On Two Classes of Extended 3-Lie Algebras
Show more
Abstract: In this paper, based on the existing research results, we obtain the unary extension 3-Lie algebras by one-dimensional extension of the known Lie algebra L. For two known 3-Lie algebras H, M, the (μ, ρ, β)-extension of H through M is given, and the necessary and sufficient conditions for the (μ, ρ, β)-extension algebra of H through M being 3-Lie algebra are obtained, and the structural characteristics and properties of these two kinds of extended 3-Lie algebras are given.

1. Introduction

In recent years, the study of 3-Lie algebra has been paid much attention because of its wide application in mathematics and physics. 3-Lie algebra is a special form of n-Lie algebra, which is an algebraic system with ternary linearly oblique symmetric multiplication table satisfying the generalized Jacobi equation [1]. 3-Lie algebra has extremely profound and rich algebraic and analytical structure. In this paper, the extension problem of 3-Lie algebra is studied on the basis of the existing research. Firstly, we define the unary extended 3-Lie algebra for a known Lie algebra L by one-dimensional extension, and study its properties. Secondly, for two known 3-Lie algebras H, M, the $\left(\mu ,\rho ,\beta \right)$ -extension of H through M is defined, and the $\left(\mu ,\rho ,\beta \right)$ -extension of H through M is given as a necessary and sufficient condition for the 3-Lie algebra. Finally, the structure and properties of this extended 3-Lie algebra are discussed. Thus, it lays a foundation for the further study of the properties of the derivatives of two kinds of 3-Lie algebras.

2. Fundamental Notions

Firstly, the basic knowledge [1] - [9] to be used in this paper is given.

Definition 2.1 Let A be a vector space over a domain F and have a 3-element linear operation $\left[\text{\hspace{0.17em}},\text{\hspace{0.17em}},\text{\hspace{0.17em}}\right]:A\wedge A\wedge A\to A$, satisfied for arbitrary, ${x}_{1},{x}_{2},{x}_{3},{y}_{2},{y}_{3}\in A$

$\left[\left[{x}_{1},{x}_{2},{x}_{3}\right],{y}_{2},{y}_{3}\right]=\underset{i=1}{\overset{3}{\sum }}\left[{x}_{1},\left[{x}_{i},{y}_{2},{y}_{3}\right],{x}_{3}\right]$, (1)

$\left(A,\left[\text{\hspace{0.17em}},\text{\hspace{0.17em}},\text{\hspace{0.17em}}\right]\right)$ is called 3-Lie algebra. Without confusion, A is called 3-Lie algebra for short.

Definition 2.2 Let A be a 3-Lie algebra, and D be a linear transformation ofA, if this equation is satisfied

$\left[D\left(x\right),y,z\right]+\left[x,D\left(y\right),z\right]+\left[x,y,D\left(z\right)\right]=D\left(\left[x,y,z\right]\right)$, $x,y,z\in A$ (2)

Then D is the derivative of A, and the set of derivatives is denoted by $Der\left(A\right)$. It is easy to prove that $Der\left(A\right)$ is a subalgebra of the general linear Lie algebra $gl\left(A\right)$.

The map

$ad\left({x}_{1},{x}_{2}\right):A\to A$, $ad\left({x}_{1},{x}_{2}\right)\left(x\right)=\left[{x}_{1},{x}_{2},x\right]$

for $x\in A$ is called the left multiplication defined by elements ${x}_{1},{x}_{2}\in A$. Obviously the left multiplication is the derivative. The linear combination of the left multiplication is called the inner derivative, denoted by $ad\left(A\right)$.

Let B be a subspace ofA, and if $\left[B,B,B\right]\subseteq B\left(\left[B,A,A\right]\subseteq B\right)$, then B be a subalgebra (ideal) of A. And if $\left[B,B,B\right]=0\left(\left[B,B,A\right]=0\right)$, then B is called a Abel subalgebra

(Abel ideal). In particular, the subalgebra spanned by $\left[{x}_{1},{x}_{2},{x}_{3}\right]\left(\forall {x}_{1},{x}_{2},{x}_{3}\in A\right)$ is called the derivative algebra of A, denoted by ${A}^{1}$. If ${A}^{1}=0$, then A is called Abel algebra. If an ideal I of A is a Abel subalgebra but not an Abel ideal, that is $\left[I,I,I\right]=0$, but $\left[I,I,A\right]\ne 0$, then I is called an hypo-abelian ideal.

