Generalized representations of 3-Hom-Lie algebras

TL;DR

This paper extends the concept of generalized representations from 3-Lie algebras to 3-Hom-Lie algebras, developing cohomology theory, semi-direct products, and exploring extensions.

Contribution

It introduces generalized representations for 3-Hom-Lie algebras, constructs their cohomology, and links extensions to semidirect products.

Findings

Developed cohomology theory for 3-Hom-Lie algebras

Computed 2-cocycles in the new cohomology

Established connections between extensions and semidirect products

Abstract

The propose of this paper is to extend generalized representations of 3-Lie algebras to Hom-type algebras. We introduce the concept of generalized representation of multiplicative 3-Hom-Lie algebras, develop the corresponding cohomology theory and study semi-direct products. We provide a key construction, various examples and computation of 2-cocycles of the new cohomology. Also, we give a connection between a split abelian extension of a 3-Hom-Lie algebra and a generalized semidirect product 3-Hom-Lie algebra.

Equations

$$[u,v,[x,y,z]]=[[u,v,x],y,z]+[x,[u,v,y],z]+[x,y,[u,v,z]].$$

$$[\alpha (u),\alpha (v),[x,y,z]]=[[u,v,x],\alpha (y),\alpha (z)]+[\alpha (x),[u,v,y],\alpha (z)]+[\alpha (x),\alpha (y),[u,v,z]].$$

$$[{\alpha }_1(u),{\alpha }_2(v),[x,y,z]]=[[u,v,x],{\alpha }_1(y),{\alpha }_2(z)]+[{\alpha }_1(x),[u,v,y],{\alpha }_2(z)]+[{\alpha }_1(x),{\alpha }_2(y),[u,v,z]].$$

$$[X,Y{]}_{\mathcal{L}}=[x_1,x_2,y_1]\wedge \alpha (y_2)+\alpha (y_1)\wedge [x_1,x_2,y_2],\forall X=x_1\wedge x_2,Y=y_1\wedge y_2,$$

$$\rho (\alpha (x_1),\alpha (x_2))\circ A$$

$$\rho (\alpha (x_1),\alpha (x_2))\rho (x_3,x_4)-\rho (\alpha (x_3),\alpha (x_4))\rho (x_1,x_2)$$

$$\rho ([x_1,x_2,x_3],\alpha (x_4))\circ A-\rho (\alpha (x_2),\alpha (x_3))\rho (x_1,x_4)$$

$$\tilde{\rho }([x_3,x_4,x_5{]}_{\alpha },\alpha (x_1))\circ A-\tilde{\rho }(\alpha (x_3),\alpha (x_4))\tilde{\rho }(x_5,x_1)-\tilde{\rho }(\alpha (x_4),\alpha (x_5))\tilde{\rho }(x_3,x_1)-\tilde{\rho }(\alpha (x_5),\alpha (x_3))\tilde{\rho }(x_4,x_1)$$

$$=A^2\circ (\rho ([x_3,x_4,x_5],x_1)-\rho (x_3,x_4)\rho (x_5,x_1)-\rho (x_4,x_5)\rho (x_3,x_1)-\rho (x_5,x_3)\rho (x_4,x_1))$$

$$=0.$$

$$[x_1+v_1,x_2+v_2,x_3+v_3{]}_{\rho }=[x_1,x_2,x_3]+\rho (x_1,x_2)v_3+\rho (x_3,x_1)v_2+\rho (x_2,x_3)v_1,$$

$$A\circ \phi (X_1\otimes ...\otimes X_{p-1},x)=\phi (\alpha (X_1)\otimes ...\otimes \alpha (X_{p-1}),\alpha (y)),\forall X_1,...,X_{p-1}\in {\wedge }^2\mathfrak{g},y\in \mathfrak{g}.$$

$$({\delta }_{\rho }\phi )(X_1,\cdots ,X_p,z)$$

$$=\sum _{1\le j<k}(-1{)}^j\phi (\alpha (X_1),\cdots ,{\hat{X}}_j,\cdots ,\alpha (X_{k-1}),[X_j,X_k{]}_{\mathcal{L}},\alpha (X_{k+1}),\cdots ,\alpha (X_p),\alpha (z))$$

$$+\sum _{j=1}^p(-1{)}^j\phi (\alpha (X_1),\cdots ,{\hat{X}}_j,\cdots ,\alpha (X_p),[X_j,z])$$

$$+\sum _{j=1}^p(-1{)}^{j+1}\rho ({\alpha }^p(X_j))\phi (X_1,\cdots ,{\hat{X}}_j,\cdots ,X_p,z)$$

$$\displaystyle+(-1)^{p+1}\Big{(}\rho(\alpha^{p}(y_{p}),\alpha^{p}(z))\varphi(X_{1},\cdots,X_{p-1},x_{p})+\rho(\alpha^{p}(z),\alpha^{p}(x_{p}))\alpha(X_{1},\cdots,X_{p-1},y_{p})\Big{)},$$

$$H_{3HL}^p(\mathfrak{g};V)=Z_{3HL}^p(\mathfrak{g};V)/B_{3HL}^p(\mathfrak{g};V).$$

$$[\phi ,\psi {]}^{3HL}=(-1{)}^{pq}{j}_{\psi }^{\alpha }(\phi )-j_{\phi }^{\alpha }(\psi )=(-1{)}^{pq}\phi {\circ }_{\alpha }\psi -\psi {\circ }_{\alpha }\phi ,$$

$$j_{\psi }^{\alpha }(\phi )(X_1,...,X_{p+q},x)=\phi {\circ }_{\alpha }\psi (X_1,...,X_{p+q},x)$$

$$=\sum _{J,j_q<i_{k+1}\le p+q}(-1{)}^{(J,I)}(\phi ({\alpha }^p(X_{i_1}),...,{\alpha }^p(X_{i_k}),\psi (X_{j_1},...,X_{j_p},){\bullet }_{\alpha }{X}_{i_{k+1}},...,{\alpha }^p(X_{i_q}),{\alpha }^p(x))$$

$$+\sum _J(-1{)}^{(J,I)}(-1{)}^q(\phi ({\alpha }^p(X_{i_1}),...,{\alpha }^p(X_{i_q}),\psi (X_{j_1},...,X_{j_p},x)),$$

$$\psi (X_{j_1},...,X_{j_q},){\bullet }_{\alpha }{X}_{i_{k+1}}=\psi (X_{j_1},...,X_{j_p},x_{i_{k+1}})\wedge {\alpha }^p(y_{i_{k+1}})+{\alpha }^p(x_{i_{k+1}})\wedge \psi (X_{j_1},...,X_{j_p},y_{i_{k+1}})$$

$$[j_{\phi }^{\alpha },j_{\psi }^{\alpha }](\xi )(X_1,X_2,...,X_{p+q+r},x)$$

$$=D_1-(-1{)}^{pq}{D}_2,$$

$$D_1=j_{\phi }^{\alpha }(j_{\psi }^{\alpha }\xi )(X_1,X_2,...,X_{p+q+r},x)\text{ and}{D}_2=j_{\psi }^{\alpha }(j_{\phi }^{\alpha }\xi ))(X_1,X_2,...,X_{p+q+r},x).$$

