On classification of (n + 6)-dimensional nilpotent n-Lie algebras of class 2 with n ≥ 4

Abstract

Purpose

The purpose of this paper is to determine the structure of nilpotent (n+6)-dimensional n-Lie algebras of class 2 when n≥4.

Design/methodology/approach

By dividing a nilpotent (n+6)-dimensional n-Lie algebra of class 2 by a central element, the authors arrive to a nilpotent (n+5) dimensional n-Lie algebra of class 2. Given that the authors have the structure of nilpotent (n+5)-dimensional n-Lie algebras of class 2, the authors have access to the structure of the desired algebras.

Findings

In this paper, for each n≥4, the authors have found 24 nilpotent (n+6) dimensional n-Lie algebras of class 2. Of these, 15 are non-split algebras and the nine remaining algebras are written as direct additions of n-Lie algebras of low-dimension and abelian n-Lie algebras.

Originality/value

This classification of n-Lie algebras provides a complete understanding of these algebras that are used in algebraic studies.

Citation

Jamshidi, M., Saeedi, F. and Darabi, H. (2021), "On classification of (n + 6)-dimensional nilpotent n-Lie algebras of class 2 with n ≥ 4", Arab Journal of Mathematical Sciences, Vol. 27 No. 2, pp. 139-150. https://doi.org/10.1108/AJMS-09-2020-0075

1. Introduction

In 1985, Filippov [1] introduced the concept of n-Lie (Filippov) algebras, as an n-ary multilinear and skew-symmetric operation [x1,…,xn], which satisfies the following generalized Jacobi identity

[[x1,…,xn], y2,…,yn]=∑i=1n[x1,…,[xi, y2,…,yn],…, xn] ].

Clearly, such an algebra becomes an ordinary Lie algebra when n=2. Beside presenting many examples of n-Lie algebras, he also extended the notions of simplicity and nilpotency and determined all (n+1)-dimensional n-Lie algebras over an algebraically closed field of characteristic zero.

The study of n-Lie algebras is important, since it is related to geometry and physics. Among other results, n-Lie algebras are classified in some cases. For example, Bai et al. [2] classified all n-Lie algebras of dimension n+1 over a field of characteristic 2. Also, they showed that there is no simple n-Lie algebra of dimension n+2. Then, Bai et al. [3] classified n-Lie algebras of dimension n+2 on the algebraically closed fields with characteristic zero. (see [4–7] for more information on the Filippov algebras).

In 1986, Kasymov [8] studied some properties of nilpotent and solvable n-Lie algebras. An n-Lie algebra A is nilpotent if As=0 for some nonnegative integer s, where Ai is defined inductively by A1=A and Ai+1=[Ai, A,…, A]. The n-Lie algebra A is nilpotent of class c, if Ac+1=0 and Ai=0 for each i≤c. The ideal A2=[A,…, A] is called the derived subalgebra of A. The center of A is defined by

Z(A)={x∈A:[x, A,…, A]=0}.

Let Z0(A)=〈0〉. Then the ith center of A is defined inductively by

Zi(A)Zi−1(A)=Z(AZi−1(A))

for all i≥1. Clearly, Z1(A)=Z(A).

The nilpotent theories of many algebras attract more and more attention. For example, in [9,10], and [11], the authors studied nilpotent Leibniz n-algebras, nilpotent Lie and Leibniz algebras, and nilpotent n-Lie algebras, respectively.

The (n+3)-dimensional nilpotent n-Lie algebras and (n+4)-dimensional nilpotent n-Lie algebras of class 2 were classified in [12]. Hoseini et al. [13] classified (n+5)-dimensional nilpotent n-Lie algebras of class 2.

In this paper, we have interest for algebras of class 2 (the minimal class for nonabelian case). The concept of filiform n-Lie algebras (maximal class) has been studied in some papers. For example, see [14].

The rest of our paper is organized as follows: Section 2 includes the results that are used frequently in the last section. In Section 3, we classify (n+6)-dimensional n-Lie algebras of class 2 when n≥4. For the case n=2, this problem is dealt with by Yan et al. [15]. Also, the case n=3 stated in [16].

