arXiv:1802.06961v1 [math.RA] 20 Feb 2018

ON CLASSIFICATION OF (n + 5)-DIMENSIONAL NILPOTENT
n-LIE ALGEBRA OF CLASS TWO
ZAHRA HOSEINI1 , FARSHID SAEEDI1 , AND HAMID DARABI2
Abstract. In this paper, we classify (n + 5)-dimensional nilpotent n-Lie algebras of class two over the arbitrary field, when n ≥ 3.

1. Introduction and Preliminaries
The classification of low dimensional Lie algebras is one of the fundamental
issues in Lie algebras Theory. The classification of Lie algebras can be found in
many books and papers. In 1950 Morozov [11] proposed a classification of sixdimensional nilpotent Lie algebras over fields of characteristic 0. The classification
of the 6-dimensional Lie algebras on the arbitrary field was shown by Cicalo et
al. [4]. Moreover, the 7-dimensional nilpotent Lie algebras over algebraically closed
fields and real number field were classified in [9]. In 1985, Filippov [8] introduced the
n-Lie algebras as a non-symmetrical linear vector space which satisfies the following
Jacobian identity:
n
X
[x1 , . . . , xi−1 , [xi , y2 , . . . , yn ] , xi+1 , . . . , xn ]
[[x1 , x2 , . . . , xn ] , y2 , . . . , yn ] =
i=1

for all xi , yj ∈ L , 1 ≤ i ≤ n, 2 ≤ j ≤ n. He also classified n-Lie algebras of
dimensions n and n + 1 on the algebraically closed field with characteristic zero.
In 2008, Bai et al. [2] classified n-Lie algebras of dimension n + 1 on fields of
characteristic two. Then, Bai [1] classified n-Lie algebras of dimension n + 2 on the
algebraically closed fields with characteristic zero.
Assume that A1 , . . . ,An are the subalgebra of n-Lie algebra A. Then, subalgebra
A generated by all vectors [x1 , . . . , xn ](xi ∈ Ai ) will be represented by the symbol
[A1 , . . . , An ]. The subalgebra A2 = [A, . . . , A] is called derived n-Lie algebra of A.
The center of n-Lie algebra A is defined as follows
Z (A) = {x ∈ A : [x, A, . . . , A] = 0} .
Assume Z0 (A) = 0, then ith center of A is defined inductively as Zi (A)/Zi−1 (A) =
Z (A/Zi−1 (A) ) for all i ≥ 1. The notion of nilpotent n-Lie algebra was defined
by Kasymov [10] as follows: We say that a n-Lie algebra A is nilpotent if As = 0
i
where, s is a non-negative
 i integer number. Note that A is defined as induction
1
i+1
by A = A , A
= A , A, . . . , A . The n-Lie algebra A is nilpotent of class c if
Ac+1 = 0, and Ai 6= 0, for each i ≤ c.
An important category of n-Lie algebras of class 2, which plays an important
role in nilpotent n-Lie algebras, is the Heisenberg n-Lie algebras. We call n-Lie
2010 Mathematics Subject Classification. Primary 17B05, 17B30; Secondary 17D99.
Key words and phrases. Nilpotent n-Lie algebra, Classification, Nilpotent n-Lie algebra of
class two.
1

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Z. HOSEINI, F. SAEEDI, AND H. DARABI

algebra A, generalized Heisenberg of rank k, if A2 = Z(A) and dim A2 = k. In
[6] the authors study the case when k = 1, which is called later special Heisenberg
n-Lie algebras.
The rest of our paper is organized as follows: Section 2 includes the results that
are used frequently in the next section. In Section 3, we classify (n+ 5)-dimensional
n-Lie algebras of class two.
2. Some Known Results
In this section, we introduce some known and necessary results.
Theorem 2.1. ([6]) Every special Heisenberg n-Lie algebras have a dimension
mn + 1 for some natural number m, and it is isomorphic to


H (n, m) = x, x1 , . . . , xnm : xn(i−1)+1 , xn(i−1)+2 , . . . , xni = x, i = 1, . . . , m .