$$j_{\phi }^{\alpha }(j_{\psi }^{\alpha }\xi )(X_1,X_2,...,X_{p+q+r},x)=(j_{\psi }^{\alpha }(\xi )){\circ }_{\alpha }\phi (X_1,X_2,...,X_{p+q+r},x)$$

$$=\sum _{J,j_q<i_{k+1}\le p+q+r}(-1{)}^{(J,I)}(j_{\psi }^{\alpha }(\xi ))({\alpha }^q(X_{i_1}),...,{\alpha }^q(X_{i_k}),\phi (X_{j_1},...,X_{j_q},){\bullet }_{\alpha }{X}_{i_{k+1}},{\alpha }^q(X_{i_{k+2}}),...,{\alpha }^q(X_{i_{p+r}}),{\alpha }^q(x))$$

$$+\sum _J(-1{)}^{(J,I)}(-1{)}^{p+r}(j_{\psi }^{\alpha }(\xi ))({\alpha }^q(X_{i_1}),...,{\alpha }^q(X_{i_{p+r}}),\phi (X_{j_1},...,X_{j_q},x))$$

$$=\sum _{J,j_q<i_{k+1}\le p+q+r}\sum _{\begin{array}{c}L,l_p<h_{m+1}\le i_{p+r},\\ i_{k+1}=h_n,n\le m\end{array}}(-1{)}^{(J,I)}(-1{)}^{(L,H)}\xi ({\alpha }^{p+q}(X_{h_1}),...,{\alpha }^{p+q}(X_{h_{n-1}}),{\alpha }^p(\phi (X_{j_1},...,X_{j_q},){\bullet }_{\alpha }{X}_{i_{k+1}}),$$

$$...,{\alpha }^{p+q}(X_{h_m}),\psi ({\alpha }^q(X_{l_1}),...,{\alpha }^q(X_{l_p}),){\bullet }_{\alpha }{\alpha }^q(X_{h_{m+1}}),{\alpha }^{p+q}(X_{h_{m+2}}),...,{\alpha }^{p+q}(X_{h_r}),{\alpha }^{p+q}(x))$$

$$+\sum _{J,j_q<i_{k+1}\le p+q+r}\sum _{\begin{array}{c}L,l_p<h_{m+1}\le i_{p+r},\\ i_{k+1}=h_{m+1}\end{array}}(-1{)}^{(J,I)}(-1{)}^{(L,H)}\xi ({\alpha }^{p+q}(X_{h_1}),...,{\alpha }^{p+q}(X_{h_m}),\psi ({\alpha }^q(X_{l_1}),...,{\alpha }^q(X_{l_p}),)$$

$${\bullet }_{\alpha }(\phi (X_{j_1},...,X_{j_q},){\bullet }_{\alpha }{X}_{i_{k+1}}),{\alpha }^{p+q}(X_{h_{m+2}}),...,{\alpha }^{p+q}(X_{h_r}),{\alpha }^{p+q}(x))$$

$$+\sum _{J,j_q<i_{k+1}\le p+q+r}\sum _{\begin{array}{c}L,l_p<h_{m+1}\le i_{p+r},\\ i_{k+1}=h_n,n>m+1\end{array}}(-1{)}^{(J,I)}(-1{)}^{(L,H)}\xi ({\alpha }^{p+q}(X_{h_1}),...,{\alpha }^{p+q}(X_{h_m}),\psi ({\alpha }^q(X_{l_1}),...,{\alpha }^q(X_{l_p}),){\bullet }_{\alpha }$$

$${\alpha }^{p+q}(X_{h_{m+1}}),{\alpha }^{p+q}(X_{h_{m+2}}),...,{\alpha }^{p+q}(X_{h_{n-1}}),{\alpha }^p(\phi (X_{j_1},...,X_{j_q},){\bullet }_{\alpha }{X}_{i_{k+1}}),{\alpha }^{p+q}(X_{h_{n+1}},...,{\alpha }^{p+q}(X_{h_r}),{\alpha }^{p+q}(x))$$

$$+\sum _{J,j_q<i_{k+1}\le p+q+r}\sum _{\begin{array}{c}L,l_p<h_{m+1}\le i_{p+r},\\ i_{k+1}=l_s,s\le p\end{array}}(-1{)}^{(J,I)}(-1{)}^{(L,H)}\xi ({\alpha }^{p+q}(X_{h_1}),...,{\alpha }^{p+q}(X_{h_m}),\psi ({\alpha }^q(X_{l_1}),...,{\alpha }^q(X_{l_{s-1}}),$$

$$\phi (X_{j_1},...,X_{j_q},X_{j_q},){\bullet }_{\alpha }{X}_{i_{k+1}},{\alpha }^q(X_{l_{s+1}}),...,{\alpha }^q(X_{l_p}),){\bullet }_{\alpha }{\alpha }^q(X_{h_{m+1}}),{\alpha }^{p+q}(X_{h_{m+2}}),...,{\alpha }^{p+q}(X_{h_r}),{\alpha }^{p+q}(x))$$

$$+\sum _{J,j_q<i_{k+1}\le p+q+r}\sum _{\begin{array}{c}L,i_{k+1}=h_n\end{array}}(-1{)}^{(J,I)}(-1{)}^{(L,H)}(-1{)}^r\xi ({\alpha }^{p+q}(X_{h_1}),...,{\alpha }^{p+q}(X_{h_{n-1}}),{\alpha }^p(\phi (X_{j_1},...,X_{j_q},){\bullet }_{\alpha }{X}_{i_{k+1}}),$$

$${\alpha }^{p+q}(X_{h_{n+1}}),...,{\alpha }^{p+q}(X_{h_r}),\psi ({\alpha }^q(X_{l_1}),...,{\alpha }^q(X_{l_p}),{\alpha }^q(x)))$$

$$+\sum _{J,j_q<i_{k+1}\le p+q+r}\sum _{\begin{array}{c}L,i_{k+1}=l_s\end{array}}(-1{)}^{(J,I)}(-1{)}^{(L,H)}(-1{)}^r\xi ({\alpha }^{p+q}(X_{h_1}),...,{\alpha }^{p+q}(X_{h_r}),\psi ({\alpha }^q(X_{l_1}),...,{\alpha }^q(X_{l_{s-1}}),$$

$$\phi (X_{j_1},...,X_{j_q},){\bullet }_{\alpha }{X}_{i_{k+1}},{\alpha }^q(X_{l_{s+1}}),...,{\alpha }^q(X_{l_p}),{\alpha }^q(x)))$$

$$+\sum _J\sum _{L,l_p<h_{m+1}\le i_{p+r}}(-1{)}^{(J,I)}(-1{)}^{p+r}(-1{)}^{(L,H)}\xi ({\alpha }^{p+q}(X_{h_1}),...,{\alpha }^{p+q}(X_{h_m}),\psi ({\alpha }^q(X_{l_1}),...,{\alpha }^q(X_{l_p}),){\bullet }_{\alpha }({\alpha }^q(X_{h_{m+1}}),$$

$${\alpha }^{p+q}(X_{h_{m+2}}),...,{\alpha }^{p+q}(X_{h_r}),{\alpha }^p(\phi (X_{j_1},...,X_{j_q},x)))$$

$$+\sum _J\sum _L(-1{)}^{(J,I)}(-1{)}^{p+r}(-1{)}^{(L,H)}(-1{)}^r\xi ({\alpha }^{p+q}(X_{h_1}),...,{\alpha }^{p+q}(X_{h_r}),\psi ({\alpha }^q(X_{l_1}),...,{\alpha }^{p+q}(X_{l_p}),\phi (X_{j_1},...,X_{j_q},x))).$$

