3-Lie algebra $A_{\omega}^{\delta}$-modules and induced modules

TL;DR

This paper explores the structure of modules over a specific 3-Lie algebra, establishing connections with induced modules of its derivation algebra and constructing infinite-dimensional modules with particular properties.

Contribution

It introduces the concept of induced modules for 3-Lie algebras and constructs explicit infinite-dimensional modules, analyzing their relation to modules of the inner derivation algebra.

Findings

Only certain modules are induced modules among the constructed infinite-dimensional modules.

Established a relationship between 3-Lie algebra modules and modules of its derivation algebra.

Constructed explicit examples of infinite-dimensional modules for the algebra.

Abstract

In this paper, we define the induced modules of Lie algebra ad$(B)$ associated with a 3-Lie algebra $B$-module, and study the relation between 3-Lie algebra $A_{\omega}^{\delta}$-modules and induced modules of inner derivation algebra ad$(A_{\omega}^{\delta})$. We construct two infinite dimensional intermediate series modules of 3-Lie algebra $A_{\omega}^{\delta}$, and two infinite dimensional modules $(V, \psi_{\lambda\mu})$ and $(V, \phi_{\mu})$ of the Lie algebra ad$(A_{\omega}^{\delta})$, and prove that only $(V, \psi_{\lambda0})$ and $(V, \psi_{\lambda1})$ are induced modules.

Equations

$$[f_1,f_2,f_3{]}_{\partial }=\frac{\partial (f_1,f_2,f_3)}{\partial (x,y,z)},$$

$$[x_1,x_2,[x_3,x_4,x_5]]=[[x_1,x_2,x_3],x_4,x_5]+[x_3,[x_1,x_2,x_4],x_5]+[x_3,x_4,[x_1,x_2,x_5]].$$

$$\text{mbox}ad(x_1,x_2)(x)=[x_1,x_2,x],\text{mbox}forallx\in B,$$

$$[\rho (x_1,x_2),\rho (x_3,x_4)]=\rho ([x_1,x_2,x_3{]}_B,x_4)+\rho (x_3,[x_1,x_2,x_4{]}_B),$$

$$\rho ([x_1,x_2,x_3{]}_B,x_4)=\rho (x_1,x_2)\rho (x_3,x_4)+\rho (x_2,x_3)\rho (x_1,x_4)+\rho (x_3,x_1)\rho (x_2,x_4),$$

$$V_{\lambda }=\{v\in V\mid \rho (h_1,h_2)v=\lambda (h_1,h_2)v,\forall h_1,h_2\in H\}\ne 0,$$

$$V=\sum _{\lambda \in (H\wedge H{)}^{\ast }}{V}_{\lambda },$$

$$L_r{L}_s=L_{r+s},M_r{M}_s=M_{r+s},L_r{M}_s=0,\forall r,s\in \mathbb{Z}.$$

$$\delta (L_r)=r{L}_r,\delta (M_r)=r{M}_r,\forall r\in \mathbb{Z},$$

$$\omega (L_r)=M_{-r},\omega (M_r)=L_{-r},\forall r\in \mathbb{Z}.$$

$$\delta \omega +\omega \delta =0.$$

$$[x,y,z{]}_{\omega }=\left|\begin{array}{ccc}\omega (x)& \omega (y)& \omega (z)\\ x& y& z\\ \delta (x)& \delta (y)& \delta (z)\end{array}\right|,\forall x,y,z\in A,$$

$$\{\begin{array}{l}[L_r,L_s,M_t{]}_{\omega }=(s-r)L_{r+s-t},r,s,t\in Z,\\ [L_r,M_s,M_t{]}_{\omega }=(t-s)M_{s+t-r},r,s,t\in Z,\\ \text{mbox}andothersarezero,\end{array}$$

