Deformations and Extensions of Modified λ-Differential 3-Lie Algebras

Abstract

In this paper, we propose the representation and cohomology of modified $\lambda$-differential 3-Lie algebras. As their applications, the linear deformations, abelian extensions and $T^{\ast }$-extensions of modified $\lambda$-differential 3-Lie algebras are also studied.

1. Introduction

3-Lie algebra plays an important role in string theory, and it is also used to study supersymmetry and gauge symmetry transformation of the world-volume theory of multiple coincident M2-branes [1,2]. The concept of 3-Lie algebra, general n-Lie algebra, was first introduced by Filippov [3] and can be regarded as a generalization of Lie algebra to higher-order algebra. 3-Lie algebra has attracted the attention of scholars from mathematics and physics [4,5,6]. Representation theory, cohomology theory, deformations, Nijenhuis operators and extension theory of n-Lie algebras have been widely studied by scholars [7,8,9,10,11,12,13,14,15,16,17].

Derivations play important roles in the study of homotopy Lie algebras [18], differential Galois theory [19], control theory and gauge theories of quantum field theory [20]. Recently, authors have studied algebras with derivations in [21,22] from the operadic point of view. In [23], Tang et al. investigated the deformation and extension of Lie algebras with derivations from the cohomological point of view. The results of [23] have been extended to 3-Lie algebras with derivations [24,25]. More research on algebraic structures with derivations has been developed; see [26,27,28,29,30,31] and references cited therein.

In recent years, scholars have increasingly focused on structures with arbitrary weights, thanks to the important work of [32,33,34,35,36,37]. The papers [38,39,40] established the cohomology, extensions and deformations of Rota–Baxter 3-Lie algebras with any weight $\lambda$, as well as the differential 3-Lie algebras with any weight $\lambda$. Additionally, the cohomology and deformation of modified Rota–Baxter algebras were studied by Das [41]. The works [42,43] provided insights into the cohomology and deformation of modified Rota–Baxter Leibniz algebras with weight $\lambda$. Furthermore, Peng et al. [44] introduced the concept of modified $\lambda$-differential Lie algebras.

Motivated by the mentioned work on the modified $\lambda$-differential operator on Lie algebras and considering the importance of 3-Lie algebra, representation and cohomology, in this paper, our main objective is to study modified $\lambda$-differential 3-Lie algebras. We develop a cohomology theory of modified $\lambda$-differential 3-Lie algebras that controls the deformation and extensions of modified $\lambda$-differential 3-Lie algebras.

The paper is organized as follows. In Section 2, we introduce the representation and cohomology of modified $\lambda$-differential 3-Lie algebras (MD${}_{\lambda }$3-LieAs). In Section 3, we consider linear deformations of MD${}_{\lambda }$3-LieAs. In Section 4, we study abelian extensions of MD${}_{\lambda }$3-LieAs. In Section 5, we study $T^{\ast }$-extensions of MD${}_{\lambda }$3-LieAs.

Throughout this paper, $\mathbf{k}$ denotes a field of characteristic zero. All the vector spaces and linear maps are taken over $\mathbf{k}$.

2. Representations and Cohomologies of MD${}_{\mathbf{\lambda }}$3-LieAs

In this section, we introduce the concept of a MD${}_{\lambda }$3-LieA and present some examples. Then, we give representations and cohomologies of MD${}_{\lambda }$3-LieAs.

Definition 1 

([3]). (i) A 3-Lie algebra is a tuple $(\mathfrak{A},[-,-,-\left]\right)$ in which $\mathfrak{A}$ is a vector space together with a skew-symmetric ternary operation $[-,-,-]:{\wedge }^3\mathfrak{A}\to \mathfrak{A}$ such that

$$\begin{array}{cc}\hfill & [a_1,a_2,[a_3,a_4,a_5]]=[[a_1,a_2,a_3],a_4,a_5]+[a_3,[a_1,a_2,a_4],a_5]+[a_3,a_3,[a_1,a_2,a_5]],\hfill \end{array}$$

(1)

for all $a_1,a_2,a_3,a_4,a_5\in \mathfrak{A}$.

(ii) 

A homomorphism between two 3-Lie algebras $({\mathfrak{A}}_1,{[-,-,-]}_1)$ and $({\mathfrak{A}}_2,{[-,-,-]}_2)$ is a linear map $\eta :{\mathfrak{A}}_1\to {\mathfrak{A}}_2$ satisfying $\eta \left({[a_1,a_2,a_3]}_1\right)={[\eta \left(a_1\right),\eta \left(a_2\right),\eta \left(a_3\right)]}_2,\forall a_1,a_2,a_3\in {\mathfrak{A}}_1.$

Definition 2. 

(i) Let $\lambda \in \mathbf{k}$ and $(\mathfrak{A},[-,-,-\left]\right)$ be a 3-Lie algebra. A modified λ-differential operator on 3-Lie algebra $\mathfrak{A}$ is a linear map $\mathrm{d}:\mathfrak{A}\to \mathfrak{A}$, such that

$$\begin{array}{c}\hfill \mathrm{d}[a_1,a_2,a_3]=[\mathrm{d}\left(a_1\right),a_2,a_3]+[a_1,\mathrm{d}\left(a_2\right),a_3]+[a_1,a_2,\mathrm{d}\left(a_3\right)]+\lambda [a_1,a_2,a_3],\forall a_1,a_2,a_3\in \mathfrak{A}.\end{array}$$

(2)

(ii) 

A modified λ-differential 3-Lie algebra (MD${}_{\lambda }$3-LieA) is a triple $(\mathfrak{A},[-,-,-],\mathrm{d})$ consisting of a 3-Lie algebra $(\mathfrak{A},[-,-,-\left]\right)$ and a modified λ-differential operator $\mathrm{d}$.

(iii) 

A homomorphism between two MD${}_{\lambda }$3-LieAs $({\mathfrak{A}}_1,{[-,-,-]}_1,{\mathrm{d}}_1)$ and $({\mathfrak{A}}_2,{[-,-,-]}_2,{\mathrm{d}}_2)$ is a 3-Lie algebra homomorphism $\eta :({\mathfrak{A}}_1,{[-,-,-]}_1)\to ({\mathfrak{A}}_2,{[-,-,-]}_2)$ such that $\eta \circ {\mathrm{d}}_1={\mathrm{d}}_2\circ \eta$. Furthermore, if η is nondegenerate, then η is called an isomorphism from ${\mathfrak{A}}_1$ to ${\mathfrak{A}}_2$.

Let $(\mathfrak{A},[-,-,-\left]\right)$ be a 3-Lie algebra; then, the elements in ${\wedge }^2\mathfrak{A}$ are called fundamental objects of the 3-Lie algebra $(\mathfrak{A},[-,-,-\left]\right)$. There is a bilinear operation $[-,-]$ on ${\wedge }^2\mathfrak{A}$, which is given by

$$\begin{array}{ccc}& & {[A,B]}_{\mathcal{F}}=[a_1,a_2,b_1]\wedge b_2+b_1\wedge [a_1,a_2,b_2],\forall A=a_1\wedge a_2,B=b_1\wedge b_2\in {\wedge }^2\mathfrak{A}.\hfill \end{array}$$

It is shown in [45] that $({\wedge }^2\mathfrak{A},{[-,-]}_{\mathcal{F}})$ is a Leibniz algebra. Furthermore, by direct calculation, we have the following result.

Proposition 1. 

Let $(\mathfrak{A},[-,-,-],\mathrm{d})$ be a MD${}_{\lambda }$3-LieA. Then, $({\wedge }^2\mathfrak{A},{[-,-]}_{\mathcal{F}},{\mathrm{d}}_{\mathcal{F}})$ is a Leibniz algebra with a derivation, where ${\mathrm{d}}_{\mathcal{F}}(a_1\wedge a_2)=\mathrm{d}\left(a_1\right)\wedge a_2+a_1\wedge \mathrm{d}\left(a_2\right)+\lambda a_1\wedge a_2$, for all $a_1\wedge a_2\in {\wedge }^2\mathfrak{A}$. See [26] for more details about Leibniz algebras with derivations.

Remark 1. 

Let $\mathrm{d}$ be a modified λ-differential operator on a 3-Lie algebra $(\mathfrak{A},[-,-,-\left]\right)$. If $\lambda =0,$ then $\mathrm{d}$ is a derivation on the 3-Lie algebra $\mathfrak{A}$. See [46] for various derivations of 3-Lie algebras.

Example 1. 

Let $(\mathfrak{A},[-,-,-\left]\right)$ be a 3-Lie algebra. Then, a linear map $\mathrm{d}:\mathfrak{A}\to \mathfrak{A}$ is a modified λ-differential operator if and only if $\mathrm{d}+\frac{\lambda }{2}{\mathrm{id}}_{\mathfrak{A}}$ is a derivation on the 3-Lie algebra $\mathfrak{A}$.

Example 2. 

Let $(\mathfrak{A},[-,-,-],\mathrm{d})$ be a MD${}_{\lambda }$3-LieA. Then, for $k\in \mathbf{k}$, $(\mathfrak{A},[-,-,-],k\mathrm{d})$ is a MD${}_{\left(k\lambda \right)}$3-LieA.

Definition 3. 

(i) (see [8]) A representation of a 3-Lie algebra $(\mathfrak{A},[-,-,-\left]\right)$ on a vector space $\mathfrak{M}$ is a skew-symmetric bilinear map $\rho :\mathfrak{A}\wedge \mathfrak{A}\to \mathrm{End}\left(\mathfrak{M}\right)$, such that

$$\begin{array}{cc}\hfill & \rho ([a_1,a_2,a_3],a_4)=\rho (a_2,a_3)\rho (a_1,a_4)+\rho (a_3,a_1)\rho (a_2,a_4)+\rho (a_1,a_2)\rho (a_3,a_4),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & \rho (a_1,a_2)\rho (a_3,a_4)=\rho (a_3,a_4)\rho (a_1,a_2)+\rho ([a_1,a_2,a_3],a_4)+\rho (a_3,[a_1,a_2,a_4]),\hfill \end{array}$$

(4)

for all $a_1,a_2,a_3,a_4\in \mathfrak{A}$. We also denote a representation of $\mathfrak{A}$ on $\mathfrak{M}$ by $(\mathfrak{M};\rho )$.

(ii) A representation of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ is a triple $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$, where $(\mathfrak{M};\rho )$ is a representation of the 3-Lie algebra $(\mathfrak{A},[-,-,-\left]\right)$ and ${\mathrm{d}}_{\mathfrak{M}}$ is a linear operator on $\mathfrak{M}$, satisfying the following equation

$$\begin{array}{c}\hfill {\mathrm{d}}_{\mathfrak{M}}\left(\rho (a,b)v\right)=\rho (\mathrm{d}\left(a\right),b)v+\rho (a,\mathrm{d}\left(b\right))v+\rho (a,b){\mathrm{d}}_{\mathfrak{M}}\left(v\right)+\lambda \rho (a,b)v,\end{array}$$

(5)

for any $a,b\in \mathfrak{A}$ and $v\in \mathfrak{M}.$

Remark 2. 

Let $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ be a representation of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$. If $\lambda =0,$ then $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ is a representation of the 3-Lie algebra with a derivation $(\mathfrak{A},[-,-,-],\mathrm{d})$. See [24,25,29] for more details about 3-Lie algebras with derivations.

Example 3. 