Theorem 2.2. ([5]) Let A be a d-dimensional nilpotent n-Lie algebra and dim A2 =
1. Then for some m ≥ 1
∼ H (n, m) ⊕ F (d − mn − 1) .
A=
Theorem 2.3. ([5]) Let A be a nilpotent n-Lie algebra of dimension d = n + k for
3 ≤ k ≤ n + 1 such that A2 = Z (A) and dim A2 = 2. Then
A∼
= he1 , . . . , en+k : [ek−1 , . . . , en+k−2 ] = en+k , [e1 , . . . , en ] = en+k−1 i .
This n-Lie algebra denoted by An,d,k .

For unification of notation, in what follows the kth d-dimmensional n-lie algebra
will be denoted by An,d,k
Theorem 2.4. ([5]) Let A be a non-abelian nilpotent n-Lie algebra of dimension
d ≤ n + 2. Then
A∼
= H (n, 1) , H (n, 1) ⊕ F (1) , An,n+2,1
where
An,n+2,1 = he1 , . . . , en+2 : [e1 , . . . , en ] = en+1 , [e2 , . . . , en+1 ] = en+2 i.
Theorem 2.5. ([4])
(1) Over a field F of characteristic different from 2, the list of the isomorphisms
types of 6-dimensional nilpotent Lie algebras is the following: L5,k ⊕ F with
k ∈ {1, . . . , 9} ; L6,k with k ∈ {10, . . . , 18, 20, 23, 25, . . . , 28}; L6,k (ε1 ) with
∗
∗
k ∈ {19, 21} and ε1 ∈ F ∗ /(∼); L6,k (ε2 ) with k ∈ {22, 24} and ε2 ∈ F/(∼).
(2) Over a field F of characteristic 2, the isomorphism types of 6-dimensional
nilpotent Lie algebras are: L5,k ⊕ F with k ∈ {1, . . . , 9}; L6,k with k ∈
∗
{10, . . . , 18, 20, 23, 25, . . . , 28}; L6,k (ε1 ) with k ∈ {19, 21} and ε1 ∈ F ∗ /(∼);
∗+
(2)
L6,k (ε2 ) with k ∈ {22, 24} and ε2 ∈ F/( ∼ ); L6,k with k ∈ {1, 2, 5, 6};
(2)

∗+

(2)

L6,k (ε3 ) with k ∈ {3, 4} and ε3 ∈ F ∗ /( ∼ ); L6,k (ε4 ) with k ∈ {7, 8} and
ε4 ∈ {0, ω}.
Eshrati et al. [7] classified n-Lie algebras of n + 3-dimensional for n > 2. For
the case n = 2, we have
H(2, 2), H(2, 1) ⊕ F (2), L =< x1 , ..., x5 |[x1 , x2 ] = x4 , [x1 , x3 ] = x5 > .
Additionally, for n + 4-dimensional n-Lie algebras we have the following theorem:

ON CLASSIFICATION OF (n+5)-DIMENSIONAL NILPOTENT n-LIE ALGEBRA OF CLASS TWO
3

Theorem 2.6. ([7]) The only (n + 4)-dimensional nilpotent n-Lie algebras of class
two are:
H(n, 1)⊕F (3), An,n+4,1 , An,n+4,2 , An,n+4,3 , H(2, 2)⊕F (1), H(3, 2), L6,22 (ε) , L26,7 (η).
Theorem 2.7. ([9]) The 7-dimensional nilpotent Lie algebras of class two over
algebraically closed fields and real number field are:
H(2, 1) ⊕ F (4), H(2, 2) ⊕ F (2), H(2, 3), L7,i ,

1 ≤ i ≤ 10.