2. Preliminaries

In this section, we introduce some known and necessary results. We denote d-dimensional abelian n-Lie algebra by F(d). An important category of n-Lie algebras of class 2, which plays an essential role in classification of nilpotent n-Lie algebras, are algebras whose derived and center are equal. We call an n-Lie algebra A, a generalized Heisenberg of rank k, if A2=Z(A) and dim A2=k. The particular case k=1, is called the special Heisenberg n-Lie algebras. The structure of this algebras defined as follows.

Theorem 2.1.

[17] Every special Heisenberg n-Lie algebra has 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.

[18] Let A be a d-dimensional nilpotent n-Lie algebra, and let dim A2=1. Then, for some m≥1, it follows that

A≅H(n, m)⊕F(d−mn−1).

Theorem 2.3.

[18] Let A be a nonabelian nilpotent n-Lie algebra of dimension d≤n+2. Then A is isomorphic to H(n, 1), H(n, 1)⊕F(1) or An,n+2,1, where An,n+2,1=〈e1,…,en+2:[e1,…, en]=en+1,[e2,…, en+1]=en+2〉.

For unification of notation in what follows, the tth d-dimensional n-Lie algebra is denoted by An,d,t.

Theorem 2.4.

[12] The (n+3)-dimensional nonabelian nilpotent n-Lie algebras for n>2 over an arbitrary field are An,n+3,i(2≤i≤8). Moreover nilpotent classes of An,n+3,2 and An,n+3,5 is two, nilpotent classes of An,n+3,3, An,n+3,4 and An,n+3,8 is three and finally, nilpotent classes of An,n+3,6 and An,n+3,7 is four (maximal class).

Theorem 2.5.

[12] The only (n+4)-dimensional nilpotent n-Lie algebras of class 2 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), and L6,72(η).

Theorem 2.6.

[13] The (n+5)-dimensional nilpotent n-Lie algebras of class 2 for n>2 over an arbitrary field are H(n, 1)⊕F(4), An,n+5,i(1≤i≤7), H(3, 2)⊕F(1) and H(4,2).

Theorem 2.7.

[19] Let A be a nilpotent n-Lie algebra of class 2. Then, there exist a generalized Heisenberg n-Lie algebra H and an abelian n-Lie algebra F such that A=H⊕F.

3. Main results

In this section, we classify (n+6)-dimensional nilpotent n-Lie algebras of class 2. If n-Lie algebra A is nilpotent of class 2, then A is nonabelian and A2⊆Z(A). The nilpotent n-Lie algebra of class 2 plays an essential role in some geometry problems such as the commutative Riemannian manifold. Additionally, the classification of nilpotent Lie algebras of class 2 is one of the most important issues in Lie algebras.

The following theorems define the structure of generalized Heisenberg n-Lie algebras of rank 2 with dimension at most 2n+3.

Theorem 3.1.

[18] Let A be a nilpotent n-Lie algebra of dimension d=n+k for 3≤k≤n+1 such that A2=Z(A) and dimA2=2. Then

A≅〈e1,…, en+k:[ek−1,…, en+k−2]=en+k,[e1,…, en]=en+k−1〉.

Remark. In the above theorem for n=2 and k=3, we obtain

A≅〈e1, e2, e3, e4, e5:[e1, e2]=e4,[e2, e3]=e5〉.

This algebra appears many times in differential geometry in the study of Pfaffian systems. It was developed by P. Libermann and introduced in [20].

Theorem 3.2.

[19] Let A be a generalized Heisenberg n-Lie algebra of rank 2 with dimension 2n+2. Then

A≅An,2n+2,1=〈e1,…, e2n+2:[e1,…, en]=e2n+1,[en+1,…, e2n]=e2n+2〉.

Theorem 3.3.

[19] Let A be a generalized Heisenberg n-Lie algebras of rank 2 with dimension 2n+3. Then, A is isomorphic to one of the following n-Lie algebras:

An,2n+3,1=〈e1,…, e2n+3:[e1,…, en]=e2n+3,[e2,…, en+1]=[en+2,…, e2n+1]=e2n+2〉.

An,2n+3,2=〈e1,…, e2n+3:[e1,…, en]=[en+1,…, e2n]=e2n+3,[e2,…, en+1]=[en+2,…, e2n+1]=e2n+2〉.