The ideal I of a 3-Lie algebra A is called s-solvable, $2\le s\le 3$, if ${I}^{\left(k,s\right)}=0$ for some $k\ge 0$, where ${I}^{\left(0,s\right)}=I$, ${I}^{\left(k+1,s\right)}$ is defined as

${I}^{\left(k+1,s\right)}\text{=}\left[\underset{s}{\underset{︸}{{I}^{\left(k,s\right)},\cdots ,{I}^{\left(k,s\right)}}},\underset{3-s}{\underset{︸}{A,\cdots ,A}}\right]$. Where 2-solvable is also called solvable, and ${I}^{\left(k,s\right)}$ is abbreviated as ${I}^{\left(k\right)}$.

An ideal I of a 3-Lie algebra A is called nilpotent if ${I}^{s}=0$ for some $s\ge 0$, where ${I}^{0}=I$ and ${I}^{s}=\left[{I}^{s-1},I,A\right]$.

The center of A is denoted by $Z\left(A\right)=\left\{x\in A|\left[x,A,A\right]=0\right\}$. Obviously $Z\left(A\right)$ is the Abel ideal of A.

Let A is a 3-Lie algebra over the field F, V is a vector space, $\rho :A\wedge A\to End\left(V\right)$ is a linear mapping, if $\rho$ satisfies for any ${x}_{1},{x}_{2},{x}_{3},{x}_{4}\in A$

$\left[\rho \left({x}_{1},{x}_{2}\right),\rho \left({x}_{3},{x}_{4}\right)\right]=\rho \left(\left[{x}_{1},{x}_{2},{x}_{3}\right]{x}_{4}\right)-\rho \left(\left[{x}_{1},{x}_{2},{x}_{4}\right]{x}_{3}\right),$ (3)

$\begin{array}{c}\rho \left(\left[{x}_{1},{x}_{2},{x}_{3}\right],{x}_{4}\right)=\rho \left({x}_{1},{x}_{2}\right)\rho \left({x}_{3},{x}_{4}\right)+\rho \left({x}_{2},{x}_{3}\right)\rho \left({x}_{1},{x}_{4}\right)\\ \text{\hspace{0.17em}}+\rho \left({x}_{3},{x}_{1}\right)\rho \left({x}_{2},{x}_{4}\right)\end{array}$ (4)

Then $\left(V,\rho \right)$ is called the representation of A (or $\left(V,\rho \right)$ is A-module).

Lemma 2.1 Let A is a 3-Lie algebra over the field F, V is a vector space, $\rho :A\wedge A\to End\left(V\right)$ is a linear mapping. If $\left(V,\rho \right)$ is an A-module, then for any $x,y,z,u\in A$, the following equation is true:

$\rho \left(\left[x,y,z\right],u\right)-\rho \left(\left[x,y,u\right],z\right)+\rho \left(\left[x,z,u\right],y\right)-\rho \left(\left[y,z,u\right],x\right)=0$, (5)

$\begin{array}{l}\rho \left(x,u\right)\rho \left(y,z\right)+\rho \left(y,z\right)\rho \left(x,u\right)+\rho \left(x,y\right)\rho \left(z,u\right)+\rho \left(z,u\right)\rho \left(x,y\right)\\ -\rho \left(x,z\right)\rho \left(y,u\right)-\rho \left(y,u\right)\rho \left(x,z\right)=0.\end{array}$ (6)

3. The Unary Extension 3-Lie Algebra of Lie Algebras

Definition 3.1 Let $\left(L,\left[\text{\hspace{0.17em}},\text{\hspace{0.17em}}\right]\right)$ be a Lie algebra over a field F, let $A=L\oplus F{x}_{0}$ ${x}_{0}\in F$, and ${x}_{0}\notin L$. Linear operation $\left[\text{\hspace{0.17em}},\text{\hspace{0.17em}},\text{\hspace{0.17em}}\right]:A\wedge A\wedge A\to A$ for all $x,y,z\in L$ that satisfy the following multiplication table:

$\left[x,y,{x}_{0}\right]=\left[x,y\right],\left[x,y,z\right]=0$. (7)

Then A is called the unary extension of Lie algebra L. If $\left(A,\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]\right)$ is a 3-Lie algebra, then $\left(A,\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]\right)$ is a unary extension 3-Lie algebra of the Lie algebra L.