Let $(\mathfrak{M};\rho )$ be a representation of a 3-Lie algebra $(\mathfrak{A},[-,-,-\left]\right)$. Then, $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ is a representation of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ if and only if $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}}+\frac{\lambda }{2}{\mathrm{id}}_{\mathfrak{M}})$ is a representation of the 3-Lie algebra with a derivation $(\mathfrak{A},[-,-,-],\mathrm{d}+\frac{\lambda }{2}{\mathrm{id}}_{\mathfrak{A}})$.

Example 4. 

Let $(\mathfrak{M};\rho )$ be a representation of a 3-Lie algebra $(\mathfrak{A},[-,-,-\left]\right)$. Then, for $k\in \mathbf{k}$, $(\mathfrak{M};\rho ,{\mathrm{id}}_{\mathfrak{M}})$ is a representation of the MD${}_{(-2k)}$3-LieA $(\mathfrak{A},[-,-,-],k{\mathrm{id}}_{\mathfrak{A}})$.

Example 5. 

Let $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ be a representation of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$. Then, for $k\in \mathbf{k}$, $(\mathfrak{M};\rho ,k{\mathrm{d}}_{\mathfrak{M}})$ is a representation of the MD${}_{\left(k\lambda \right)}$3-LieA $(\mathfrak{A},[-,-,-],k\mathrm{d})$.

Proposition 2. 

Let $(\mathfrak{A},[-,-,-\left]\right)$ be a 3-Lie algebra, and $(\mathfrak{M};\rho )$ be a representation of it. Then, $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ is a representation of MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ if and only if $\mathfrak{A}\oplus \mathfrak{M}$ is a MD${}_{\lambda }$3-LieA under the following maps:

$$\begin{array}{cc}\hfill {[a_1+u_1,a_2+u_2,a_3+u_3]}_{\rho }:=& [a_1,a_2,a_3]+\rho (a_1,a_2)u_3+\rho (a_3,a_1)u_2+\rho (a_2,a_3)u_1,\hfill \\ \hfill \mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}}(a_1+u_1):=& \mathrm{d}\left(a_1\right)+{\mathrm{d}}_{\mathfrak{M}}\left(u_1\right),\hfill \end{array}$$

for all $a_1,a_2,a_3\in \mathfrak{A}$ and $u_1,u_2,u_3\in \mathfrak{M}$.

Proof. 

Assume that $(\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho },\mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}})$ is a MD${}_{\lambda }$3-LieA, for any $a_1,a_2\in \mathfrak{A}$ and $u_3\in \mathfrak{M}$; we have

$$\begin{array}{cc}\hfill & \mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}}{[a_1+0,a_2+0,0+u_3]}_{\rho }\hfill \\ \hfill =& {[\mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}}(a_1+0),a_2+0,0+u_3]}_{\rho }+{[a_1+0,\mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}}(a_2+0),0+u_3]}_{\rho }\hfill \\ \hfill & +{[a_1+0,a_2+0,\mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}}(0+u_3)]}_{\rho }+\lambda {[a_1+0,a_2+0,0+u_3]}_{\rho },\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill {\mathrm{d}}_{\mathfrak{M}}\left(\rho (a_1,a_2)u_3\right)=\rho (\mathrm{d}\left(a_1\right),a_2)u_3+\rho (a_1,\mathrm{d}\left(a_2\right))u_3+\rho (a_1,a_2){\mathrm{d}}_{\mathfrak{M}}\left(u_3\right)+\lambda \rho (a_1,a_2)u_3.\end{array}$$

Therefore, $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ is a representation of $(\mathfrak{A},[-,-,-],\mathrm{d})$.

The converse can be proved similarly. □

Let $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ be a representation of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ and ${\mathfrak{M}}^{\ast }:=\mathrm{Hom}(\mathfrak{M},\mathbf{k})$ be a dual space of $\mathfrak{M}$. We define a bilinear map ${\rho }^{\ast }:{\wedge }^2\mathfrak{A}\to \mathrm{End}\left({\mathfrak{M}}^{\ast }\right)$ and a linear map ${\mathrm{d}}_{\mathfrak{M}}^{\ast }:{\mathfrak{M}}^{\ast }\to {\mathfrak{M}}^{\ast }$ respectively by

$$\begin{array}{ccc}\hfill & \hfill & \hfill \langle {\rho }^{\ast }(a_1,a_2)u^{\ast },v\rangle =-\langle u^{\ast },\rho (a_1,a_2)v\rangle \mathrm{and}\langle {\mathrm{d}}_{\mathfrak{M}}^{\ast }{u}^{\ast },v\rangle =\langle u^{\ast },{\mathrm{d}}_{\mathfrak{M}}\left(v\right)\rangle ,\end{array}$$

(6)

for any $a_1,a_2\in \mathfrak{A},v\in \mathfrak{M}$ and $u^{\ast }\in {\mathfrak{M}}^{\ast }.$

Proposition 3. 

With the above notations, $({\mathfrak{M}}^{\ast };{\rho }^{\ast },-{\mathrm{d}}_{\mathfrak{M}}^{\ast })$ is a representation of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$.

Proof. 

First, It has been proved that [47] $({\mathfrak{M}}^{\ast };{\rho }^{\ast })$ is a representation of the 3-Lie algebra $(\mathfrak{A},[-,-,-\left]\right)$. Furthermore, for any $a_1,a_2\in \mathfrak{A},v\in \mathfrak{M}$ and $u^{\ast }\in {\mathfrak{M}}^{\ast },$ by Equations (5) and (6), we have

$$\begin{array}{cc}\hfill & \langle {\rho }^{\ast }(\mathrm{d}\left(a_1\right),a_2)u^{\ast },v\rangle +\langle {\rho }^{\ast }(a_1,\mathrm{d}\left(a_2\right))u^{\ast },v\rangle +\langle {\rho }^{\ast }(a_1,a_2)(-{\mathrm{d}}_{\mathfrak{M}}^{\ast })u^{\ast },v\rangle +\langle \lambda {\rho }^{\ast }(a_1,a_2)u^{\ast },v\rangle \hfill \\ \hfill & -\langle (-{\mathrm{d}}_{\mathfrak{M}}^{\ast }){\rho }^{\ast }(a_1,a_2)u^{\ast },v\rangle \hfill \\ \hfill =& -\langle u^{\ast },\rho (\mathrm{d}\left(a_1\right),a_2)v\rangle -\langle u^{\ast },\rho (a_1,\mathrm{d}\left(a_2\right))v\rangle -\langle (-{\mathrm{d}}_{\mathfrak{M}}^{\ast })u^{\ast },\rho (a_1,a_2)v\rangle -\langle u^{\ast },\lambda \rho (a_1,a_2)v\rangle \hfill \\ \hfill & +\langle {\rho }^{\ast }(a_1,a_2)u^{\ast },{\mathrm{d}}_{\mathfrak{M}}\left(v\right)\rangle \hfill \\ \hfill =& -\langle u^{\ast },\rho (\mathrm{d}\left(a_1\right),a_2)v\rangle -\langle u^{\ast },\rho (a_1,\mathrm{d}\left(a_2\right))v\rangle +\langle u^{\ast },{\mathrm{d}}_{\mathfrak{M}}\left(\rho (a_1,a_2)v\right)\rangle -\langle u^{\ast },\lambda \rho (a_1,a_2)v\rangle \hfill \\ \hfill & -\langle u^{\ast },\rho (a_1,a_2){\mathrm{d}}_{\mathfrak{M}}\left(v\right)\rangle \hfill \\ \hfill =& -\langle u^{\ast },\rho (\mathrm{d}\left(a_1\right),a_2)v+\rho (a_1,\mathrm{d}\left(a_2\right))v-{\mathrm{d}}_{\mathfrak{M}}\left(\rho (a_1,a_2)v\right)+\lambda \rho (a_1,a_2)v+\rho (a_1,a_2){\mathrm{d}}_{\mathfrak{M}}\left(v\right)\rangle \hfill \\ \hfill =& 0,\hfill \end{array}$$

which implies that ${\rho }^{\ast }(\mathrm{d}\left(a_1\right),a_2)u^{\ast }+{\rho }^{\ast }(a_1,\mathrm{d}\left(a_2\right))u^{\ast }+{\rho }^{\ast }(a_1,a_2)(-{\mathrm{d}}_{\mathfrak{M}}^{\ast })u^{\ast }+\lambda {\rho }^{\ast }(a_1,a_2)u^{\ast }-(-{\mathrm{d}}_{\mathfrak{M}}^{\ast }){\rho }^{\ast }(a_1,a_2)u^{\ast }=0$. So, we obtain the result. □

Example 6. 

Let $(\mathfrak{A},[-,-,-],\mathrm{d})$ be a MD${}_{\lambda }$3-LieA and define $ad:\mathfrak{A}\wedge \mathfrak{A}\to \mathrm{End}\left(\mathfrak{A}\right)$ by $ad(a_1,a_2)\left(a\right)=[a_1,a_2,a],\forall a_1,a_2,a\in \mathfrak{A}$. Then, $(\mathfrak{A};ad,\mathrm{d})$ is a representation of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$, which is called the adjoint representation of $(\mathfrak{A},[-,-,-],\mathrm{d})$. Furthermore, $({\mathfrak{A}}^{\ast };a{d}^{\ast },-{\mathrm{d}}^{\ast })$ is a dual adjoint representation of $(\mathfrak{A},[-,-,-],\mathrm{d})$, which is called the coadjoint representation of $(\mathfrak{A},[-,-,-],\mathrm{d})$.

Next, we will study the cohomology of a MD${}_{\lambda }$3-LieA with coefficients in its representation.

Recall from [14] that let $(\mathfrak{M};\rho )$ be a representation of a 3-Lie algebra $(\mathfrak{A},[-,-,-\left]\right)$. Denote the n—cochains of $\mathfrak{A}$ with coefficients in representation $(\mathfrak{M};\rho )$ by

$$\begin{array}{c}\hfill {\mathcal{C}}_{3\mathrm{Lie}}^n(\mathfrak{A},\mathfrak{M}):=\mathrm{Hom}({\left({\wedge }^2\mathfrak{A}\right)}^{\otimes n-1}\wedge \mathfrak{A},\mathfrak{M}),n\ge 1.\end{array}$$

The coboundary operator $\delta :{\mathcal{C}}_{3\mathrm{Lie}}^n(\mathfrak{A},\mathfrak{M})\to {\mathcal{C}}_{3\mathrm{Lie}}^{n+1}(\mathfrak{A},\mathfrak{M})$, for $A_i=a_i\wedge b_i\in {\wedge }^2\mathfrak{A},a_{n+1}\in \mathfrak{A}$ and $f\in {\mathcal{C}}_{3\mathrm{Lie}}^n(\mathfrak{A},\mathfrak{M})$, as

$$\begin{array}{cc}\hfill & \delta f(A_1,\cdots ,A_n,a_{n+1})\hfill \\ \hfill =& {(-1)}^{n+1}\left(\rho (b_n,a_{n+1})f(A_1,\cdots ,A_{n-1},a_n)+\rho (a_{n+1},a_n)f(A_1,\cdots ,A_{n-1},b_n)\right)\hfill \\ \hfill & +\sum _{i=1}^n{(-1)}^{i+1}\rho (a_i,b_i)f(A_1,\cdots ,A_{i-1},A_{i+1},\cdots ,A_n,a_{n+1})\hfill \\ \hfill & +\sum _{i=1}^n{(-1)}^{i}f(A_1,\cdots ,A_{i-1},A_{i+1},\cdots ,A_n,[a_i,b_i,a_{n+1}])\hfill \\ \hfill & +\sum _{1\le i<k\le n}{(-1)}^{i}f(A_1,\cdots ,A_{i-1},A_{i+1},\cdots ,A_{k-1},[a_i,b_i,a_k]\wedge b_k+a_k\wedge [a_i,b_i,b_k],\cdots ,A_n,a_{n+1}),\hfill \end{array}$$

it was proved that $\delta \circ \delta =0.$

Lemma 1. 