Full text

Introduction

The first instances of ternary Lie algebras appeared first in Nambu’s generalization of Hamiltonian mechanics [23], which was formulated algebraically by Takhtajan [29]. The structure of $n$-Lie algebras was studied by Filippov [15] then completed by Kasymov in [21].

The representation theory of $n$-Lie algebras was first introduced by Kasymov in [21]. The adjoint representation is defined by the ternary bracket in which two elements are fixed. Through fundamental objects one may also represent a $3$-Lie algebra and more generally an $n$-Lie algebra by a Leibniz algebra ([11]). The cohomology of $n$-Lie algebras, generalizing the Chevalley-Eilenberg Lie algebras cohomology, was introduced by Takhtajan [30] in its simplest form, later a complex adapted to the study of formal deformations was introduced by Gautheron [17], then reformulated by Daletskii and Takhtajan [11] using the notion of base Leibniz algebra of an $n$-Lie algebra. In [2, 3], the structure and cohomology of $3$-Lie algebras induced by Lie algebras has been investigated.

The concept of generalized representation of a $3$-Lie algebra was introduced by Liu, Makhlouf and Sheng in [19]. They study the corresponding generalized semidirect product $3$-Lie algebra and cohomology theory. Furthermore, they describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements. Non-abelian extensions was explored in [26].

The aim of this paper is to extend the concept of generalized representation of $3$-Lie algebras to Hom-type algebras. The notion of Hom-Lie algebras was introduced by Hartwig, Larsson, and Silvestrov in [18] as part of a study of deformations of the Witt and the Virasoro algebras. The $n$-Hom-Lie algebras and various generalizations of $n$-ary algebras were considered in [4]. In a Hom-Lie algebra, the Jacobi identity is twisted by a linear map, called the Hom-Jacobi identity. In particular, representations and cohomologies of Hom-Lie algebras were studied in [25], while the representations and cohomology of $n$-Hom-Lie algebras were first studied in [1].

The paper is organized as follows. in Section 1, we provide some basics about $3$-Hom-Lie algebras, representations and cohomology. The second Section includes the new concept of generalized representation of a $3$-Hom-Lie algebra, extending to Hom-type algebras the notion and results obtained in [19]. We define a corresponding semi-direct product and provide a twist procedure leading generalized representations of $3$-Hom-Lie algebras starting from generalized representations of $3$-Hom-Lie algebras and algebra maps. In Section 3, we construct a new cohomology corresponding the generalized representations and show examples. In the last section we discuss abelian extensions of multiplicative $3$-Hom-Lie algebras.

1. Representation of $3$-Hom-Lie algebras

The aim of this section is to recall some basics about $3$-Lie algebras and $3$-Hom-Lie algebras. We refer mainly to [15] and [4]. In this paper, all vector spaces are considered over a field $\mathbb{K}$ of characteristic [math].

Definition 1.1**.**

A $3$-Lie algebra is a pair $(\mathfrak{g},[\cdot,\cdot,\cdot])$ consisting of a $\mathbb{K}$-vector space $\mathfrak{g}$ and a trilinear skew-symmetric multiplication $[\cdot,\cdot,\cdot]$ satisfying the Filippov-Jacobi identity: for $x,y,z,u,v$ in $\mathfrak{g}$

[TABLE]

In this paper, we are dealing with $3$-Hom-Lie algebras corresponding to the following definition.

Definition 1.2**.**

A $3$-Hom-Lie algebra is a triple $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ consisting of a $\mathbb{K}$-vector space $\mathfrak{g}$, a trilinear skew-symmetric multiplication $[\cdot,\cdot,\cdot]$ and an algebra map $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ satisfying the Hom-Filippov-Jacobi identity: for $x,y,z,u,v$ in $\mathfrak{g}$

[TABLE]

Remark 1.3*.*

There is more general definition of $3$-Hom-Lie algebras which are given by a quadruple $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha_{1},\alpha_{2})$ consisting of a $\mathbb{K}$-vector space $\mathfrak{g}$, two linear maps $\alpha_{1},\alpha_{2}:\mathfrak{g}\rightarrow\mathfrak{g}$ and a trilinear skew-symmetric multiplication $[\cdot,\cdot,\cdot]$ satisfying the following generalized Hom-Filippov-Jacobi identity: for $x,y,z,u,v$ in $\mathfrak{g}$

[TABLE]

We get our class of $3$-Hom-Lie algebras when $\alpha_{1}=\alpha_{2}=\alpha$ and where alpha is an algebra map.This kind of algebras are usually called multiplicative $3$-Hom-Lie algebras.

Proposition 1.4**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot])$ be a $3$-Lie algebra and $\alpha:\mathfrak{g}\to\mathfrak{g}$ be a $3$-Lie algebra morphism. Then $(\mathfrak{g},[\cdot,\cdot,\cdot]_{\alpha}:=\alpha\circ[\cdot,\cdot,\cdot],\alpha)$ is a $3$-Hom-Lie algebra.

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ be a $3$-Hom-Lie algebra, elements in $\wedge^{2}\mathfrak{g}$ are called fundamental objects of the $3$-Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$. There is a bilinear operation $[\cdot,\cdot]_{\mathcal{L}}$ on $\wedge^{2}\mathfrak{g}$, which is given by

[TABLE]

and a linear map $\overline{\alpha}$ on $\wedge^{2}\mathfrak{g}$ defined by $\overline{\alpha}(X)=\alpha(x_{1})\wedge\alpha(x_{2})$, for simplicity, we will write $\overline{\alpha}(X)=\alpha(X)$. It is well-known that $(\wedge^{2}\mathfrak{g},[\cdot,\cdot]_{\mathcal{L}},\overline{\alpha})$ is a Hom-Leibniz algebra ([11]).

Definition 1.5**.**

A representation of a $3$-Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ on a vector space $V$ with respect to $A\in gl(V)$ is a skew-symmetric linear map $\rho:\wedge^{2}\mathfrak{g}\rightarrow End(V)$ such that

[TABLE]

for $x_{1},x_{2},x_{3}$ and $x_{4}$ in $\mathfrak{g}$.