For n=2, we obtain also a Lie algebra of the previous type.

Now we are going to classify (n+6)-dimensional nilpotent n-Lie algebras of class 2.

According to Theorem 2.7, we can write A=H⊕F, where H is a generalized Heisenberg n-Lie algebra of rank 2 and F is abelian. Therefore, first we classify the generalized Heisenberg n-Lie algebra of rank 2.

By the classification of nilpotent n-Lie algebras of class 2, we have the following theorem. All the algebras defined in theorem 3.4 and follow are in Table 1 at the end of the paper.

Theorem 3.4.

1. The only (n+4)-dimensional generalized Heisenberg n-Lie algebra of rank 3 is An,n+4,3.

2. The only (n+5)-dimensional generalized Heisenberg n-Lie algebras of rank 3 are An,n+5,4 and An,n+5,5.

3. The only (n+5)-dimensional generalized Heisenberg n-Lie algebra of rank 4 is An,n+5,6.

The following lemma defines the structure of (n+6)-dimensional generalized Heisenberg n-Lie algebras of rank 2.

Theorem 3.5.

Let A be a generalized Heisenberg n-Lie algebra of rank 2 with dimension n+6. Then

A≅An,n+6,1=〈e1,…, en+6:[e1,…, en]=en+5,[e5,…, en+4]=en+6〉.

Proof. For n=4, we have n+6=2n+2. Thus by Theorem 3.2, if n≥5, then n+3<n+6≤2n+1. Applying Theorem 3.1 completes the proof. ▪

Theorem 3.6.

The only (n+6)-dimensional generalized Heisenberg n-Lie algebras of rank 3 are

An,n+6,2, An,n+6,3, An,n+6,4, An,n+6,5, and An,n+6,6(ε).

Proof. Suppose that A is an (n+6)-dimensional generalized Heisenberg n-Lie algebra of rank 3 with basis {e1,…, en+6}, which n≥4. Also, suppose that A2=〈en+4, en+5, en+6〉. In this case, A/〈en+6〉 is an (n+5)-dimensional nilpotent n-Lie algebra of class 2 with derived algebra of dimension 2. By Theorem 2.6, we have three possibilities for A/〈en+6〉: Case 1: Let A/〈en+6〉≅An,n+5,1. Then the brackets in A can be written as

{[e1,…, en]=en+4+αen+6,[e2,…, en+1]=en+5+βen+6,[e1,…, eˆi,…, en, en+1]=αien+6,2≤i≤n,[e1,…, eˆi,…, en, en+2]=βien+6,1≤i≤n,[e1,…, eˆi,…, en, en+3]=γien+6,1≤i≤n,[e1,…, eˆi,…, eˆj,…, en, en+1, en+2]=χijen+6,1≤i<j≤n,[e1,…, eˆi,…, eˆj,…, en, en+1, en+3]=δijen+6,1≤i<j≤n,[e1,…, eˆi,…, eˆj,…,en, en+2, en+3]=λijen+6,1≤i<j≤n,[e1,…, eˆi,…, eˆj,…, eˆk,…, en,en+1, en+2, en+3]=φijken+6,1≤i<j<k≤n.

Regarding a suitable change of basis, one can assume that α=β=0.

Since dim(A/〈en+4, en+5〉)2=1, we have A/〈en+4, en+5〉≅H(n, 1)⊕F(3). According to the structure of n-Lie algebras, we conclude that one of the coefficients

λij(1≤i<j≤n), φijk(1≤i<j<k≤n)

is equal to one, and the others are zero. We have four possibilities:

1. λ12=1, λij=0(1≤i<j≤n,(i, j)≠(1, 2)), and  φijk=0(1≤i<j<k≤n). In this case, the brackets in A can be written as

[e1,…,en]=en+4, [e2,…,en+1]=en+5, [e3,…,en,en+2,en+3]=en+6,

which we denote it by An,n+6,2.

1. λ23=1, λij=0(1≤i<j≤n,(i, j)≠(2, 3)), and  φijk=0(1≤i<j<k≤n). In this case, the brackets in A can be written as

[e1,…, en]=en+4, [e2,…, en+1]=en+5, [e1, e4,…, en, en+2, en+3]=en+6,

which we denote it by An,n+6,3.

