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 JAMP  Vol.9 No.4 , April 2021
On Two Classes of Extended 3-Lie Algebras
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 ( μ , ρ , β ) -extension of H through M is defined, and the ( μ , ρ , β ) -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 [ , , ] : A A A A , satisfied for arbitrary, x 1 , x 2 , x 3 , y 2 , y 3 A

[ [ x 1 , x 2 , x 3 ] , y 2 , y 3 ] = i = 1 3 [ x 1 , [ x i , y 2 , y 3 ] , x 3 ] , (1)

( A , [ , , ] ) 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

[ D ( x ) , y , z ] + [ x , D ( y ) , z ] + [ x , y , D ( z ) ] = D ( [ x , y , z ] ) , x , y , z A (2)

Then D is the derivative of A, and the set of derivatives is denoted by D e r ( A ) . It is easy to prove that D e r ( A ) is a subalgebra of the general linear Lie algebra g l ( A ) .

The map

a d ( x 1 , x 2 ) : A A , a d ( x 1 , x 2 ) ( x ) = [ x 1 , x 2 , x ]

for x A is called the left multiplication defined by elements x 1 , x 2 A . Obviously the left multiplication is the derivative. The linear combination of the left multiplication is called the inner derivative, denoted by a d ( A ) .

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

(Abel ideal). In particular, the subalgebra spanned by [ x 1 , x 2 , x 3 ] ( x 1 , x 2 , x 3 A ) 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 [ I , I , I ] = 0 , but [ I , I , A ] 0 , then I is called an hypo-abelian ideal.

The ideal I of a 3-Lie algebra A is called s-solvable, 2 s 3 , if I ( k , s ) = 0 for some k 0 , where I ( 0 , s ) = I , I ( k + 1 , s ) is defined as

I ( k + 1 , s ) = [ I ( k , s ) , , I ( k , s ) s , A , , A 3 s ] . Where 2-solvable is also called solvable, and I ( k , s ) is abbreviated as I ( k ) .

An ideal I of a 3-Lie algebra A is called nilpotent if I s = 0 for some s 0 , where I 0 = I and I s = [ I s 1 , I , A ] .

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

Let A is a 3-Lie algebra over the field F, V is a vector space, ρ : A A E n d ( V ) is a linear mapping, if ρ satisfies for any x 1 , x 2 , x 3 , x 4 A

[ ρ ( x 1 , x 2 ) , ρ ( x 3 , x 4 ) ] = ρ ( [ x 1 , x 2 , x 3 ] x 4 ) ρ ( [ x 1 , x 2 , x 4 ] x 3 ) , (3)

ρ ( [ x 1 , x 2 , x 3 ] , x 4 ) = ρ ( x 1 , x 2 ) ρ ( x 3 , x 4 ) + ρ ( x 2 , x 3 ) ρ ( x 1 , x 4 ) + ρ ( x 3 , x 1 ) ρ ( x 2 , x 4 ) (4)

Then ( V , ρ ) is called the representation of A (or ( V , ρ ) is A-module).

Lemma 2.1 Let A is a 3-Lie algebra over the field F, V is a vector space, ρ : A A E n d ( V ) is a linear mapping. If ( V , ρ ) is an A-module, then for any x , y , z , u A , the following equation is true:

ρ ( [ x , y , z ] , u ) ρ ( [ x , y , u ] , z ) + ρ ( [ x , z , u ] , y ) ρ ( [ y , z , u ] , x ) = 0 , (5)

ρ ( x , u ) ρ ( y , z ) + ρ ( y , z ) ρ ( x , u ) + ρ ( x , y ) ρ ( z , u ) + ρ ( z , u ) ρ ( x , y ) ρ ( x , z ) ρ ( y , u ) ρ ( y , u ) ρ ( x , z ) = 0. (6)

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

Definition 3.1 Let ( L , [ , ] ) be a Lie algebra over a field F, let A = L F x 0 x 0 F , and x 0 L . Linear operation [ , , ] : A A A A for all x , y , z L that satisfy the following multiplication table:

[ x , y , x 0 ] = [ x , y ] , [ x , y , z ] = 0 . (7)

Then A is called the unary extension of Lie algebra L. If ( A , [ , , ] ) is a 3-Lie algebra, then ( A , [ , , ] ) 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 F x 0 x 0 F , x 0 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 ( m ) = L ( m ) , A ( m , 2 ) = L ( m , 2 ) = L ( m ) , A ( 2 , 3 ) = 0.

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

A 1 = [ A , A , A ] = [ L , L , L ] + [ L , L , F x 0 ] = L 1 ,

A 2 = [ A 1 , A , A ] = [ L 1 , L , F x 0 ] = L 2 ,

Assume A m 1 = L m 1 , then

A m = [ A m 1 , A , A ] = [ L m 1 , L + F x 0 , L + F x 0 ] = [ L m 1 , L ] = L m .

similarly, A ( m , 2 ) = L ( m , 2 ) = L ( m ) and A ( 2 , 3 ) = 0 . The conclusion is proved.