Let $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ be a representation of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$. For any $n\ge 1$, we define a linear map $\mathrm{\Phi }:{\mathcal{C}}_{3\mathrm{Lie}}^n(\mathfrak{A},\mathfrak{M})\to {\mathcal{C}}_{3\mathrm{Lie}}^n(\mathfrak{A},\mathfrak{M})$ by

$$\begin{array}{cc}\hfill \mathrm{\Phi }f(A_1,\cdots ,A_{n-1},a_n)=& \sum _{i=1}^{n-1}f(A_1,\cdots ,A_{i-1},\mathrm{d}\left(a_i\right)\wedge b_i+a_i\wedge \mathrm{d}\left(b_i\right),A_{i+1},\cdots ,A_{n-1},a_n)\hfill \\ \hfill & +f(A_1,\cdots ,A_{n-1},\mathrm{d}\left(a_n\right))+(n-1)\lambda f(A_1,\cdots ,A_{n-1},a_n)\hfill \\ \hfill & -{\mathrm{d}}_{\mathfrak{M}}\left(f(A_1,\cdots ,A_{n-1},a_n)\right),\hfill \end{array}$$

for any $f\in {\mathcal{C}}_{3\mathrm{Lie}}^n(\mathfrak{A},\mathfrak{M})$ and $A_i=a_i\wedge b_i\in {\wedge }^2\mathfrak{A},i=1,\cdots ,n-1,a_n\in \mathfrak{A}$. Then, Φ is a cochain map; i.e., $\mathrm{\Phi }\circ \delta =\delta \circ \mathrm{\Phi }.$

Proof. 

It follows by a straightforward tedious calculations. □

Let $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ be a representation of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$; we define n-cochains for MD${}_{\lambda }$3-LieA as follows:

$${\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^n(\mathfrak{A},\mathfrak{M}):=\left\{\begin{array}{cc}{\mathcal{C}}_{3\mathrm{Lie}}^n(\mathfrak{A},\mathfrak{M})\oplus {\mathcal{C}}_{3\mathrm{Lie}}^{n-1}(\mathfrak{A},\mathfrak{M}),\hfill & n\ge 2,\hfill \\ \mathrm{Hom}(\mathfrak{A},\mathfrak{M}),\hfill & n=1.\hfill \end{array}\right.$$

We define a linear map $\partial :{\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^n(\mathfrak{A},\mathfrak{M})\to {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^{n+1}(\mathfrak{A},\mathfrak{M})$ by

$$\begin{array}{cc}\hfill \partial \left(f\right)& =(\delta f,-\mathrm{\Phi }f),\mathrm{if}f\in {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^1(\mathfrak{A},\mathfrak{M});\hfill \\ \hfill \partial (f,g)& =(\delta f,\delta g+{(-1)}^n\mathrm{\Phi }f),\mathrm{if}(f,g)\in {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^n(\mathfrak{A},\mathfrak{M}),n\ge 2.\hfill \end{array}$$

In view of Lemma 1, we have the following theorem.

Theorem 1. 

The linear map ∂ is a coboundary operator; that is, $\partial \circ \partial =0.$

Therefore, we obtain a cochain complex $({\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^{\ast }(\mathfrak{A},\mathfrak{M}),\partial )$, for $n\ge 2$, and we denote the set of n-cocycles by ${\mathcal{Z}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^n(\mathfrak{A},\mathfrak{M})=\{(f,g)\in {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^n(\mathfrak{A},\mathfrak{M})|$ $\partial (f,g)=0\}$, the set of n-coboundaries by ${\mathcal{B}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^n(\mathfrak{A},\mathfrak{M})=\{\partial (f,g)|(f,g)\in {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^{n-1}(\mathfrak{A},\mathfrak{M})\}$ and the n-th cohomology group of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ with coefficients in the representation $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ by ${\mathcal{H}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^n(\mathfrak{A},\mathfrak{M})={\mathcal{Z}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^n(\mathfrak{A},\mathfrak{M})/{\mathcal{B}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^n(\mathfrak{A},\mathfrak{M})$.

Lastly, we calculate the 1-cocycle and 2-cocycle.

For $f\in {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^1(\mathfrak{A},\mathfrak{M})$, f is a 1-cocycle if $\partial \left(f\right)=(\delta f,-\mathrm{\Phi }f)=0,$ i.e.,

$$\rho (b_1,a_2)f\left(a_1\right)+\rho (a_2,a_1)f\left(b_1\right)+\rho (a_1,b_1)f\left(a_2\right)-f\left([a_1,b_1,a_2]\right)=0$$

and

$${\mathrm{d}}_{\mathfrak{M}}\left(f\left(a_1\right)\right)-f\left(\mathrm{d}\left(a_1\right)\right)=0.$$

For $(f,g)\in {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{M})$, $(f,g)$ is a 2-cocycle if $\partial (f,g)=(\delta f,\delta g+\mathrm{\Phi }f)=0,$ i.e.,

$$\begin{array}{cc}\hfill & -\rho (b_2,a_3)f(a_1,b_1,a_2)-\rho (a_3,a_2)f(a_1,b_1,b_2)+\rho (a_1,b_1)f(a_2,b_2,a_3)-\rho (a_2,b_2)f(a_1,b_1,a_3)\hfill \\ \hfill & -f(a_2,b_2,[a_1,b_1,a_3])+f(a_1,b_1,[a_2,b_2,a_3])-f([a_1,b_1,a_2],b_2,a_3)-f(a_2,[a_1,b_1,b_2],a_3)=0\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \rho (b_1,a_2)f\left(a_1\right)+\rho (a_2,a_1)f\left(b_1\right)+\rho (a_1,b_1)f\left(a_2\right)-f\left([a_1,b_1,a_2]\right)\hfill \\ \hfill & +f(\mathrm{d}\left(a_1\right),b_1,a_2)+f(a_1,\mathrm{d}\left(b_1\right),a_2)+f(a_1,b_1,d\left(a_2\right))+\lambda f(a_1,b_1,a_2)-{\mathrm{d}}_{\mathfrak{M}}\left(f(a_1,b_1,a_2)\right)=0.\hfill \end{array}$$

3. Linear Deformations of MD${}_{\mathbf{\lambda }}$3-LieAs

In this section, we study linear deformations of MD${}_{\lambda }$3-LieAs.

Let $(\mathfrak{A},[-,-,-],\mathrm{d})$ be a MD${}_{\lambda }$3-LieA. Denote ${\nu }_0=[-,-,-]$ and ${\mathrm{d}}_0=\mathrm{d}$. Consider a family of linear maps:

$${\nu }_t={\nu }_0+t{\nu }_1+t^2{\nu }_2,{\nu }_1,{\nu }_2\in {\mathcal{C}}_{3\mathrm{Lie}}^2(\mathfrak{A},\mathfrak{A}),{\mathrm{d}}_t={\mathrm{d}}_0+t{\mathrm{d}}_1,{\mathrm{d}}_1\in {\mathcal{C}}_{3\mathrm{Lie}}^1(\mathfrak{A},\mathfrak{A}).$$

Definition 4. 

A linear deformation of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ is a pair $({\nu }_t,{\mathrm{d}}_t)$ which endows $(\mathfrak{A}\left[\left[t\right]\right],{\nu }_t,{\mathrm{d}}_t)$ with the MD${}_{\lambda }$3-LieA.

Proposition 4. 

The pair $({\nu }_t,{\mathrm{d}}_t)$ generates a linear deformation of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ if and only if the following equations hold:

$$\begin{array}{cc}\hfill \sum _{i+j=n}{\nu }_i(a_1,a_2,{\nu }_j(a_3,a_4,a_5))=& \sum _{i+j=n}{\nu }_i({\nu }_j(a_1,a_2,a_3),a_4,a_5)+\sum _{i+j=n}{\nu }_i(a_3,{\nu }_j(a_1,a_2,a_4),a_5)\hfill \\ \hfill & +\sum _{i+j=n}{\nu }_i(a_3,a_4,{\nu }_j(a_1,a_2,a_5)),\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i+l=n}}{\mathrm{d}}_l\left({\nu }_i(a_1,a_2,a_3)\right)=& {\displaystyle \sum _{i+l=n}}{\nu }_i({\mathrm{d}}_l\left(a_1\right),a_2,a_3)+{\displaystyle \sum _{i+l=n}}{\nu }_i(a_1,{\mathrm{d}}_l\left(a_2\right),a_3)\hfill \\ \hfill & +{\displaystyle \sum _{i+l=n}}{\nu }_i(a_1,a_2,{\mathrm{d}}_l\left(a_3\right))+\lambda {\nu }_n(a_1,a_2,a_3),\hfill \end{array}$$

(8)

for any $a_1,a_2,a_3,a_4,a_5\in \mathfrak{A}$ and $i,j=0,1,2,l=0,1$.

Proof. 

$(\mathfrak{A}\left[\left[t\right]\right],{\nu }_t,{\mathrm{d}}_t)$ is a MD${}_{\lambda }$3-LieA if and only if

$$\begin{array}{cc}\hfill & {\nu }_t(a_1,a_2,{\nu }_t(a_3,a_4,a_5))\hfill \\ \hfill & ={\nu }_t({\nu }_t(a_1,a_2,a_3),a_4,a_5)+{\nu }_t(a_3,{\nu }_t(a_1,a_2,a_4),a_5)+{\nu }_t(a_3,a_4,{\nu }_t(a_1,a_2,a_5)),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\mathrm{d}}_t\left({\nu }_t(a_1,a_2,a_3)\right)\hfill \\ \hfill & ={\nu }_t({\mathrm{d}}_t\left(a_1\right),a_2,a_3)+{\nu }_t(a_1,{\mathrm{d}}_t\left(a_2\right),a_3)+{\nu }_t(a_1,a_2,{\mathrm{d}}_t\left(a_3\right))+\lambda {\nu }_t(a_1,a_2,a_3).\hfill \end{array}$$

(10)

Comparing the coefficients of $t^n$ on both sides of the above equations, Equations (9) and (10) are equivalent to Equations (7) and (8), respectively. □

Corollary 1. 

Let $(\mathfrak{A}\left[\left[t\right]\right],{\nu }_t,{\mathrm{d}}_t)$ be a linear deformation of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},{\nu }_0,\mathrm{d})$. Then, $({\nu }_1,{\mathrm{d}}_1)$ is a 2-cocycle of $(\mathfrak{A},[-,-,-],\mathrm{d})$ with the coefficient in the adjoint representation $(\mathfrak{A};ad,\mathrm{d})$.

Proof. 