$$V=\sum _{\alpha \in \mathbb{F}}\mathbb{F}{v}_{\alpha },$$

$$U(\alpha )=\sum _{m\in \mathbb{Z}}\mathbb{F}{v}_{\alpha +m}$$

$$\{\begin{array}{l}{\rho }_{\lambda \mu }(L_r,M_s)v_{\alpha }=(\lambda +\alpha +(s-r)\mu )v_{\alpha +s-r},r,s\in \mathbb{Z},\alpha \in \mathbb{F},\\ {\rho }_{\lambda \mu }(L_r,L_s)v_{\alpha }=0,r,s\in \mathbb{Z},\alpha \in \mathbb{F},\\ {\rho }_{\lambda \mu }(M_r,M_s)v_{\alpha }=0,r,s\in \mathbb{Z},\alpha \in \mathbb{F}.\end{array}$$

$$[{\rho }_{\lambda \mu }(L_r,M_m),{\rho }_{\lambda \mu }(L_s,M_l)]v_{\alpha }={\rho }_{\lambda \mu }(L_r,M_m){\rho }_{\lambda \mu }(L_s,M_l)v_{\alpha }-{\rho }_{\lambda \mu }(L_s,M_l){\rho }_{\lambda \mu }(L_r,M_m)v_{\alpha }$$

$${\rho }_{\lambda \mu }([L_r,M_m,L_s],M_l)+{\rho }_{\lambda \mu }(L_s,[L_r,M_m,M_l])v_{\alpha }$$

$$=(r-s+l-m)(\lambda +\alpha +(l-r-s+m)\mu )v_{\alpha -s+m+l-r}.$$

$$[{\rho }_{\lambda \mu }(L_r,M_m),{\rho }_{\mu \nu }(L_s,M_l)]={\rho }_{\mu \nu }([L_r,M_m,L_s],M_l)+{\rho }_{\mu \nu }(L_s,[L_r,M_m,M_l]).$$

$$[{\rho }_{\lambda \mu }(L_r,L_m),{\rho }_{\mu \nu }(L_s,L_l)]={\rho }_{\lambda \mu }([L_r,L_m,L_s],L_l)+{\rho }_{\lambda \mu }(L_s,[L_r,L_m,L_l])=0,$$

$$[{\rho }_{\lambda \mu }(M_r,M_m),{\rho }_{\lambda \mu }(M_s,M_l)]={\rho }_{\lambda \mu }([M_r,M_m,M_s],M_l)+{\rho }_{\lambda \mu }(M_s,[M_r,M_m,M_l])=0.$$

$${\rho }_{\lambda \mu }([L_r,L_s,M_m],M_l)v_{\alpha }=(s-r)(\lambda +\alpha +(l-r-s+m)\mu )v_{\alpha +l-r-s+m},$$

$$({\rho }_{\lambda \mu }(L_r,L_s){\rho }_{\lambda \mu }(M_m,M_l)+{\rho }_{\lambda \mu }(L_s,M_m){\rho }_{\lambda \mu }(L_r,M_l)+{\rho }_{\lambda \mu }(M_m,L_r){\rho }_{\lambda \mu }(L_s,M_l))v_{\alpha }$$

$$=(\lambda +\alpha +(l-r)\mu )(\lambda +\alpha +l-r+(m-s)\mu )v_{\alpha +l-r+m-s}$$

$$-(\lambda +\alpha +(l-s)\mu )(\lambda +\alpha +l-s+(m-r)\mu )v_{\alpha +l-r+m-s}$$

$$=(s-r)(\lambda +\alpha +(l-r-s+m)\mu +(l-m)\mu (1-\mu ))v_{\alpha +l-r-s+m};$$

$${\rho }_{\lambda \mu }([M_m,M_l,L_r],L_s)=(m-l)(\lambda +\alpha +(m+l-r-s)\mu )v_{\alpha +m+l-r-s},$$

$${\rho }_{\lambda \mu }(M_l,L_r){\rho }_{\lambda \mu }(M_m,L_s)+{\rho }_{\lambda \mu }(L_r,M_m){\rho }_{\lambda \mu }(M_l,L_s)+{\rho }_{\lambda \mu }(M_m,M_l){\rho }_{\lambda \mu }(L_r,L_s))v_{\alpha }$$

$$=(m-l)(\lambda +\alpha +(m+l-2s)\mu +(s-r){\mu }^2)v_{\alpha +m+l-r-s}.$$

Full text

1. Introduction

n-Lie algebras were introduced in 1985 ([12]), and their structures were studied extensively [5, 6, 7, 13, 15, 16, 17, 18, 20, 21, 22]. Later, 3-Lie algebras were applied in noncommutative geometry and the quantum geometry of branes in M-theory ([3, 4, 11, 2]).

