+[Rx,\delta y]+[\delta x,Ry]+[x,\delta\circ R]+\lambda [\delta x,y]+\lambda [x,\delta y]\\ &\overset{\eqref{condition132RBLieDer32pair}}{=}[R\circ \delta x,y]+[Rx,\delta y]+[\delta x,Ry]+[x,R\circ \delta]+\lambda [\delta x,y]+\lambda [x,\delta y]\\ &=\Big([R\circ \delta x,y]+ [\delta x,Ry]+\lambda [\delta x,y]\Big)+\Big([Rx,\delta y]+[x,R\circ \delta]+\lambda [x,\delta y] \Big)\\ &=[\delta x,y]_R+[x,\delta y]_R. \end{align}\] This complete the proof. ◻
Proposition 10. The triple \((\mathfrak{L}_R,\delta,R)\) is a weighted Rota-Baxter LieDer pair and the map \(R:L_R\rightarrow L\) is a morphism of weighted Rota-Baxter LieDer pair.
Proof. We have that \(R\) is a \(\lambda\)-weighted Rota-Baxter operator on L, it follows then from 1 that \[R([x,y]_R)=[Rx,Ry],\quad \forall x,y\in L.\] This complete the proof. ◻
Theorem 11. Let \((\mathfrak{L},\delta,R)\) be a weighted Rota-Baxter LieDer pair and \((\mathcal{V},\delta_{\mathrm{V}},T)\) be a representation of it. Define a map \[\label{rep32of32new32Rota-Baxter32LieDer32pairs} \widetilde{\rho}(x)(a)=\rho(Rx)(a)-T(\rho(x)(a)),\quad \text{for }x\in L , a\in V\qquad{(2)}\] Then \(\widetilde{\rho}\) defines a representation of the LieDer pair \((L_{\mathrm{R}},\delta)\) on \((\widehat{\mathcal{V}},\delta_{\mathrm{V}})=((V;\widetilde{\rho}),\delta_{\mathrm{V}})\) if and only if the too conditions 5 and 11 are satisfied. Moreover, \((\widehat{\mathcal{V}},\delta_{\mathrm{V}},T)\) is a representation of the weighted Rota-Baxter LieDer pair \((L_{\mathrm{R}},\delta,R)\).
Proof. We have already, from [28], that \(\widetilde{\rho}\) is a representation of \(L_\mathrm{R}\) on \(V\) in the context of Lie algebra structure. So we need just to prove that equation 9 holds for the representation \(\widetilde{\rho}\). Let \(x\in L\) and \(a\in V\) \[\begin{align} \delta_{\mathrm{V}}\circ \widetilde{\rho}(x)a&=\delta_{\mathrm{V}}\circ (\rho(Rx)a-T(\rho(x)a))\\ &=\delta_{\mathrm{V}}\circ \rho(Rx)a-\delta_{\mathrm{V}}(T(\rho(x)a))\\ &\overset{\ref{Rep32of32RBDer32pair1}}{=}\rho(\delta\circ Rx)a+\rho(Rx)\circ \delta_{\mathrm{V}}a-\delta_{\mathrm{V}}(T(\rho(x)a))\\ &\overset{\ref{Rep32of32RBDer32pair3}}{=}\rho(\delta\circ Rx)a+\rho(Rx)\circ \delta_{\mathrm{V}}a-T(\delta_{\mathrm{V}}(\rho(x)a))\\ &\overset{\ref{Rep32of32RBDer32pair1}}{=}\rho(\delta\circ Rx)a+\rho(Rx)\circ \delta_{\mathrm{V}}a-T(\rho(\delta x)a+\rho(x)\circ \delta_{\mathrm{V}}a)\\ &\overset{\ref{condition132RBLieDer32pair}}{=}\rho(R\circ\delta x)a-T(\rho(\delta x)a)+\rho(Rx)\circ \delta_{\mathrm{V}}a-T(\rho(x)\circ \delta_{\mathrm{V}}a)\\ &=\widetilde{\rho}(\delta x)a+\widetilde{\rho}(x)\circ \delta_{\mathrm{V}}a. \end{align}\] This means that \((\widetilde{\mathcal{V}},\delta_{\mathrm{V}})\) is a representation of the LieDer pair \((L_\mathrm{R},\delta)\). For the next result see (Theorem 2.17 in [28] ). ◻
In this section we introduce the cohomology of weighted Rota-Baxter LieDer and AssDer pairs.
In this subsection, we first recall the Chevally-Eilenberg cohomology of Lie algebras, the cohomology of weighted Rota-Baxter Lie algebras [28] and the cohomology
of LieDer pairs [19] with coefficients in an arbitrary representation. Then we define the cohomology of \(\lambda\)-weighted
Rota-Baxter LieDer pairs.
Let \(\mathfrak{L}=(L,[\cdot,\cdot])\) be a Lie algebra, the Chevally-Eilenberg cohomology of \(\mathfrak{L}\) with cofficents in the representation \(\mathcal{V}\) is given by the cohomology of the cochain complex \(\{\mathrm{C}^{\star}(L;\mathcal{V}),\mathrm{d}\}\) where \(\mathrm{C}^{n}(L;\mathcal{V})=\mathrm{Hom}(\wedge^nL,V)\) for \(n\geq0\) and the coboundary map \[\mathrm {d}:C^n(L;\mathcal{V})\rightarrow
C^{n+1}(L;\mathcal{V})\] is given by \[\begin{align} (\mathrm {d}(f_n))(x_1,\ldots,x_{n+1})&=\displaystyle\sum_{i=1}^{n+1}(-1)^{i+n}\rho(x_i)f_n(x_1,\ldots,\hat{x_i},\ldots,x_{n+1})\\ &+\displaystyle\sum_{1\leq
i<j\leq n+1}(-1)^{i+j+n+1}f_n([x_i,x_j],x_1,\ldots,\hat{x_i},\ldots,\hat{x_j},\ldots,x_{n+1})
\end{align}\] for \(f_n\in C^n({L;\mathcal{V}})\) and \(x_1,\ldots,x_{n+1}\in L\).
Let \((\mathfrak{L},\delta)\) be a LieDer pair. Recall that the cohomology of LieDer pair \((\mathfrak{L},\delta)\) with coefficents in a representation \((\mathcal{V},\delta_{\mathrm{V}})\) is given as follow:
The set of LieDer pair \(0\)-cochains is \(0\) and the set of LieDer pair \(1\)-cochains is \(\mathfrak{C}^1_{\mathrm{LieDer}}(L;\mathcal{V})=\mathrm{Hom}(L,V)\). For \(n\geq2\), the set of LieDer pair \(n\)-cochains is given by \[\mathfrak{C}^n_{\mathrm{LieDer}}(L;\mathcal{V}):=C^n(L;\mathcal{V})\times C^{n-1}(L;\mathcal{V})\] For \(n\geq1\), define the following operator \(\partial:C^n(L;\mathcal{V})\rightarrow C^n(L;\mathcal{V})\) by \[\partial f_n=\displaystyle\sum_{i=1}^nf_n\circ (\mathbf{1}\otimes \cdots\otimes \underbrace{\delta}_{\text{i-th place}}\otimes
\cdots\otimes \mathbf{1})-\delta_{\mathrm{V}}\circ f_n.\] Define \(\mathrm{D}:\mathfrak{C}^1_{\mathrm{LieDer}}(L;\mathcal{V})\rightarrow \mathfrak{C}^2_{\mathrm{LieDer}}(L;\mathcal{V})\) by \[\mathrm{D}f_1=(\mathrm {d}(f_1),(-1)^1\partial f_1),\quad \forall f_1\in \mathrm{Hom}(L,V).\] And for \(n\geq2\), define \(\mathrm{D}:\mathfrak{C}^n_{\mathrm{LieDer}}(L;\mathcal{V})\rightarrow \mathfrak{C}^{n+1}_{\mathrm{LieDer}}(L;\mathcal{V})\) by \[\mathrm{D}(f_n,g_{n-1})=(\mathrm{d}(f_n),\mathrm{d}(g_{n-1})+(-1)^n\partial f_n)\] for all \(f_n\in C^n(L;\mathcal{V})\) and \(g_{n-1}\in
C^{n-1}(L;\mathcal{V})\).