1. λij=0(1≤i<j≤n), φ123=1, and φijk=0(1≤i<j<k≤n,(i, j, k)≠(1, 2, 3)). In this case, the brackets in A can be written as

[e1,…, en]=en+4, [e2,…, en+1]=en+5, [e4,…, en+3]=en+6.

One can easily see that this algebra is isomorphic to An+6,3.

1. λij=0(1≤i<j≤n),φ234=1,φijk=0(1≤i<j<k≤n),(i,j,k)≠(2,3,4)). In this case, the brackets in A can be written as

[e1,…, en]=en+4, [e2,…, en+1]=en+5, [e1, e5,…, en+3]=en+6,

which we denote it by An,n+6,4.

Case 2.

Let A/〈en+6〉≅An,n+5,2. Then the brackets in A can be written as

{[e1,…,en]=en+4+αen+6,[e2,…,en+1]=en+5+βen+6,[e1,…,eˆi,…,en,en+1]=αien+6,1≤i≤n,[e1,…,eˆi,…,en,en+2]=βien+6,1≤i≤n,[e1,…,eˆi,…,en,en+3]=γien+6,1≤i≤n,[e1,…,eˆi,…,eˆj,…,en,en+1,en+2]=χijen+6,1≤i<j≤n,(i,j)≠(1,2),[e1,…,eˆi,…,eˆj,…,en,en+1,en+3]=δijen+6,1≤i<j≤n,[e1,…,eˆi,…,eˆj,…,en,en+2,en+3]=λijen+6,1≤i<j≤n,[e1,…,eˆi,…,eˆj,…,eˆk,…,en,en+1,en+2,en+3]=φijken+6,1≤i<j<k≤n.

Regarding a suitable change of basis, one can assume that α=β=0.

Since dim(A/〈en+4, en+5〉)2=1, we have A/〈en+4,en+5〉≅H(n, 1)⊕F(3). According to the structure of n-Lie algebras and Z(A)=〈en+4, en+5, en+6〉, we conclude that one of the coefficients

γi(1≤i≤n),  δij(1≤i<j≤n),

λij(1≤i<j≤n), φijk(1≤i<j<k≤n)

is equal to one, and the others are zero. Similar to case 1, up to isomorphism, we have the following algebras:

[e1,…, en]=en+4,[e2,…, en+1]=en+5,[e2,…, en, en+3]=en+6,[e1,…, en]=en+4,[e2,…, en+1]=en+5,[e1, e2, e4,…, en, en+3]=en+6,[e1,…, en]=en+4,[e2,…, en+1]=en+5,[e2, e4,…, en+1, en+3]=en+6,[e1,…, en]=en+4,[e2,…, en+1]=en+5,[e1, e2, e5,…, en+1, en+3]=en+6.

One can easily see that the first and second algebras are isomorphic to An,n+6,2 and An,n+6,3, respectively. The third and fourth algebras are denoted by An,n+6,5 and An,n+6,6, respectively, that is,

An,n+6,5=〈e1,…, en+6:[e1,…, en]=en+4,[e2,…, en+1]=en+5,[e2, e4,…, en+1, en+3]=en+6〉,An,n+6,6=〈e1,…, en+6:[e1,…, en]=en+4,[e2,…, en+1]=en+5,[e1, e2, e5,…, en+1, en+3]=en+6〉,

Case 3.

Let A/〈en+6〉≅An,n+5,3. Then the brackets in A can be written as

{[e1,…,en]=en+4+αen+6,[e2,…,en+1]=en+5+βen+6,[e1,…,eˆi,…,en,en+1]=αien+6,1≤i≤n,[e1,…,eˆi,…,en,en+2]=βien+6,1≤i≤n,[e1,…,eˆi,…,en,en+3]=γien+6,1≤i≤n,[e1,…,eˆi,…,eˆj,…,en,en+1,en+2]=χijen+6,1≤i<j≤n,[e1,…,eˆi,…,eˆj,…,en,en+1,en+3]=δijen+6,1≤i<j≤n,[e1,…,eˆi,…,eˆj,…,en,en+2,en+3]=λijen+6,1≤i<j≤n,[e1,…,eˆi,…,eˆj,…,eˆk,…,en,en+1,en+2,en+3]=φijken+6,1≤i<j<k≤n,(i,j,k)≠(1,2,3).