For $n=1$, Equations (7) and (8) are equivalent to

$$\begin{array}{cc}\hfill & {\nu }_1(a_1,a_2,[a_3,a_4,a_5])+[a_1,a_2,{\nu }_1(a_3,a_4,a_5)]\hfill \\ \hfill =& {\nu }_1([a_1,a_2,a_3],a_4,a_5)+[{\nu }_1(a_1,a_2,a_3),a_4,a_5]+{\nu }_1(a_3,[a_1,a_2,a_4],a_5)+[a_3,{\nu }_1(a_1,a_2,a_4),a_5]\hfill \\ \hfill & +{\nu }_1(a_3,a_4,[a_1,a_2,a_5])+[a_3,a_4,{\nu }_1(a_1,a_2,a_5)],\hfill \\ \hfill & {\mathrm{d}}_1\left([a_1,a_2,a_3]\right)+\mathrm{d}\left({\nu }_1(a_1,a_2,a_3)\right)\hfill \\ \hfill =& [{\mathrm{d}}_1\left(a_1\right),a_2,a_3]+{\nu }_1(\mathrm{d}\left(a_1\right),a_2,a_3)+[a_1,{\mathrm{d}}_1\left(a_2\right),a_3]+{\nu }_1(a_1,\mathrm{d}\left(a_2\right),a_3)+[a_1,a_2,{\mathrm{d}}_1\left(a_3\right)]\hfill \\ \hfill & +{\nu }_1(a_1,a_2,\mathrm{d}\left(a_3\right))+\lambda {\nu }_1(a_1,a_2,a_3),\hfill \end{array}$$

which imply that $\delta {\nu }_1=0,\delta {\mathrm{d}}_1+\mathrm{\Phi }{\nu }_1=0$, respectively. Hence, $({\nu }_1,{\mathrm{d}}_1)$ is a 2-cocycle of $(\mathfrak{A},[-,-,-],\mathrm{d})$ with the coefficient in the adjoint representation $(\mathfrak{A};ad,\mathrm{d})$. □

Definition 5. 

The 2-cocycle $({\nu }_1,{\mathrm{d}}_1)$ is called the infinitesimal of the linear deformation $(\mathfrak{A}\left[\left[t\right]\right],{\nu }_t,{\mathrm{d}}_t)$ of $(\mathfrak{A},[-,-,-],\mathrm{d})$.

Definition 6. 

(i) Two linear deformations $(\mathfrak{A}\left[\left[t\right]\right],{\nu }_t,{\mathrm{d}}_t)$ and $(\mathfrak{A}\left[\left[t\right]\right],{\nu }_t^{\prime },{\mathrm{d}}_t^{\prime })$ of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ are said to be equivalent if there exists a linear map $N:\mathfrak{A}\to \mathfrak{A}$, such that $N_t={\mathrm{id}}_{\mathfrak{A}}+tN$ satisfying

$$\begin{array}{cc}\hfill N_t\left({\mathrm{d}}_t\left(a_1\right)\right)=& {\mathrm{d}}_t^{\prime }\left(N_t\left(a_1\right)\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill N_t{\nu }_t(a_1,a_2,a_3)=& {\nu }_t^{\prime }(N_t\left(a_1\right),N_t\left(a_2\right),N_t\left(a_3\right)),\hfill \end{array}$$

(12)

for any $a_1,a_2,a_3\in \mathfrak{A}$.

(ii) A linear deformation $(\mathfrak{A}\left[\left[t\right]\right],{\nu }_t,{\mathrm{d}}_t)$ of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ is said to be trivial if $(\mathfrak{A}\left[\left[t\right]\right],{\nu }_t,{\mathrm{d}}_t)$ is equivalent to $(\mathfrak{A},[-,-,-],\mathrm{d})$.

Comparing the coefficients of t on both sides of the above Equations (11) and (12), we have

$$\begin{array}{cc}\hfill {\nu }_1(a_1,a_2,a_3)-{\nu }_1^{\prime }(a_1,a_2,a_3)& =[N{a}_1,a_2,a_3]+[a_1,N{a}_2,a_3]+[a_1,a_2,N{a}_3]-N[a_1,a_2,a_3],\hfill \\ \hfill {\mathrm{d}}_1\left(a\right)-{\mathrm{d}}_1^{\prime }\left(a\right)& =\mathrm{d}\left(N_{1}a\right)-N_1\mathrm{d}\left(a\right).\hfill \end{array}$$

Thus, we have the following theorem.

Theorem 2. 

The infinitesimals of two equivalent linear deformations of $(\mathfrak{A},[-,-,-],\mathrm{d})$ are in the same cohomological class in ${\mathcal{H}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{A})$.

Let $(\mathfrak{A}\left[\left[t\right]\right],{\nu }_t,{\mathrm{d}}_t)$ be a trivial deformation of $(\mathfrak{A},[-,-,-],\mathrm{d})$. Then, there exists a linear map $N:\mathfrak{A}\to \mathfrak{A}$, such that $N_t={\mathrm{id}}_{\mathfrak{A}}+tN$ satisfying

$$\begin{array}{cc}\hfill N_t\left({\mathrm{d}}_t\left(a_1\right)\right)=& \mathrm{d}\left(N_t\left(a_1\right)\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill N_t{\nu }_t(a_1,a_2,a_3)=& [N_t\left(a_1\right),N_t\left(a_2\right),N_t\left(a_3\right)].\hfill \end{array}$$

(14)

Compare the coefficients of $t^i(1\le i\le 3)$ on both sides of Equations (13) and (14), and we can obtain

$$\begin{array}{cc}\hfill N\mathrm{d}\left(a_1\right)=& \mathrm{d}\left(N{a}_1\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\nu }_1(a_1,a_2,a_3)+N[a_1,a_2,a_3]=& [N{a}_1,a_2,a_3]+[a_1,N{a}_2,a_3]+[a_1,a_2,N{a}_3],\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\nu }_2(a_1,a_2,a_3)+N{\nu }_1(a_1,a_2,a_3)=& [N{a}_1,N{a}_2,a_3]+[a_1,N{a}_2,N{a}_3]+[N{a}_1,a_2,N{a}_3],\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill N{\nu }_2(a_1,a_2,a_3)=& [N{a}_1,N{a}_2,N{a}_3].\hfill \end{array}$$

(18)

Thus, from a trivial deformation, we can obtain the following definition of Nijenhuis operator.

Definition 7. 

Let $(\mathfrak{A},[-,-,-],\mathrm{d})$ be a MD${}_{\lambda }$3-LieA. A linear map $N:\mathfrak{A}\to \mathfrak{A}$ is called a Nijenhuis operator if the following equations hold:

$$\begin{array}{cc}\hfill N\circ \mathrm{d}=& \mathrm{d}\circ N,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill [N{a}_1,N{a}_2,N{a}_3]=& N([a_1,N{a}_2,N{a}_3]+[N{a}_1,a_2,N{a}_3]+[N{a}_1,N{a}_2,a_3])\hfill \\ \hfill & -N^2([N{a}_1,a_2,a_3]+[a_1,N{a}_2,a_3]+[a_1,a_2,N{a}_3])+N^3[a_1,a_2,a_3],\hfill \end{array}$$

(20)

for any $a_1,a_2,a_3\in \mathfrak{A}.$

Proposition 5. 

Let $(\mathfrak{A},[-,-,-],\mathrm{d})$ be a MD${}_{\lambda }$3-LieA, and $N:\mathfrak{A}\to \mathfrak{A}$ a Nijenhuis operator. Then, $(\mathfrak{A},{[-,-,-]}_N,\mathrm{d})$ is a MD${}_{\lambda }$3-LieA, where

$$\begin{array}{cc}\hfill {[a_1,a_2,a_3]}_N=& [a_1,N{a}_2,N{a}_3]+[N{a}_1,a_2,N{a}_3]+[N{a}_1,N{a}_2,a_3]\hfill \\ \hfill & -N([N{a}_1,a_2,a_3]+[a_1,N{a}_2,a_3]+[a_1,a_2,N{a}_3])+N^2[a_1,a_2,a_3].\hfill \end{array}$$

Proof. 

In the light of [9], $(\mathfrak{A},{[-,-,-]}_N)$ is a 3-Lie algebra. Next, we prove that $\mathrm{d}$ is a modified $\lambda$-differential operator of $(\mathfrak{A},{[-,-,-]}_N)$, for any $a_1,a_2,a_3\in \mathfrak{A},$ by Equations (2) and (19), and we have

$$\begin{array}{cc}\hfill & \mathrm{d}{[a_1,a_2,a_3]}_N\hfill \\ \hfill =& \mathrm{d}[a_1,N{a}_2,N{a}_3]+\mathrm{d}[N{a}_1,a_2,N{a}_3]+\mathrm{d}[N{a}_1,N{a}_2,a_3]\hfill \\ \hfill & -N(\mathrm{d}[N{a}_1,a_2,a_3]+\mathrm{d}[a_1,N{a}_2,a_3]+\mathrm{d}[a_1,a_2,N{a}_3])+N^2\mathrm{d}[a_1,a_2,a_3]\hfill \\ \hfill =& [\mathrm{d}\left(a_1\right),N{a}_2,N{a}_3]+[a_1,N\mathrm{d}\left(a_2\right),N{a}_3]+[a_1,N{a}_2,N\mathrm{d}\left(a_3\right)]+\lambda [a_1,N{a}_2,N{a}_3]\hfill \\ \hfill & +[N\mathrm{d}\left(a_1\right),a_2,N{a}_3]+[N{a}_1,\mathrm{d}\left(a_2\right),N{a}_3]+[N{a}_1,a_2,N\mathrm{d}\left(a_3\right)]+\lambda [N{a}_1,a_2,N{a}_3]\hfill \\ \hfill & +[N\mathrm{d}\left(a_1\right),N{a}_2,a_3]+[N{a}_1,N\mathrm{d}\left(a_2\right),a_3]+[N{a}_1,N{a}_2,\mathrm{d}\left(a_3\right)]+\lambda [N{a}_1,N{a}_2,a_3]\hfill \\ \hfill & -N([N\mathrm{d}\left(a_1\right),a_2,a_3]+[N{a}_1,\mathrm{d}\left(a_2\right),a_3]+[N{a}_1,a_2,\mathrm{d}\left(a_3\right)]+\lambda [N{a}_1,a_2,a_3])\hfill \\ \hfill & -N([\mathrm{d}\left(a_1\right),N{a}_2,a_3]+[a_1,N\mathrm{d}\left(a_2\right),a_3]+[a_1,N{a}_2,\mathrm{d}\left(a_3\right)]+\lambda [a_1,N{a}_2,a_3])\hfill \\ \hfill & -N([\mathrm{d}\left(a_1\right),a_2,N{a}_3]+[a_1,\mathrm{d}\left(a_2\right),N{a}_3]+[a_1,a_2,N\mathrm{d}\left(a_3\right)]+\lambda [a_1,a_2,N{a}_3])\hfill \\ \hfill & +N^2([\mathrm{d}\left(a_1\right),a_2,a_3]+[a_1,\mathrm{d}\left(a_2\right),a_3]+[a_1,a_2,\mathrm{d}\left(a_3\right)]+\lambda [a_1,a_2,a_3])\hfill \\ \hfill =& {[\mathrm{d}\left(a_1\right),a_2,a_3]}_N+{[a_1,\mathrm{d}\left(a_2\right),a_3]}_N+{[a_1,a_2,\mathrm{d}\left(a_3\right)]}_N+\lambda {[a_1,a_2,a_3]}_N.\hfill \end{array}$$

So, we obtain the conclusion. □

Definition 8. 