A canonical Nambu 3-Lie algebra ([19]) is a triple of classical observables on a three-dimensional phase space with coordinates $x,y,z,$

[TABLE]

which is related to Nambu mechanics (generalizing Hamiltonian mechanics by using more than one hamiltonian). The algebraic formulation of this theory is due to Takhtajan ([24]), see also [11, 14]. Through years of work, 3-Lie algebras can be easily obtained from Lie algebras, commutative algebras, and Pre-Lie algebras ([1, 8, 9, 10]).

The representation theory of n-Lie algebras was first introduced in [17]. The adjoint representation $(A,$ ad) of any n-Lie algebra $A$ is defined by ad$:A\wedge A\rightarrow gl(A)$, for $\forall x,y\in A$, ad$(x,y):A\rightarrow A$, $\forall z\in A$, $ad(x,y)z$ is the ternary bracket. In [10], the infinite-dimensional simple 3-Lie algebra $A_{\omega}^{\delta}$ was introduced; its adjoint representation was investigated there as well. It is proved that the regular representation of $A_{\omega}^{\delta}$ is a Harish-Chandra module, and the natural module of the inner derivation algebra is an intermediate series module. In [23], authors provided a new approach to representation theory of 3-Lie algebras, which is a generalized representation of a 3-Lie algebra. They also gave examples of generalized representations of 3-Lie algebras, and computed 2-cocycles of the new cohomology.

In this paper, we continue to study the representations of 3-Lie algebras. First we construct two infinite dimensional 3-Lie algebra $A_{\omega}^{\delta}$-modules $T_{\lambda 0}$ and $T_{\lambda 1}$. Then we study the induced module of the Lie algebra ad$(B)$ associated with a 3-Lie algebra $B$ module. We also construct two modules $(V,\psi_{\lambda\mu}),(V,\phi_{\mu})$ of inner derivation algebra ad$(A_{\omega}^{\delta})$, and we prove that $(V,\psi_{\lambda 0})$ and $(V,\psi_{\lambda 1})$ are induced modules, others are not.

Unless otherwise stated, algebras and vector spaces are over a field $\mathbb{F}$ of characteristic zero, and $Z$ is the set of integers.

2. Preliminaries

In this section we collect some basic notions of $3$-Lie algebras.

A 3-Lie algebra [12] is a vector space $B$ with a linear multiplication (or a $3$-Lie bracket) $[~{},~{},~{}]:B\wedge B\wedge B\rightarrow B$ satisfying, $\forall x_{1},x_{2},x_{3},x_{4},x_{5}\in B,$

[TABLE]

For $x_{1},x_{2}\in B,$ the linear mapping $\mbox{ad}(x_{1},x_{2}):B\rightarrow B$,

[TABLE]

is called a left multiplication defined by $x_{1}$, $x_{2}\in B$.

Thanks to (2.1), left multiplications and their linear combinations are derivations, which are called inner derivations. The set ad$(B)$ of all the inner derivations of $B$ is an ideal of the derivation algebra Der$(B)$, and ad$(B)$ is called the inner derivation algebra of $B$.

Let $B$ be a 3-Lie algebra, $V$ be a vector space and $\rho:B\wedge B\rightarrow gl(V)$ be a linear mapping. If $\rho$ satisfies for all $x_{1},x_{2},x_{3},x_{4}\in B,$

[TABLE]

[TABLE]

then $(V,\rho)$ is called a representation[17] of the 3-Lie algebra $B$ (or is simply called a 3-Lie algebra $B$-module). If there does not exist non-trivial modules of $V$, then $V$ is called an irreducible module of $B$.