Recall also the following equation \[\label{coboundary1} \mathrm{d}\circ \partial =\partial \circ \mathrm{d}\tag{12}\]
Let \((\mathfrak{L},R)\) be a Rota-Baxter Lie algebra, the cohomology of weighted Rota-Baxter Lie algebra with coefficients in a representation \((\mathcal{V},T)\) is given as follow as
follow :
For each \(n\geq 0\), define an abelian group \(\mathfrak{C}^n_{\mathrm{R}}(L;\mathcal{V})\) by \[\mathfrak{C}^n_{\mathrm{R}}(L;V)=
\left\{
\begin{array}{ll} C^0(L;\mathcal{V})=V,& \text{ if } n=0,\\ C^n(L;\mathcal{V})\oplus C^{n-1}(L_{\mathrm{R}};\widetilde{\mathcal{V}})=\mathrm{Hom}(\wedge^nL,V)\otimes\mathrm{Hom}(\wedge^{n-1}L,V),& \text{ if } n\geq1.
\end{array}
\right.\] Where \(\mathfrak{L}_{\mathrm{R}}=(L,[\cdot,\cdot]_{\mathrm{R}},R)\) is a Rota-Baxter Lie algebra with the bracket is defined in ?? and \(\widetilde{\mathcal{V}}=(V,\widetilde{\rho})\) is a representation of it with \(\widetilde{\rho}\) is defined in ?? .
The coboundary map is defined as \(\mathfrak{d}_{\mathrm{R}}:\mathfrak{C}^n_{\mathrm{R}}(L;\mathcal{V})\rightarrow \mathfrak{C}^{n+1}_{\mathrm{R}}(L;\mathcal{V})\) by \[\left\{
\begin{array}{ll} \mathfrak{d}_{\mathrm{R}}(v)=(\mathrm{d}(v),-v),& \text{ for } v\in \mathfrak{C}^0_{\mathrm{R}}(L;\mathcal{V})=V,\\ \mathfrak{d}_{\mathrm{R}}(f_n,g_{n-1})=(\mathrm{d}(f_n),-\mathrm{d}_{\mathrm{R}}(g_{n-1})-\Phi^n(f_n)),& \text{
for } (f_n,g_{n-1})\in\mathfrak{C}^n_{\mathrm{R}}(L;\mathcal{V}) .
\end{array}
\right.\] With \(\mathrm{d}_{\mathrm{R}}:C^n(L_{\mathrm{R}};\widetilde{\mathcal{V}})\rightarrow C^{n+1}(L_{\mathrm{R}};\widetilde{\mathcal{V}})\) is given by \[\begin{align}
(\mathrm{d}_{\mathrm{R}}f_n)(x_1,\ldots,x_{n+1})&=\displaystyle\sum_{i=1}^{n+1}(-1)^{i+n} \;\widetilde{\rho}(x_i)f_n(x_1,\ldots,\hat{x_i},\ldots,x_{n+1})\\ &+\displaystyle\sum_{1\leq i<j\leq
n+1}(-1)^{i+j+n+1}f_n([x_i,x_j]_{\mathrm{R}},x_1,\ldots,\hat{x_i},\ldots,\hat{x_j},\ldots,x_{n+1})
\end{align}\] and \(\Phi:C^n(L;\mathcal{V})\rightarrow C^n(L_{\mathrm{R}};\widetilde{\mathcal{V}})\), is inspired from [14],
defined by
\[\label{cohomology32of32RBO} \left\{ \begin{array}{ll} \Phi=\mathbf{1}_{\mathrm{V}},& \\ \Phi(f_n)(x_1,\cdots,x_n)=f_n(Rx_1,\cdots,Rx_n)-\displaystyle\sum_{k=0}^{n-1}\lambda^{n-k-1}\displaystyle\sum_{i_1<\ldots<i_k}T\circ f_n(x_1,\cdots,R(x_{i_1}),\cdots,R(x_{i_k}),\cdots,x_n).& \end{array} \right.\tag{13}\]
is a morphism of cochain complex from \(\{C^{\star}(L;\mathcal{V}),\mathrm{d}\}\) to \(\{C^{\star}(L_{\mathrm{R}};\widetilde{\mathcal{V}}),\mathrm{d}\}\) i.e.
\[\label{coboundary2} \mathrm{d}_{\mathrm{R}}\circ\Phi=\Phi\circ \mathrm{d},\quad \forall n\geq0.\tag{14}\] Also since \(L_{\mathrm{R}}\) is a
Lie algebra and \(\mathrm{d}_{\mathrm{R}}\) its coboundary with respect to the representation \(\widetilde{\mathcal{V}}=(V;\widetilde{\rho})\) and \((L_{\mathrm{R}},\delta)\) is a LieDer pair then, from the cohomology of LieDer pair ([19]. Lemma(3.1)), we get
\[\label{coboundary3}
\partial\circ\mathrm{d}_{\mathrm{R}}=\mathrm{d}_{\mathrm{R}}\circ\partial\tag{15}\] Using all those tools we are in position to define the cohomology of \(\lambda\)-weighted Rota-Baxter LieDer pair \((\mathfrak{L},\delta,R)\) with coefficients in a representation \((\mathcal{V},\delta_\mathrm{V},T)\). The set of \(\lambda\)-weighted Rota-Baxter LieDer pair
\(0\)-cochains is \(0\) and the set of \(\lambda\)-weighted Rota-Baxter LieDer pair \(1\)-cochains to be \(\mathfrak{C}^1_{\mathrm{RBLieDer}}(L,\mathcal{V})=\mathrm{Hom}(L,V)\).
For \(n\geq2\) define \(\lambda\)-weighted Rota-Baxter LieDer pair \(n\)-cochains by \[\mathfrak{C}^n_{\mathrm{RBLieDer}}(L;\mathcal{V}):=\mathfrak{C}^n_{\mathrm{LieDer}}(L;\mathcal{V})\otimes C^{n-1}(L_R;\widetilde{\mathcal{V}})\] Define \[\mathfrak{D}_{\mathrm{RBLieDer}}:\mathfrak{C}^n_{\mathrm{RBLieDer}}(L,\mathcal{V})\rightarrow \mathfrak{C}^{n+1}_{\mathrm{RBLieDer}}(L,\mathcal{V})\] as follow
For \(n=1\) \[\begin{align} \mathfrak{D}_{\mathrm{RBLieDer}}&:&\mathfrak{C}^1_{\mathrm{RBLieDer}}(L,\mathcal{V})\rightarrow \mathfrak{C}^2_{\mathrm{RBLieDer}}(L,\mathcal{V}) \text{ is given by}\\ &&\mathfrak{D}_{\mathrm{RBLieDer}}(f)=(d(f),-\partial (f),-\Phi (f)),\quad \forall f\in \mathrm{Hom}(L,V). \end{align}\]
For \(n\geq2\) \[\begin{align} \mathfrak{D}_{\mathrm{RBLieDer}}&:&\mathfrak{C}^n_{\mathrm{RBLieDer}}(L,\mathcal{V})\rightarrow \mathfrak{C}^{n+1}_{\mathrm{RBLieDer}}(L,\mathcal{V}) \text{ is given by}\\ &&\mathfrak{D}_{\mathrm{RBLieDer}}((f,g),h)=(d(f),d(g)+(-1)^n\partial (f),-d_R(h)-\Phi f),\quad \forall ((f,g),h)\in \mathrm{Hom}(L,V). \end{align}\]
Next in the following theorem we are in position to prove that \(\mathfrak{D}_{\mathrm{RBLieDer}}\) is a coboundary map.