Regarding a suitable change of basis, one can assume that α=β=0.

Since dim(A/〈en+4, en+5〉)2=1, we have A/〈en+4, en+5〉≅H(n, 1)⊕F(3). According to the structure of n-Lie algebra, we conclude that one of the coefficients

αi,βi,γi(1≤i≤n),  χij,δij,λij(1≤i<j≤n),φijk(1≤i<j<k≤n,(i,j,k)≠(1,2,3))

is equal to one, and the others are zero. Similar to case 1, up to isomorphism, we have the following algebras:

[e1,…, en]=en+5,[e4,…, en+3]=en+4,[e2, e3,…, en+1]=en+6,[e1,…, en]=en+5,[e4,…, en+3]=en+4,[e1, e2, e3, e5,…, en+1]=en+6,[e1,…,en]=en+5,[e4,…, en+3]=en+4,[e2, e3, e5,…, en+2]=en+6.

One can easily see that these algebras are isomorphic to An,n+6,3, An,n+6,4 and An,n+6,6, respectively. Therefore, there is no new algebra in this case. ▪

Theorem 3.7.

The only (n+6)-dimensional generalized Heisenberg n-Lie algebras of rank 4 are

An,n+6,7, An,n+6,8, An,n+6,9, An,n+6,10, An,n+6,11, An,n+6,12 and An,n+6,13.

Proof. Suppose that A is an (n+6)-dimensional generalized Heisenberg n-Lie algebra of rank 4 with basis {e1,…, en+6}, which n≥4. Also, suppose that A2=〈en+3, en+4, en+5, en+6〉. In this case, A/〈en+6〉 is an (n+5)-dimensional nilpotent n-Lie algebra of class 2 with derived algebra of dimension 3. By Theorem LABEL:?, we have three possibilities for A/〈en+6〉:

Case 4.

Let A/〈en+6〉≅An,n+5,4. Then the brackets in A can be written as

{[e1,…,en]=en+3+αen+6,[e2,…,en+1]=en+4+βen+6,[e1,e3,…,en+1]=en+5+γen+6,[e1,…,eˆi,…,en,en+1]=αien+6,3≤i≤n,[e1,…,eˆi,…,en,en+2]=βien+6,1≤i≤n,[e1,…,eˆi,…,eˆj,…,en,en+1,en+2]=χijen+6,1≤i<j≤n.

Regarding a suitable change of basis, one can assume that α=β=γ=0.

Since dim(A/〈en+3,en+4,en+5〉)2=1, we have A/〈en+3, en+4, en+5〉≅H(n, 1)⊕F(2). According to the structure of n-Lie algebras, we conclude that one of the coefficients

αi (3≤i≤n), βi (1≤i≤n), χij (1≤i<j≤n)

is equal to one, and the others are zero. According to Z(A)={en+3, en+4, en+5,en+6}, the coefficients αi(3≤i≤n) cannot be equal to one. We have three possibilities:

1. β1=1, βi=0 (2≤i≤n), αi=0 (3≤i≤n), χij=0 (1≤i<j≤n).

In this case, the brackets in A can be written as

[e1,…, en]=en+3, [e2,…, en+1]=en+4,[e1, e3,…, en+1]=en+5, [e2,…, en, en+2]=en+6,

which we denote it by An,n+6,7.

1. β3=1, βi=0 (1≤i≤n, n≠3), αi=0 (3≤i≤n), χij=0 (1≤i<j≤n).

In this case, the brackets in A can be written as

[e1,…, en]=en+3, [e2,…, en+1]en+4,[e1,e3,…, en+1]=en+5, [e1, e2, e4,…, en, en+2]=en+6,

which we denote it by An,n+6,8.

1. Only one of χijs(1≤i<j≤n) is equal to one and the others are zero. Up to isomorphism, we have the following algebras:

{[e1,…, en]=en+3,[e2,…,en+1]=en+4,[e1, e3,…, en+1]=en+5,[e3,…, en+2]=en+6,{[e1,…, en]=en+3,[e2,…, en+1]=en+4,[e1, e3,…, en+1]=en+5,[e2, e4,…, en+2]=en+6,{[e1,…, en]=en+3,[e2,…, en+1]=en+4,[e1, e3,…, en+1]=en+5,[e1, e2, e5,…, en+2]=en+6.