Lemma 3.1 let L be a Lie algebra over a field F. If let $A=L\oplus F{x}_{0}$ ${x}_{0}\in F$, ${x}_{0}\notin L$ and the multiplication of is defined by (7), then A is a 3-Lie algebra, and for positive integers m, the following equation holds

${A}^{\left(m\right)}={L}^{\left(m\right)},{A}^{\left(m,2\right)}={L}^{\left(m,2\right)}={L}^{\left(m\right)},{A}^{\left(2,3\right)}=0.$

Proof: By multiplication (7), direct calculation A is 3-Lie algebra. Due to the

${A}^{1}=\left[A,A,A\right]=\left[L,L,L\right]+\left[L,L,F{x}_{0}\right]={L}^{1}$,

${A}^{2}=\left[{A}^{1},A,A\right]=\left[{L}^{1},L,F{x}_{0}\right]={L}^{2}$,

Assume ${A}^{m-1}={L}^{m-1}$, then

${A}^{m}=\left[{A}^{m-1},A,A\right]=\left[{L}^{m-1},L+F{x}_{0},L+F{x}_{0}\right]=\left[{L}^{m-1},L\right]={L}^{m}$.

similarly, ${A}^{\left(m,2\right)}={L}^{\left(m,2\right)}={L}^{\left(m\right)}$ and ${A}^{\left(2,3\right)}=0$. The conclusion is proved.

Theorem 3.1 Let L be a Lie algebra on the field F and $A=L\oplus F{x}_{0}$ be a unary extension 3-Lie algebra, where ${x}_{0}\in F$ and ${x}_{0}\notin L$, then

1) A is 2-solvable if and only if L is a solvable Lie algebra.

2) A is nilpotent if and only if L is a nilpotent Lie algebra.

3) A is 3-solvable.

4) $Z\left(A\right)=Z\left(L\right)$.

Proof: According to lemma 3.1, (1), (2) and (3) can be obtained directly. It is proved below that (4) is true. If ${L}^{1}=0$, then ${A}^{1}={L}^{1}=0$ and $Z\left(A\right)=Z\left(L\right)$. If ${L}^{1}\ne 0$, then exists $y,z\in L$ such that $\left[y,z\right]\ne 0$. For any $x\in L$, $\lambda \in F$, $x+\lambda {x}_{0}\in Z\left(A\right)$, because of $\left[x+\lambda {x}_{0},y,z\right]=\lambda \left[y,z\right]=0$, therefore $\lambda =0$.And because $\left[x+\lambda {x}_{0},A,{x}_{0}\right]=\left[x,L\right]=0$, so $x\in Z\left(L\right)$. Therefore $Z\left(A\right)\subseteq Z\left(L\right)$. Obviously, the conclusion of $Z\left(L\right)\subseteq Z\left(A\right)$ is true.

Theorem 3.2 Let L be a Lie algebra on the field F and I be a subspace ofL:

1) I is an ideal of A if and only if I is an ideal of L.

2) Let $J=I\oplus F{x}_{0}$, then J is ideal of A if and only if ${L}^{1}\subseteq I$.

3) If ${L}^{1}\subseteq I$, then for positive integers m, ${J}^{\left(m,2\right)}\subseteq {I}^{\left(m-1\right)}$. If I is a solvable ideal of L, then J is a 2-solvable ideal of A.

4) If L is a simple Lie algebra, then L is hypo-abelian ideal of A.

Proof: From $\left[I,A,A\right]=\left[I,L,{x}_{0}\right]=\left[I,L\right]$, we can get (1). From Equation (7),

$\left[J,A,A\right]=\left[I,L,{x}_{0}\right]+\left[{x}_{0},L,L\right]=\left[I,L\right]+\left[L,L\right]$,

So $\left[J,A,A\right]\subseteq J$ if and only if $\left[L,L\right]\subseteq I$. That means (2) is true.

If I is the ideal of L and ${L}^{1}\subseteq I$, then

${J}^{\left(1,2\right)}=\left[J,J,A\right]=\left[I,I\right]+\left[I,L\right]={I}^{\left(1\right)}+\left[I,L\right]\subseteq {I}^{\left(1\right)}+I\subseteq I={I}^{\left(0\right)}$,

${J}^{\left(2,2\right)}=\left[{J}^{\left(1,2\right)},{J}^{\left(1,2\right)},A\right]=\left[I,I,L+F{x}_{0}\right]\subseteq {I}^{\left(1\right)}$,

Assuming ${J}^{\left(m-1,2\right)}\subseteq {I}^{\left(m-2\right)}$ is true, then

${J}^{\left(m,2\right)}=\left[{J}^{\left(m-1,2\right)},{J}^{\left(m-1,2\right)},A\right]\subseteq \left[{I}^{\left(m-2\right)},{I}^{\left(m-2\right)},L+F{x}_{0}\right]\subseteq {I}^{\left(m-1\right)}$.