A linear map $R:\mathfrak{M}\to \mathfrak{A}$ is called an $\mathcal{O}$-operator on the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],$$\mathrm{d})$ with respect to the representation $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ if the following equations hold:

$$\begin{array}{cc}\hfill R\circ {\mathrm{d}}_{\mathfrak{M}}=& \mathrm{d}\circ R,\hfill \\ \hfill [R{v}_1,R{v}_2,R{v}_3]=& R(\rho (R{v}_1,R{v}_2)v_3+\rho (R{v}_2,R{v}_3)v_1+\rho (R{v}_3,R{v}_1)v_2),\hfill \end{array}$$

for any $v_1,v_2,v_3\in \mathfrak{M}.$

Remark 3. 

Obviously, an invertible linear map $R:\mathfrak{M}\to \mathfrak{A}$ is an $\mathcal{O}$-operator if and only if $R^{-1}$ is a 1-cocycle of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ with coefficients in the representation $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$.

Proposition 6. 

Let $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ be a representation of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$. Then, $R:\mathfrak{M}\to \mathfrak{A}$ is an $\mathcal{O}$-operator if and only if $\overline{R}=\left(\begin{array}{cc}0& R\\ 0& 0\end{array}\right):\mathfrak{A}\oplus \mathfrak{M}\to \mathfrak{A}\oplus \mathfrak{M}$ is a Nijenhuis operator on semidirect product MD${}_{\lambda }$3-LieA $(\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho },\mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}})$.

Proof. 

For any $a_1,a_2,a_3\in \mathfrak{A}$ and $u_1,u_2,u_3\in \mathfrak{M}$, by ${\overline{R}}^2=0,$ we have

$$\begin{array}{cc}\hfill & (\mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}})\overline{R}(a_1+u_1)=(\mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}})(R{u}_1+0)=\mathrm{d}\left(R{u}_1\right)+0,\hfill \\ \hfill & \overline{R}(\mathrm{d}\oplus {\mathrm{d}}_{\mathfrak{M}})(a_1+u_1)=\overline{R}(\mathrm{d}\left(a_1\right)+{\mathrm{d}}_{\mathfrak{M}}\left(u_1\right))=R{\mathrm{d}}_{\mathfrak{M}}\left(u_1\right)+0,\hfill \\ \hfill & \overline{R}({[a_1+u_1,\overline{R}(a_2+u_2),\overline{R}(a_3+u_3)]}_{\rho }+{[\overline{R}(a_1+u_1),a_2+u_2,\overline{R}(a_3+u_3)]}_{\rho }\hfill \\ \hfill & +{[\overline{R}(a_1+u_1),\overline{R}(a_2+u_2),a_3+u_3]}_{\rho })-{[\overline{R}(a_1+u_1),\overline{R}(a_2+u_2),\overline{R}(a_3+u_3)]}_{\rho }\hfill \\ \hfill =& \overline{R}({[a_1+u_1,R{u}_2+0,R{u}_3+0]}_{\rho }+{[R{u}_1+0,a_2+u_2,R{u}_3+0]}_{\rho }+{[R{u}_1+0,R{u}_2+0,a_3+u_3]}_{\rho })\hfill \\ \hfill & -{[R{u}_1+0,R{u}_2+0,R{u}_3+0]}_{\rho }\hfill \\ \hfill =& \overline{R}([a_1,R{u}_2,R{u}_3]+\rho (R{u}_2,R{u}_3)u_1+[R{u}_1,a_2,R{u}_3]+\rho (R{u}_3,R{u}_1)u_2+[R{u}_1,R{u}_2{a}_3]+\rho (R{u}_1,R{u}_2)u_3)\hfill \\ \hfill & -[R{u}_1,R{u}_2,R{u}_3]+0\hfill \\ \hfill =& R(\rho (R{u}_2,R{u}_3)u_1+\rho (R{u}_3,R{u}_1)u_2+\rho (R{u}_1,R{u}_2)u_3)-[R{u}_1,R{u}_2,R{u}_3]+0,\hfill \end{array}$$

which implies that R is an $\mathcal{O}$-operator if and only if $\overline{R}$ is a Nijenhuis operator. □

4. Abelian Extensions of MD${}_{\mathbf{\lambda }}$3-LieAs

In this section, we study abelian extensions of MD${}_{\lambda }$3-LieAs and show that they are classified by the second cohomology groups.

Notice that a vector space $\mathfrak{M}$ together with a linear map ${\mathrm{d}}_{\mathfrak{M}}$ is naturally a MD${}_{\lambda }$3-LieA where the bracket on $\mathfrak{M}$ is defined to be ${[-,-,-]}_{\mathfrak{M}}=0.$

Definition 9. 

An abelian extension of $(\mathfrak{A},[-,-,-],\mathrm{d})$ by $(\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})$ is a short exact sequence of homomorphisms of MD${}_{\lambda }$3-LieAs

$$0\to (\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})\stackrel{i}{⟶}(\widehat{\mathfrak{A}},{[-,-,-]}_{\widehat{\mathfrak{A}}},\widehat{\mathrm{d}})\stackrel{p}{⟶}(\mathfrak{A},[-,-,-],\mathrm{d})\to 0,$$

i.e., there exists a commutative diagram:

$$\begin{array}{ccccccccc}0& \stackrel{}{\to }& \mathfrak{M}& \stackrel{i}{\to }& \widehat{\mathfrak{A}}& \stackrel{p}{\to }& \mathfrak{A}& \stackrel{}{\to }& 0\\ & & {\displaystyle {\mathrm{d}}_{\mathfrak{M}}}\downarrow \phantom{{\displaystyle {\mathrm{d}}_{\mathfrak{M}}}}& & {\displaystyle \widehat{\mathrm{d}}}\downarrow \phantom{{\displaystyle \widehat{\mathrm{d}}}}& & {\displaystyle \mathrm{d}}\downarrow \phantom{{\displaystyle \mathrm{d}}}& & & & \\ 0& \stackrel{}{\to }& \mathfrak{M}& \stackrel{i}{\to }& \widehat{\mathfrak{A}}& \stackrel{p}{\to }& \mathfrak{A}& \stackrel{}{\to }& 0,\end{array}$$

where the MD${}_{\lambda }$3-LieA $(\widehat{\mathfrak{A}},{[-,-,-]}_{\widehat{\mathfrak{A}}},\widehat{\mathrm{d}})$ satisfies ${[-,u,v]}_{\widehat{\mathfrak{A}}}=0$, for all $u,v\in \mathfrak{M}$.

We will call $(\widehat{\mathfrak{A}},{[-,-,-]}_{\widehat{\mathfrak{A}}},\widehat{\mathrm{d}})$ an abelian extension of $(\mathfrak{A},[-,-,-],\mathrm{d})$ by $(\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},$${\mathrm{d}}_{\mathfrak{M}})$.

A section of an abelian extension $(\widehat{\mathfrak{A}},{[-,-,-]}_{\widehat{\mathfrak{A}}},\widehat{\mathrm{d}})$ of $(\mathfrak{A},[-,-,-],\mathrm{d})$ by $(\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},$${\mathrm{d}}_{\mathfrak{M}})$ is a linear map $s:\mathfrak{A}\to \widehat{\mathfrak{A}}$ such that $p\circ s={\mathrm{id}}_{\mathfrak{A}}$.

Let $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$ be a representation of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$. Assume that $(f,g)\in {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{M})$. Define ${[-,-,-]}_{\rho f}:{\wedge }^3(\mathfrak{A}\oplus \mathfrak{M})\to \mathfrak{A}\oplus \mathfrak{M}$ and ${\mathrm{d}}_g:\mathfrak{A}\oplus \mathfrak{M}\to \mathfrak{A}\oplus \mathfrak{M}$, respectively, by

$$\begin{array}{cc}\hfill & {[a_1+u_1,a_2+u_2,a_3+u_3]}_{\rho f}\hfill \\ \hfill & =[a_1,a_2,a_3]+\rho (a_2,a_3)u_1+\rho (a_3,a_1)u_2+\rho (a_1,a_2)u_3+f(a_1,a_2,a_3),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & {\mathrm{d}}_g(a_1+u_1)=\mathrm{d}\left(a_1\right)+{\mathrm{d}}_{\mathfrak{M}}\left(u_1\right)+g\left(a_1\right),\forall a_1,a_2,a_3\in \mathfrak{A},u_1,u_2,u_3\in \mathfrak{M}.\hfill \end{array}$$

(22)

Proposition 7. 

The triple $(\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho f},{\mathrm{d}}_g)$ is a MD${}_{\lambda }$3-LieA if and only if $(f,g)$ is a 2-cocycle in the cohomology of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ with the coefficient in $(\mathfrak{M};\rho ,{\mathrm{d}}_{\mathfrak{M}})$. In this case,

$$0\to (\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})\hookrightarrow (\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho f},{\mathrm{d}}_g)\stackrel{p}{⟶}(\mathfrak{A},[-,-,-],\mathrm{d})\to 0$$

is an abelian extension.

Proof. 

$(\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho f},{\mathrm{d}}_g)$ is a MD${}_{\lambda }$3-LieA if and only if

$$\begin{array}{cc}\hfill & {[{[a_1+u_1,a_2+u_2,a_3+u_3]}_{\rho f},a_4+u_4,a_5+u_5]}_{\rho f}+{[a_3+u_3,{[a_1+u_1,a_2+u_2,a_4+u_4]}_{\rho f},a_5+u_5]}_{\rho f}\hfill \\ \hfill & +{[a_3+u_3,a_4+u_4,{[a_1+u_1,a_2+u_2,a_5+u_5]}_{\rho f}]}_{\rho f}-{[a_1+u_1,a_2+u_2,{[a_3+u_3,a_4+u_4,a_5+u_5]}_{\rho f}]}_{\rho f}\hfill \\ \hfill & =0,\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & {[{\mathrm{d}}_g(a_1+u_1),a_2+u_2,a_3+u_3]}_{\rho f}+{[a_1+u_1,{\mathrm{d}}_g(a_2+u_2),a_3+u_3]}_{\rho f}+{[a_1+u_1,a_2+u_2,{\mathrm{d}}_g(a_3+u_3)]}_{\rho f}\hfill \\ \hfill & +\lambda {[a_1+u_1,a_2+u_2,a_3+u_3]}_{\rho f}-{\mathrm{d}}_g{[a_1+u_1,a_2+u_2,a_3+u_3]}_{\rho f}\hfill \\ \hfill & =0,\hfill \end{array}$$

(24)

for any $a_1,a_2,a_3,a_4,a_5\in \mathfrak{A},u_1,u_2,u_3,u_4,u_5\in \mathfrak{M}$. Furthermore, Equations (23) and (24) are equivalent to

$$\begin{array}{cc}\hfill & \rho (a_4,a_5)f(a_1,a_2,a_3)+f([a_1,a_2,a_3],a_4,a_5)+\rho (a_5,a_3)f(a_1,a_2,a_4)+f(a_3,[a_1,a_2,a_4],a_5)\hfill \\ \hfill & +\rho (a_3,a_4)f(a_1,a_2,a_5)+f(a_3,a_4,[a_1,a_2,a_5])-\rho (a_1,a_2)f(a_3,a_4,a_5)-f(a_1,a_2,[a_3,a_4,a_5])\hfill \\ \hfill & =0,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & \rho (a_2,a_3)g\left(a_1\right)+f(\mathrm{d}\left(a_1\right),a_2,a_3)+\rho (a_3,a_1)g\left(a_2\right)+f(a_1,\mathrm{d}\left(a_2\right),a_3)+\rho (a_1,a_2)g\left(a_3\right)\hfill \\ \hfill & +f(a_1,a_2,\mathrm{d}\left(a_3\right))+\lambda f(a_1,a_2,a_3)-{\mathrm{d}}_{\mathfrak{M}}\left(f(a_1,a_2,a_3)\right)-g\left([a_1,a_2,a_3]\right)=0,\hfill \end{array}$$

(26)

using Equations (25) and (26), we obtain $\delta f=0$ and $\delta g+\mathrm{\Phi }f=0$, respectively. Therefore, $\partial (f,g)=(\delta f,\delta g+\mathrm{\Phi }f)=0,$ which implies that $(f,g)$ is a 2-cocycle.