Theorem 12. The map \(\mathfrak{D}_{\mathrm{RBLieDer}}\) is a coboundary operator, means that \[\mathfrak{D}_{\mathrm{RBLieDer}}\circ \mathfrak{D}_{\mathrm{RBLieDer}}=0.\]
Proof. For \(n\geq1\), using equations 12 and 14 \[\begin{align} \mathfrak{D}_{\mathrm{RBLieDer}}\circ\mathfrak{D}_{\mathrm{RBLieDer}}((f,g),h)&=\mathfrak{D}_{\mathrm{RBLieDer}}(d(f),d(g)+(-1)^n\partial (f),-d_R(h)-\Phi f)\\ &=(d^2(f),d^2(g)+(-1)^nd\circ\partial(f)+(-1)^{n+1}(f),d^2_R(h)+d_R\circ\Phi(f)-\Phi\circ d(f))\\ &=(0,0+(-1)^nd\circ\partial(f)+(-1)^{n+1}(f),0+d_R\circ\Phi(f)-\Phi\circ d(f))\\ &=(0,0,0) \end{align}\] This complete the proof. ◻
With respect to the representation \((\mathcal{V},\delta_V,T)\) we obtain a complex \(\{\mathfrak{C}^{\star}_{\mathrm{RBLieDer}}(L,\mathcal{V}),\mathfrak{D}_{\mathrm{RBLieDer}}\}\). Let
\(\mathcal{Z}^n_{\mathrm{RBLieDer}}(L,\mathcal{V})\) and \(\mathcal{B}^n_{\mathrm{RBLieDer}}(L,\mathcal{V})\) denote the space of \(n\)-cocycles and \(n\)-coboundaries, respectively. Then we define the corresponding cohomology groups by \[\mathcal{H}^n_{\mathrm{RBLieDer}}(L,\mathcal{V}):=\frac{\mathcal{Z}^n_{\mathrm{RBLieDer}}(L,\mathcal{V})}{\mathcal{B}^n_{\mathrm{RBLieDer}}(L,\mathcal{V})},\quad \text{for } n\geq0.\] They are called the cohomolgy of \(\lambda\)-weighted Rota-Baxter LieDer pair \((\mathfrak{L},\delta,R)\) with coefficients in the representation \((\mathcal{V},\delta_{\mathrm{V}},T)\).
Remark 2. Recall that from [28], if \((\mathfrak{L},R)\) is a Rota-Baxter Lie algebra and \((\mathcal{V},T)\) is a representation of it. An element \(v\in V\) is in \(\mathrm{Z}^0_{RB}(L,\mathcal{V})\) if and only if \((d_{\mathrm{R}}(v),-v)=0\), this holds when \(v=0\) which means that \(\mathrm{H}^0_{RB}(L,\mathcal{V})=0\) and it coincide in our paper with \(\mathrm{H}^0_{RBLieDer}(L,\mathcal{V})=0=\mathrm{H}^0_{RB}(L,\mathcal{V})\).
Also \(\mathrm{H}^1_{RBLieDer}(L,\mathcal{V})\) coincide with \(\mathrm{H}^1_{RB}(L,\mathcal{V})=\frac{\mathrm{Der}(L,\mathcal{V})}{\mathrm{InnDer}(L,\mathcal{V})}\).
Proposition 13. Let \((\mathcal{V},\delta_{\mathrm{V}},T)\) be a representation of a \(\lambda\)-weighted Rota-Baxter LieDer pair \((\mathfrak{L},\delta,R)\). Then we have \[\mathcal{H}^1_{\mathrm{RBLieDer}}(L,\mathcal{V})=\{f;\quad f\in\mathcal{Z}^1(L,\mathcal{V}),\quad f\circ\delta=\delta_V\circ f,\quad f\circ R=T\circ f\}.\]
In [14] authors defined the cohomology of a weighted Rota-Baxter associative algebra with coefficients in a Rota-Baxter bimodule. Later Das.A studied the
associative algebra with derivation and denote it by AssDer pairs [21].
In this subsection we study the cohomology of \(\lambda\)-weighted Rota-Baxter AssDer pair and we show that this cohomolgy is related to the cohomology of Rota-Baxter LieDer pairs by suitable skew-symmetrization.
Let \((\mathfrak{A}=(A,\mu))\) be an associative algebra and \(\delta:A\rightarrow A\) be a derivation on it. Recall that an AssDer pair \((\mathfrak{A},\delta)\) is an associative algebra equipped with the derivation \(\delta\).
A linear map \(R:A \rightarrow A\) is said to be a \(\lambda\)-weighted Rota-Baxter operator on \(\mathfrak{A}\) if it satisfies
\[\label{RBO32on32Ass32Algebras} \mu(Rx,Ry)=R(\mu(Rx,y)+\mu(x,Ry)+\lambda \mu(x,y)),\quad \forall x,y\in L.\tag{16}\] A bimodule over \(\mathfrak{A}\) consists of a vector space \(M\) together with two linear maps \(l:A\otimes M\rightarrow M\) and \(r:M\otimes
A\rightarrow M\) such that for all \(x,y\in A\) and \(m\in M\) \[\begin{align} l(\mu(x,y),m)&=&l(x,l(y,m)),\\ r(l(x,m),y)&=&l(x,r(m,y))\\
r(r(m,x),y)&=&r(m,\mu(x,y)).
\end{align}\] We will write \(xm\) instead of \(l(x,m)\) and \(mx\) instead of \(r(m,x)\) when there are no
confusions.
Definition 8. A weighted Rota-Baxter AssDer pair consisits of an AssDer pair \((\mathfrak{A},\delta)\) together with a weighted Rota-Baxter operator such that \[R\circ \delta=\delta\circ R.\]
Definition 9. Let \((\mathfrak{A}_1,\delta_1,R_1)\) and \((\mathfrak{A}_2,\delta_2,R_2)\) be two \(\lambda\)-weighted Rota-Baxter AssDer pairs. Then the linear map \(f:A_1\rightarrow A_2\) is said to be a \(\lambda\)-weighted Rota-Baxter homomorphism if is satisfies the following, for \(x,y\in A\) \[\begin{align} f\circ\mu_1(x,y)&=&\mu_2(f(x),f(y)),\\ f\circ\delta_1&=&\delta_1\circ f,\\ f\circ R_1&=&R_1\circ f. \end{align}\]
Definition 10. Let \((\mathfrak{A},\delta,R)\) be a \(\lambda\)-weighted Rota-Baxter AssDer pair. A bimodule (representation) over it is a triple \((M,\delta_\mathrm{M},T)\) which is both a left and right module on \((\mathfrak{A},\delta,R)\) and \(M\) is an \(A\)-bimodule. This means the followings, for all \(x,y\in A\) and \(m\in M\) \[\begin{align} \delta_\mathrm{M}(xm)&=&\delta(x)m+x\delta_\mathrm{M}(m),\tag{17}\\ \delta_\mathrm{M}(mx)&=&\delta_\mathrm{M}(m)x+m\delta(x)\tag{18}\\ R(x)T(m)&=&T(R(x)m+xT(m)+\lambda xm)\\ T(m)R(x)&=&T(T(m)x+mR(x)++\lambda mx)\\ \delta_\mathrm{M}\circ T&=&T\circ \delta_\mathrm{M} \tag{19}. \end{align}\]
Proposition 14. Let \((\mathfrak{A},\delta,R)\) be a \(\lambda\)-weighted Rota-Baxter AssDer pair and \((M,\delta_\mathrm{M},T)\) be a representation of it. Then \((A\oplus M,\delta\oplus\delta_\mathrm{M},R\oplus T)\) is a \(\lambda\)-weighted Rota-Baxter AssDer pair where its structure is given by for all \(x,y\in A\) and \(m,n\in M\) \[\begin{align} \mu_\ltimes(x+m,y+n)&=&\mu(x,y)+xn+my,\\ (\delta\oplus\delta_\mathrm{M})(x+m)&=&\delta(x)+\delta_\mathrm{M}(m)\\ (R\oplus T)(x+m)&=&R(x)+T(m). \end{align}\]
Proof. the proof is easy to check because it is known that \(A\oplus M\) with the product \(\mu_\ltimes\) and the linear map \(\delta\oplus\delta_\mathrm{M}\) is a LieDer pair, see [21]. And it is well know that \(R\oplus T\) is a \(\lambda\)-weighted Rota-Baxter operator. ◻
Proposition 15. Let \((\mathfrak{A},\delta,R)\) be a weighted Rota-Baxter AssDer pair. Define a new binary operation as follow : \[\mu_\mathrm{R}:=\mu(x,Ry)+\mu(Rx,y)+\lambda \mu(x,y),\quad \forall x,y\in A.\] Then
The operation \(\mu_\mathrm{R}\) forms an AssDer pair \((A,\mu_\mathrm{R},\delta)\) together with the derivation \(\delta\).