For example, for any 3-Lie algebra $B$, $(B,\mbox{ad})$ is a 3-Lie algebra $B$-module, which is called the adjoint module of $B$, where ad$:B\wedge B\rightarrow gl(B)$, for all $x_{1},x_{2}\in B$, ad$(x_{1},x_{2})$ is the left multiplication.

A Cartan subalgebra of a 3-Lie algebra $B$ is a nilpotent subalgebra of $B$ such that $\forall y\in B$, if $y$ satisfies $[y,H,B]\subseteq H$, then $y\in H$.

Let $(V,\rho)$ be a 3-Lie algebra $B$-module, $\lambda\in(H\wedge H)^{*}$. If

[TABLE]

then $\lambda$ is called a weight associated with the Cartan subalgebra $H$, and $V_{\lambda}$ is called a weight space of $\lambda$.

A module $(V,\rho)$ of a $3$-Lie algebra $B$ is referred to as a weight module if it admits a weight space decomposition

[TABLE]

where $H$ is a Cartan subalgebra of $B$.

A module of $B$ is called a Harish-Chandra module if it is a weight module and every weight space is finite-dimensional. A Harish-Chandra module of $B$ is an intermediate series module if the dimension of every weight space is equal to one.

Let A be a commutative associative algebra over $\mathbb{F}$ with a basis $\{L_{r},M_{r}|r\in\mathbb{Z}\}$, where

[TABLE]

Define linear mappings $\omega,\delta:A\rightarrow A$ by

[TABLE]

[TABLE]

Then $\delta$ is a derivation of $A$, $\omega$ is an involution and satisfy

[TABLE]

Thanks to Theorem 3.3 in [10], $A$ is a simple canonical Nambu 3-Lie algebra with the multiplication

[TABLE]

that is,

[TABLE]

which is denoted by $A_{\omega}^{\delta}$.

3. Modules and induced modules of $A_{\omega}^{\delta}$

In [10], authors discussed the structure of the infinite dimensional simple 3-Lie algebra $A_{\omega}^{\delta}$. It is proved that the adjoint representation $(A_{\omega}^{\delta},\mbox{ad})$ is a Harish-Chandra module, and $A_{\omega}^{\delta}$ (as vector space) as the natural module of the inner derivation algebra ad$(A_{\omega}^{\delta})$ is an intermediate series module.

In this section we construct infinite dimensional 3-Lie algebra $A_{\omega}^{\delta}$-module $T_{\lambda\mu}$, and discuss the relation between 3-Lie algebra $A_{\omega}^{\delta}$-modules and induced modules of inner derivation algebra ad$(A_{\omega}^{\delta})$.

In the following of the paper, suppose

[TABLE]

is a vector space over $\mathbb{F}$ with a basis $\{v_{\alpha}~{}~{}|~{}~{}\alpha\in\mathbb{F}\}$, and for $\forall\alpha\in\mathbb{F}$, denotes

[TABLE]

the subspace of $V$.

3.1. 3-Lie algebra $A_{\omega}^{\delta}$-module $T_{\lambda\mu}$

For some $\lambda,\mu\in\mathbb{F}$, define the linear mapping $\rho_{\lambda\mu}:A_{\omega}^{\delta}\wedge A_{\omega}^{\delta}\rightarrow gl(V)$ by

[TABLE]

The pair $(V,\rho_{\lambda\mu})$ is denoted by $T_{\lambda\mu}$.

Theorem 3.1**.**

By the above notations, $T_{\lambda\mu}$ is a 3-Lie algebra $A_{\omega}^{\delta}$-module if and only if $\mu=0$ or $\mu=1$. And in the case $\mu=0$ or $\mu=1$,

1) $T_{\lambda\mu}$ is a reducible module with submodules $U(\alpha)$  for all $\alpha\in\mathbb{F}$;

2) if $\mu=0$ , then $\mathbb{F}v_{-\lambda}$ is a trivial submodule of $U(-\lambda)$;

3) if $\mu=1$, $\alpha+\lambda\in Z$, $\alpha\in\mathbb{F}$, then $\overline{U_{\alpha}}=\sum\limits_{m\in Z_{\neq-\lambda-\alpha}}\mathbb{F}v_{\alpha+m}$ is an irreducible submodule of $U(\alpha)$;

4) for $\alpha\in\mathbb{F}$, $U(\alpha)$  is irreducible if and only if $\alpha+\lambda\notin Z$.