3. Classification of (n+5)-dimensional nilpotent n-Lie algebras of class
two
In this section, we classify (n + 5)-dimensional nilpotent n-Lie algebras of class
two. The n-Lie algebra A is nilpotent of class two, when A is non-abelian and
A2 ⊆ Z (A). The nilpotent n-Lie algebra of class two plays an essential role in some
geometry problems such as the commutative Riemannian manifold. Additionally,
the classification of nilpotent Lie algebras of class two is one of the important issues
in Lie algebras.
We first prove a lemma for 3-Lie algebras.
Lemma 3.1. Let A be a 3-Lie algebra of dimension 8 such that A2 = Z (A) and
dim A2 = 2. Then
A = he1 , . . . , e8 : [e1 , e2 , e3 ] = e7 , [e4 , e5 , e6 ] = e8 i .
Proof. Let A = he1 , . . . , e8 i and A2 = Z (A) = he7 , e8 i. We may assume that
[e4 , e5 , e6 ] = e8 . So, there are α0 , β0 , αi,j,k , βi,j,k of F such that

[e1 , e2 , e3 ] = α0 e8 + β0 e7
[ei , ej , ek ] = αi,j,k e8 + βi,j,k e7 , 1 ≤ i < j < k ≤ 6, (i, j, k) 6= (1, 2, 3) , (4, 5, 6) .
Taking I = he7 i, thus dim (A/I )2 = 1. Therefore, by Theorem 2.2, A/I is
isomorphic to H (3, 1) ⊕ F (3) or H (3, 2) .
(i) Assume that A/I ∼
= H (3, 1) ⊕ F (3). In this case, according to the structure
of A/I , we have α0 = αi,j,k = 0. Therefore, the brackets of A are as follows:
(
[e4 , e5 , e6 ] = e8 ,
[e1 , e2 , e3 ] = β0 e7 ,
[ei , ej , ek ] = βi,j,k e7 ,
1 ≤ i < j < k ≤ 6, (i, j, k) 6= (1, 2, 3), (4, 5, 6).
Now, by choosing J = he8 i, and by taking into account dim (A/J )2 = 1, we have
A/J ∼
= he1 , . . . , e7 : [e1 , e2 , e3 ] = β0 e7 , [ei , ej , ek ] = βi,j,k e7 , 1 ≤ i < j < k ≤
6, (i, j, k) 6= (1, 2, 3), (4, 5, 6)i. According to the above brackets and special Heisenberg n-Lie algebra, the above algebra is isomorphic to H (3, 1) ⊕ F (3). So, only
one of β0 , βi,j,k is equal to one and the other coefficients are zero.
If one of the coefficients βi,j,k equal to one, then the condition A2 = Z (A) will
be false. So, we conclude βi,j,k = 0 for each 1 ≤ i < j < k ≤ 6, (i, j, k) 6=
(1, 2, 3) , (4, 5, 6). Thus
A3,8,1 =< e1 , ..., e8 |[e1 , e2 , e3 ] = e8 , [e4 , e5 , e6 ] = e7 > .

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Z. HOSEINI, F. SAEEDI, AND H. DARABI

(ii) Consider A/I ∼
= H (3, 2). According to the structure of A/I , we have α0 = 1
and αi,j,k = 0. Therefore, the brackets of A:
(
[e4 , e5 , e6 ] = e8 ,
[e1 , e2 , e3 ] = e8 + β0 e7 ,
[ei , ej , ek ] = βi,j,k e7 ,
1 ≤ i < j < k ≤ 6, (i, j, k) 6= (1, 2, 3), (4, 5, 6).
2

Now, by choosing J = he8 i, we have dim (A/J ) = 1. Therefore, with respect
to the structure of special Heisenberg n-Lie algebras, this algebra is isomorphic to
H (3, 1) ⊕ F (3). So, only one of coefficients β0 , βi,j,k is equal to one and the other
coefficients are zero. Of course, if one of the coefficients βi,j,k is equal to one, we
have a contradiction. So, we have βi,j,k = 0 for each 1 ≤ i < j < k ≤ 6, (i, j, k) 6=
(1, 2, 3) , (4, 5, 6). Therefore, the brackets of A are as follows:
[e4 , e5 , e6 ] = e8 , [e1 , e2 , e3 ] = e8 + e7 .
By interchanging
e′i = ei ,

1 ≤ i ≤ 8, i 6= 7,

e′7 = e8 + e7

this algebra is isomorphic to A3,8,1 . so, the proof is completed.