The quadruple \((A,\mu_\mathrm{R},\delta,R)\) forms a weighted Rota-Baxter AssDer pair and it is denoted \((\mathfrak{A}_\mathrm{R},\delta,R)\).
The map \(R:(A,\mu_\mathrm{R},\delta,R)\rightarrow(A,\mu,\delta,R)\) is a morphism of weighted Rota-Baxter AssDer pairs.
Proof. All we need to prove is that \(\delta\) is a derivation on the operation \(\mu_\mathrm{R}\), the rest of the proof see ([27],Theorem 1.1.17).
Let \(x,y\in A\) \[\begin{align} \delta(\mu_\mathrm{R}(x,y))&=\delta(\mu(x,Ry)+\mu(Rx,y)+\lambda \mu(x,y))\\ &=\mu(\delta x,Ry)+\mu(x,\delta\circ Ry)+\mu(\delta\circ Rx,y)+\mu(Rx,\delta
y)+\lambda \mu(\delta x,y)+\lambda \mu(x,\delta y)\\ &\overset{\ref{condition132RBLieDer32pair}}{=}\mu(\delta x,Ry)+\mu(R\circ \delta x,y)+\lambda \mu(\delta x,y)+\mu(x,R\circ\delta y)+\mu(Rx,\delta y)+\lambda \mu(x,\delta y)\\
&=\mu_\mathrm{R}(\delta x,y)+\mu_\mathrm{R}(x,\delta y).
\end{align}\] This complete the proof. ◻
Proposition 16. Let \((\mathfrak{A},\delta,R)\) be a weighted Rota-Baxter AssDer pair and \((M,\delta_\mathrm{M},T)\) be weighted Rota-Baxter bimodule over it. Define a left action \(l_\mathrm{R}\) and a right action \(r_\mathrm{R}\) of \(A\) on \(M\) as follows, for all \(x\in A,m\in M\) \[\begin{align} l_\mathrm{R}(x,m)&=&l(Rx,m)-T(L(x,m))\\ r_\mathrm{R}(m,x)&=&r(m,Rx)-T(r(m,x)). \end{align}\] Then these actions makes \(M\) into weighted Rota-Baxter bimodule over \((\mathfrak{A}_\mathrm{R},\delta,R)\) and denote it this new bimodule by \((M_\mathrm{R},\delta_M,T)\).
Proof. It is known that \(l_\mathrm{R}\) (respectively \(r_\mathrm{R}\)) is a left action (respectively right action) of \(A\) on \(M_\mathrm{R}\). So we need just to prove equations 17 and 18 .
Let \(x\in A,m\in M\) \[\begin{align} \delta_\mathrm{M}\circ l_\mathrm{R}&=\delta_\mathrm{M}(l(R(x),m)-T(l(x,m)))\\ &\overset{\ref{condition132RBLieDer32pair}}{=}l(R\circ \delta
x,m)+l(Rx,\delta_\mathrm{M}(m))-T\circ \delta_\mathrm{M}(l(x,m))\\ &\overset{\ref{AssDer32rep325}}{=}l(R\circ \delta x,m)-T(l(\delta x,m))+l(Rx,\delta_\mathrm{M}(m))-T(l(x,\delta_\mathrm{M}(m)))\\ &=l_\mathrm{R}(\delta
x,m)+l_\mathrm{R}(x,\delta_\mathrm{M}(m))
\end{align}\]Similarly to \(r_\mathrm{R}\). ◻
Next we will define a cohomology for weighted Rota-Baxter AssDer pairs.
Let \((\mathfrak{A},\delta)\) be an AssDer pair and \((M,\delta_M)\) be a representation of it. Recall from [14], the Hochshild cochain complex of \(\mathfrak{A}_\mathrm{R}\) with coefficients in \(M_\mathrm{R}\) is given as follow \[\mathrm{d_{R.Hoch}}:C^n(\mathfrak{A}_\mathrm{R},M_\mathrm{R})\rightarrow C^{n+1}(\mathfrak{A}_\mathrm{R},M_\mathrm{R}) \text{ by }\] \[\begin{align} \mathrm{d_{R.Hoch}}(f)(x_1,\ldots,x_{n+1})&=
l_\mathrm{R}(x_1,f(x_2,\ldots,x_{n+1}))+\displaystyle\sum_{i=1}^n(-1)^if(x_1,\ldots,x_{i-1},\mu_\mathrm{R}(x_i,x_{i+1}),\ldots,x_{n+1})\\ &+(-1)^{n+1}r_\mathrm{R}(f(x_1,\ldots,x_n),x_{n+1}).
\end{align}\] Define \(\partial\) the coboundary operator from \(C^n(A,M)\) to \(C^n(A,M)\) and it is given by \[\partial
(f)=\displaystyle\sum_{i=1}^n f\circ (\mathbf{1}\otimes \ldots \delta \otimes \ldots \otimes \mathbf{1})-\delta_\mathrm{M}\circ f.\] First step let’s define the cohomology, in context of AssDer pair structure, of the weighted Rota-Baxter operator
\(R\) with coefficients in the representation \((M_\mathrm{R},\delta_\mathrm{M},T)\).
Consider the cochain complex of the AssDer pair \((\mathfrak{A}_\mathrm{R},\delta)\) with coefficients in the representation \((M_\mathrm{R},\delta_M)\) \[\mathfrak{C}_\mathrm{AssDer}^\star(\mathfrak{A}_\mathrm{R},M_\mathrm{R}):=\oplus_{n\geq0}\mathfrak{C}_\mathrm{AssDer}^n(\mathfrak{A}_\mathrm{R},M_\mathrm{R})\] The space \(\mathfrak{C}_\mathrm{AssDer}^n(\mathfrak{A}_\mathrm{R},M_\mathrm{R})\) of \(n\)-cochains is defined as follow \[\left\{
\begin{array}{ll} \mathfrak{C}_\mathrm{AssDer}^0(\mathfrak{A}_\mathrm{R},M_\mathrm{R})=0,&\\ \mathfrak{C}_\mathrm{AssDer}^1(\mathfrak{A}_\mathrm{R},M_\mathrm{R})=\mathrm{Hom}(\mathfrak{A}_\mathrm{R},M_\mathrm{R}),&\\
\mathfrak{C}_\mathrm{AssDer}^n(\mathfrak{A}_\mathrm{R},M_\mathrm{R})=C^n(\mathfrak{A}_\mathrm{R},M_\mathrm{R})\oplus C^{n-1}(\mathfrak{A}_\mathrm{R},M_\mathrm{R}),& \text{for } n\geq2.