One can easily see that the first and second algebras are isomorphic to An,n+6,7 and An,n+6,8, respectively. The third algebras is denoted by An,n+6,9.

Case 5.

Let A/〈en+6〉≅An,n+5,5. Then the brackets in A can be written as

{[e1,…,en]=en+3+αen+6,[e2,…,en+1]=en+4+βen+6,[e2,…,en,en+2]=en+5+γen+6,[e1,…,eˆi,…,en,en+1]=αien+6,2≤i≤n,[e1,…,eˆi,…,en,en+2]=βien+6,2≤i≤n,[e1,…,eˆi,…,eˆj,…,en,en+1,en+2]=χijen+6,1≤i<j≤n.

Regarding a suitable change of basis, one can assume that α=β=γ=0.

Since dim(A/〈en+3,en+4,en+5〉)2=1, we have A/〈en+3, en+4, en+5〉≅H(n, 1)⊕F(2). According to the structure of n-Lie algebras, we conclude that one of the coefficients

αi (3≤i≤n), βi (1≤i≤n), χij (1≤i<j≤n)

is equal to one, and the others are zero. We have two possibilities:

1. Only one of αi(3≤i≤n) and βi(1≤i≤n) is equal to one and the others are zero. Without loss of generality, we assume α2=1. Thus, the brackets in A can be written as

[e1,…,en]=en+3, [e2,…,en+1]=en+4,[e2,…,en,en+2]=en+5, [e1,e3,…,en+1]=en+6.

One can easily see that this algebra is isomorphic to An,n+6,7.

1. Only one of χijs(1≤i<j≤n) is equal to one and the others are zero. Up to isomorphism, we have the following algebras:

{[e1,…, en]=en+3,[e2,…, en+1]=en+4,[e2,…, en, en+2]=en+5,[e3,…, en+2]=en+6,{[e1,…, en]=en+3,[e2,…, en+1]=en+4,[e2,…, en, en+2]=en+5,[e1, e4,…, en+2]=en+6.

One can easily see that the first algebra is isomorphic to An,n+6,7. The second algebra is denoted by An,n+6,10.

Case 6.

Let A/〈en+6〉≅An,n+5,6. Then the brackets in A can be written as

{[e1,…,en]=en+3+αen+6,[e2,…,en+1]=en+4+βen+6,[e3,…,en+2]=en+5+γen+6,[e1,…,eˆi,…,en,en+1]=αien+6,2≤i≤n,[e1,…,eˆi,…,en,en+2]=βien+6,1≤i≤n,[e1,…,eˆi,…,eˆj,…,en,en+1,en+2]=χijen+6,1≤i<j≤n,(i,j)≠(1,2).

Regarding a suitable change of basis, one can assume that α=β=γ=0.

Since dim(A/〈en+3, en+4,en+5〉)2=1, we have A/〈en+3, en+4, en+5〉≅H(n, 1)⊕F(2). According to the structure of n-Lie algebras, we conclude that one of the coefficients

αi (2≤i≤n), βi (1≤i≤n), χij (1≤i<j≤n,(i, j)≠(1, 2))

is equal to one, and the others are zero. Up to isomorphism, we have the following algebras:

{[e1,…, en]=en+3,[e2,…, en+1]=en+4,[e3,…, en+2]=en+5,[e1,e3,…, en+1]=en+6,{[e1,…, en]=en+3,[e2,…, en+1]=en+4,[e3,…, en+2]=en+5,[e1,e2,e4,…, en+1]=en+6,{[e1,…, en]=en+3,[e2,…, en+1]=en+4,[e3,…, en+2]=en+5,[e1,e3,…, en, en+2]=en+6,{[e1,…, en]=en+3,[e2,…, en+1]=en+4,[e3,…, en+2]=en+5,[e1,e2,e4,…, en, en+2]=en+6,{[e1,…, en]=en+3,[e2,…, en+1]=en+4,[e3,…, en+2]=en+5,[e1,e2,e5,…, en+2]=en+6.