Now we are going to classify (n + 5)-dimensional nilpotent n-Lie algebras of class
two.
Assume that A is a (n + 5)-dimensional nilpotent n-Lie algebras of class two
where n ≥ 3 and A = he1 , . . . , en+5 i (see Theorem 2.7 for the case n = 2). If
dim A2 = 1, then by Theorem 2.2, A is isomorphic to one of the following algebras:
H (n, 1) ⊕ F (4) , H (3, 2) ⊕ F (1) , H (4, 2) .
Now, assume that dim A2 ≥ 2 and hen+4 , en+5 i ⊆ A2 . Therefore, A/hen+5 i is a
(n + 4)-dimensional nilpotent n-Lie algebras of class 2. It follows from Theorem 2.6
that A/hen+5 i is one of the following forms:
H (n, 1) ⊕ F (3) , An,n+4,1 , An,n+4,2 , An,n+4,3 , H (3, 2) .
Case 1: A/hen+5 i ∼
= H(n, 1) ⊕ F (3).
= he1 , . . . , en+4 : [e1 , . . . , en ] = en+4 i ∼
The brackets of A are as follows:

[e1 , e2 , . . . , en ] = en+4 + αen+5




[e

1 , . . . , êi , . . . , en , en+1 ] = αi en+5



[e

1 , . . . , êi , . . . , en , en+2 ] = βi en+5 ,


[e1 , . . . , êi , . . . , en , en+3 ] = γi en+5 ,
[e1 , . . . , êi , . . . , êj , . . . , en , en+1 , en+2 ] = θij en+5 ,




[e1 , . . . , êi , . . . , êj , . . . , en , en+1 , en+3 ] = µij en+5 ,




 [e1 , . . . , êi , . . . , êj , . . . , en , en+2 , en+3 ] = λij en+5 ,


[e1 , . . . , êi , . . . , êj , . . . , êk , . . . , en , en+1 , en+2 , en+3 ] = ζijk en+5 ,

1≤i≤n
1≤i≤n
1≤i≤n
.
1≤i<j≤n
1≤i<j≤n
1≤i<j≤n
1≤i<j<k≤n

By changing the base, we can result α = 0. Since dim A2 ≥ 2, we get dim Z (A) ≤ 4.
First, we assume that dim Z (A) = 4. In this case, without lose of generality,
we can assume that Z (A) = hen+2 , en+3 , en+4 , en+5 i. Consequently, the non-zero
brackets of A are as follows:


[e1 , e2 , . . . , en ] = en+4
[e1 , . . . , êi , . . . , en , en+1 ] = αi en+5 ,

1≤i≤n

.

ON CLASSIFICATION OF (n+5)-DIMENSIONAL NILPOTENT n-LIE ALGEBRA OF CLASS TWO
5

At least one of the αi ’s is non-zero. Without lose of generality, we can assume that
α1 6= 0. By interchanging
e′1 = e1 +

n
X

(−1)i−1

i=2

we have

αi
ei ,
α1

e′j = ej ,

2 ≤ j ≤ n + 4,

e′n+5 = α1 en+5

[e1 , . . . , en ] = en+4 , [e2 , . . . , en+1 ] = en+5 .
We denote this algebra by An,n+5,1 . Now, suppose dim Z (A) = 3. Without lose of
generality, we assume Z (A) = hen+3 , en+4 , en+5 i. Therefore, the brackets of A are
as follows:

[e1 , e2 , . . . , en ] = en+4



[e1 , . . . , êi , . . . , en , en+1 ] = αi en+5 ,
1≤i≤n
.
1≤i≤n
 [e1 , . . . , êi , . . . , en , en+2 ] = βi en+5 ,


[e1 , . . . , êi , . . . , êj , . . . , en , en+1 , en+2 ] = θij en+5 , 1 ≤ i < j ≤ n
2

