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 $\left(n+6\right)$-dimensional n-Lie algebras of class 2 when $n\ge 4$.

Design/methodology/approach

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

Findings

In this paper, for each $n\ge 4$, the authors have found 24 nilpotent $\left(n+6\right)$ 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 $\left[x_1,\dots ,x_n\right]$, which satisfies the following generalized Jacobi identity

$$[\left[x_1,\dots ,x_n\right],y_2,\dots ,y_n]={\displaystyle \sum _{i=1}^n[x_1,\dots ,[x_i,y_2,\dots ,y_n],\dots ,x_n]]}.$$

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 $\left(n+1\right)$-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 $A^s=0$ for some nonnegative integer s, where $A^i$ is defined inductively by $A^1=A$ and $A^{i+1}=\left[A^i,A,\dots ,A\right]$. The n-Lie algebra A is nilpotent of class c, if $A^{c+1}=0$ and $A^i=0$ for each $i\le c$. The ideal $A^2=\left[A,\dots ,A\right]$ is called the derived subalgebra of A. The center of A is defined by

$$Z(A)=\left\{x\in A:\left[x,A,\dots ,A\right]=0\right\}.$$

Let $Z_0(A)=〈0〉$. Then the ith center of A is defined inductively by

$$\frac{Z_i(A)}{Z_{i-1}(A)}=Z\left(\frac{A}{Z_{i-1}(A)}\right)$$

for all $i\ge 1$. Clearly, $Z_1(A)=Z(A)$.

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 $\left(n+3\right)$-dimensional nilpotent n-Lie algebras and $\left(n+4\right)$-dimensional nilpotent n-Lie algebras of class 2 were classified in [12]. Hoseini et al. [13] classified $\left(n+5\right)$-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 $\left(n+6\right)$-dimensional n-Lie algebras of class 2 when $n\ge 4$. 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 $A^2=Z(A)$ and $\text{dim}{A}^2=k$. 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

$$H\left(n,m\right)=〈x,x_1,\dots ,x_{nm}:\left[x_{n\left(i-1\right)+1},x_{n\left(i-1\right)+2},\dots ,x_{ni}\right]=x,i=1,\dots ,m〉\text{.}$$

Theorem 2.2.

[18] Let A be a d-dimensional nilpotent n-Lie algebra, and let $\dim A^2=1$. Then, for some $m\ge 1$, it follows that

$$A\cong H\left(n,m\right)\oplus F\left(d-mn-1\right)\text{.}$$

Theorem 2.3.

[18] Let A be a nonabelian nilpotent n-Lie algebra of dimension $d\le n+2$. Then A is isomorphic to $H\left(n,1\right),H\left(n,1\right)\oplus F(1)$ or $A_{n,n+\mathrm{2,1}}$, where $A_{n,n+\mathrm{2,1}}=〈e_1,\dots ,e_{n+2}:[e_1,\dots ,e_n]=e_{n+1},[e_2,\dots ,e_{n+1}]=e_{n+2}〉$.

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

Theorem 2.4.

[12] The $\left(n+3\right)$-dimensional nonabelian nilpotent n-Lie algebras for $n>2$ over an arbitrary field are $A_{n,n+3,i}\left(2\le i\le 8\right)\text{.}$ Moreover nilpotent classes of $A_{n,n+\mathrm{3,2}}$ and $A_{n,n+\mathrm{3,5}}$ is two, nilpotent classes of $A_{n,n+\mathrm{3,3}},A_{n,n+\mathrm{3,4}}$ and $A_{n,n+\mathrm{3,8}}$ is three and finally, nilpotent classes of $A_{n,n+\mathrm{3,6}}$ and $A_{n,n+\mathrm{3,7}}$ is four (maximal class).

Theorem 2.5.

[12] The only $\left(n+4\right)$-dimensional nilpotent n-Lie algebras of class 2 are $H\left(n,1\right)\oplus F(3),A_{n,n+\mathrm{4,1}},A_{n,n+\mathrm{4,2}},A_{n,n+\mathrm{4,3}},H\left(2,2\right)\oplus F(1),H\left(3,2\right),L_{\mathrm{6,22}}(\epsilon )\text{,}$ and $L_{\mathrm{6,7}}^2(\eta )\text{.}$

Theorem 2.6.

[13] The $\left(n+5\right)$-dimensional nilpotent n-Lie algebras of class 2 for $n>2$ over an arbitrary field are $H\left(n,1\right)\oplus F(4),A_{n,n+5,i}\left(1\le i\le 7\right),H\left(3,2\right)\oplus F(1)$ and $H\left(\mathrm{4,2}\right).$

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 $A=H\oplus F$.