\end{array}
\right.\] With the coboundary map is given by \(\mathfrak{d}_{R.A.D}:\mathfrak{C}_\mathrm{R.AssDer}^n(\mathfrak{A}_\mathrm{R},M_\mathrm{R})\rightarrow
\mathfrak{C}_\mathrm{R.AssDer}^{n+1}(\mathfrak{A}_\mathrm{R},M_\mathrm{R})\) as follow
\[\label{coboundary32AssDer32of32d46R} \mathfrak{d}_{R.A.D}(f,g)=(d_\mathrm{R.Hoch}(f),d_\mathrm{R.Hoch}(g)+(-1)^n\partial (f)),\quad \forall (f,g)\in \mathfrak{C}_\mathrm{R.AssDer}^n(\mathfrak{A}_\mathrm{R},M_\mathrm{R}).\tag{20}\]
By [19] and since \(d_\mathrm{R.Hoch}\) is Hochschild coboundary [14], it is easy to check that \[\begin{align} \mathfrak{d}_{R.A.D}\circ \mathfrak{d}_{R.A.D}&=&0,\\ d_\mathrm{R.Hoch}\circ \partial&=&\partial \circ d_\mathrm{R.Hoch}. \end{align}\] with the previous result we are able to define the cohomology of Rota-Baxter operator \(R\) with coefficients in \((M_\mathrm{R},\delta_\mathrm{M},T)\) in the context of AssDer pair structures.
Definition 11. Let \((\mathfrak{A},\delta,R)\) be a weighted Rota-Baxter AssDer pair and \((M,\delta_\mathrm{M},T)\) be a representation of it. Then the cocchain complex \((\mathfrak{C}_\mathrm{R.B.AssDer}(\mathfrak{A}_\mathrm{R},M_\mathrm{R}),\mathfrak{d}_{R.A.D})\) is called the cochain complex of weighted Rota-Baxter operator \(R\) with coefficients in \((M,\delta_\mathrm{M},T)\), denoted by \(\mathfrak{C}_\mathrm{R.AssDer}^\star(A,M)\) and its comology is denoted by \(\mathcal{H}_\mathrm{R.B.AssDer}(A,M)\), are called the cohomology of weighted Rota-Baxter operator \(R\) with coefficients in \((M,\delta_\mathrm{M},T)\).
Remark 3. When \((M,\delta_\mathrm{M},T)=(A,\delta,R)\) is the adjoint representation, then we denote the cochain complex of weighted Rota-Baxter operator \(R\) by \(\mathfrak{C}^\star_\mathrm{R.B.AssDer}(A)\) and its cohomology groups are denoted simply by \(\mathcal{H}^\star_\mathrm{R.B.AssDer}(A)\).
At the final step, we will combine the cohomology of AssDer pairs with the cohomology of weighted Rota-Bxater operators to define a cohomology theory for weighted Rota-Baxter AssDer pair.
Define the map \(\Phi^\star:C^\star(A,M)\rightarrow C^\star(\mathfrak{A}_\mathrm{R},M_\mathrm{R})\) by
\(\Phi^0=Id_\mathrm{M}\) and for \(n\geq1\) and \(f\in C^n(A,M)\) define \(\Phi^n(f)\in
C^n(\mathfrak{A}_\mathrm{R},M_\mathrm{R})\) as follow \[\Phi^n(f)(x_1,\cdots,x_n)=f(Rx_1,\cdots,Rx_n)-\displaystyle\sum_{k=0}^{n-1}\lambda^{n-k-1}\displaystyle\sum_{i_1<\ldots<i_k}T\circ
f(x_1,\cdots,R(x_{i_1}),\cdots,R(x_{i_k}),\cdots,x_n)\] From ([14], proposition 5.2), the map \(\Phi^\star\) is a chain
map.
Definition 12. Let \((M,\delta_\mathrm{M},T)\) be a representation of a weighted Rota-Baxter AssDer pair \((\mathfrak{A},\delta,R)\). Define the cochain complex \((\mathfrak{C}^\star_\mathrm{R.B.AssDer}(A,M),\mathfrak{D}_\mathrm{R.B.AssDer})\) of weighted Rota-Baxter AssDer pair \((\mathfrak{A},\delta,R)\) with coeffecients in \((M,\delta_\mathrm{M},T)\) such that \[\mathfrak{C}^0_\mathrm{R.B.AssDer}(A,M)=0 \text{ and } \mathfrak{C}^n_\mathrm{R.B.AssDer}(A,M):=\mathfrak{C}^n_\mathrm{AssDer}(A,M)\oplus
C^{n-1}(\mathfrak{A}_\mathrm{R},M_\mathrm{R}),\quad \forall n\geq1.\] With \(\mathfrak{C}^n_\mathrm{R.B.AssDer}(A,M)=(C^n(A,M)\oplus C^{n-1}(A,M))\oplus C^{n-1}(\mathfrak{A}_\mathrm{R},M_\mathrm{R})\).
And the differential \[\mathfrak{D}_\mathrm{R.B.AssDer}:\mathfrak{C}^n_\mathrm{R.B.AssDer}(A,M)\rightarrow \mathfrak{C}^{n+1}_\mathrm{R.B.AssDer}(A,M) \text{ is given by }\] \[\mathfrak{D}_\mathrm{R.B.AssDer}((f,g),h):=(d_\mathrm{Hoch}(f),d_\mathrm{Hoch}(g)+(-1)^n\partial f,-d_\mathrm{R.Hoch}(h)-\Phi^n(f)).\] Where \(d_\mathrm{Hoch}:C^n(A,M)\rightarrow C^{n+1}(A,M)\)
is the coboundary map associated to the Hochschild cohomology [30].
The cohomology of weighted Rota-Baxter AssDer pair \((\mathfrak{C}^\star_\mathrm{R.B.AssDer}(\mathfrak{A}_\mathrm{R},M_\mathrm{R}),\mathfrak{D}_\mathrm{R.B.AssDer})\) of \((\mathfrak{A},\delta,R)\) with coefficients in \((M,\delta_\mathrm{M},T)\) is denoted by \(\mathcal{H}^\star_\mathrm{R.B.AssDer}(A)\)
In this section, we show that the cohomology of Rota-Baxter LieDer pair is related to the cohomology of Rota-Baxter AssDer pairs via a suitable skew-symmetrization.
Proposition 17. Let \((\mathfrak{A},\delta,R)\) be a weighted Rota-Baxter AssDer pair. Then \((\mathfrak{A}_\mathrm{c},\delta,R)\) is a weighted Rota-Baxter LieDer pair, where \(\mathfrak{A}_\mathrm{c}=(A,[\cdot,\cdot]_\mathrm{c})\) such that \[_\mathrm{c}=\mu(x,y)-\mu(y,x),\quad \forall x,y \in A.\] (is called the skew-symmetrization).
Remark 4. Moreover, if \((M,\delta_\mathrm{M},T)\) is a representation of \((\mathfrak{A},\delta,R)\) then \((M_\mathrm{c},\delta_\mathrm{M},T)\) is a representation of the weighted Rota-Baxter LieDer pair \((\mathfrak{A}_\mathrm{c},\delta,R)\) with the representation is given as follow \[\rho(x)(m)=l(x,m)-r(m,x),\quad \forall x\in A, m\in M.\]
It is known that the standard skew-symmetrization gives rise to a morphism from the Hochschild cochain complex of an associative algebra to the Chevalley-Eilenberg cochain complex of the corresponding skewsymmetrized Lie algebra. Means that there is a morphism \(S_n\) from the \(n\)-th cochain group (in context of Hochschild cohomology) \(C^n(\mathfrak{A},M)\) to the \(n\)-th cochain group \(C^n(\mathfrak{A}_\mathrm{c},M_\mathrm{c})\) (in the context of Chevalley-Eilenberg cohomology), more general \(S_\star\) are called the skew-symmetrization maps.