Since dim (A/hen+4 i ) = 1, we have A/hen+4 i ∼
= H (n, 1) ⊕ F (3). According to
the structure of n-Lie algebras, we conclude that one of the coefficients
θij (1 ≤ i < j ≤ n) , βi (1 ≤ i ≤ n) , αi (1 ≤ i ≤ n)

is equal to one, and the others are zero. According to Z (A) = hen+3 , en+4 , en+5 i,
the coefficients of θij (1 ≤ i < j ≤ n) , βi (1 ≤ i ≤ n) , αi (1 ≤ i ≤ n) can not be equal
to one. Without loss of generality, assume that
θ12 = 1, θij = 0 (1 ≤ i < j ≤ n, (i, j) 6= (1, 2)) , βi = 0 (1 ≤ i ≤ n) , αi = 0 (1 ≤ i ≤ n) .
So, the brackets of A are as follows:
[e1 , . . . , en ] = en+4 , [e3 , . . . , en+2 ] = en+5 .
We denote this algebra by An,n+5,2 .
Now, assume that dim Z (A) = 2. Therefore Z (A) = A2 . In the case n ≥ 4, by
Theorem 2.3 we reach the algebra An,n+5,3 .
In the case n = 3, according to Lemma 3.1, the desired algebra is
A = he1 , . . . , e8 : [e1 , e2 , e3 ] = e7 , [e4 , e5 , e6 ] = e8 i .
Which is denoted by A3,8,7 .
Case 2: A/hen+5 i ∼
= he1 , . . . , en+4 : [e1 , . . . , en ] = en+3 , [e2 , . . . , en+1 ] = en+4 i.
In this case dim A2 = 3, so A2 = hen+3 , en+4 , en+5 i. Thus 3 ≤ dim Z (A) ≤ 4.
Hence the bracket of A are as follows:

[e1 , e2 , . . . , en ] = en+3 + αen+5




[e
 2 , . . . , en+1 ] = en+4 + βen+5
[e1 , . . . , êi , . . . , en , en+1 ] = αi en+5 ,
2≤i≤n
.


[e
,
.
.
.
,
ê
,
.
.
.
,
e
,
e
]
=
β
e
,
1
≤
i
≤
n

1
i
n n+2
i n+5


[e1 , . . . , êi , . . . , êj , . . . , en , en+1 , en+2 ] = θij en+5 , 1 ≤ i < j ≤ n

By changing the base, we can take α = β = 0. First, we assume that dim Z (A) = 4.
It follows that Z (A) = hen+2 , en+3 , en+4 , en+5 i. Therefore

 [e1 , e2 , . . . , en ] = en+3
[e2 , . . . , en+1 ] = en+4
.

[e1 , . . . , êi , . . . , en , en+1 ] = αi en+5 , 2 ≤ i ≤ n

6

Z. HOSEINI, F. SAEEDI, AND H. DARABI

Due to the derived dimension, we conclude that at least one of the αi ′ s (2 ≤ i ≤ n)
is non-zero. Without loss of generality, we assume that α2 6= 0. Taking
n
X
αi
e′2 = e2 +
(−1)i ei , e′j = ej , j = 1, 3, . . . , n + 4, e′n+5 = α2 en+5
α2
i=3
Hence, the non-zero brackets of this algebra are as follows:

[e1 , . . . , en ] = en+3 , [e2 , . . . , en+1 ] = en+4 , [e1 , e3 , . . . , en+1 ] = en+5 .
We denote this algebra by An,n+5,4 .
Now, assume that dim Z (A) = 3, thus A2 = Z (A) = hen+3 , en+4 , en+5 i. Hence

[e1 , e2 , . . . , en ] = en+3




 [e2 , . . . , en+1 ] = en+4
[e1 , . . . , êi , . . . , en , en+1 ] = αi en+5 ,
2≤i≤n
.