Theorem 1.6**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot])$ be a $3$-Lie algebra, $(V,\rho)$ be a representation, $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be a $3$-Lie algebra morphism and $A:V\rightarrow V$ be a linear map such that $A\circ\rho(x_{1},x_{2})=\rho(\alpha(x_{1}),\alpha(x_{2}))\circ A$. Then $(V,\widetilde{\rho}:=A\circ\rho,A)$ is a representation of the $3$-Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot,\cdot]_{\alpha}:=\alpha\circ[\cdot,\cdot,\cdot],\alpha)$.

Proof.

Let $x_{i}\in\mathfrak{g}$, where $1\leq i\leq 5$. Then we have

[TABLE]

The second condition (1.6) is obtained similarly.

∎

The previous result allows to twist along morphisms a 3-Lie algebra with a representation to a $3$-Hom-Lie algebra with a corresponding representation.

Proposition 1.7**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ be a $3$-Hom-Lie algebra, $V$ be a vector space, $A\in gl(V)$ and $\rho:\wedge^{2}\mathfrak{g}\rightarrow gl(V)$ be a skew-symmetric linear map. Then $(V;\rho,A)$ is a representation of $3$-Hom-Lie algebra $\mathfrak{g}$ if and only if there is a $3$-Hom-Lie algebra structure $(\mathfrak{g}\oplus V,[\cdot,\cdot,\cdot]_{\rho},\alpha_{\mathfrak{g}\oplus V})$ on the direct sum of vector spaces $\mathfrak{g}\oplus V$, defined by

[TABLE]

and $\alpha_{\mathfrak{g}\oplus V}=\alpha+A$, for all $x_{i}\in\mathfrak{g},v_{i}\in V,1\leq i\leq 3$. The obtained $3$-Hom-Lie algebra is denoted by $\mathfrak{g}\ltimes_{\rho}V$ and called semidirect product.

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ be a $3$-Hom-Lie algebra and $(V,\rho,A)$ be a representation of $\mathfrak{g}$. We denote by $C_{\alpha,A}^{p}(\mathfrak{g},V)$ the space of all linear maps $\varphi:\underbrace{\wedge^{2}\mathfrak{g}\otimes...\otimes\wedge^{2}\mathfrak{g}}_{(p-1)}\wedge\mathfrak{g}\longrightarrow V$ satisfying:

[TABLE]

Let $\varphi$ be a $(p-1)$-cochain, the coboundary operator $\delta_{\rho}:C_{\alpha,A}^{p-1}(\mathfrak{g},V)\longrightarrow C_{\alpha,A}^{p}(\mathfrak{g},V)$ is given by

[TABLE]

for all $X_{i}=(x_{i},y_{i})\in\wedge^{2}\mathfrak{g}$, $z\in\mathfrak{g}$ and where $[X_{i},z]=[x_{i},y_{i},z]$. An element $\varphi\in{C}_{\alpha,A}^{p-1}(\mathfrak{g},V)$ is called a $p$-cocycle if $\delta_{\rho}\varphi=0$. It is called a $p$-coboundary if there exists some $f\in{C}_{\alpha,A}^{p-2}(\mathfrak{g},V)$ such that $\varphi=\delta_{\rho}f$. Denote by $Z_{3HL}^{p}(\mathfrak{g};V)$ and $B_{3HL}^{p}(\mathfrak{g};V)$ the sets of $p$-cocycles and $p$-coboundaries respectively. Then the $p$-th cohomology group is

[TABLE]

In [28], the authors constructed a graded Lie algebra structure by which one can describe an $n$-Leibniz algebra structure as a canonical structure. Here, we give the precise formulas for the $3$-Hom-Lie algebra case, generalizing the result in [19].

Set $C_{\alpha,\alpha}(\mathfrak{g},\mathfrak{g})=\oplus_{p\geq 0}C_{\alpha,\alpha}^{p}(\mathfrak{g},\mathfrak{g})$. Let $\varphi\in C_{\alpha,\alpha}^{q}(\mathfrak{g},\mathfrak{g})$, $\psi\in C_{\alpha,\alpha}^{p}(\mathfrak{g},\mathfrak{g})$, $p,q\geq 0$, $X_{i}=x_{i}\wedge y_{i}\in\wedge^{2}\mathfrak{g}$ for $i=1,2,\cdots,p+q$ and $x\in\mathfrak{g}$. For each subset $J=\{j_{1},\cdots,j_{p}\}_{j_{1}<\cdots<j_{p}}\subset N\triangleq\{1,2,\cdots,p+q\}$, let $I=\{i_{1},\cdots,i_{q}\}_{i_{1}<\cdots<i_{q}}=N/J$.

Define on the graded vector space $C_{\alpha,\alpha}(\mathfrak{g},\mathfrak{g})$ the graded commutator bracket

[TABLE]

where $\varphi\circ_{\alpha}\psi\in C_{\alpha,\alpha}^{p+q}(\mathfrak{g},\mathfrak{g})$ is defined by

[TABLE]

where

[TABLE]

where $k$ is uniquely determined by the condition $j_{p}\leq i_{k+1}$ and if $j_{p}\leq i_{1}$ then $j_{p}=p,$ $i_{1}=p+1$ and $(-1)^{(J,I)}$ is the sign of the permutation $(J,I)=(j_{1},...,j_{p},i_{1},...,i_{q})$ of $N$.

We need the following lemma to establish a structure of graded Lie algebra on $C_{\alpha,\alpha}(\mathfrak{g},\mathfrak{g})$ .

Lemma 1.8**.**

We have $j_{[\varphi,\psi]^{3HL}}=-[j_{\varphi}^{\alpha},j_{\psi}^{\alpha}]$ for all $\varphi,\psi\inžC_{\alpha,\alpha}(\mathfrak{g},\mathfrak{g})$, where $[\cdot,\cdot]$ is the graded commutator on $End(C_{\alpha,\alpha}(\mathfrak{g},\mathfrak{g}))$.

Proof.

Let $\varphi\in C_{\alpha,\alpha}^{q}(\mathfrak{g},\mathfrak{g})$, $\psi\in C_{\alpha,\alpha}^{p}(\mathfrak{g},\mathfrak{g})$ $\xi\in C_{\alpha,\alpha}^{r}(\mathfrak{g},\mathfrak{g})$, $X_{1},...,X_{p+q+r}\in\wedge^{2}{\mathfrak{g}}$ and $x\in\mathfrak{g}$

[TABLE]

where

[TABLE]

For each subset $J=\{j_{1},\cdots,j_{q}\}_{j_{1}<\cdots<j_{q}}\subset N\triangleq\{1,2,\cdots,p+q+r\}$, let $I=\{i_{1},\cdots,i_{p+r}\}_{i_{1}<\cdots<i_{p+r}}=N\setminus J$ and $L=\{l_{1},\cdots,l_{p}\}_{l_{1}<\cdots<l_{p}}\subset I\triangleq\{i_{1},i_{2},\cdots,i_{p+r}\}$, let $H=\{h_{1},\cdots,h_{r}\}_{h_{1}<\cdots<h_{r}}=I\setminus L$, we have for $D_{1}$