Conversely, if $(f,g)$ satisfying Equations (25) and (26), then $(\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho f},{\mathrm{d}}_g)$ is a MD${}_{\lambda }$3-LieA. □

Let $(\widehat{\mathfrak{A}},{[-,-,-]}_{\widehat{\mathfrak{A}}},\widehat{\mathrm{d}})$ be an abelian extension of $(\mathfrak{A},[-,-,-],\mathrm{d})$ by $(\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})$ and $s:\mathfrak{A}\to \widehat{\mathfrak{A}}$ a section. Define $\varrho :{\wedge }^2\mathfrak{A}\to \mathrm{End}\left(\mathfrak{M}\right)$, $\upsilon :{\wedge }^3\mathfrak{A}\to \mathfrak{M}$ and $\mu :\mathfrak{A}\to \mathfrak{M}$, respectively, by

$$\begin{array}{cc}\hfill \varrho (a_1,a_2)u:& ={[s\left(a_1\right),s\left(a_2\right),u]}_{\widehat{\mathfrak{A}}},\hfill \\ \hfill \upsilon (a_1,a_2,a_3):& ={[s\left(a_1\right),s\left(a_2\right),s\left(a_3\right)]}_{\widehat{\mathfrak{A}}}-s\left([a_1,a_2,a_3]\right),\hfill \\ \hfill \mu \left(a_1\right):& =\widehat{\mathrm{d}}\left(s\left(a_1\right)\right)-s\left(\mathrm{d}\left(a_1\right)\right),\forall a_1,a_2,a_3\in \mathfrak{A},u\in \mathfrak{M}.\hfill \end{array}$$

Note that $\varrho$ is independent on the choice of s.

Proposition 8. 

With the above notations, $(\mathfrak{M},\varrho ,{\mathrm{d}}_{\mathfrak{M}})$ is a representation of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ and $(\upsilon ,\mu )$ is a 2-cocycle in the cohomology of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ with the coefficient in $(\mathfrak{M};\varrho ,{\mathrm{d}}_{\mathfrak{M}})$. Furthermore, the cohomological class of the 2-cocycle $\left[(\upsilon ,\mu )\right]\in {\mathcal{H}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{M})$ is independent of the choice of sections of p.

Proof. 

First, for any $a_1,a_2,a_3,a_4\in \mathfrak{A},u\in \mathfrak{M}$, by Equation (1), we obtain

$$\begin{array}{cc}\hfill & \varrho (a_2,a_3)\varrho (a_1,a_4)u+\varrho (a_3,a_1)\varrho (a_2,a_4)u+\varrho (a_1,a_2)\varrho (a_3,a_4)u-\varrho ([a_1,a_2,a_3],a_4)u\hfill \\ \hfill =& {[s\left(a_2\right),s\left(a_3\right),{[s\left(a_1\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}+{[s\left(a_3\right),s\left(a_1\right),{[s\left(a_2\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill & +{[s\left(a_1\right),s\left(a_2\right),{[s\left(a_3\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}-{[s\left([a_1,a_2,a_3]\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill =& {[s\left(a_2\right),s\left(a_3\right),{[s\left(a_1\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}+{[s\left(a_3\right),s\left(a_1\right),{[s\left(a_2\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill & +{[s\left(a_1\right),s\left(a_2\right),{[s\left(a_3\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}-{[{[s\left(a_1\right),s\left(a_2\right),s\left(a_3\right)]}_{\widehat{\mathfrak{A}}}-\upsilon (a_1,a_2,a_3),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill =& 0,\hfill \\ \hfill & \varrho (a_3,a_4)\varrho (a_1,a_2)u+\varrho ([a_1,a_2,a_3],a_4)u+\varrho (a_3,[a_1,a_2,a_4])u-\varrho (a_1,a_2)\varrho (a_3,a_4)u\hfill \\ \hfill =& {[s\left(a_3\right),s\left(a_4\right),{[s\left(a_1\right),s\left(a_2\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}+{[s\left([a_1,a_2,a_3]\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}+{[s\left(a_3\right),s\left([a_1,a_2,a_4]\right),u]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill & -{[s\left(a_1\right),s\left(a_2\right),{[s\left(a_3\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill =& {[s\left(a_3\right),s\left(a_4\right),{[s\left(a_1\right),s\left(a_2\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}+{[[s\left(a_1\right),s\left(a_2\right),s\left(a_3\right)]-\upsilon (a_1,a_2,a_3),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill & +{[s\left(a_3\right),[s\left(a_1\right),s\left(a_2\right),s\left(a_4\right)]-\upsilon (a_1,a_2,a_4),u]}_{\widehat{\mathfrak{A}}}-{[s\left(a_1\right),s\left(a_2\right),{[s\left(a_3\right),s\left(a_4\right),u]}_{\widehat{\mathfrak{A}}}]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill =& 0.\hfill \end{array}$$

In addition, by Equation (2), we have

$$\begin{array}{cc}\hfill & {\mathrm{d}}_{\mathfrak{M}}\left(\varrho (a_1,a_2)u\right)={\mathrm{d}}_{\mathfrak{M}}\left({[s\left(a_1\right),s\left(a_2\right),u]}_{\widehat{\mathfrak{A}}}\right)\hfill \\ \hfill =& {[\widehat{\mathrm{d}}\left(s\left(a_1\right)\right),s\left(a_2\right),u]}_{\widehat{\mathfrak{A}}}+{[s\left(a_1\right),\widehat{\mathrm{d}}\left(s\left(a_2\right)\right),u]}_{\widehat{\mathfrak{A}}}+{[s\left(a_1\right),s\left(a_2\right),{\mathrm{d}}_{\mathfrak{M}}\left(u\right)]}_{\widehat{\mathfrak{A}}}+\lambda {[s\left(a_1\right),s\left(a_2\right),u]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill =& {[s\left(\mathrm{d}\left(a_1\right)\right)+\mu \left(a_1\right),s\left(a_2\right),u]}_{\widehat{\mathfrak{A}}}+{[s\left(a_1\right),s\left(\mathrm{d}\left(a_2\right)\right)+\mu \left(a_2\right),u]}_{\widehat{\mathfrak{A}}}+{[s\left(a_1\right),s\left(a_2\right),{\mathrm{d}}_{\mathfrak{M}}\left(u\right)]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill & +\lambda {[s\left(a_1\right),s\left(a_2\right),u]}_{\widehat{\mathfrak{A}}}\hfill \\ \hfill =& \varrho (\mathrm{d}\left(a_1\right),a_2)u+\varrho (a_1,\mathrm{d}\left(a_2\right))u+\varrho (a_1,a_2){\mathrm{d}}_{\mathfrak{M}}\left(u\right)+\lambda \varrho (a_1,a_2)u.\hfill \end{array}$$

Hence, $(\mathfrak{M},\varrho ,{\mathrm{d}}_{\mathfrak{M}})$ is a representation over $(\mathfrak{A},[-,-,-],\mathrm{d})$.

Since $(\widehat{\mathfrak{A}},{[-,-,-]}_{\widehat{\mathfrak{A}}},\widehat{\mathrm{d}})$ is an abelian extension of $(\mathfrak{A},[-,-,-],\mathrm{d})$ by $(\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},$${\mathrm{d}}_{\mathfrak{M}})$, by Proposition 7, $(\upsilon ,\mu )$ is a 2-cocycle. Moreover, let $s_1,s_2:\mathfrak{A}\to \widehat{\mathfrak{M}}$ be two distinct sections providing 2-cocycles $({\upsilon }_1,{\mu }_1)$ and $({\upsilon }_2,{\mu }_2)$, respectively. Define linear map $\iota :\mathfrak{A}\to \mathfrak{M}$ by $\iota \left(a_1\right)=s_1\left(a_1\right)-s_2\left(a_1\right)$. Then,

$$\begin{array}{cc}\hfill & {\upsilon }_1(a_1,a_2,a_3)\hfill \\ \hfill =& {[s_1\left(a_1\right),s_1\left(a_2\right),s_1\left(a_3\right)]}_{{\widehat{\mathfrak{A}}}_1}-s_1\left([a_1,a_2,a_3]\right)\hfill \\ \hfill =& {[s_2\left(a_1\right)+\iota \left(a_1\right),s_2\left(a_2\right)+\iota \left(a_2\right),s_2\left(a_3\right)+\iota \left(a_3\right)]}_{{\widehat{\mathfrak{A}}}_1}-(s_2\left([a_1,a_2,a_3]\right)+\iota \left([a_1,a_2,a_3]\right))\hfill \\ \hfill =& {[s_2\left(a_1\right),s_2\left(a_2\right),s_2\left(a_3\right)]}_{{\widehat{\mathfrak{A}}}_2}+\varrho (a_2,a_3)\iota \left(a_1\right)+\varrho (a_3,a_1)\iota \left(a_2\right)+\varrho (a_1,a_2)\iota \left(a_3\right)\hfill \\ \hfill & -s_2\left([a_1,a_2,a_3]\right)-\iota \left([a_1,a_2,a_3]\right)\hfill \\ \hfill =& {[s_2\left(a_1\right),s_2\left(a_2\right),s_2\left(a_3\right)]}_{{\widehat{\mathfrak{A}}}_2}-s_2\left([a_1,a_2,a_3]\right)\hfill \\ \hfill & +\varrho (a_2,a_3)\iota \left(a_1\right)+\varrho (a_3,a_1)\iota \left(a_2\right)+\varrho (a_1,a_2)\iota \left(a_3\right)-\iota \left([a_1,a_2,a_3]\right)\hfill \\ \hfill =& {\upsilon }_2(a_1,a_2,a_3)+\delta \iota (a_1,a_2,a_3)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mu }_1\left(a_1\right)& =\widehat{d}\left(s_1\left(a_1\right)\right)-s_1\left(\mathrm{d}\left(a_1\right)\right)\hfill \\ \hfill & =\widehat{\mathrm{d}}(s_2\left(a_1\right)+\iota \left(a_1\right))-\left(s_2\left(\mathrm{d}\left(a_1\right)\right)+\iota \left(\mathrm{d}\left(a_1\right)\right)\right)\hfill \\ \hfill & =\left(\widehat{\mathrm{d}}\left(s_2\left(a_1\right)\right)-s_2\left(\mathrm{d}\left(a_1\right)\right)\right)+\widehat{\mathrm{d}}\left(\iota \left(a_1\right)\right)-\iota \left(\mathrm{d}\left(a_1\right)\right)\hfill \\ \hfill & ={\mu }_2\left(a_1\right)+{\mathrm{d}}_{\mathfrak{M}}\left(\iota \left(a_1\right)\right)-\iota \left(\mathrm{d}\left(a_1\right)\right)\hfill \\ \hfill & ={\mu }_2\left(a_1\right)-\mathrm{\Phi }\iota \left(a_1\right),\hfill \end{array}$$

which implies that $({\upsilon }_1,{\mu }_1)-({\upsilon }_2,{\mu }_2)=(\delta \iota ,-\mathrm{\Phi }\iota )=\partial \left(\iota \right)\in {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{M})$. So $\left[({\upsilon }_1,{\mu }_1)\right]=\left[({\upsilon }_2,{\mu }_2)\right]\in {\mathcal{H}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{M})$. □

Definition 10. 