Therefore (3) holds. If L is a simple Lie algebra, then L is ideal of A, and $\left[L,L,L\right]=0$, $\left[L,L,A\right]=\left[L,L,{x}_{0}\right]={L}^{1}\ne 0$. Therefore, L is hypo-abelian ideal of A. That’s the end of the argument.

4. $\left(\mu ,\rho ,\beta \right)$ -Extension of 3-Lie Algebras

Definition 4.1 Let $\left(H,{\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{H}\right)$ and $\left(M,{\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{M}\right)$ be two 3-Lie algebras over the field F, $A=M\oplus H$, and

$\rho :M\wedge M\to Der\left(H\right)$, $\beta :M\wedge H\to Der\left(H\right)$, $\mu :M\wedge M\wedge M\to H$

is linear mappings. Define a linear operation ${\left[\text{\hspace{0.17em}},\text{\hspace{0.17em}},\text{\hspace{0.17em}}\right]}_{\mu \rho \beta }:A\wedge A\wedge A\to A$, for any $x,y,z\in M$, $h,{h}_{1},{h}_{2}\in H$ that satisfies the multiplication table:

${\left[x,y,z\right]}_{\mu \rho \beta }={\left[x,y,z\right]}_{M}+\mu \left(x,y,z\right)$, ${\left[x,y,h\right]}_{\mu \rho \beta }=\rho \left(x,y\right)h$ (8)

${\left[{h}_{1},{h}_{2},{h}_{3}\right]}_{\mu \rho \beta }={\left[{h}_{1},{h}_{2},{h}_{3}\right]}_{H}$, ${\left[x,{h}_{1},{\begin{array}{c}h\end{array}}_{2}\right]}_{\mu \rho \beta }=\beta \left(x,{h}_{1}\right){h}_{2}$.

Then $\left(A,{\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{\mu \rho \beta }\right)$ is called the $\left(\mu ,\rho ,\beta \right)$ -extension of H through M. If $\left(A,{\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{\mu \rho \beta }\right)$ is a 3-Lie algebra, then $\left(A,{\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{\mu \rho \beta }\right)$ is $\left(\mu ,\rho ,\beta \right)$ -extension algebra of 3-Lie algebra. If $\beta =0$, then A is called $\left(\mu ,\rho \right)$ -extension of H through M, and ${\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{\mu \rho \beta }$ denoted as ${\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{\mu \rho }$. For convenience, we will abbreviate ${\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{M}$ and ${\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{H}$ as $\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]$ and ${\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{\mu \rho \beta }$ as ${\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{A}$.

Lemma 4.1 Let $\left(H,{\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{H}\right)$ and $\left(M,{\left[\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ },\text{\hspace{0.17em}}\text{ }\right]}_{M}\right)$ be two 3-Lie algebras over the field F, and A be the $\left(\mu ,\rho ,\beta \right)$ -extension of H through M, and for all ${x}_{1},{x}_{2},{x}_{3},{x}_{4}\in M$ satisfy

$\begin{array}{c}\rho \left({x}_{4},\left[{x}_{1},{x}_{2},{x}_{3}\right]\right)=\rho \left({x}_{3},{x}_{1}\right)\rho \left({x}_{4},{x}_{2}\right)-\rho \left({x}_{2},{x}_{1}\right)\rho \left({x}_{4},{x}_{3}\right)\\ \text{\hspace{0.17em}}+\rho \left({x}_{2},{x}_{3}\right)\rho \left({x}_{4},{x}_{1}\right)-\beta \left({x}_{4},\mu \left({x}_{1},{x}_{2},{x}_{3}\right)\right).\end{array}$ (9)