[TABLE]

Similarly one can compute $D_{2}.$

Let $A=\{a_{1},a_{2},...,a_{p+q}\}_{a_{1}<a_{2}<...<a_{p+q}}\subseteq N=\{1,...,p+q+r\}$, $H=\{h_{1},...,h_{r}\}_{h_{1}<h_{2}<...<h_{r}}=N\setminus A$, and $J=\{j_{1},...,j_{q}\}_{j_{1}<...<j_{q}}\subseteq A$ and $L=\{l_{1},...,l_{p}\}_{l_{1}<...<l_{p}}=A\setminus J$. We have

[TABLE]

By a straightforward verification, we obtain $D_{1}-(-1)^{pq}D_{2}=j_{[\varphi,\psi]^{3HL}}^{\alpha}(\xi)(X_{1},...,X_{p+q+r},x).$ Hence the proof. ∎

Theorem 1.9**.**

The pair $(C_{\alpha,\alpha}(\mathfrak{g},\mathfrak{g}),[\cdot,\cdot]^{3HL})$ is a graded Lie algebra.

Proof.

Let $\varphi\in C_{\alpha,\alpha}^{q}(\mathfrak{g},\mathfrak{g})$, $\psi\in C_{\alpha,\alpha}^{p}(\mathfrak{g},\mathfrak{g})$ and $\phi\in C_{\alpha,\alpha}^{r}(\mathfrak{g},\mathfrak{g})$.

1. (1)

skew-symmetry

[TABLE] 2. (2)

Graded Jacobi identity

[TABLE]

Organizing these terms leads to

[TABLE]

Using the previous lemma we get

[TABLE]

∎

Remark 1.10*.*

The pair $(C_{\alpha,\alpha}(\mathfrak{g},\mathfrak{g}),\circ_{\alpha})$ is a right symmetric graded algebra.

The previous structure of graded Lie algebra is useful to describe 3-Hom-Lie algebra structures as well as coboundary operators.

Corollary 1.11**.**

The maps $\pi:\wedge^{3}\mathfrak{g}\longrightarrow\mathfrak{g}$ and $\alpha:\mathfrak{g}\longrightarrow\mathfrak{g}$ define a $3$-Hom-Lie structure if and only if $[\pi,\pi]^{3HL}=0$.

Let $\mathfrak{g}$ be a $3$-Hom-Lie algebra. Given $x_{1},x_{2}\in\mathfrak{g}$, define $ad:\wedge^{2}\mathfrak{g}\longrightarrow gl(\mathfrak{g})$ by

[TABLE]

Then, the pair $(ad,\alpha)$ defines a representation of the $3$-Hom-Lie algebra $\mathfrak{g}$ on itself, which we call adjoint representation of $\mathfrak{g}$. The coboundary operator associated to this representation is denoted by $\delta_{\mathfrak{g}}$.

Corollary 1.12**.**

If $\pi:\wedge^{3}\mathfrak{g}\longrightarrow\mathfrak{g}$ is a $3$-Hom-Lie bracket, then we have

[TABLE]

2. Generalized representations of $3$-Hom-Lie algebras

In this section, we provide the Hom-type version of generalized representation of a $3$-Lie algebras introduced in [19]. First, we show that a representation of a $3$-Hom-Lie algebra will give rise to a canonical structure.

Let $\mathfrak{g}$ be a $3$-Hom-Lie algebra and $V$ be a vector space. Let $\rho:\wedge^{2}\mathfrak{g}\longrightarrow gl(V)$ be a linear map. Then, it induces a linear map $\overline{\rho}:\wedge^{3}(\mathfrak{g}\oplus V)\longrightarrow\mathfrak{g}\oplus V$ defined by 

[TABLE]

Consider the graded Lie algebra given in Theorem 1.9 associated to the vector space $\mathfrak{g}\oplus V$.

Proposition 2.1**.**

A linear map $\rho:\wedge^{2}\mathfrak{g}\longrightarrow gl(V)$ is a representation on a vector space $V$ of the $3$-Hom-Lie algebra $\mathfrak{g}$ with respect to $A\in gl(V)$ if and only if $\pi+\overline{\rho}$ is a canonical structure in the graded Lie algebra associated to $\mathfrak{g}\oplus V$, i.e.

[TABLE]

Proof.

By Proposition 1.7, $\rho:\wedge^{2}\mathfrak{g}\longrightarrow gl(V)$ is a representation of $\mathfrak{g}$ if and only if $\mathfrak{g}\oplus V$ is a $3$-Hom-Lie algebra, where the $3$-Hom-Lie structure is exactly given by

[TABLE]

and $\alpha_{\mathfrak{g}\oplus V}=\alpha+A$. Thus, by Lemma 1.11, $\rho:\wedge^{2}\mathfrak{g}\longrightarrow gl(V)$ is a representation of $\mathfrak{g}$ if and only if $\pi+\bar{\rho}$ is a canonical structure. ∎

The concept of representation of $3$-Lie algebras introduced by Liu, Makhlouf and Sheng ([19]) is generalized to Hom-type algebras as follows.

Definition 2.2**.**

A **generalized representation ** of a $3$-Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ with respect to $A\in gl(V)$ consists of linear maps $\rho:\wedge^{2}\mathfrak{g}\longrightarrow gl(V)$, $\nu:\mathfrak{g}\longrightarrow Hom(\wedge^{2}V,V)$, such that

[TABLE]

where $\overline{\nu}:\wedge^{3}(\mathfrak{g}\oplus V)\longrightarrow(\mathfrak{g}\oplus V)$ is induced by $\nu$ via 

[TABLE]

We will refer to a generalized representation by $(V;\rho,\nu,A)$.

Remark 2.3*.*

If $\nu=0$,  then we recover the usual definition of a representation of a $3$-Hom-Lie algebra on a vector space $V$. If the dimension of the vector space$V$ is 1, then $\nu$ must be zero. In this case, we only have the usual representation.

Given linear maps $\rho:\wedge^{2}\mathfrak{g}\longrightarrow End(V)$, $\nu:\mathfrak{g}\longrightarrow Hom(\wedge^{2}V,V)$, and $A:V\longrightarrow V$ define a trilinear bracket operation on $\mathfrak{g}\oplus V$ by

[TABLE]

Theorem 2.4**.**

*Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ be a $3$-Hom-Lie algebra and $(V;\rho,\nu,A)$ a generalized representation of $\mathfrak{g}$ with respect to $A$. Then $(\mathfrak{g}\oplus V,[\cdot,\cdot,\cdot]_{(\rho,\nu)},\alpha_{\mathfrak{g}\oplus V}=\alpha+A)$ is a $3$-Hom-Lie algebra, where $[\cdot,\cdot,\cdot]_{(\rho,\nu)}$is given by (2.3).

We call the $3$-Hom-Lie algebra $(\mathfrak{g}\oplus V,[\cdot,\cdot,\cdot]_{(\rho,\nu)},\alpha_{\mathfrak{g}\oplus V}=\alpha+A)$ **the generalized semidirect product *of $\mathfrak{g}$ and $V$.