Let $({\widehat{\mathfrak{A}}}_1,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_1},{\widehat{\mathrm{d}}}_1)$ and $({\widehat{\mathfrak{A}}}_2,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_2},{\widehat{\mathrm{d}}}_2)$ be two abelian extensions of $(\mathfrak{A},[-,-,-],\mathrm{d})$ by $(\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})$. They are said to be equivalent if there is an isomorphism of MD${}_{\lambda }$3-LieA $\eta :({\widehat{\mathfrak{A}}}_1,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_1},{\widehat{\mathrm{d}}}_1)\to ({\widehat{\mathfrak{A}}}_2,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_2},{\widehat{\mathrm{d}}}_2)$ such that the following diagram is commutative:

$$\begin{array}{ccccccccc}0& \stackrel{}{\to }& (\mathfrak{M},{\mathrm{d}}_{\mathfrak{V}})& \stackrel{i_1}{\to }& ({\widehat{\mathfrak{A}}}_1,{\widehat{\mathrm{d}}}_1)& \stackrel{p_1}{\to }& (\mathfrak{A},\mathrm{d})& \stackrel{}{\to }& 0\\ & & \parallel & & \eta \downarrow \phantom{\eta }& & \parallel & & & & \\ 0& \stackrel{}{\to }& (\mathfrak{M},{\mathrm{d}}_{\mathfrak{V}})& \stackrel{i_2}{\to }& ({\widehat{\mathfrak{A}}}_2,{\widehat{\mathrm{d}}}_2)& \stackrel{p_2}{\to }& (\mathfrak{A},\mathrm{d})& \stackrel{}{\to }& 0.\end{array}$$

Now, we are ready to classify abelian extensions of a MD${}_{\lambda }$3-LieA.

Theorem 3. 

There is a one-to-one correspondence between equivalence classes of abelian extensions of a MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ by $(\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})$ and the second cohomology group ${\mathcal{H}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{M})$ of $(\mathfrak{A},[-,-,-],\mathrm{d})$ with coefficients in the representation $(\mathfrak{M},\varrho ,{\mathrm{d}}_{\mathfrak{M}})$.

Proof. 

Assume that $({\widehat{\mathfrak{A}}}_1,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_1},{\widehat{\mathrm{d}}}_1)$ and $({\widehat{\mathfrak{A}}}_2,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_2},{\widehat{\mathrm{d}}}_2)$ are two equivalent abelian extensions of $(\mathfrak{A},$ $[-,-,-],\mathrm{d})$ by $(\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})$ with the associated isomorphism $\eta :({\widehat{\mathfrak{A}}}_1,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_1},{\widehat{\mathrm{d}}}_1)\to ({\widehat{\mathfrak{A}}}_2,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_2},{\widehat{\mathrm{d}}}_2)$. Let $s_1$ be a section of $({\widehat{\mathfrak{A}}}_1,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_1},{\widehat{\mathrm{d}}}_1)$. As $p_2\circ \eta =p_1$, we have

$$p_2\circ (\eta \circ s_1)=p_1\circ s_1={\mathrm{id}}_{\mathfrak{A}}.$$

That is, $\eta \circ s_1$ is a section of $({\widehat{\mathfrak{A}}}_2,{[-,-,-]}_{{\widehat{\mathfrak{A}}}_2},{\widehat{\mathrm{d}}}_2)$. Denote $s_2:=\eta \circ s_1$. Since $\eta$ is an isomorphism of MD${}_{\lambda }$3-LieAs such that ${\eta |}_{\mathfrak{M}}={\mathrm{id}}_{\mathfrak{M}}$, we obtain

$$\begin{array}{cc}\hfill {\upsilon }_2(a_1,a_2,a_3)& ={[s_2\left(a_1\right),s_2\left(a_2\right),s_2\left(a_3\right)]}_{{\widehat{\mathfrak{A}}}_2}-s_2\left([a_1,a_2,a_3]\right)\hfill \\ \hfill & ={[\eta \left(s_1\left(a_1\right)\right),\eta \left(s_1\left(a_2\right)\right),\eta \left(s_1\left(a_3\right)\right)]}_{{\widehat{\mathfrak{A}}}_2}-\eta \left(s_1\left([a_1,a_2,a_3]\right)\right)\hfill \\ \hfill & =\eta \left({[s_1\left(a_1\right),s_1\left(a_2\right),s_1\left(a_3\right)]}_{{\widehat{\mathfrak{A}}}_1}-s_1\left([a_1,a_2,a_3]\right)\right)\hfill \\ \hfill & =\eta \left({\upsilon }_1(a_1,a_2,a_3)\right)\hfill \\ \hfill & ={\upsilon }_1(a_1,a_2,a_3)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mu }_2\left(a_1\right)& ={\widehat{\mathrm{d}}}_2\left(s_2\left(a_1\right)\right)-s_2\left(\mathrm{d}\left(a_1\right)\right)={\widehat{\mathrm{d}}}_2\left(\eta \left(s_1\left(a_1\right)\right)\right)-\eta \left(s_1\left(\mathrm{d}\left(a_1\right)\right)\right)\hfill \\ \hfill & =\eta \left({\widehat{\mathrm{d}}}_1\left(s_1\left(a_1\right)\right)-s_1\left(\mathrm{d}\left(a_1\right)\right)\right)\hfill \\ \hfill & =\eta \left({\mu }_1\left(a_1\right)\right)\hfill \\ \hfill & ={\mu }_1\left(a_1\right).\hfill \end{array}$$

Hence, all equivalent abelian extensions give rise to the same element in ${\mathcal{H}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{M})$.

Conversely, suppose that $\left[(f_1,g_1)\right]=\left[(f_2,g_2)\right]\in {\mathcal{H}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{M})$, and we can construct two abelian extensions $0\to (\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})\hookrightarrow (\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho f_1},{\mathrm{d}}_{g_1})\stackrel{p_1}{⟶}(\mathfrak{A},$$[-,-,-],\mathrm{d})\to 0$ and $0\to (\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})\hookrightarrow (\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho f_2},{\mathrm{d}}_{g_2})\stackrel{p_2}{⟶}(\mathfrak{A},$$[-,-,-],\mathrm{d})\to 0$ via Equations (21) and (22). Then, there exists a linear map $\iota :\mathfrak{A}\to \mathfrak{M}$ such that

$$(f_2,g_2)=(f_1,g_1)+\partial \left(\iota \right).$$

Define linear map ${\eta }_{\iota }:\mathfrak{A}\oplus \mathfrak{M}\to \mathfrak{A}\oplus \mathfrak{M}$ by ${\eta }_{\iota }(a_1+u_1):=a_1+\iota \left(a_1\right)+u_1,a_1\in \mathfrak{A},u_1\in \mathfrak{M}.$ Then, ${\eta }_{\iota }$ is an isomorphism of these two abelian extensions $(\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho f_1},{\mathrm{d}}_{g_1})$ and $(\mathfrak{A}\oplus \mathfrak{M},{[-,-,-]}_{\rho f_2},{\mathrm{d}}_{g_2})$. □

Remark 4. 

In particular, any vector space $\mathfrak{M}$ with linear transformation ${\mathrm{d}}_{\mathfrak{M}}$ can serve as a trivial representation of $(\mathfrak{A},[-,-,-],\mathrm{d})$. In this situation, central extensions of $(\mathfrak{A},[-,-,-],\mathrm{d})$ by $(\mathfrak{M},{[-,-,-]}_{\mathfrak{M}},{\mathrm{d}}_{\mathfrak{M}})$ are classified by the second cohomology group ${\mathcal{H}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},\mathfrak{M})$ of $(\mathfrak{A},$$[-,-,-],\mathrm{d})$ with the coefficient in the trivial representation $(\mathfrak{M},\rho =0,{\mathrm{d}}_{\mathfrak{M}}).$

5. ${\mathit{T}}^{\ast }$-Extensions of MD${}_{\mathbf{\lambda }}$3-LieAs

The $T^{\ast }$-extension of a 3-Lie algebra was studied in [11]. In this section, we consider $T^{\ast }$-extensions of MD${}_{\lambda }$3-LieAs by the second cohomology groups with the coefficient in a coadjoint representation.

Let $(\mathfrak{A},[-,-,-],\mathrm{d})$ be a MD${}_{\lambda }$3-LieA and ${\mathfrak{A}}^{\ast }$ be the dual space of $\mathfrak{A}$. By Example 6, $({\mathfrak{A}}^{\ast };a{d}^{\ast },-{\mathrm{d}}^{\ast })$ is a coadjoint representation of $(\mathfrak{A},[-,-,-],\mathrm{d})$. Suppose that $(f,g)\in {\mathcal{C}}_{\mathrm{md}3{\mathrm{Lie}}_{˘}}^2(\mathfrak{A},{\mathfrak{A}}^{\ast })$. Define a trilinear map ${[-,-,-]}_f:{\wedge }^3(\mathfrak{A}\oplus {\mathfrak{A}}^{\ast })\to \mathfrak{A}\oplus {\mathfrak{A}}^{\ast }$ and a linear map ${\mathrm{d}}_g:\mathfrak{A}\oplus {\mathfrak{A}}^{\ast }\to \mathfrak{A}\oplus {\mathfrak{A}}^{\ast }$, respectively, by

$$\begin{array}{cc}\hfill & {[a_1+{\alpha }_1,a_2+{\alpha }_2,a_3+{\alpha }_3]}_f\hfill \\ \hfill & =[a_1,a_2,a_3]+a{d}^{\ast }(a_2,a_3){\alpha }_1+a{d}^{\ast }(a_3,a_1){\alpha }_2+a{d}^{\ast }(a_1,a_2){\alpha }_3+f(a_1,a_2,a_3),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & {\mathrm{d}}_g(a_1+{\alpha }_1)=\mathrm{d}\left(a_1\right)-{\mathrm{d}}^{\ast }\left({\alpha }_1\right)+g\left(a_1\right),\forall a_1,a_2,a_3\in \mathfrak{A},{\alpha }_1,{\alpha }_2,{\alpha }_3\in {\mathfrak{A}}^{\ast }.\hfill \end{array}$$

(28)

Similar to Proposition 7, we have the following result.

Proposition 9. 

With the above notations, $(\mathfrak{A}\oplus {\mathfrak{A}}^{\ast },{[-,-,-]}_f,{\mathrm{d}}_g)$ is a MD${}_{\lambda }$3-LieA if and only if $(f,g)$ is a 2-cocycle in the cohomology of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$ with the coefficient in the representation $({\mathfrak{A}}^{\ast };a{d}^{\ast },-{\mathrm{d}}^{\ast })$.