One can easily see that the first and second algebras are isomorphic to An,n+6,7 and An,n+6,8, respectively. The third, fourth and fifth algebras are denoted by An,n+6,11, An,n+6,12 and An,n+6,13, respectively. ∎

Theorem 3.8.

The only (n+6)-dimensional nilpotent n-Lie algebras of class 2 where n≥4 are

{H(n, 1)⊕F(5)(n≥4), H(4, 2)⊕F(1), H(5, 2),An,n+5,1⊕F(1), An,n+4,1⊕F(2), An,n+3,1⊕F(3), An,n+5,4⊕F(1)An,n+5,5⊕F(1), An,n+4,3⊕F(2), An,n+5,6⊕F(1), and An,n+6,i(1≤i≤14).

Proof. Assume that A is an (n+6)-dimensional nilpotent n-Lie algebra of class 2, where n≥4 and A=〈e1, …, en+6〉. If dim A2=1, then by Theorem 2.2, A is isomorphic to one of the following algebras:

H(n, 1)⊕F(5)(n≥4), H(4, 2)⊕F(1), H(5, 2).

Now, assume that dim A2≥2 and that 〈en+5, en+6〉⊂A2. Therefore, A/〈en+6〉 is an (n+5)-dimensional nilpotent n-Lie algebra of class 2. It follows from Theorem 2.5 that A/〈en+6〉 is one of the following forms:

H(n, 1)⊕F(4), H(4, 2), An,n+5,i(1≤i≤7).

If A/〈en+6〉 is isomorphic to H(n, 1)⊕F(4) or H(4, 2), then dim A2=2. According to Lemma 2.7, we can write A=H⊕F, where H is a generalized Heisenberg n-Lie algebra of rank 2 and F is abelian. The center of A has a dimension at most 5; thus the possible cases of A are H0, H1⊕F(1), H2⊕F(2), H3⊕F(3), where H0, H1, H2, H3 are generalized Heisenberg n-Lie algebras of rank 2 with dimensions n+6, n+5, n+4, n+3, respectively. These algebras read as follows:

An,n+6,1, An,n+5,1⊕F(1), An,n+4,1⊕F(2), An,n+3,1⊕F(3).

If A/〈en+6〉 is isomorphic to An,n+5,1, An,n+5,2 or An,n+5,3, then dim A2=3. According to Lemma 2.7, we can write A=H⊕F, where H is a generalized Heisenberg n-Lie algebra of rank 3 and F is abelian. According to ?, these algebras read as follows:

An,n+6,2, An,n+6,3, An,n+6,4, An,n+6,5, An,n+6,6,An,n+5,4⊕F(1), An,n+5,5⊕F(1), An,n+4,3⊕F(2).

Also, If A/〈en+6〉 is isomorphic to An,n+5,4, An,n+5,5 or An,n+5,6, then dim A2=4. According to Lemma 2.7, we can write A=H⊕F, where H is a generalized Heisenberg n-Lie algebra of rank 4 and F is abelian. According to ?, these algebras read as follows:

An,n+6,7, An,n+6,8, An,n+6,9, An,n+6,10,An,n+6,11, An,n+6,12, An,n+6,13, An,n+5,6⊕F(1).

Finally, If A/〈en+6〉≅An,n+5,7, then A2=Z(A)=〈en+1, en+3, en+4, en+5, en+6〉. The brackets in A can be written as

{[e1,…, en]=en+1+αen+6,[e2,…, en, en+2]=en+3+βen+6,[e1, e3,…, en, en+2]=en+4+γen+6,[e1, e2, e4,…, en, en+2]=en+5+φen+6,[e1,…, eˆi,…, en, en+2]=βien+6,4≤i≤n.

With a suitable change of basis, one can assume that α=β=γ=φ=0. Thus, the brackets in A are

{[e1,…, en]=en+1,[e2,…, en, en+2]=en+3,[e1, e3,…, en, en+2]=en+4,[e1, e2, e4,…, en, en+2]=en+5,[e1,…, eˆi,…, en, en+2]=βien+6,4≤i≤n.