[e
,
.
.
.
,
ê
,
.
.
.
,
e
,
e
]
=
β
e
,
1
≤
i
≤
n

1
i
n
n+2
i
n+5


[e1 , . . . , êi , . . . , êj , . . . , en , en+1 , en+2 ] = θij en+5 , 1 ≤ i < j ≤ n

2
Since dim (A/hen+3 , en+4 i ) = 1, we have A/hen+3 , en+4 i ∼
= H (n, 1) ⊕ F (2).
According to the structure of n-Lie algebras, we conclude that only one of the
coefficients θij (1 ≤ i < j ≤ n), βi (1 ≤ i ≤ n), αi (2 ≤ i ≤ n) is equal to one and
the other are zero. On account of Z (A) = hen+3 , en+4 , en+5 i, the coefficients
αi (2 ≤ i ≤ n) can not be equal to one. We have two cases.
(i) Only one of the coefficients βi (1 ≤ i ≤ n) is equal to one and the others are
zero. Without loss of generality, we assume that β1 = 1 and the others are zero.
So the non-zero brackets of algebra are as follows:

[e1 , . . . , en ] = en+3 , [e2 , . . . , en+1 ] = en+4 , [e2 , . . . , en , en+2 ] = en+5
We denote this algebra by An,n+5,5 .
(ii) Only one of the coefficients θij (1 ≤ i < j ≤ n) is equal one and the others
are zero.
Without loss of generality, we assume that θ12 = 1, and the others are zero. So,
the non-zero brackets of algebra are as follows:
[e1 , . . . , en ] = en+3 , [e2 , . . . , en+1 ] = en+4 , [e3 , . . . , en+2 ] = en+5 .
We denote this algebra byAn,n+5,6 .
Case 3: A/hen+5 i ∼
= he1 , . . . , en+4 : [e1 , . . . , en ] = en+3 , [e3 , . . . , en+2 ] = en+4 i.
In this case dim A2 = 3, and so A2 = Z (A) = hen+3 , en+4 , en+5 i. The brackets
of A are as follows:

[e1 , e2 , . . . , en ] = en+3 + αen+5




 [e3 , . . . , en+2 ] = en+4 + βen+5
[e1 , . . . , êi , . . . , en , en+1 ] = αi en+5 ,
1≤i≤n


[e
,
.
.
.
,
ê
,
.
.
.
,
e
,
e
]
=
β
e
,
1≤i≤n

1
i
n
n+2
i
n+5


[e1 , . . . , êi , . . . , êj , . . . , en , en+1 , en+2 ] = θij en+5 , 1 ≤ i < j ≤ n and (i, j) 6= (1, 2)
By changing the base, we can take α = β = 0. So

[e1 , e2 , . . . , en ] = en+3




 [e3 , . . . , en+2 ] = en+4
[e1 , . . . , êi , . . . , en , en+1 ] = αi en+5 ,


[e1 , . . . , êi , . . . , en , en+2 ] = βi en+5 ,



[e1 , . . . , êi , . . . , êj , . . . , en , en+1 , en+2 ] = θij en+5 ,

1≤i≤n
1≤i≤n
1 ≤ i < j ≤ n and (i, j) 6= (1, 2)

ON CLASSIFICATION OF (n+5)-DIMENSIONAL NILPOTENT n-LIE ALGEBRA OF CLASS TWO
7
2
Since dim (A/hen+3 , en+4 i ) = 1, we get A/hen+3 , en+4 i ∼
= H (n, 1) ⊕ F (2). According to the structure of n-Lie algebras, we conclude that at least one of the
coefficients θij (1 ≤ i < j ≤ n, (i, j) 6= (1, 2)), βi (1 ≤ i ≤ n), αi (1 ≤ i ≤ n) is equal
one and the others are zero. Since Z (A) = hen+3 , en+4 , en+5 i, we have three cases.
(i) Only one of the coefficients αi (1 ≤ i ≤ n) is equal to one and the others are
zero. Without loss of generality, we assume that α1 = 1, and the others are zero.
So, we have