Proof.

It follows from $[x+u,y+v,z+w](\rho,\nu)=(\pi+\overline{\rho}+\overline{\nu})(x+u,y+v,z+w)$ and Lemma 1.11. ∎

In the following, we give a characterization of a generalized representation of a $3$-Hom-Lie algebra.

Proposition 2.5**.**

Let $\rho:\wedge^{2}\mathfrak{g}\longrightarrow End(V)$, $\nu:\mathfrak{g}\longrightarrow Hom(\wedge^{2}V,V)$, $A:V\longrightarrow V$ be linear maps. They give rise to a generalized representation of a $3$-Hom-Lie algebra $\mathfrak{g}$ with respect to $A$ if and only if for all $x_{i}\in\mathfrak{g}$, $v_{j}\in V$, the following equalities hold:

[TABLE]

Proof.

the quadruple $(V;\rho,\nu,A)$ is a generalized representation if and only if $[\pi+\overline{\rho}+\overline{\nu},\pi+\overline{\rho}+\overline{\nu}]^{3HL}=0$. By straightforward computations,

[TABLE]

is equivalent to (2.4); and

[TABLE]

is equivalent to (2.5). Other identities can be proved similarly. The details are omited.∎

Remark 2.6*.*

By (2.4) and (2.5), the map $\rho$ in a generalized representation $(V;\rho,\nu,A)$ gives rise to a usual representation in the sense of Definition 1.5. Conversely, for any representation $\rho$, $(V;\rho,\nu=0,A)$ is a generalized representation.

Definition 2.7**.**

Let $(V_{1};\rho_{1},\nu_{1},A_{1})$ and $(V_{2};\rho_{2},\nu_{2},A_{2})$ be two generalized representations of a $3$-Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$. They are said to be equivalent if there exists an isomorphism of vector spaces $T:V_{1}\longrightarrow V_{2}$ such that

[TABLE]

In terms of diagrams, we have

[TABLE]

In the following, we provide a series of examples to illustrate the new concept of generalized representation and also a procedure to twist a generalized representation along linear maps.

Example 2.8**.**

Let $\mathfrak{g}$ be an abelian $3$-Hom-Lie algebra. Define $\rho=0$, and $\nu=\xi\otimes\pi$, where $\xi\in\mathfrak{g}^{*}$ and $\pi\in Hom(\wedge^{2}V\otimes V)$ is a Hom-Lie algebra structure on $(V,\pi,A)$. Then $(V;\rho,\nu,A)$ is a generalized representation. In fact, since $\mathfrak{g}$ is abelian and $\rho=0$, (2.4)-(2.7) hold naturally. Since $\pi$ satisfies the Hom-Jacobi identity, (2.8) and (2.9) also hold.

Proposition 2.9**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ be a $3$-Hom-Lie algebra, $(V,\rho,\nu,A)$ be a generalized representation, $\beta:\mathfrak{g}\rightarrow\mathfrak{g}$ be an algebra morphism and $B:V\rightarrow V$ a linear map such that

[TABLE]

Then $(V,\widetilde{\rho},\widetilde{\nu},B)$ is a generalized representation of $3$-Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot,\cdot]_{\beta},\beta\circ\alpha)$ where

[TABLE]

Proof.

We have to show that $\widetilde{\rho}$ and $\widetilde{\nu}$ satisfy Eqs.(2.4)-(2.9).

Let $x_{1},x_{2},x_{3}\in\mathfrak{g}$ and $v_{1},v_{2}\in V$

[TABLE]

[TABLE]

Then identities (2.6) and (2.7) are proved. One similarly proves identities (2.8) and (2.9). ∎

Corollary 2.10**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot])$ be a $3$-Lie algebra, $(V,\rho,\nu)$ be a generalized representation, $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ be an algebra morphism and $A:V\rightarrow V$ be a linear map such that for all $x_{1},x_{2}\in\mathfrak{g}$ and $v_{1},v_{2}\in V$,

[TABLE]

Then $(V,\widetilde{\rho}:=A\circ\rho,\widetilde{\nu}:=A\circ\nu,A)$ is a generalized representation of $3$-Hom-Lie algebra $(L,[\cdot,\cdot,\cdot]_{\alpha}:=\alpha\circ[\cdot,\cdot,\cdot],\alpha)$.

Example 2.11**.**

Let $\mathfrak{g}$ be the $3$-dimensional $3$-Lie algebra defined with respect to a basis $\{e_{1},e_{2},e_{3}\}$ by the skew-symmetric bracket $[e_{1},e_{2},e_{3}]=e_{1}$. Let $V$ be a $2$-dimensional vector space and $\{v_{1},v_{2}\}$ its basis. We have a representation defined by the following maps $(\rho,\mu)$, given with respect to previous bases by

[TABLE]

where $r_{1},r_{2},s$ are parameters in $\mathbb{K}$.

Let $\alpha:\mathfrak{g}\longrightarrow\mathfrak{g}$ be a algebra morphism and $A\in gl(V)$ defined respectively by:

[TABLE]

where $\lambda$ is a parameter in $\mathbb{K}$. They satisfy

[TABLE]

where $x,x_{1},x_{2}$ are in $\mathfrak{g}$ and $u_{1},u_{2}$ in $V$.

Then, using the Twist procedure, $(V;\widetilde{\rho},\widetilde{\nu},A)$ is a generalized representation of the $3$-Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot,\cdot]_{\alpha},\alpha)$. More precisely, we have

[TABLE]

[TABLE]

Example 2.12**.**

Let $\mathfrak{g}$ be the $4$-dimensional $3$-Lie algebra defined, with respect to a basis $\{e_{1},e_{2},e_{3},e_{4}\}$, by the skew-symmetric brackets

[TABLE]

Every generalized representation $(V;\rho,\nu)$, on a 2-dimensional vector space $V$ with trivial $\rho$, of $\mathfrak{g}$ is given by one of the following maps $\nu$ defined, with respect to a basis $\{v_{1},v_{2}\}$ of $V$, by

1. (1)

$\nu(e_{1})(v_{1},v_{2})=0,\;\nu(e_{2})(v_{1},v_{2})=0,\;\nu(e_{3})(v_{1},v_{2})=0,\;\nu(e_{4})(v_{1},v_{2})=s_{1}v_{1}+s_{2}v_{2},$**

*where $s_{1},s_{2}$ are parameters in $\mathbb{K}$, and $s_{1}s_{2}\neq 0$.

Let $\alpha:\mathfrak{g\longrightarrow\mathfrak{g}}$ be a $3$-Lie algebra morphism and $A:V\longrightarrow V$ be a linear map, defined respectively by*

[TABLE]

[TABLE]

*where $a_{1},a_{2},s_{1},s_{2}$ are parameters in $\mathbb{K}$ such that, $a_{1}s_{1}s_{2}\neq 0$.