3. Main results

In this section, we classify $\left(n+6\right)$-dimensional nilpotent n-Lie algebras of class 2. If n-Lie algebra A is nilpotent of class 2, then A is nonabelian and $A^2\subseteq Z(A)$. 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 $3\le k\le n+1$ such that $A^2=Z(A)$ and $\dim A^2=2$. Then

$$A\cong 〈e_1,\dots ,e_{n+k}:[e_{k-1},\dots ,e_{n+k-2}]=e_{n+k},[e_1,\dots ,e_n]=e_{n+k-1}〉\text{.}$$

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

$$A\cong 〈e_1,e_2,e_3,e_4,e_5:[e_1,e_2]=e_4,[e_2,e_3]=e_5〉\text{.}$$

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

$$A\cong A_{n,2n+\mathrm{2,1}}=〈e_1,\dots ,e_{2n+2}:[e_1,\dots ,e_n]=e_{2n+1},[e_{n+1},\dots ,e_{2n}]=e_{2n+2}〉\text{.}$$

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:

$$A_{n,2n+\mathrm{3,1}}=〈e_1,\dots ,e_{2n+3}:\left[e_1,\dots ,e_n\right]=e_{2n+3},\left[e_2,\dots ,e_{n+1}\right]=\left[e_{n+2},\dots ,e_{2n+1}\right]=e_{2n+2}〉\text{.}$$

$$A_{n,2n+\mathrm{3,2}}=〈e_1,\dots ,e_{2n+3}:\left[e_1,\dots ,e_n\right]=\left[e_{n+1},\dots ,e_{2n}\right]=e_{2n+3},\left[e_2,\dots ,e_{n+1}\right]=\left[e_{n+2},\dots ,e_{2n+1}\right]=e_{2n+2}〉\text{.}$$

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

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

According to Theorem 2.7, we can write $A=H\oplus F$, 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 $\left(n+4\right)$-dimensional generalized Heisenberg n-Lie algebra of rank 3 is $A_{n,n+\mathrm{4,3}}$.

2. The only $\left(n+5\right)$-dimensional generalized Heisenberg n-Lie algebras of rank 3 are $A_{n,n+\mathrm{5,4}}$ and $A_{n,n+\mathrm{5,5}}$.

3. The only $\left(n+5\right)$-dimensional generalized Heisenberg n-Lie algebra of rank 4 is $A_{n,n+\mathrm{5,6}}$.

The following lemma defines the structure of $\left(n+6\right)$-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

$$A\cong A_{n,n+\mathrm{6,1}}=〈e_1,\dots ,e_{n+6}:\left[e_1,\dots ,e_n\right]=e_{n+5},\left[e_5,\dots ,e_{n+4}\right]=e_{n+6}〉\text{.}$$

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

Theorem 3.6.

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

$$A_{n,n+\mathrm{6,2}},A_{n,n+\mathrm{6,3}},A_{n,n+\mathrm{6,4}},A_{n,n+\mathrm{6,5}},and{A}_{n,n+\mathrm{6,6}}(\epsilon ).$$

Proof. Suppose that A is an $\left(n+6\right)$-dimensional generalized Heisenberg n-Lie algebra of rank 3 with basis $\left\{e_1,\dots ,e_{n+6}\right\}$, which $n\ge 4$. Also, suppose that $A^2=〈e_{n+4},e_{n+5},e_{n+6}〉$. In this case, $A/〈e_{n+6}〉$ is an $\left(n+5\right)$-dimensional nilpotent n-Lie algebra of class 2 with derived algebra of dimension 2. By Theorem 2.6, we have three possibilities for $A/〈e_{n+6}〉$: Case 1: Let $A/〈e_{n+6}〉\cong A_{n,n+\mathrm{5,1}}$. Then the brackets in A can be written as

$$\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+4}+\alpha e_{n+6}\text{,}& \\ \left[e_2,\dots ,e_{n+1}\right]=e_{n+5}+\beta e_{n+6}\text{,}& \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+1}\right]={\alpha }_i{e}_{n+6}\text{,}& 2\le i\le n\text{,}\\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+2}\right]={\beta }_i{e}_{n+6}\text{,}& 1\le i\le n\text{,}\\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+3}\right]={\gamma }_i{e}_{n+6}\text{,}& 1\le i\le n\text{,}\\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+1},e_{n+2}\right]={\chi }_{ij}{e}_{n+6}\text{,}& 1\le i<j\le n\text{,}\\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+1},e_{n+3}\right]={\delta }_{ij}{e}_{n+6}\text{,}& 1\le i<j\le n\text{,}\\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+2},e_{n+3}\right]={\lambda }_{ij}{e}_{n+6}\text{,}& 1\le i<j\le n\text{,}\\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,{\hat{e}}_k,\dots ,e_n,e_{n+1},e_{n+2},e_{n+3}\right]={\phi }_{ijk}{e}_{n+6}\text{,}& 1\le i<j<k\le n\text{.}\end{array}$$

Regarding a suitable change of basis, one can assume that $\alpha =\beta =0$.

Since $\dim {\left(A/〈e_{n+4},e_{n+5}〉\right)}^2=1$, we have $A/〈e_{n+4},e_{n+5}〉\cong H\left(n,1\right)\oplus F(3)$. According to the structure of n-Lie algebras, we conclude that one of the coefficients

$${\lambda }_{ij}\left(1\le i<j\le n\right),{\phi }_{ijk}\left(1\le i<j<k\le n\right)$$

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

1. ${\lambda }_{12}=1,{\lambda }_{ij}=0(1\le i<j\le n,\left(i,j\right)\ne \left(1,2\right)),\text{and} {\phi }_{ijk}=0\left(1\le i<j<k\le n\right)$. In this case, the brackets in A can be written as

$$\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_3,\dots ,e_n,e_{n+2},e_{n+3}\right]=e_{n+6},$$

which we denote it by $A_{n,n+\mathrm{6,2}}$.