[e1 , . . . , en ] = en+3 , [e3 , . . . , en+2 ] = en+4 , [e2 , . . . , en , en+1 ] = en+5 .
One can check that, this algebra is isomorphic to An,n+5,6 .
(ii) Only one of the coefficients βi (1 ≤ i ≤ n) is equal to one and the others are
zero. Without loss of generality, we assume that β1 = 1, and the others are zero.
Hence, the non-zero bracket of algebra are
[e1 , . . . , en ] = en+3 , [e3 , . . . , en+2 ] = en+4 , [e2 , . . . , en , en+2 ] = en+5 .
One can easily see that, this algebra is isomorphic to An,n+5,6 .
(iii) Only one of the coefficients of θij (1 ≤ i < j ≤ n, (i, j) 6= (1, 2)) is equal to
one and the others are zero. Without loss of generality, we assume that θ13 = 1,
and the others are zero. Hence, the non-zero bracket of algebra are
[e1 , . . . , en ] = en+3 , [e2 , . . . , en+1 ] = en+4 , [e2 , e4 , . . . , en+2 ] = en+5 .
One can easily see that, this algebra is isomorphic to An,n+5,5 .
∼ he1 , . . . , en+4 : [e1 , . . . , en ] = en+1 , [e2 , . . . , en , en+2 ] =
Case 4: A/hen+5 i =
en+3 i.
In this case dim A2 = 4, thus A2 = Z (A) = hen+1 , en+3 , en+4 , en+5 i. The
brackets of this algebra are as follows:

[e1 , e2 , . . . , en ] = en+1 + αen+5



[e2 , . . . , en , en+2 ] = en+3 + βen+5
[e1 , e3 , . . . , en , en+2 ] = en+4 + γen+5



[e1 , . . . , êi , . . . , en , en+2 ] = βi en+5 ,
3≤i≤n
We use the change of base. In fact, by letting α = β = γ = 0, we get

[e1 , e2 , . . . , en ] = en+1



[e2 , . . . , en , en+2 ] = en+3
.
[e1 , e3 , . . . , en , en+2 ] = en+4



[e1 , . . . , êi , . . . , en , en+2 ] = βi en+5 , 3 ≤ i ≤ n

Due to the derived dimension, we conclude that at least one of the βi (3 ≤ i ≤ n)
is non-zero. Without loss of generality, we assume that β3 6= 0. Applying the
following transformations
n
X
βi
(−1)i−1 ei , e′j = ej , j = 1, 2, 4, . . . , n + 4, e′n+5 = β3 en+5
e′3 = e3 +
β3
i=4

the non-zero brackets of algebra are as follows
[e1 , e2 , . . . , en ] = en+1 ,
[e1 , e3 , . . . , en , en+2 ] = en+4 ,

[e2 , . . . , en , en+2 ] = en+3
[e1 , e2 , e4 , . . . , en , en+2 ] = en+5 .

We denote this algebra by An,n+5,7 .
Case 5: A/hen+5 i ∼
= he1 , . . . , e7 : [e1 , e2 , e3 ] = [e4 , e5 , e6 ] = e7 i.
= H(3, 2) ∼

8

Z. HOSEINI, F. SAEEDI, AND H. DARABI

In this case, according to the structure of algebra, we get A2 = Z (A) = he7 , e8 i.
According to Lemma 3.1, we have A = he1 , . . . , e8 : [e1 , e2 , e3 ] = e7 , [e4 , e5 , e6 ] = e8 i.
Therefore, this algebra does not satisfy our conditions.
Theorem 3.2. The (n + 5)-dimensional nilpotent n-Lie algebras of class two, for
n > 2, over the arbitrary field are:
H(n, 1) ⊕ F (4), H(3, 2) ⊕ F (1), H(4, 2), A3,8,7 , An,n+5,i ,

1 ≤ i ≤ 7.