Proof.

$\bullet$ First, we prove that $T_{\lambda\mu}$ is an $A_{\omega}^{\delta}$-module if and only if $\mu=0$ or $1$.

For $\forall r,m,s,l\in Z,\alpha,\lambda\in\mathbb{F}$, and $\mu=0$ or $1$, thanks to Eqs (2.8) and (3.3), and a direct computation,

[TABLE]

$\hskip 150.79968pt=(\lambda+\alpha+(l-s+m-r)\mu)(l-s-m+r)v_{\alpha-s+l-r+m},$

[TABLE]

[TABLE]

Therefore,

[TABLE]

It is clear that

[TABLE]

[TABLE]

Therefore, Eq (2.3) holds.

By Eqs (2.8) and (3.3), for $\forall m,l,r,s\in Z,\alpha\in\mathbb{F}$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From the above discussion, $\forall m,l,r,s\in Z,\alpha\in\mathbb{F}$, the identities

$\rho_{\lambda\mu}([L_{r},L_{s},M_{m}],M_{l})=\rho_{\lambda\mu}(L_{r},L_{s})\rho_{\lambda\mu}(M_{m},M_{l})+\rho_{\lambda\mu}(L_{s},M_{m})\rho_{\lambda\mu}(L_{r},M_{l})$

$+\rho_{\lambda\mu}(M_{m},L_{r})\rho_{\lambda\mu}(L_{s},M_{l})$  and

$\rho_{\lambda\mu}([M_{m},M_{l},L_{r}],L_{s})=\rho_{\lambda\mu}(M_{m},M_{l})\rho_{\lambda\mu}(L_{r},L_{s})+\rho_{\lambda\mu}(M_{l},L_{r})\rho_{\lambda\mu}(M_{m},L_{s})$

$+\rho_{\lambda\mu}(L_{r},M_{m})\rho_{\lambda\mu}(M_{l},L_{s})$

hold if and only if $\mu=0$ or $\mu=1$, that is, (2.4) holds if and only $\mu=0$ or $\mu=1$.

Therefore, $T_{\lambda\mu}$ is an $A_{\omega}^{\delta}$-module if and only if $\mu=0$ or $\mu=1$.

$\bullet\bullet$ Thanks to (3.3), for $\forall\alpha\in\mathbb{F}$, $U(\alpha)=\sum\limits_{m\in\mathbb{Z}}\mathbb{F}v_{\alpha+m}$ is a submodule of $T_{\lambda\mu}$, we get 1).

If $\mu=0$, then

[TABLE]

it follows that $\mathbb{F}v_{-\lambda}$ is a trivial submodule. It follows 2).

If $\mu=1$, then for all $v_{\alpha+m},v_{\alpha+m^{\prime}}\in U(\alpha)$, $m\neq m^{\prime}\in Z$, there are distinct $r,s\in Z$, such that $m^{\prime}=m+s-r$. By (3.3)

[TABLE]

Then in the case  $\alpha+\lambda\in Z$, for $\forall m^{\prime}\neq-\lambda-\alpha$, $\lambda+\alpha+m^{\prime}\neq 0$. Thanks to (3.5), $\overline{U_{\alpha}}=\sum\limits_{m\in Z_{\neq-\lambda-\alpha}}\mathbb{F}v_{\alpha+m}$ is an irreducible submodule of $U(\alpha)$. We get 3).