They satisfy $A\circ\nu(x)=\nu(\alpha(x))\circ(A\otimes A)$. Therefore, using the Twist procedure, $(V;\widetilde{\rho},\widetilde{\nu},A)$ is a generalized representation of the $3$-Hom-Lie algebra $(L,[\cdot,\cdot,\cdot]_{\alpha},\alpha)$ with trivial $\widetilde{\rho}$. Namely, we have*

[TABLE]

and

[TABLE]

3. New cohomology complex of $3$-Hom-Lie algebras

Based on the generalized representations defined in the previous section, we introduce a new type of cohomology for $3$-Hom-Lie algebras.

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ be a $3$-Hom-Lie algebra and $(V;\rho,\nu,A)$ be a generalized representation of $\mathfrak{g}$. We set ${\widetilde{C}}_{\alpha+A,A}^{p}(\mathfrak{g}\oplus V,V)$ to be the set of $(p+1)$-Hom-cochains, which are defined as a subset of ${C}_{\alpha+A,A}^{p}(\mathfrak{g}\oplus V,V)$ such that

[TABLE]

Elements of ${C}_{\alpha+A,A}^{p}(\mathfrak{g}\oplus V,V)$ are of the form $\varphi:\wedge^{2}(\mathfrak{g}\oplus V)\otimes\stackrel{{\scriptstyle(p\text{ times})}}{{\cdots}}\otimes\wedge^{2}(\mathfrak{g}\oplus V)\wedge(\mathfrak{g}\oplus V)\longrightarrow V.$

By direct calculation, we have

[TABLE]

Define $d:{\widetilde{C}}_{\alpha+A,A}^{p}(\mathfrak{g}\oplus V,V)\longrightarrow{\widetilde{C}}_{\alpha+A,A}^{p+1}(\mathfrak{g}\oplus V,V)$ by

[TABLE]

Theorem 3.1**.**

Let $(V;\rho,\nu,A)$ be a generalized representation of a $3$-Hom-Lie algebra $\mathfrak{g}$. Then $d\circ d=0.$ Thus, we obtain a new cohomology complex, where the space of $p$-Hom-cochains is given by ${\widetilde{C}}_{\alpha+A,A}^{p-1}(\mathfrak{g}\oplus V,V)$.

Proof.

By the graded Jacobi identity, for any $\varphi\in{\widetilde{C}}_{\alpha+A,A}^{p-1}(\mathfrak{g}\oplus V,V)$, one obtains

[TABLE]

∎

An element $\varphi\in{\widetilde{C}}_{\alpha+A,A}^{p-1}(\mathfrak{g}\oplus V,V)$ is called a $p$-cocycle if $d(\varphi)=0$; It is called a $p$-coboundary if there exists $f\in{\widetilde{C}}_{\alpha+A,A}^{p-2}(\mathfrak{g}\oplus V,V)$ such that $\varphi=d(f)$.

Denote by $\mathcal{Z}_{3HL}^{p}(\mathfrak{g};V)$ and $\mathcal{B}_{3HL}^{p}(\mathfrak{g};V)$ the sets of $p$-cocycles and $p$-coboundaries respectively. By Theorem 3.1, we have $\mathcal{B}_{3HL}^{p}(\mathfrak{g};V)\subset\mathcal{Z}^{p}(\mathfrak{g};V)$. We define the $p$-th cohomolgy group $\mathcal{H}_{3HL}^{p}(\mathfrak{g};V)$ to be $\mathcal{Z}_{3HL}^{p}(\mathfrak{g};V)/\mathcal{B}_{3HL}^{p}(\mathfrak{g};V)$.

The following proposition provides a relationship between this new cohomology and the one given by (1.10).

Proposition 3.2**.**

There is a forgetful map from $\mathcal{H}_{3HL}^{p}(\mathfrak{g};V)$ to $H_{3HL}^{p}(\mathfrak{g};V)$.

Proof.

It is obvious that ${C}_{\alpha,A}^{p}(\mathfrak{g},V)\subseteq{\widetilde{C}}_{\alpha+A,A}^{p}(\mathfrak{g}\oplus V,V)$. By direct calculation, for $X_{i}\in\wedge^{2}\mathfrak{g},z\in\mathfrak{g}$, we have

[TABLE]

where $\delta_{\rho}$ is the coboundary operator given by (1.9). Thus, the natural projection from ${\widetilde{C}}_{\alpha+A,A}^{p}(\mathfrak{g}\oplus V,V)$ to ${C}_{\alpha,A}^{p}(\mathfrak{g},V)$ induces a forgetful map from $\mathcal{H}_{3HL}^{p}(\mathfrak{g};V)$ to $H_{3HL}^{p}(\mathfrak{g};V)$. ∎

In the sequel, we give some characterization of low dimensional cocycles.

Proposition 3.3**.**

A linear map $\varphi\in Hom(\mathfrak{g},V)$ is a $1$-cocycle if only if for all $x_{1},x_{2},x_{3}\in\mathfrak{g},v\in V$, the following identities hold :

[TABLE]

Proof.

For $\varphi\in Hom(\mathfrak{g},V)$ satisfying $\varphi\circ\alpha=A\circ\varphi$, we have

[TABLE]

and

[TABLE]

∎

Proposition 3.4**.**

A $2$-cochain $\varphi_{1}+\varphi_{2}+\varphi_{3}\in{\widetilde{C}}_{\alpha,A}^{1}(\mathfrak{g}\oplus V,V)$, where $\varphi_{1}\in Hom(\wedge^{2}V\wedge\mathfrak{g},V),~{}\varphi_{2}\in Hom(\wedge^{2}\mathfrak{g}\wedge V,V),~{}\varphi_{3}\in Hom(\wedge^{3}\mathfrak{g},V)$, is a $2$-cocycle if and only if for all $x_{i}\in\mathfrak{g},v_{j}\in V$ and $v\in V$, the following identities hold:

[TABLE]

Proof.

For $\varphi_{3}\in Hom(\wedge^{3}\mathfrak{g},V)$, we have

[TABLE]

For $\varphi_{2}\in Hom(\wedge^{2}\mathfrak{g}\wedge V,V)$, we have

[TABLE]

[TABLE]

For $\varphi_{1}\in Hom(\wedge^{2}V\wedge\mathfrak{g},V)$, we have

[TABLE]

Thus, $d(\varphi_{1}+\varphi_{2}+\varphi_{3})=0$ if and only if Eqs (3.3)-(3.10) hold. ∎

In the following we provide two examples of computation of $2$-cocycles of the $3$-dimensional $3$-Hom-Lie algebra.

Example 3.5**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot],\alpha)$ be the $3$-dimensional $3$-Hom-Lie algebra defined, with respect to a basis $\{e_{1},e_{2},e_{3}\}$, by $[e_{1},e_{2},e_{3}]=a_{1}e_{1},~{}\alpha(e_{1})=a_{1}e_{1},~{}\alpha(e_{2})=a_{2}e_{2},~{}\alpha(e_{3})=\frac{1}{a_{2}}e_{3}$. Let $V$ be a 2-dimensional vector space, $\{v_{1},v_{2}\}$ its basis and $A\in gl(V)$ defined by: $A(v_{1})=a_{1}v_{1},A(v_{2})=\frac{a_{2}a_{3}}{a_{1}}v_{2},$ where $a_{1},a_{2},a_{3}$ are parameters in $\mathbb{K}$.