Then Equation (6) is true if and only if the following equation

$\begin{array}{l}\rho \left({x}_{4},\left[{x}_{1},{x}_{2},{x}_{3}\right]\right)\\ =\rho \left({x}_{3},\left[{x}_{1},{x}_{2},{x}_{4}\right]\right)-\beta \left({x}_{4},\mu \left({x}_{1},{x}_{2},{x}_{3}\right)\right)+\beta \left({x}_{3},\mu \left({x}_{1},{x}_{2},{x}_{4}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\rho \left({x}_{1},{x}_{2}\right)\rho \left({x}_{3},{x}_{4}\right)+\rho \left({x}_{3},{x}_{4}\right)\rho \left({x}_{1},{x}_{2}\right).\end{array}$ (10)

Proof: From Equation (9), we can get

$\begin{array}{c}\rho \left({x}_{3},\left[{x}_{1},{x}_{2},{x}_{4}\right]\right)=\rho \left({x}_{2},{x}_{4}\right)\rho \left({x}_{3},{x}_{1}\right)-\rho \left({x}_{1},{x}_{4}\right)\rho \left({x}_{3},{x}_{2}\right)\\ \text{\hspace{0.17em}}+\rho \left({x}_{1},{x}_{2}\right)\rho \left({x}_{3},{x}_{4}\right)-\beta \left({x}_{3},\mu \left({x}_{1},{x}_{2},{x}_{4}\right)\right),\end{array}$

$\begin{array}{l}\rho \left({x}_{4},\left[{x}_{1},{x}_{2},{x}_{3}\right]\right)-\rho \left({x}_{3},\left[{x}_{1},{x}_{2},{x}_{4}\right]\right)\\ =\rho \left({x}_{1},{x}_{3}\right)\rho \left({x}_{2},{x}_{4}\right)-\rho \left({x}_{1},{x}_{2}\right)\rho \left({x}_{3},{x}_{4}\right)-\rho \left({x}_{2},{x}_{3}\right)\rho \left({x}_{1},{x}_{4}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\beta \left({x}_{4},\mu \left({x}_{1},{x}_{2},{x}_{3}\right)\right)+\rho \left({x}_{2},{x}_{4}\right)\rho \left({x}_{1},{x}_{3}\right)-\rho \left({x}_{1},{x}_{4}\right)\rho \left({x}_{2},{x}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\rho \left({x}_{1},{x}_{2}\right)\rho \left({x}_{3},{x}_{4}\right)+\beta \left({x}_{3},\mu \left({x}_{1},{x}_{2},{x}_{4}\right)\right)\end{array}$

$\begin{array}{l}=\rho \left({x}_{1},{x}_{3}\right)\rho \left({x}_{2},{x}_{4}\right)+\rho \left({x}_{2},{x}_{4}\right)\rho \left({x}_{1},{x}_{3}\right)-\rho \left({x}_{2},{x}_{3}\right)\rho \left({x}_{1},{x}_{4}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\rho \left({x}_{1},{x}_{4}\right)\rho \left({x}_{2},{x}_{3}\right)-2\rho \left({x}_{1},{x}_{2}\right)\rho \left({x}_{3},{x}_{4}\right)-\beta \left({x}_{4},\mu \left({x}_{1},{x}_{2},{x}_{3}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\beta \left({x}_{3},\mu \left({x}_{1},{x}_{2},{x}_{4}\right)\right).\end{array}$

So Equation (10) holds. On the other hand, if

$\begin{array}{c}\rho \left({x}_{4},\left[{x}_{1},{x}_{2},{x}_{3}\right]\right)=\rho \left({x}_{2},{x}_{4}\right)\rho \left({x}_{3},{x}_{1}\right)-\rho \left({x}_{1},{x}_{4}\right)\rho \left({x}_{3},{x}_{2}\right)\\ \text{\hspace{0.17em}}+\rho \left({x}_{1},{x}_{2}\right)\rho \left({x}_{3},{x}_{4}\right)-\beta \left({x}_{3},\mu \left({x}_{1},{x}_{2},{x}_{4}\right)\right)\\ \text{\hspace{0.17em}}-\beta \left({x}_{4},\mu \left({x}_{1},{x}_{2},{x}_{3}\right)\right)+\beta \left({x}_{3},\mu \left({x}_{1},{x}_{2},{x}_{4}\right)\right)\\ \text{\hspace{0.17em}}-\rho \left({x}_{1},{x}_{2}\right)\rho \left({x}_{3},{x}_{4}\right)+\rho \left({x}_{3},{x}_{4}\right)\rho \left({x}_{1},{x}_{2}\right).\end{array}$

Through Equation (9), it can be concluded that Equation (6) holds.