Definition 11. 

The MD${}_{\lambda }$3-LieA $(\mathfrak{A}\oplus {\mathfrak{A}}^{\ast },{[-,-,-]}_f,{\mathrm{d}}_g)$ is called the $T^{\ast }$-extension of the MD${}_{\lambda }$3-LieA $(\mathfrak{A},[-,-,-],\mathrm{d})$. Denote the $T^{\ast }$-extension by $T_{(f,g)}^{\ast }\left(\mathfrak{A}\right)=(T^{\ast }\left(\mathfrak{A}\right)=\mathfrak{A}\oplus {\mathfrak{A}}^{\ast },{[-,-,-]}_f,{\mathrm{d}}_g)$.

Definition 12. 

Let $(\mathfrak{A},[-,-,-],\mathrm{d})$ be a MD${}_{\lambda }$3-LieA. $(\mathfrak{A},[-,-,-],\mathrm{d})$ is said to be metrised if it has a non-degenerate symmetric bilinear form ${\varpi }_{\mathfrak{A}}\in {\otimes }^2{\mathfrak{A}}^{\ast }$ which satisfies

$$\begin{array}{cc}\hfill & {\varpi }_{\mathfrak{A}}([a_1,a_2,a_3],a_4)+{\varpi }_{\mathfrak{A}}(a_3,[a_1,a_2,a_4])=0,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & {\varpi }_{\mathfrak{A}}(\mathrm{d}\left(a_1\right),a_2)+{\varpi }_{\mathfrak{A}}(a_1,\mathrm{d}\left(a_2\right))=0,\forall a_1,a_2,a_3,a_4\in \mathfrak{A}.\hfill \end{array}$$

(30)

We may also say that $(\mathfrak{A},[-,-,-],\mathrm{d},{\varpi }_{\mathfrak{A}})$ is a metric MD${}_{\lambda }$3-LieA.

Define a bilinear map $\varpi :{\wedge }^2{T}^{\ast }\left(\mathfrak{A}\right)\to \mathfrak{A}$ by

$$\begin{array}{cc}\hfill & \varpi (a_1+{\alpha }_1,a_2+{\alpha }_2)={\alpha }_1\left(a_2\right)+{\alpha }_2\left(a_1\right),\forall a_1,a_2\in \mathfrak{A},{\alpha }_1,{\alpha }_2\in {\mathfrak{A}}^{\ast }\hfill \end{array}$$

(31)

Proposition 10. 

With the above notations, $(T_{(f,g)}^{\ast }\left(\mathfrak{A}\right),\varpi )$ is a metric MD${}_{\lambda }$3-LieA if and only if

$$f(a_1,a_2,a_3)\left(a_4\right)+f(a_1,a_2,a_4)\left(a_3\right)=0,g\left(a_1\right)\left(a_2\right)+g\left(a_2\right)\left(a_1\right)=0,\forall a_1,a_2,a_3,a_4\in \mathfrak{A}.$$

Proof. 

For any $a_1,a_2,a_3,a_4\in \mathfrak{A},{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4\in {\mathfrak{A}}^{\ast }$, using Equations (6), (27)–(31) we have

$$\begin{array}{cc}\hfill & \varpi ({[a_1+{\alpha }_1,a_2+{\alpha }_2,a_3+{\alpha }_3]}_f,a_4+{\alpha }_4)+\varpi (a_3+{\alpha }_3,{[a_1+{\alpha }_1,a_2+{\alpha }_2,a_4+{\alpha }_4]}_f)\hfill \\ \hfill =& \varpi ([a_1,a_2,a_3]+a{d}^{\ast }(a_2,a_3){\alpha }_1+a{d}^{\ast }(a_3,a_1){\alpha }_2+a{d}^{\ast }(a_1,a_2){\alpha }_3+f(a_1,a_2,a_3),a_4+{\alpha }_4)\hfill \\ \hfill & +\varpi (a_3+{\alpha }_3,[a_1,a_2,a_4]+a{d}^{\ast }(a_2,a_4){\alpha }_1+a{d}^{\ast }(a_4,a_1){\alpha }_2+a{d}^{\ast }(a_1,a_2){\alpha }_4+f(a_1,a_2,a_4))\hfill \\ \hfill =& {\alpha }_4\left([a_1,a_2,a_3]\right)+a{d}^{\ast }(a_2,a_3){\alpha }_1\left(a_4\right)+a{d}^{\ast }(a_3,a_1){\alpha }_2\left(a_4\right)+a{d}^{\ast }(a_1,a_2){\alpha }_3\left(a_4\right)+f(a_1,a_2,a_3)\left(a_4\right)\hfill \\ \hfill & +{\alpha }_3\left([a_1,a_2,a_4]\right)+a{d}^{\ast }(a_2,a_4){\alpha }_1\left(a_3\right)+a{d}^{\ast }(a_4,a_1){\alpha }_2\left(a_3\right)+a{d}^{\ast }(a_1,a_2){\alpha }_4\left(a_3\right)+f(a_1,a_2,a_4)\left(a_3\right)\hfill \\ \hfill =& {\alpha }_4\left([a_1,a_2,a_3]\right)-{\alpha }_1([a_2,a_3,a_4]-{\alpha }_2\left([a_3,a_1,a_4]\right)-{\alpha }_3\left([a_1,a_2,a_4]\right)+f(a_1,a_2,a_3)\left(a_4\right)\hfill \\ \hfill & +{\alpha }_3\left([a_1,a_2,a_4]\right)-{\alpha }_1\left([a_2,a_4,a_3]\right)-{\alpha }_2\left([a_4,a_1,a_3]\right)-{\alpha }_4(a_1,a_2,a_3)+f(a_1,a_2,a_4)\left(a_3\right)\hfill \\ \hfill =& f(a_1,a_2,a_3)\left(a_4\right)+f(a_1,a_2,a_4)\left(a_3\right)\hfill \\ \hfill =& 0,\hfill \\ \hfill & \varpi ({\mathrm{d}}_g(a_1+{\alpha }_1),a_2+{\alpha }_2)+\varpi (a_1+{\alpha }_1,{\mathrm{d}}_g(a_2+{\alpha }_2))\hfill \\ \hfill =& \varpi (\mathrm{d}\left(a_1\right)-{\mathrm{d}}^{\ast }\left({\alpha }_1\right)+g\left(a_1\right),a_2+{\alpha }_2)+\varpi (a_1+{\alpha }_1,\mathrm{d}\left(a_2\right)-{\mathrm{d}}^{\ast }\left({\alpha }_2\right)+g\left(a_2\right))\hfill \\ \hfill =& -{\mathrm{d}}^{\ast }\left({\alpha }_1\right)\left(a_2\right)+g\left(a_1\right)\left(a_2\right)+{\alpha }_2\left(\mathrm{d}\left(a_1\right)\right)+{\alpha }_1\left(\mathrm{d}\left(a_2\right)\right)-{\mathrm{d}}^{\ast }\left({\alpha }_2\right)\left(a_1\right)+g\left(a_2\right)\left(a_1\right)\hfill \\ \hfill =& -{\alpha }_1\left(\mathrm{d}\left(a_2\right)\right)+g\left(a_1\right)\left(a_2\right)+{\alpha }_2\left(\mathrm{d}\left(a_1\right)\right)+{\alpha }_1\left(\mathrm{d}\left(a_2\right)\right)-{\alpha }_2\left(\mathrm{d}\left(a_1\right)\right)+g\left(a_2\right)\left(a_1\right)\hfill \\ \hfill =& g\left(a_1\right)\left(a_2\right)+g\left(a_2\right)\left(a_1\right)\hfill \\ \hfill =& 0.\hfill \end{array}$$

Thus, we obtain the result. □

Let $(\mathfrak{A},[-,-,-],\mathrm{d},{\varpi }_{\mathfrak{A}})$ be a metric MD${}_{\lambda }$3-LieA, then ${\varpi }_{\mathfrak{A}}$ induces an isomorphism ${\varpi }_{\mathfrak{A}}^{♮}:\mathfrak{A}\to {\mathfrak{A}}^{\ast }$ defined by

$$\langle {\varpi }_{\mathfrak{A}}^{♮}\left(a_1\right),a_2\rangle ={\varpi }_{\mathfrak{A}}(a_1,a_2),\forall a_1,a_2\in \mathfrak{A}.$$

Proposition 11. 

With the above notations, ${\varpi }_{\mathfrak{A}}^{♮}$ is an isomorphism from the adjoint representation $(\mathfrak{A};ad,\mathrm{d})$ to the coadjoint representation $({\mathfrak{A}}^{\ast };a{d}^{\ast },-{\mathrm{d}}^{\ast }).$

Proof. 

For any $a_1,a_2,a_3,a_4\in \mathfrak{A},$ by Equations (6) and (29), we have

$$\begin{array}{cc}\hfill \langle {\varpi }_{\mathfrak{A}}^{♮}\left(ad(a_1,a_2)a_3\right),a_4\rangle & ={\varpi }_{\mathfrak{A}}([a_1,a_2,a_3],a_4)=-{\varpi }_{\mathfrak{A}}(a_3,[a_1,a_2,a_4])\hfill \\ \hfill & =-\langle {\varpi }_{\mathfrak{A}}^{♮}\left(a_3\right),[a_1,a_2,a_4]\rangle =-\langle {\varpi }_{\mathfrak{A}}^{♮}\left(a_3\right),ad(a_1,a_2)a_4\rangle \hfill \\ \hfill & =\langle {\left(ad(a_1,a_2)\right)}^{\ast }{\varpi }_{\mathfrak{A}}^{♮}\left(a_3\right),a_4\rangle ,\hfill \end{array}$$

which implies that ${\varpi }_{\mathfrak{A}}^{♮}\left(ad(a_1,a_2)a_3\right)={\left(ad(a_1,a_2)\right)}^{\ast }{\varpi }_{\mathfrak{A}}^{♮}\left(a_3\right).$

In addition, for any $a_1,a_2\in \mathfrak{A},$ by Equations (6) and (30), we have

$$\begin{array}{cc}\hfill \langle {\varpi }_{\mathfrak{A}}^{♮}\left(\mathrm{d}\left(a_1\right)\right),a_2\rangle & ={\varpi }_{\mathfrak{A}}(\mathrm{d}\left(a_1\right),a_2)=-{\varpi }_{\mathfrak{A}}(a_1,\mathrm{d}\left(a_2\right))=-\langle {\varpi }_{\mathfrak{A}}^{♮}\left(a_1\right),\mathrm{d}\left(a_2\right)\rangle \hfill \\ \hfill & =-\langle {\mathrm{d}}^{\ast }{\varpi }_{\mathfrak{A}}^{♮}\left(a_1\right),a_2\rangle ,\hfill \end{array}$$

which implies that ${\varpi }_{\mathfrak{A}}^{♮}\left(\mathrm{d}\left(a_1\right)\right)=-{\mathrm{d}}^{\ast }{\varpi }_{\mathfrak{A}}^{♮}\left(a_1\right).$ Therefore, ${\varpi }_{\mathfrak{A}}^{♮}$ is an isomorphism from $(\mathfrak{A};ad,\mathrm{d})$ to $({\mathfrak{A}}^{\ast };a{d}^{\ast },-{\mathrm{d}}^{\ast })$. □