According to the above theorem and Theorem 2.7, the main theorem of this
paper is as follows:
Corollary 3.3. The (n + 5)-dimensional nilpotent n-Lie algebras of class two are
as following
H(n, 1) ⊕ F (4), H(2, 2) ⊕ F (2), H(2, 3), H(3, 2) ⊕ F (1), H(4, 2), A3,8,7 ,
An,n+5,i ,

1 ≤ i ≤ 6,

L7,i ,

1 ≤ i ≤ 10.

These algebras are valid for the case n = 2 on the integers field and algebraically
closed field, and n > 2 on the arbitrary field.
In the following table we list the algebras which are presented in this article.
Nilpotent n-Lie algebra
An,n+4,1
An,n+4,2
An,n+4,3
An,n+5,1
An,n+5,2
An,n+5,3
An,n+5,4
An,n+5,5
An,n+5,6
An,n+5,7
A3,8,7
L7,1
L7,2
L7,3
L7,4
L7,5
L7,6
L7,7
L7,8
L7,9
L7,10

Table I
Non-zero multiplications
[e1 , . . . , en ] = en+3 , [e2 , . . . , en+1 ] = en+4
[e1 , . . . , en ] = en+3 , [e3 , . . . , en+2 ] = en+4 (n ≥ 3)
[e1 , . . . , en ] = en+1 , [e2 , . . . , en , en+2 ] = en+3 ,
[e1 , e3 , . . . , en , en+2 ] = en+4
[e1 , . . . , en ] = en+4 , [e2 , . . . , en+1 ] = en+5
[e1 , . . . , en ] = en+4 , [e3 , . . . , en+2 ] = en+5 (n ≥ 3)
[e1 , . . . , en ] = en+5 , [e4 , . . . , en+3 ] = en+4 (n ≥ 4)
[e1 , . . . , en ] = en+3 , [e2 , . . . , en+1 ] = en+4 ,
[e1 , e3 , . . . , en+1 ] = en+5
[e1 , . . . , en ] = en+3 , [e2 , . . . , en+1 ] = en+4 ,
[e2 , . . . , en , en+2 ] = en+5
[e1 , . . . , en ] = en+3 , [e2 , . . . , en+1 ] = en+4 ,
[e3 , . . . , en+2 ] = en+5
[e1 , . . . , en ] = en+1 , [e1 , e2 , e4 , . . . , en , en+2 ] = en+5 ,
[e1 , e3 , . . . , en , en+2 ] = en+4 , [e2 , . . . , en , en+2 ] = en+3
[e1 , e2 , e3 ] = e7 , [e4 , e5 , e6 ] = e8
[e1 , e2 ] = e4 , [e1 , e3 ] = e5
[e1 , e2 ] = e4 , [e1 , e3 ] = e5 , [e2 , e3 ] = e6
[e1 , e2 ] = [e3 , e4 ] = e5 , [e3 , e4 ] = e6
[e1 , e2 ] = e5 , [e3 , e4 ] = e6
[e1 , e2 ] = e5 , [e2 , e3 ] = e6 , [e2 , e4 ] = e7
[e1 , e2 ] = e5 , [e2 , e3 ] = e6 , [e3 , e4 ] = e7
[e1 , e2 ] = [e3 , e4 ] = e5 , [e2 , e3 ] = e6 , [e2 , e4 ] = e7
[e1 , e2 ] = [e3 , e4 ] = e5 , [e1 , e3 ] = e6 , [e2 , e4 ] = e7
[e1 , e5 ] = [e3 , e4 ] = e6 , [e2 , e5 ] = e7
[e1 , e2 ] = [e3 , e4 ] = e6 , [e1 , e5 ] = [e2 , e3 ] = e7

ON CLASSIFICATION OF (n+5)-DIMENSIONAL NILPOTENT n-LIE ALGEBRA OF CLASS TWO
9

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1 Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad,
Iran.
2 Department of Mathematics, Esfarayen Branch, Islamic Azad University, Esfarayen,
Iran.
E-mail address: zhosseini822@gmail.com
E-mail address: saeedi@mshdiau.ac.ir
E-mail address: darabi@iauesf.ac.ir