Lemma 4.2. Let A be the $\left(\mu ,\rho ,\beta \right)$ -extension of H through M, for all ${x}_{1},{x}_{2}\in M$, ${h}_{1},{h}_{2},h\in H$ satisfies

$\begin{array}{l}\beta \left(y,{h}_{2}\right)\beta \left(x,{h}_{1}\right)h-\beta \left(y,h\right)\beta \left(x,{h}_{1}\right){h}_{2}-\beta \left(x,{h}_{1}\right)\beta \left(y,{h}_{2}\right)h\\ =\left[\rho \left(x,y\right){h}_{1},{h}_{2},h\right]\end{array}$ (11)

There are

$\begin{array}{l}\rho \left(x,y\right)\left[{h}_{1},{h}_{2},h\right]+\beta \left(y,{h}_{1}\right)\beta \left(x,{h}_{2}\right)h-\beta \left(x,{h}_{1}\right)\beta \left(y,{h}_{2}\right)h\\ =\left[\rho \left(x,y\right){h}_{1},{h}_{2},h\right]\end{array}$ (12)

Proof: From Equation (11) and the $\rho \left(x,y\right)$ is derivative of H, we can get

$\begin{array}{l}\left[{h}_{1},\rho \left(x,y\right){h}_{2},h\right]\\ =\beta \left(y,{h}_{1}\right)\beta \left(x,h\right){h}_{2}+\beta \left(y,h\right)\beta \left(x,{h}_{2}\right){h}_{1}+\beta \left(x,{h}_{2}\right)\beta \left(y,{h}_{1}\right)h\end{array}$

$\begin{array}{l}\left[{h}_{1},{h}_{2},\rho \left(x,y\right)h\right]\\ =\beta \left(y,{h}_{2}\right)\beta \left(x,{h}_{1}\right)h+\beta \left(y,{h}_{1}\right)\beta \left(x,h\right){h}_{2}+\beta \left(x,h\right)\beta \left(y,{h}_{2}\right){h}_{1}\end{array}$

$\begin{array}{l}\rho \left(x,y\right)\left[{h}_{1},{h}_{2},h\right]\\ =2\left(\beta \left(y,{h}_{1}\right)\beta \left(x,h\right){h}_{2}+\beta \left(y,{h}_{2}\right)\beta \left(x,{h}_{1}\right)h+\beta \left(y,h\right)\beta \left(x,{h}_{2}\right){h}_{1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\beta \left(x,{h}_{1}\right)\beta \left(y,h\right){h}_{2}+\beta \left(x,{h}_{2}\right)\beta \left(y,{h}_{1}\right)h+\beta \left(x,h\right)\beta \left(y,{h}_{2}\right){h}_{1},\end{array}$

so

$\begin{array}{l}\beta \left(y,{h}_{1}\right)\beta \left(y,h\right){h}_{2}+\beta \left(y,{h}_{2}\right)\beta \left(x,{h}_{1}\right)h+\beta \left(y,h\right)\beta \left(x,{h}_{2}\right){h}_{1}\\ +\beta \left(x,{h}_{1}\right)\beta \left(y,h\right){h}_{2}+\beta \left(x,{h}_{2}\right)\beta \left(y,{h}_{1}\right)h+\beta \left(y,h\right)\beta \left(y,{h}_{2}\right){h}_{1}=0,\end{array}$

$\begin{array}{l}\rho \left(x,y\right)\left[{h}_{1},{h}_{2},h\right]\\ =\beta \left(y,{h}_{1}\right)\beta \left(x,h\right){h}_{2}+\beta \left(y,{h}_{2}\right)\beta \left(x,{h}_{1}\right)h+\beta \left(y,h\right)\beta \left(x,{h}_{2}\right){h}_{1}\end{array}$

Namely

$\begin{array}{l}\beta \left(y,{h}_{2}\right)\beta \left(x,{h}_{1}\right)h-\beta \left(y,h\right)\beta \left(x,{h}_{1}\right){h}_{2}\\ =\rho \left(x,y\right)\left[{h}_{1},{h}_{2},h\right]+\beta \left(y,{h}_{1}\right)\beta \left(x,{h}_{2}\right)h.\end{array}$

Using Equation (11) again, Equation (12) can be obtained.

Lemma 4.3. Let A be the $\left(\mu ,\rho ,\beta \right)$ -extension of H through M. If for all $x\in M$, satisfies

(13)

Then

(14)

Proof: According to Equation (13),

Because of, therefore

Hence, Equation (14) holds.