Theorem 18. Let \((\mathfrak{A},\delta,R)\) be a weighted Rota-Baxter AssDer pair and \((M,\delta_\mathrm{M},T)\) be a representation of it. Then the collection of maps \[\mathcal{S}_\mathrm{n}:\mathfrak{C}^n_\mathrm{R.B.AssDer}(\mathfrak{A},M)\rightarrow \mathfrak{C}^n_\mathrm{R.B.LieDer}(\mathfrak{A}_\mathrm{c},M_\mathrm{c}),\quad \mathcal{S}_\mathrm{n}=(S_n,S_{n-1},S_{n-1})\] induces a morphism from the cohomology of \((\mathfrak{A},\delta,R)\) with coefficients in \((M,\delta_\mathrm{M},T)\) to the cohomology of \((\mathfrak{A}_\mathrm{c},\delta,R)\) with coefficients in \((M_\mathrm{c},\delta_\mathrm{M},T)\).
Proof. For \(((f,g),h)\in \mathfrak{C}^n_\mathrm{R.B.AssDer}(\mathfrak{A},M)\) we have \[\begin{align} \mathfrak{D}_\mathrm{R.B.LieDer}\circ \mathcal{S}_\mathrm{n} ((f,g),h)&=\mathfrak{D}_\mathrm{R.B.LieDer}(S_n(f),S_{n-1}(g),S_{n-1}(h))\\ &=(d\circ S_n(f),d\circ S_{n-1}(g)+(-1)^n\partial \circ S_n(f),-d_\mathrm{R}\circ S_{n-1}(h)-\Phi\circ S_n(f))\\ &=(S_{n+1}\circ d_\mathrm{Hoch}(f),S_n\circ d_\mathrm{Hoch}(g)+(-1)^nS_{n+1}\circ \partial (f),-S_n\circ d_\mathrm{R.Hoch}(h)-S_{n+1}\circ \Phi^n(f))\\ &=\mathcal{S}_{n+1}\circ \mathfrak{D}_\mathrm{R.B.AssDer}((f,g),h) \end{align}\] ◻
In this section, we will study formal deformation of weighted Rota-Baxter LieDer and AssDer pairs.
In this subsection we will deform the Lie bracket on \(L\), the Rota-Baxter operator \(R\) and the derivation \(\delta\). We investigate the relation
between such deformation and cohomology of \((L,\delta,R)\) with coefficients in the adjoint representation.
Let \((\mathfrak{L},\delta,R)\) be a Rota-Baxter LieDer pair, let \(\gamma\in C^2(L,L)=\mathrm{Hom}(\wedge^2L,L)\) be the element that correspponds to the Lie bracket on \(L\), i.e., \(\gamma(x,y)=[x,y]\) for \(x,y\in L\). Consider the space \(L[[t]]\) of the formal power series in \(t\) with coefficients from \(L\). Then \(L[[t]]\) is a \(\mathbb{K}[[t]]\)-module.
Definition 13. A formal \(1\)-parameter deformation of \((\mathfrak{L},\delta,R)\) consists of a triple \((\gamma_\mathrm{t},\delta_\mathrm{t},R_\mathrm{t})\) of three formal power series \[\begin{align} \gamma_\mathrm{t}&=&\displaystyle\sum_{i\geq0}\gamma_\mathrm{i}t^\mathrm{i},\quad \text{where } \gamma_\mathrm{i}\in \mathrm{Hom}(\wedge^2L,L) \text{ with } \gamma_0=\gamma,\\ \delta_\mathrm{t}&=&\displaystyle\sum_{i\geq0}\delta_\mathrm{i}t^\mathrm{i},\quad \text{where } \delta_\mathrm{i}\in \mathrm{Hom}(L,L) \text{ with } \delta_0=\delta,\\ R_\mathrm{t}&=&\displaystyle\sum_{i\geq0}R_\mathrm{i}t^\mathrm{i},\quad \text{where } R_\mathrm{i}\in \mathrm{Hom}(L,L) \text{ with } R_0=R. \end{align}\] Such that the \(\mathbb{K}[[t]]\)-bilinear map \(\gamma_\mathrm{t}\) defines a Lie algebra structure on \(L[[t]]\) and the two \(\mathbb{K}[[t]]\)-linear maps \(\delta_\mathrm{t},R_\mathrm{t}:L[[t]] \rightarrow L[[t]]\) are respectively a derivation and a Rota-Baxter opperator. This means that \((L[[t]]=(L[[t]],\gamma_\mathrm{t}),\delta_\mathrm{t,R_\mathrm{t}})\) is a Rota-Baxter LieDer pair over \(\mathbb{K}[[t]]\).
Thus, \((\gamma_\mathrm{t},\delta_{\mathrm{t}},R_\mathrm{t})\) is a formal \(1\)-parameter deformation of \((\mathfrak{L},\delta,R)\) if and only if, for \(x,y,z\in L\) \[\begin{align} \circlearrowleft_{x,y,z}\gamma_\mathrm{t}(x,\gamma_\mathrm{t}(y,z))&=&0,\\ \gamma_\mathrm{t}(R_\mathrm{t}x,R_\mathrm{t}y)&=&R_\mathrm{t}(\gamma_\mathrm{t}(R_\mathrm{t}x,y)+\gamma_\mathrm{t}(x,R_{\mathrm{t}}y)+\lambda\gamma_\mathrm{t}(x,y) ),\\ \delta_{\mathrm{t}}(\gamma_\mathrm{t}(x,y))&=&\gamma_\mathrm{t}(\delta_\mathrm{t}x,y)+\gamma_\mathrm{t}(x,\delta_{\mathrm{t}}y). \end{align}\] They are equivalent to the following systems of identities, for \(n\geq0\) and for all \(x,y,z\in L\), \[\begin{align} \displaystyle\sum_{i+j=n}\gamma_\mathrm{i}(x,\gamma_\mathrm{j}(y,z))+\gamma_\mathrm{i}(y,\gamma_\mathrm{j}(z,x))+\gamma_\mathrm{i}(z,\gamma_\mathrm{j}(x,y))&=&0,\tag{21}\\ \displaystyle\sum_{i+j=n}\Big(\delta_{\mathrm{i}}(\gamma_\mathrm{j}(x,y))-\gamma_\mathrm{j}(\delta_{\mathrm{i}}x,y)-\gamma_\mathrm{j}(x,\delta_{\mathrm{i}}y) \Big)&=&0,\tag{22}\\ \displaystyle\sum_{i+j+k=n}\Big(\gamma_\mathrm{i}(R_\mathrm{j}x,R_\mathrm{k}y) -R_\mathrm{i}(\gamma_\mathrm{j}(R_\mathrm{k}x,y)+\gamma_\mathrm{j}(x,R_\mathrm{k}y)+\lambda\gamma_\mathrm{j}(x,y))\Big)&=&0.\tag{23} \end{align}\]
Definition 14. Two formal deformations \((\gamma_\mathrm{t},\delta_{\mathrm{t}},R_\mathrm{t})\) and \((\gamma^{\prime}_\mathrm{t},\delta^\prime_{\mathrm{t}},R^\prime_\mathrm{t})\) of a Rota-Baxter LieDer pair \((L,\delta,R)\) are said to be equivalent if there is a formal isomorphism \[\varphi_\mathrm{t}=\displaystyle\sum_{i\geq0}\varphi_\mathrm{i}t^\mathrm{i}:L[[t]]\rightarrow L[[t]],\quad \text{where } \varphi_\mathrm{i}\in \mathrm{Hom}(L,L) \text{ and } \varphi_0=\mathrm{id}_L.\] Such that the \(\mathbb{K}[[t]]\)-linear map \(\varphi_\mathrm{t}\) is a morphism of Rota-Baxter LieDer pairs from \((L[[t]]^\prime,\delta^\prime_{\mathrm{t}},R^\prime_\mathrm{t})\) to \((L[[t]],\delta_{\mathrm{t}},R_\mathrm{t})\)
It means that the following identities holds \[\varphi_\mathrm{t}(\gamma^{\prime}_\mathrm{t}(x,y))=\gamma_\mathrm{t}(\varphi_\mathrm{t}(x),\varphi_\mathrm{t}(y)) \text{ and } \varphi_\mathrm{t}\circ \delta^\prime_{\mathrm{t}}=\delta_{\mathrm{t}} \circ \varphi_\mathrm{t} \text{ and } \varphi_\mathrm{t}\circ R^\prime_\mathrm{t}=R_\mathrm{t}\circ \varphi_\mathrm{t}.\] They can be expressed as follow, for \(n\geq0\) \[\begin{align} \displaystyle\sum_{i+j=n}\varphi_\mathrm{i}(\gamma^\prime_\mathrm{j}(x,y))&=&\displaystyle\sum_{i+j+k=n}\gamma_\mathrm{i}(\varphi_\mathrm{j}(x),\varphi_\mathrm{k}(y)),\tag{24}\\ \displaystyle\sum_{i+j=n}\varphi_\mathrm{i}\circ \delta^\prime_{\mathrm{j}}&=&\displaystyle\sum_{i+j=n}\delta_\mathrm{i}\circ \varphi_\mathrm{j},\tag{25}\\ \displaystyle\sum_{i+j=n}\varphi_\mathrm{i}\circ R^\prime_{\mathrm{j}}&=&\displaystyle\sum_{i+j=n}R_\mathrm{i}\circ \varphi_\mathrm{j}.\tag{26} \end{align}\]
Theorem 19. Let \((\gamma_\mathrm{t},\delta_{\mathrm{t}},R_\mathrm{t})\) be a formal \(1\)-parameter deformation of the Rota-Baxter LieDer pair \((L,\delta,R)\). Then \((\gamma_1,\delta_1,R_1)\in \mathfrak{C}^2_\mathrm{RBLieDer}(L,L)\) is a \(2\)-cocycle in the cohmology of \((L,\delta,R)\) with coefficients in the adjoint representation.