By dim Z(A), we must have βi≠0 for some 4≤i≤n. Without loss of generality, assume that β4≠0. By applying the transformations

e4′=e4+∑j=5n(−1)jβiβ4ej, ei′=ei (1≤i≤n+5, i≠4), en+6′=β4en+6,

we conclude that

A=〈e1,…, en+6:[e1,…, en]=en+1, [e2,…, en, en+2]=en+3,=en+4, [e1, e2, e4,…, en, en+2]=en+5,=en+6〉,

which we denote it by An,n+6,14. ∎

In Table 1, we show all (n+4)-dimensional and (n+5)-dimensional nilpotent n-Lie algebras of class 2.

In Table 2, we show all n-Lie algebras obtained in this paper.

Table 1

Nilpotent n-Lie algebras of class 2	Nonzero multiplications
An,n+4,1	[e1,…, en]=en+3, [e2,…, en+1]=en+4
An,n+4,2	[e1,…, en]=en+3, [e3,…, en+2]=en+4 (n≥3)
An,n+4,3	[e1,…, en]=en+1, [e2,…, en, en+2]=en+3,[e1,e3,…, en, en+2]=en+4
An,n+5,1	[e1,…, en]=en+4, [e2,…, en+1]=en+5
An,n+5,2	[e1,…, en]=en+4, [e3,…, en+2]=en+5 (n≥3)
An,n+5,3	[e1,…, en]=en+5, [e4,…, en+3]=en+4 (n≥3)
An,n+5,4	[e1,…, en]=en+3, [e2,…, en+1]=en+4,[e1,e3,…, en+1]=en+5
An,n+5,5	[e1,…, en]=en+3, [e2,…,en+1]=en+4,[e2,…, en, en+2]=en+5
An,n+5,6	[e1,…, en]=en+3, [e2,…, en+1]=en+4,[e3,…, en+2]=en+5
An,n+5,7	[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

Table 2

Nilpotent n-Lie algebras of class 2	Nonzero multiplications
An,n+6,1	[e1,…, en]=en+5, [e5,…, en+4]=en+6
An,n+6,2	[e1,…, en]=en+4, [e2,…, en+1]=en+5,[e3,…, en, en+2, en+3]=en+6
An,n+6,3	[e1,…, en]=en+4, [e2,…, en+1]=en+5,[e1, e4,…, en, en+2, en+3]=en+6
An,n+6,4	[e1,…, en]=en+4, [e2,…, en+1]=en+5,[e1, e5,…, en+3]=en+6
An,n+6,5	[e1,…, en]=en+4, [e3,…, en+2]=en+5,[e2, e4,…, en+1, en+3]=en+6
An,n+6,6	[e1,…, en]=en+4, [e3,…, en+2]=en+5,[e1, e2, e5,…, en+1, en+3]=en+6
An,n+6,7	[e1,…, en]=en+3, [e2,…, en+1]=en+4,[e1, e3,…, en+1]=en+5, [e3,…, en+2]=en+6
An,n+6,8	[e1,…, en]=en+3, [e2,…, en+1]=en+4,[e1, e3,…, en+1]=en+5, [e2, e4,…, en+2]=en+6
An,n+6,9	[e1,…, en]=en+3, [e2,…,en+1]=en+4,[e1, e3,…, en+1]=en+5, [e1, e2, e5,…, en+2]=en+6
An,n+6,10	[e1,…, en]=en+3, [e2,…, en+1]=en+4,[e2,…, en, en+2]=en+5, [e1, e4,…, en+2]=en+6
An,n+6,11	[e1,…, en]=en+3, [e2,…, en+1]=en+4,[e3,…, en+2]=en+5, [e1, e3,…, en, en+2]=en+6
An,n+6,12	[e1,…, en]=en+3, [e2,…, en+1]=en+4,[e3,…, en+2]=en+5, [e1, e2, e4,…, en, en+2]=en+6
An,n+6,13	[e1,…,en]=en+3,[e2,…,en+1]=en+4,[e3,…, en+2]=en+5, [e1, e2, e5,…, en+2]=en+6
An,n+6,14	[e1,…, en]=en+1, [e2,…, en, en+2]=en+3,[e1, e3,…, en, en+2]=en+4, [e3,…,en+2]=en+5,[e1, e3,…, en, en+2]=en+6