Theorem 3.1 Let L be a Lie algebra on the field F and A = L F x 0 be a unary extension 3-Lie algebra, where x 0 F and x 0 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 ( A ) = Z ( L ) .

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 ( A ) = Z ( L ) . If L 1 0 , then exists y , z L such that [ y , z ] 0 . For any x L , λ F , x + λ x 0 Z ( A ) , because of [ x + λ x 0 , y , z ] = λ [ y , z ] = 0 , therefore λ = 0 .And because [ x + λ x 0 , A , x 0 ] = [ x , L ] = 0 , so x Z ( L ) . Therefore Z ( A ) Z ( L ) . Obviously, the conclusion of Z ( L ) Z ( A ) 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 F x 0 , then J is ideal of A if and only if L 1 I .

3) If L 1 I , then for positive integers m, J ( m , 2 ) I ( m 1 ) . 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 [ I , A , A ] = [ I , L , x 0 ] = [ I , L ] , we can get (1). From Equation (7),

[ J , A , A ] = [ I , L , x 0 ] + [ x 0 , L , L ] = [ I , L ] + [ L , L ] ,

So [ J , A , A ] J if and only if [ L , L ] I . That means (2) is true.

If I is the ideal of L and L 1 I , then

J ( 1 , 2 ) = [ J , J , A ] = [ I , I ] + [ I , L ] = I ( 1 ) + [ I , L ] I ( 1 ) + I I = I ( 0 ) ,

J ( 2 , 2 ) = [ J ( 1 , 2 ) , J ( 1 , 2 ) , A ] = [ I , I , L + F x 0 ] I ( 1 ) ,

Assuming J ( m 1 , 2 ) I ( m 2 ) is true, then

J ( m , 2 ) = [ J ( m 1 , 2 ) , J ( m 1 , 2 ) , A ] [ I ( m 2 ) , I ( m 2 ) , L + F x 0 ] I ( m 1 ) .

Therefore (3) holds. If L is a simple Lie algebra, then L is ideal of A, and [ L , L , L ] = 0 , [ L , L , A ] = [ L , L , x 0 ] = L 1 0 . Therefore, L is hypo-abelian ideal of A. That’s the end of the argument.

4. ( μ , ρ , β ) -Extension of 3-Lie Algebras

Definition 4.1 Let ( H , [ , , ] H ) and ( M , [ , , ] M ) be two 3-Lie algebras over the field F, A = M H , and

ρ : M M D e r ( H ) , β : M H D e r ( H ) , μ : M M M H

is linear mappings. Define a linear operation [ , , ] μ ρ β : A A A A , for any x , y , z M , h , h 1 , h 2 H that satisfies the multiplication table:

[ x , y , z ] μ ρ β = [ x , y , z ] M + μ ( x , y , z ) , [ x , y , h ] μ ρ β = ρ ( x , y ) h (8)

[ h 1 , h 2 , h 3 ] μ ρ β = [ h 1 , h 2 , h 3 ] H , [ x , h 1 , h 2 ] μ ρ β = β ( x , h 1 ) h 2 .

Then ( A , [ , , ] μ ρ β ) is called the ( μ , ρ , β ) -extension of H through M. If ( A , [ , , ] μ ρ β ) is a 3-Lie algebra, then ( A , [ , , ] μ ρ β ) is ( μ , ρ , β ) -extension algebra of 3-Lie algebra. If β = 0 , then A is called ( μ , ρ ) -extension of H through M, and [ , , ] μ ρ β denoted as [ , , ] μ ρ . For convenience, we will abbreviate [ , , ] M and [ , , ] H as [ , , ] and [ , , ] μ ρ β as [ , , ] A .

Lemma 4.1 Let ( H , [ , , ] H ) and ( M , [ , , ] M ) be two 3-Lie algebras over the field F, and A be the ( μ , ρ , β ) -extension of H through M, and for all x 1 , x 2 , x 3 , x 4 M satisfy

ρ ( x 4 , [ x 1 , x 2 , x 3 ] ) = ρ ( x 3 , x 1 ) ρ ( x 4 , x 2 ) ρ ( x 2 , x 1 ) ρ ( x 4 , x 3 ) + ρ ( x 2 , x 3 ) ρ ( x 4 , x 1 ) β ( x 4 , μ ( x 1 , x 2 , x 3 ) ) . (9)

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

ρ ( x 4 , [ x 1 , x 2 , x 3 ] ) = ρ ( x 3 , [ x 1 , x 2 , x 4 ] ) β ( x 4 , μ ( x 1 , x 2 , x 3 ) ) + β ( x 3 , μ ( x 1 , x 2 , x 4 ) ) ρ ( x 1 , x 2 ) ρ ( x 3 , x 4 ) + ρ ( x 3 , x 4 ) ρ ( x 1 , x 2 ) . (10)