Proof. Since \((\gamma_\mathrm{t},\delta_{\mathrm{t}},R_\mathrm{t})\) is a formal \(1\)-parameter deformation, we have from 21 , 22 and 23 , for \(n\geq0\), that \[\circlearrowleft_{x,y,z}[x,\gamma_1(y,z)]+\circlearrowleft_{x,y,z}\gamma_1(x,[y,z])=0,\] and \[\gamma_1(Rx,Ry)+[R_1x,Ry]+[Rx,R_1y]=R_1([Rx,y]+[x,Ry])+R([R_1x,y]+[x,R_1y]+\gamma_1(Rx,y)+\gamma_1(x,Ry)),\] and \[\delta_1([x,y])+\delta(\gamma_1(x,y))-\gamma_1(\delta x,y)-[\delta_1 x,y]-\gamma_1[x,\delta y]-[x,\delta_1y]=0.\] The first identity means that \(\mathrm{d}\gamma_1=0\) and the second identity means that \(\mathrm{d}_\mathrm{R}(R_1)+\Phi^2(\gamma_1)=0\) and the last one means that \(\mathrm{d}(\delta_1)+\partial^2(\gamma_1)=0\) This implies that \[\begin{align} \mathfrak{D}_\mathrm{RBLieDer}((\gamma_1,\delta_1),R_1)&=(\mathrm{d}(\gamma_1),\mathrm{d}(\delta_1)+\partial(\gamma_1),-\partial(R_1)-\Phi(R_1))\\ &=(0,0,-\partial(R_1)-\Phi(R_1))\\ &=(0,0,0). \end{align}\] This complete the proof. ◻
Theorem 20. Let \((L,\delta,R)\) be a weighted Rota-Baxter LieDer pair. If \(\mathcal{H}_\mathrm{R.B.LieDer}(L,L)=0\) then any formal \(1\)-parameter deformation of \((\mathfrak{L},\delta,R)\) is equivalent to the trivial one \((\gamma^\prime_{\mathrm{t}}=\gamma,R^\prime_{\mathrm{t}}=R,\delta^\prime_{\mathrm{t}}=\delta)\).
Proof. Let \((\gamma_\mathrm{t},\delta_\mathrm{t},R_\mathrm{t})\) be a formal \(1\)-parameter deformation of the weighted Rota-Baxter LieDer pair \((\mathfrak{L},\delta,R)\). From theorem 19 we have that \((\gamma1,\delta1,R_1)\) is a \(2\)-cocycle. Then from the hypothesis there exists \(f\in \mathrm{Hom}(L,L)\) such that \[\label{equation32deformation} \mathfrak{D}_\mathrm{R.B.LieDer}(f)=(d(f),-\partial(f),-\Phi(f)).\tag{27}\] Means that \((\gamma1,\delta1,R_1)=(d(f),-\partial(f),-\Phi(f))\).
Let \(\phi_\mathrm{t}:L\rightarrow L\) be the map \(\phi_\mathrm{t}=id+\phi_1 t\). Then \((\overline{\gamma_\mathrm{t}}=\phi_\mathrm{t}^{-1}\circ
\gamma_\mathrm{t}\circ(\phi_\mathrm{t}\otimes\phi_\mathrm{t}),\overline{\delta_\mathrm{t}}=\phi_\mathrm{t}^{-1}\circ\delta_\mathrm{t}\circ\phi_\mathrm{t},\overline{R_\mathrm{t}}=\phi_\mathrm{t}^{-1}\circ R_\mathrm{t}\circ\phi_\mathrm{t} )\) is a
formal \(1\)-deformation of \((\mathfrak{L},\delta,R)\) equivalent to \((\gamma_\mathrm{t},\delta_\mathrm{t},R_\mathrm{t})\). By 27 we can easily check that \((\overline{\gamma_1}=\overline{\delta_1}=\overline{R_1}=0)\), means that \[\begin{align}
\overline{\gamma_\mathrm{t}}&=&\gamma+\gamma_2t^2+...\\ \overline{\delta_\mathrm{t}}&=&\delta+\delta_2 t^2+...\\ \overline{R_\mathrm{t}}&=&R+R_2t^2+...
\end{align}\] and by repeating the same argument we conculde that \((\gamma_\mathrm{t},\delta_\mathrm{t},R_\mathrm{t})\) is equivalent to \((\gamma^\prime_{\mathrm{t}}=\gamma,R^\prime_{\mathrm{t}}=R,\delta^\prime_{\mathrm{t}}=\delta)\). ◻
In this section we study formal deformation of weighted Rota-Baxter AssDer pairs.
Let \((\mathfrak{A},\delta,R)\) be a Rota-Baxter AssDer pair. A formal \(1\)-parameter deformation of \((\mathfrak{A},\delta,R)\) consists of three formal
power series \[\begin{align} \mu_\mathrm{t}&=&\displaystyle\sum_{i\geq0}\mu_\mathrm{i}t^\mathrm{i},\quad \text{where } \mu_\mathrm{i}\in \mathrm{Hom}(\wedge^2L,L) \text{ with } \mu_0=\mu,\\
\delta_\mathrm{t}&=&\displaystyle\sum_{i\geq0}\delta_\mathrm{i}t^\mathrm{i},\quad \text{where } \delta_\mathrm{i}\in \mathrm{Hom}(L,L) \text{ with } \delta_0=\delta,\\ R_\mathrm{t}&=&\displaystyle\sum_{i\geq0}R_\mathrm{i}t^\mathrm{i},\quad
\text{where } R_\mathrm{i}\in \mathrm{Hom}(L,L) \text{ with } R_0=R.