$\bullet\bullet\bullet$ For $\forall\alpha\in\mathbb{F}_{\neq-\lambda}$, $\mu=0,$ and  $v_{\alpha+m},v_{\alpha+m^{\prime}}\in U(\alpha)$, then $m\neq m^{\prime}\in Z$, and there are distinct $r,s\in Z$ such that $m^{\prime}=m+s-r$. By (3.3)

[TABLE]

If $\alpha+\lambda\notin Z$, then $\lambda+\alpha+m\neq 0$, and $U(\alpha)=\rho_{\lambda 0}(A_{p},A_{p})v_{\alpha+m}$, it follows that  $U(\alpha)$ is irreducible.

If  $\alpha+\lambda\in Z$, then there are distinct $r,s\in Z$, such that $\alpha+\mu=r-s$. Thanks to (3.3)

[TABLE]

Therefore,  $v_{-\lambda}\in U(\alpha)$. Thanks to (3.4), $\mathbb{F}v_{-\lambda}$ is a trivial submodule of $U(\alpha)$.

If $\mu=1$ and $\alpha+\lambda\notin Z$, then for $\forall v_{\alpha+m},v_{\alpha+m^{\prime}}\in U(\alpha)$, $m\neq m^{\prime}\in Z$, $\lambda+\alpha+m^{\prime}\neq 0$,

[TABLE]

it follows that  $U(\alpha)$ is irreducible.

Summarizing the above discussion, we get 4). The proof is complete. ∎

Theorem 3.2**.**

The module $T_{\lambda 0}$ and $T_{\lambda 1}$ are intermediate series modules of $A_{\omega}^{\delta}$.

Proof.

By (2.8),   $H=\mathbb{F}L_{0}+\mathbb{F}M_{0}$    is a Cartan subalgebra of the 3-Lie algebra $A_{\omega}^{\delta}$. Thanks to (3.3), for $\forall\alpha\in\mathbb{F},$

[TABLE]

Therefore, for $\forall\alpha\in\mathbb{F}$, $V_{\lambda+\alpha}=\mathbb{F}v_{\alpha}$ is the $\lambda+\alpha$-weight space with dimension one, and $V=\sum\limits_{\alpha\in\mathbb{F}}V_{\lambda+\alpha}$. The result follows. ∎

3.2. Induced modules of 3-Lie algebras

Let $B$ be a 3-Lie algebra over $\mathbb{F}$, $(W,\rho)$ be a 3-Lie algebra $B$-module, where

[TABLE]

Thanks to (2.3) and (2.4),

[TABLE]

is a subalgebra of $gl(W)$.

For $x,y,x^{\prime},y^{\prime}\in B$, if ad$(x,y)$=ad$(x^{\prime},y^{\prime})$, that is,

[TABLE]

then, for all $s,t\in M$,

[TABLE]

it follows that $\rho(x,y)-\rho(x^{\prime},y^{\prime})$ is in the center of the Lie algebra $L_{\rho}(W)$, that is,

[TABLE]

Therefore, in the case $Z(L_{\rho}(W))=0$, if ad$(x,y)$=ad$(x^{\prime},y^{\prime})$, then \rho(x,y)$$=\rho(x^{\prime},y^{\prime}).

Therefore, if $Z(L_{\rho}(W))=0$, we can define $\mathbb{F}$-linear mapping

[TABLE]

We get the following result.

Theorem 3.3**.**

Let $B$ be a 3-Lie algebra over $\mathbb{F}$, $(W,\rho)$ be a 3-Lie algebra $B$-module, and $Z(L_{\rho}(W))=0$. Then $(W,\bar{\rho})$ is a Lie algebra ad$(L)$-module, where $\bar{\rho}$ is defined by (3.6).

Proof.

Thanks to (2.3) and (2.4), for $\forall x,y,x^{\prime},y^{\prime}\in B$,

$[\bar{\rho}(\mbox{ad}(x,y)),\bar{\rho}(\mbox{ad}(x^{\prime},y^{\prime}))]$

$=\bar{\rho}(\mbox{ad}(x,y))\bar{\rho}(\mbox{ad}(x^{\prime},y^{\prime}))-\bar{\rho}(\mbox{ad}(x^{\prime},y^{\prime}))\bar{\rho}(\mbox{ad}(x,y))$

$=\rho(x,y)\rho(x^{\prime},y^{\prime})-\rho(x^{\prime},y^{\prime})\rho(x,y)=[\rho(x,y),\rho(x^{\prime},y^{\prime})]$

$=\rho([x,y,x^{\prime}],y^{\prime})+\rho(x^{\prime},[x,y,y^{\prime}])$

=\bar{\rho}(\mbox{ad}([x,y,x^{\prime}],y^{\prime}))+\bar{\rho}(\mbox{ad}(x^{\prime},[x,y,y^{\prime}]))=\bar{\rho}\big{(}[\mbox{ad}(x,y),\mbox{ad}(x^{\prime},y^{\prime})]\big{)}.