Theorem 4.1. Let A be the -extension of H through M, then A is a 3-Lie algebra if and only if for any, , Equations (6), (9), (11), (13) and the following are true,

(15)

(16)

(17)

Proof: If A is a 3-Lie algebra, the Equations (11), (15), (16) and (17) are obtained from the Equations (1). The following proves that Equations (6), (9) and (13) are true.

For, , according to (8),

As a result,

In the above formula, is replaced by, and Equation (9) can be obtained.

Because,

So Equation (10) holds. Equation (6) is obtained from lemma 4.1.

For arbitrary, , it can be known from (8) that,

As a result,

Because of, therefore

Equation (13) holds.

Conversely, to prove that A is a 3-Lie algebra, it is only necessary to prove that (8) satisfies Equation (1).

Case 1. For all, known by (8)

From Equation (17), we can get

Case 2. For all, , know from (8)

In Equation (9), by substitution for, we can get

As a result,

Due to the,

Through lemma 4.1 and Equation (9), we can get

According to Equations (6) and (10),

Case 3. For all, , it is obtained from Equations (15), (16)

Because,

Through the direct calculation of Equations (15) and (16),

As a result,

Case 4. For all, , due to the, it can be concluded from Equation (11) that,

Because of,

Then

According to lemma 4.2,

Using Equation (11) again, we can get

namely

.

Case 5. For all, , because, through Equation (13),

To sum up, (8) satisfies Equation (1). The conclusion is proved.

Theorem 4.2 Let be the -extension of 3-Lie algebra H through M. So is M-module if and only if.

Proof: If, obviously is an M-module.

On the other hand, to any, by theorem 4.1 and Equation (9), (10),

As a result,

According to Equation (9),. And the theorem is proved.

Theorem 4.3 Let be the -extension of 3-Lie algebra H through M and be an M-module. So A is a 3-Lie algebra if and only if, and Equation (17) is true.

Proof: If A is a 3-Lie algebra, Equation (17) holds by theorem 4.1. Since is M-module, then. And, can be obtained by Equations (9) and (13). Conversely, from theorems 4.1 and 4.2, A is a 3-Lie algebra.

The above conclusions about 3-Lie algebras will be helpful for further study of their derivation algebras.

Funding

Science and Technology Research Project of Higher Education Department of Hebei Province (Z2015009).

Cite this paper: Cheng, Y. and Gao, Y. (2021) On Two Classes of Extended 3-Lie Algebras. Journal of Applied Mathematics and Physics, 9, 834-845. doi: 10.4236/jamp.2021.94056.
References

[1]   Filippov, V.T. (1985) n-Lie Algebras. Siberian Mathematical Journal, 26, 126-140.
https://doi.org/10.1007/BF00969110

[2]   Lu, F.Y. (2007) Lie Triple Derivations on Nest Algebras. Mathematische Nachrichten, 280, 882-887.
https://doi.org/10.1002/mana.200410520

[3]   Papageorgakis, C. and Samann, C. (2011) The 3-Lie Algebra (2, 0) Tensor Multiplet and Equations of Motion on Loop Space. Journal of High Energy Physics, 2011, Article No. 99.
https://doi.org/10.1007/JHEP05(2011)099

[4]   Bremner, M. and Elgendy, H. (1987) Universal Associative Envelopes of (n + 1)-Dimensional n-Lie Algebras.

[5]   Bai, R.P., Bai, C.M. and Wang, J.X. (2010) Realizations of 3-Lie Algebras. Journal of Mathematical Physics, 51, Article ID: 063505.
https://doi.org/10.1063/1.3436555

[6]   Bai, R. and Wu, Y. (2015) Constructions of 3-Lie Algebras. Linear and Multilinear Algebra, 63, 2171-2186.
https://doi.org/10.1080/03081087.2014.986121

[7]   Bai, R., Wu, W. and Li, Z. (2012) Some Results on Metric n-Lie Algebras. Acta Mathematics Sinica, English Series, 28, 1209-1220.
https://doi.org/10.1007/s10114-011-0231-4

[8]   Bai, R., Chen, S. and Cheng, R. (2016) Symplectic Structures on 3-Lie Algebras.

[9]   Bai, R., Wu, W. and Li, Y. (2012) Module Extensions of 3-Lie Algebras. Linear and Multilinear Algebra, 60, 433-447.
https://doi.org/10.1080/03081087.2011.603728

Top