\end{align}\] Then we say that \((\mu_\mathrm{t},\delta_\mathrm{t},R_\mathrm{t})\) is a formal \(1\)-parameter deformation of \((\mathfrak{A},\delta,R)\) if and only if \[\begin{align} \mu_\mathrm{t}(\mu_\mathrm{t}(x,y),z)&=&\mu_\mathrm{t}(x,\mu_\mathrm{t}(y,z)),\\
\mu_\mathrm{t}(R_\mathrm{t}x,R_\mathrm{t}y)&=&R_\mathrm{t}(\mu_\mathrm{t}(R_\mathrm{t}x,y)+\mu_\mathrm{t}(x,R_{\mathrm{t}}y)+\lambda\mu_\mathrm{t}(x,y) ),\\
\delta_{\mathrm{t}}(\mu_\mathrm{t}(x,y))&=&\mu_\mathrm{t}(\delta_\mathrm{t}x,y)+\mu_\mathrm{t}(x,\delta_{\mathrm{t}}y).
\end{align}\] And they are equivalent to the followings \[\begin{align} \displaystyle\sum_{i+j=n}\mu_\mathrm{i}(\mu_\mathrm{j}(x,y),z)-\mu_\mathrm{i}(x,\mu_\mathrm{j}(y,z))&=&0,\tag{28}\\
\displaystyle\sum_{i+j=n}\Big(\delta_{\mathrm{i}}(\mu_\mathrm{j}(x,y))-\mu_\mathrm{j}(\delta_{\mathrm{i}}x,y)-\mu_\mathrm{j}(x,\delta_{\mathrm{i}}y) \Big)&=&0,\tag{29}\\ \displaystyle\sum_{i+j+k=n}\Big(\mu_\mathrm{i}(R_\mathrm{j}x,R_\mathrm{k}y)
-R_\mathrm{i}(\mu_\mathrm{j}(R_\mathrm{k}x,y)+\mu_\mathrm{j}(x,R_\mathrm{k}y)+\lambda\mu_\mathrm{j}(x,y))\Big)&=&0.\tag{30}
\end{align}\] For \(n=1\)
we obtain \[\mu_1(\mu(x,y),z)+\mu(\mu_1(x,y),z)=\mu(x,\mu_1(y,z))+\mu_1(x,\mu(y,z))\] which means that \(d_\mathrm{Hoch}(\mu_1)=0\). And the second equation \[\delta(\mu_1(x,y))+\delta_1(\mu(x,y))=\mu(\delta_1x,y)+\mu_1(\delta x,y)+\mu(x,\delta_1y)+\mu_1(x,\delta y)\] means that \(d_\mathrm{Hoch}(\delta_1)+\partial \mu_1=0\). And the third equation
\[\begin{align} &&\mu_1(Rx,Ry)-R_1(\mu(Rx,y)+\mu(x,Ry)+\lambda \mu(x,y))\\ &&+\mu(R_1x,Ry)-R(\mu_1(Rx,y)+\mu_1(x,Ry)+\lambda \mu_1(x,y))\\ &&+\mu(Rx,Ry)-R(\mu(R_1x,y)+\mu(x,R_1y)+\lambda \mu(x,y))=0
\end{align}\] which means that \(-d_\mathrm{R.Hoch}-\Phi^2(\mu_1)=0\). which leads us to the following
Proposition 21. Let \((\mu_\mathrm{t},\delta_\mathrm{t},R_\mathrm{t})\) be a formal deformation of a weighted Rota-Baxter AssDer pair \((\mathfrak{A},\delta,R)\). Then the linear term \((\mu_1,\delta_1,R_1)\) is a \(2\)-cocycle in the cohomology of the weighted Rota-Baxter AssDer pair \((\mathfrak{A},\delta,R)\) with coefficients in itself.
Definition 15. Two formal deformations \((\mu_\mathrm{t},\delta_{\mathrm{t}},R_\mathrm{t})\) and \((\mu^{\prime}_\mathrm{t},\delta^\prime_{\mathrm{t}},R^\prime_\mathrm{t})\) of a weighted Rota-Baxter AssDer pair \((\mathfrak{A},\delta,R)\) are said to be equivalent if there is a formal isomorphism \[\varphi_\mathrm{t}=\displaystyle\sum_{i\geq0}\varphi_\mathrm{i}t^\mathrm{i}:A[[t]]\rightarrow A[[t]],\quad \text{where } \varphi_\mathrm{i}\in \mathrm{Hom}(L,L) \text{ and } \varphi_0=\mathrm{id}_L.\] Such that the \(\mathbb{K}[[t]]\)-linear map \(\varphi_\mathrm{t}\) is a morphism of Rota-Baxter LieDer pairs from \((A[[t]]^\prime,\delta^\prime_{\mathrm{t}},R^\prime_\mathrm{t})\) to \((A[[t]],\delta_{\mathrm{t}},R_\mathrm{t})\)
It means that the following identities holds \[\varphi_\mathrm{t}(\mu^{\prime}_\mathrm{t}(x,y))=\mu_\mathrm{t}(\varphi_\mathrm{t}(x),\varphi_\mathrm{t}(y)) \text{ and } \varphi_\mathrm{t}\circ \delta^\prime_{\mathrm{t}}=\delta_{\mathrm{t}} \circ \varphi_\mathrm{t} \text{ and } \varphi_\mathrm{t}\circ R^\prime_\mathrm{t}=R_\mathrm{t}\circ \varphi_\mathrm{t}.\] It is equivalent to the followings \[\begin{align} \displaystyle\sum_{i+j=n}\varphi_\mathrm{i}(\mu^\prime_\mathrm{j}(x,y))&=&\displaystyle\sum_{i+j+k=n}\mu_\mathrm{i}(\varphi_\mathrm{j}(x),\varphi_\mathrm{k}(y)),\\ \displaystyle\sum_{i+j=n}\varphi_\mathrm{i}\circ \delta^\prime_{\mathrm{j}}&=&\displaystyle\sum_{i+j=n}\delta_\mathrm{i}\circ \varphi_\mathrm{j},\\ \displaystyle\sum_{i+j=n}\varphi_\mathrm{i}\circ R^\prime_{\mathrm{j}}&=&\displaystyle\sum_{i+j=n}R_\mathrm{i}\circ \varphi_\mathrm{j}. \end{align}\] For \(n=0\) we have \(\varphi_0=\mathrm{Id_\mathrm{A}}\) and for \(n=1\) we obtain \[\begin{align} \varphi_1\circ \mu^\prime+\mu_1^\prime&=&\mu_1+\mu\circ (\varphi_1\otimes \mathrm{Id_A})+\mu\circ (\mathrm{Id_A}\otimes \varphi_1),\tag{31}\\ \varphi_1\circ\delta^\prime+\delta^\prime_1&=&\delta_1+\delta\circ\varphi_1,\tag{32}\\ \varphi_1\circ R^\prime+R^\prime_1&=&R_1+R\circ\varphi_1.\tag{33} \end{align}\] Then equations 31 ,32 and 33 we obtain that \[(\mu^\prime_1,\delta^\prime_1,R^\prime_1)-(\mu_1,\delta_1,R_1):=\mathfrak{D}_\mathrm{R.B.AssDer}(\varphi_1)\] This leads us to the following result
Theorem 22. Tow formal \(1\)-parameter deformations of a weighted Rota-Baxter assDer pair \((\mathfrak{A},\delta,R)\) are cohomologous. Therefore, they correspond to the same cohomology class.
Definition 16. A formal deformation \((\mu_\mathrm{t},\delta_\mathrm{t},R_\mathrm{t})\) of a weighted Rota-Baxter AssDer pair \((\mathfrak{A},\delta,R)\) is said trivial if it is equivalent to \((\mu^\prime_\mathrm{t}=\mu;\delta^\prime_\mathrm{t}=\delta)\).
Theorem 23. If \(\mathcal{H}^2_\mathrm{R.B.AssDer}(A,A)=0\) then every formal deformation of the weighted Rota-Baxter AssDer pair \((\mathfrak{A},\delta,R)\) is trivial.
The authors would like to thank the referee for valuable comments and suggestions on this article.