Therefore, $(W,\bar{\rho})$ is the inner derivation algebra ad$(B)$-module. ∎

The inner derivation algebra ad$(B)$-module $(W,\bar{\rho})$ is called the induced module associated with the 3-Lie algebra $B$-module $(W,\rho)$, where $\bar{\rho}$ is defined by (3.6), and $(W,\bar{\rho})$ is simply called the induced module.

Lemma 3.4**.**

[10]** The inner derivation algebra ad$(A_{\omega}^{\delta})$ of the 3-Lie algebra $A_{\omega}^{\delta}$ has a basis $\{~{}~{}p_{r},q_{r},x_{r},z_{r}|r\in Z~{}~{}\}$, where

[TABLE]

and the multiplication of ad$(A_{\omega}^{\delta})$ in the basis is as follows:

[TABLE]

Let $V$ be the vector space defined by (3.1), for arbitrary $\lambda,\mu\in\mathbb{F}$, define $\mathbb{F}$-linear mapping

[TABLE]

[TABLE]

Theorem 3.5**.**

For $\forall\lambda,\mu\in\mathbb{F}$, $(V,\psi_{\lambda\mu})$ is a reducible $\mbox{ad}(A_{\omega}^{\delta})$-module with submodules $U(\alpha)=\sum\limits_{m\in Z}\mathbb{F}v_{\alpha+m}$, $\forall\alpha\in\mathbb{F}$, and

1) if $\mu=0$, then $W=\mathbb{F}v_{-\lambda}$ is a trivial submodule; if  $\mu=1$, then $U^{\prime}(-\lambda)=\sum\limits_{m\in Z_{\neq 0}}\mathbb{F}v_{-\lambda+m}$ is irreducible;

2) $U(-\lambda)=\sum\limits_{m\in Z}\mathbb{F}v_{-\lambda+m}$ is irreducible if and only if $\mu\neq 1$ and $\mu\neq 0$.

Proof.

By Eqs (3.8) and (3.9), for $\forall r,s\in Z$, $\lambda,\mu,\alpha\in\mathbb{F}$,

[TABLE]

[TABLE]

Then we get

[TABLE]

It follows that $(V,\psi_{\lambda\mu})$ is an ad$(A_{\omega}^{\delta})$-module.

By Eqs (3.8) and (3.9), and the complete similar discussion to Theorem 3.1, $(V,\psi_{\mu\nu})$ satisfies 1) and 2). ∎

Theorem 3.6**.**

In the case $\mu=0$ and $\mu=1$, the Lie algebra ad$(A_{\omega}^{\delta})$-module $(V,\psi_{\lambda\mu})$ is the induced module associated with $T_{\lambda\mu}$.

Proof.

From Theorem 3.1 and Eq (3.3), the center $Z(L_{\rho_{\lambda\mu}}(V))=0$. Thanks to Theorem 3.3, ad$(A_{\omega}^{\delta})$-module $(V,\overline{\rho_{\lambda\mu}})$ is the induced module of $T_{\lambda\mu}$, where $\overline{\rho_{\lambda\mu}}$ is defined by (3.6). By (3.3) , $\forall r\in Z,\alpha,\lambda,\mu\in\mathbb{F}$,

[TABLE]

[TABLE]

Thanks to (3.7) and (3.8),

[TABLE]

[TABLE]

[TABLE]

Follows from (3.9), we get

[TABLE]

It follows the result. ∎

Remark 3.7**.**

From Theorem 3.6, in the case $\mu=0,1$, the inner derivation algebra ad$(A_{\omega}^{\delta})$-module $(V,\psi_{\lambda\mu})$ is the induced module. But in the case $\mu\neq 0,1$, by Theorem 3.1 and Theorem 3.6, the inner derivation algebra ad$(A_{\omega}^{\delta})$-module $(V,\psi_{\lambda\mu})$ can not be induced by $A_{\omega}^{\delta}$-modules.