Proof: From Equation (9), we can get

ρ ( x 3 , [ x 1 , x 2 , x 4 ] ) = ρ ( x 2 , x 4 ) ρ ( x 3 , x 1 ) ρ ( x 1 , x 4 ) ρ ( x 3 , x 2 ) + ρ ( x 1 , x 2 ) ρ ( x 3 , x 4 ) β ( x 3 , μ ( x 1 , x 2 , x 4 ) ) ,

ρ ( x 4 , [ x 1 , x 2 , x 3 ] ) ρ ( x 3 , [ x 1 , x 2 , x 4 ] ) = ρ ( x 1 , x 3 ) ρ ( x 2 , x 4 ) ρ ( x 1 , x 2 ) ρ ( x 3 , x 4 ) ρ ( x 2 , x 3 ) ρ ( x 1 , x 4 ) β ( x 4 , μ ( x 1 , x 2 , x 3 ) ) + ρ ( x 2 , x 4 ) ρ ( x 1 , x 3 ) ρ ( x 1 , x 4 ) ρ ( x 2 , x 3 ) ρ ( x 1 , x 2 ) ρ ( x 3 , x 4 ) + β ( x 3 , μ ( x 1 , x 2 , x 4 ) )

= ρ ( x 1 , x 3 ) ρ ( x 2 , x 4 ) + ρ ( x 2 , x 4 ) ρ ( x 1 , x 3 ) ρ ( x 2 , x 3 ) ρ ( x 1 , x 4 ) ρ ( x 1 , x 4 ) ρ ( x 2 , x 3 ) 2 ρ ( x 1 , x 2 ) ρ ( x 3 , x 4 ) β ( x 4 , μ ( x 1 , x 2 , x 3 ) ) + β ( x 3 , μ ( x 1 , x 2 , x 4 ) ) .

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

ρ ( x 4 , [ x 1 , x 2 , x 3 ] ) = ρ ( x 2 , x 4 ) ρ ( x 3 , x 1 ) ρ ( x 1 , x 4 ) ρ ( x 3 , x 2 ) + ρ ( x 1 , x 2 ) ρ ( x 3 , x 4 ) β ( x 3 , μ ( x 1 , x 2 , x 4 ) ) β ( x 4 , μ ( x 1 , x 2 , x 3 ) ) + β ( x 3 , μ ( x 1 , x 2 , x 4 ) ) ρ ( x 1 , x 2 ) ρ ( x 3 , x 4 ) + ρ ( x 3 , x 4 ) ρ ( x 1 , x 2 ) .

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

Lemma 4.2. Let A be the ( μ , ρ , β ) -extension of H through M, for all x 1 , x 2 M , h 1 , h 2 , h H satisfies

β ( y , h 2 ) β ( x , h 1 ) h β ( y , h ) β ( x , h 1 ) h 2 β ( x , h 1 ) β ( y , h 2 ) h = [ ρ ( x , y ) h 1 , h 2 , h ] (11)

There are

ρ ( x , y ) [ h 1 , h 2 , h ] + β ( y , h 1 ) β ( x , h 2 ) h β ( x , h 1 ) β ( y , h 2 ) h = [ ρ ( x , y ) h 1 , h 2 , h ] (12)

Proof: From Equation (11) and the ρ ( x , y ) is derivative of H, we can get

[ h 1 , ρ ( x , y ) h 2 , h ] = β ( y , h 1 ) β ( x , h ) h 2 + β ( y , h ) β ( x , h 2 ) h 1 + β ( x , h 2 ) β ( y , h 1 ) h

[ h 1 , h 2 , ρ ( x , y ) h ] = β ( y , h 2 ) β ( x , h 1 ) h + β ( y , h 1 ) β ( x , h ) h 2 + β ( x , h ) β ( y , h 2 ) h 1

ρ ( x , y ) [ h 1 , h 2 , h ] = 2 ( β ( y , h 1 ) β ( x , h ) h 2 + β ( y , h 2 ) β ( x , h 1 ) h + β ( y , h ) β ( x , h 2 ) h 1 ) + β ( x , h 1 ) β ( y , h ) h 2 + β ( x , h 2 ) β ( y , h 1 ) h + β ( x , h ) β ( y , h 2 ) h 1 ,

so

β ( y , h 1 ) β ( y , h ) h 2 + β ( y , h 2 ) β ( x , h 1 ) h + β ( y , h ) β ( x , h 2 ) h 1 + β ( x , h 1 ) β ( y , h ) h 2 + β ( x , h 2 ) β ( y , h 1 ) h + β ( y , h ) β ( y , h 2 ) h 1 = 0 ,

ρ ( x , y ) [ h 1 , h 2 , h ] = β ( y , h 1 ) β ( x , h ) h 2 + β ( y , h 2 ) β ( x , h 1 ) h + β ( y , h ) β ( x , h 2 ) h 1

Namely

β ( y , h 2 ) β ( x , h 1 ) h β ( y , h ) β ( x , h 1 ) h 2 = ρ ( x , y ) [ h 1 , h 2 , h ] + β ( y , h 1 ) β ( x , h 2 ) h .

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

Lemma 4.3. Let A be the ( μ , ρ , β ) -extension of H through M. If for all x 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.
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