We consider the generalized representation $(V,\rho,\nu,A)$, where $\rho$ and $\nu$ are defined with respect to the basis by

[TABLE]

with $s,r_{1},r_{2}$ parameters in $\mathbb{K}$ and $a_{1}a_{2}s\neq 0$.

We have the following $2$-cocycles : $\varphi_{1}=0$, $\varphi_{3}=0$ and $\varphi_{2}$ defined as

[TABLE]

where $c_{1},c_{2},c_{3}$ are parameters in $\mathbb{K}$. **

4. Abelian extensions of $3$-Hom-Lie algebras

In this section, we show that associated to any abelian extension, there is a generalized representation and a $2$-cocycle.

Definition 4.1**.**

Let $(\mathfrak{g},[\cdot,\cdot,\cdot]_{\mathfrak{g}},\alpha)$,  $(V,[\cdot,\cdot,\cdot]_{V},A)$,  and $(\mathfrak{\hat{g}},[\cdot,\cdot,\cdot]_{\mathfrak{\hat{g}}},\alpha_{\mathfrak{\hat{g}}})$ be $3$-Hom-Lie algebras and $i:V\longrightarrow\mathfrak{\hat{g}}$, $p:\mathfrak{\hat{g}}\longrightarrow\mathfrak{g}$ be morphisms of $3$-Hom-Lie algebras. The following sequence of $3$-Hom-Lie algebras is a short exact sequence if $Im(i)=Ker(p)$, $Ker(i)=0$ and $Im(p)=\mathfrak{g}$:

[TABLE]

where $A(V)=\alpha_{\mathfrak{\hat{g}}}(V)$. In this case, we call  $\mathfrak{\hat{g}}$ an extension of $\mathfrak{g}$ by $V$, and denote it by $E_{\mathfrak{\hat{g}}}$ . It is called an abelian extension if $V$ is an abelian ideal of $\mathfrak{\hat{g}}$, i.e., $[u,v]_{V}=0$ for all $u,v\in V$. A section $\sigma$ of $p:\mathfrak{\hat{g}}\longrightarrow\mathfrak{g}$ consists of linear maps $\sigma:\mathfrak{g\longrightarrow}\mathfrak{\hat{g}}$ such that $p\circ\sigma=id_{\mathfrak{g}}$ and $\sigma\circ\alpha=\alpha_{\mathfrak{\hat{g}}}\circ\sigma.$

Definition 4.2**.**

Two extensions of $3$-Hom-Lie algebras

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 are equivalent if there exists a morphism of $3$-Hom-Lie algebras $\phi:\mathfrak{\hat{g}}\longrightarrow\mathfrak{\tilde{g}}$ such that the following diagram commutes:

[TABLE]

A linear map $\sigma:\mathfrak{g}\longrightarrow\mathfrak{\hat{g}}$ is called a splitting of $\mathfrak{g}$ if it satisfies $p\circ\sigma=id_{\mathfrak{g}}$. If there exists a splitting which is also a homomorphism between $3$-Hom-Lie algebras, we say that the abelian extension is split. Let $\hat{\mathfrak{g}}$ be a split abelian extension and $\sigma:\mathfrak{g}\longrightarrow\hat{\mathfrak{g}}$ the corresponding splitting. Define $\rho:\wedge^{2}\mathfrak{g}\longrightarrow\mathfrak{gl}(V)$ and $\nu:\mathfrak{g}\longrightarrow Hom(\wedge^{2}V,V)$ by

[TABLE]

Then, we can transfer the $3$-Hom-Lie algebra structure on $\hat{\mathfrak{g}}$ to that on $\mathfrak{g}\oplus V$ in terms of $\rho$ and $\nu$:

Note that the Hom-Filippov-Jacobi identity gives the character of $\rho$ and $\nu$.

[TABLE]

However, by Theorem 2.4, it is straightforward to obtain the following proposition.

Proposition 4.3**.**

Any split abelian extension of $3$-Hom-Lie algebras is isomorphic to a generalized semidirect of product $3$-Hom-Lie algebra.

Now, for non-split abelian extensions, we can further define $\omega:\wedge^{3}\mathfrak{g}\longrightarrow V$ by

[TABLE]

Then, we also transfer the $3$-Hom-Lie algebra structure on $\hat{\mathfrak{g}}$ to that on $\mathfrak{g}\oplus V$ in terms of $\rho,\nu$ and $\omega:$

[TABLE]

Theorem 4.4**.**

With above notations, $(\mathfrak{g}\oplus V,[\cdot,\cdot,\cdot]_{(\rho,\nu,\omega)},\alpha_{\mathfrak{g}\oplus V})$ is a $3$-Hom-Lie algebra if and only if for all $x_{1},x_{2},x_{3},x_{4},x_{5}\in\mathfrak{g}$ and $v,v_{1},v_{2},v_{3}\in V$, Eqs. (2.6)-(2.9) and the following identities hold:

[TABLE]

The Fundamental Identity gives the character of $\rho$, $\nu$ and $\omega$.

Proof.

The pair $(\mathfrak{g}\oplus V,[\cdot,\cdot,\cdot]_{(\rho,\nu,\omega)},\alpha_{\mathfrak{g}\oplus V})$ defines a $3$-Hom-Lie algebra if and only if the Hom-Filippov-Jacobi identity holds on all elements of $\mathfrak{g}\oplus V$. Condition (4.1) is obtained using the Hom-Filippov-Jacobi identity on $\{x_{1},x_{2},x_{3},x_{4},x_{5}\}$ elements of $\mathfrak{g}$.

Similarly, elements $\{x_{1},v,x_{2},x_{3},x_{4}\}$ gives Eq. (4.2), $\{x_{1},x_{2},v,x_{3},x_{4}\}$ gives Eq. (4.3), $\{x_{1},x_{2},v_{1},v_{2},x_{3}\}$ gives Eq. (2.6), $\{v_{1},x_{1},v_{2},x_{2},x_{3}\}$ gives Eq. (2.7), $\{v_{1},x_{1},v_{2},v_{3},x_{2}\}$ gives Eq. (2.8) and $\{v_{1},v_{2},v_{3},x_{1},x_{2}\}$ gives Eq. (2.9).

Conversely, if Eqs. (2.6)-(2.9) and Eqs. (4.1)-(4.3) hold, it is straightforward to see that for all $e_{1},\cdots,e_{5}\in\mathfrak{g}\oplus V$, the Hom-Filippov-Jacobi identity holds. Thus, $(\mathfrak{g}\oplus V,[\cdot,\cdot,\cdot]_{(\rho,\nu,\omega)},\alpha_{\mathfrak{g}\oplus V})$ is a $3$-Hom-Lie algebra. ∎

Acknowledgment : The authors would like to thank Yunhe Sheng for his comments and suggestions.