At last of the paper, we construct an ad$(A_{\omega}^{\delta})$-module $(V,\phi_{\mu})$, which is not an induced module.

Define $\phi_{\mu}:\mbox{ad}(A_{\omega}^{\delta})\rightarrow gl(V)$:

[TABLE]

Theorem 3.8**.**

For any $\mu\in\mathbb{F},$ $(V,\phi_{\mu})$ is a reducible ad$(A_{\omega}^{\delta})$-module, where $\phi_{\mu}$ is defined by (3.10), and

1) for $\forall\alpha\in\mathbb{F}-Z,U(\alpha)$ is an irreducible submodule;

2) $U(0)$ is a indecomposable submodule containing an irreducible submodule $\bar{U}=\sum\limits_{m\in Z_{\neq 0}}\mathbb{F}v_{m}$;

3) $(V,\phi_{\mu})$ is not an induced module.

Proof.

Thanks to (3.8) and (3.10), for $\forall r,s\in Z$, $\mu\in\mathbb{F}$,

[TABLE]

[TABLE]

[TABLE]

Therefore,

[TABLE]

and

[TABLE]

It follows that $(V,\phi_{\mu})$ is an ad$(A_{\omega}^{\delta})$-module.

For any  $\alpha\in\mathbb{F}-Z$, and arbitrary distinct $m,m^{\prime}\in Z$, there are nonzero $r\in Z$ such that  $m^{\prime}=m-r$. By (3.10), we nave

[TABLE]

Thanks to  $\alpha\in\mathbb{F}-Z$, $m^{\prime}+\alpha\neq 0$, and U(\alpha)=\phi_{\mu}\big{(}ad(A_{\omega}^{\delta})\big{)}(v_{\alpha+m}). We get 1).

If $\alpha\in Z$, then $U(\alpha)=U(0)$.

For arbitrary distinct $m,m^{\prime}\in Z_{\neq 0}$, there is  nonzero $r_{0}\in Z$, such that $m^{\prime}=m-r_{0}$. Thanks to (3.10),

[TABLE]

Therefore, $\bar{U}=\sum\limits_{m\in Z_{\neq 0}}\mathbb{F}v_{m}$ is an irreducible submodule.

Since for any nonzero $r\in Z$,

[TABLE]

Follows from $\bar{U}=\sum\limits_{m\in Z_{\neq 0}}\mathbb{F}v_{m}$ is irreducible, $U(0)$ is an indecomposable. It follows 2).

If $(V,\phi_{\mu})$ is an induced module, then by (3.6), $\phi_{\mu}=\bar{\varphi}$, where

[TABLE]

and $(V,\varphi)$ is a 3-Lie algebra $A_{\omega}^{\delta}$-module.

Thanks to Eqs (3.6), (3.7) and (3.10), $\forall r,s\in Z$, $\alpha\in\mathbb{F}$, $\varphi$ satisfies that

[TABLE]

Therefore, $\forall r,m,s,l\in Z,\alpha,\mu\in\mathbb{F}$,

[TABLE]

[TABLE]

[TABLE]

$\varphi(L_{r},L_{s})\varphi(M_{m},M_{l})v_{\alpha}=0.$

Then for $r=4$, $s=3$, $m=2$, $l=1$, $\alpha=0$, and $\forall\mu\in\mathbb{F},$

[TABLE]

[TABLE]

that is, for some $l,r,s,m\in Z,$ we have

[TABLE]

Contradiction. It follows 3). The proof is complete. ∎

**Acknowledgements. ** The first author was supported in part by the Natural Science Foundation of Hebei Province (A2018201126).