1. ${\lambda }_{23}=1,{\lambda }_{ij}=0(1\le i<j\le n,(i,j)\ne \left(2,3\right)),\text{and} {\phi }_{ijk}=0\left(1\le i<j<k\le n\right)$. In this case, the brackets in A can be written as

$$\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_1,e_4,\dots ,e_n,e_{n+2},e_{n+3}\right]=e_{n+6}\text{,}$$

which we denote it by $A_{n,n+\mathrm{6,3}}$.

1. ${\lambda }_{ij}=0\left(1\le i<j\le n\right),{\phi }_{123}=1,\text{and}{\phi }_{ijk}=0(1\le i<j<k\le n,(i,j,k)\ne \left(1,2,3\right))$. In this case, the brackets in A can be written as

$$\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_4,\dots ,e_{n+3}\right]=e_{n+6}\text{.}$$

One can easily see that this algebra is isomorphic to $A_{n+\mathrm{6,3}}$.

1. ${\lambda }_{ij}=0(1\le i<j\le n),{\phi }_{234}=1,{\phi }_{ijk}=0(1\le i<j<k\le n),(i,j,k)\ne (\mathrm{2,3,4}))$. In this case, the brackets in A can be written as

$$\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_1,e_5,\dots ,e_{n+3}\right]=e_{n+6}\text{,}$$

which we denote it by $A_{n,n+\mathrm{6,4}}\text{.}$

Case 2.

Let $A/〈e_{n+6}〉\cong A_{n,n+\mathrm{5,2}}$. Then the brackets in A can be written as

$$\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+4}+\alpha e_{n+6},\hfill & \hfill \\ \left[e_2,\dots ,e_{n+1}\right]=e_{n+5}+\beta e_{n+6},\hfill & \hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+1}\right]={\alpha }_i{e}_{n+6},\hfill & 1\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+2}\right]={\beta }_i{e}_{n+6},\hfill & 1\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+3}\right]={\gamma }_i{e}_{n+6},\hfill & 1\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+1},e_{n+2}\right]={\chi }_{ij}{e}_{n+6},\hfill & 1\le i<j\le n,\hfill \\ \text{mtd}\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+1},e_{n+3}\right]={\delta }_{ij}{e}_{n+6},\hfill & 1\le i<j\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+2},e_{n+3}\right]={\lambda }_{ij}{e}_{n+6},\hfill & 1\le i<j\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,{\hat{e}}_k,\dots ,e_n,e_{n+1},e_{n+2},e_{n+3}\right]={\phi }_{ijk}{e}_{n+6},\hfill & 1\le i<j<k\le n.\hfill \end{array}$$

Regarding a suitable change of basis, one can assume that $\alpha =\beta =0$.

Since $\dim {\left(A/〈e_{n+4},e_{n+5}〉\right)}^2=1$, we have $A/〈e_{n+4},e_{n+5}〉\cong H\left(n,1\right)\oplus F(3)$. According to the structure of n-Lie algebras and $Z(A)=〈e_{n+4},e_{n+5},e_{n+6}〉$, we conclude that one of the coefficients

$${\gamma }_i\left(1\le i\le n\right),{\delta }_{ij}\left(1\le i<j\le n\right)\text{,}$$

$${\lambda }_{ij}\left(1\le i<j\le n\right),{\phi }_{ijk}\left(1\le i<j<k\le n\right)$$

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

$$\begin{array}{l}\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_2,\dots ,e_n,e_{n+3}\right]=e_{n+6}\text{,}\\ \left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_1,e_2,e_4,\dots ,e_n,e_{n+3}\right]=e_{n+6}\text{,}\\ \left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_2,e_4,\dots ,e_{n+1},e_{n+3}\right]=e_{n+6}\text{,}\\ \left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_1,e_2,e_5,\dots ,e_{n+1},e_{n+3}\right]=e_{n+6}\text{.}\end{array}$$

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

$$\begin{array}{l}{A}_{n,n+\mathrm{6,5}}=〈e_1,\dots ,e_{n+6}:\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_2,e_4,\dots ,e_{n+1},e_{n+3}\right]=e_{n+6}〉\text{,}\\ A_{n,n+\mathrm{6,6}}=〈e_1,\dots ,e_{n+6}:\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5},\left[e_1,e_2,e_5,\dots ,e_{n+1},e_{n+3}\right]=e_{n+6}〉\text{,}\end{array}$$

Case 3.

Let $A/〈e_{n+6}〉\cong A_{n,n+\mathrm{5,3}}$. Then the brackets in A can be written as

$$\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+4}+\alpha e_{n+6},\hfill & \hfill \\ \left[e_2,\dots ,e_{n+1}\right]=e_{n+5}+\beta e_{n+6},\hfill & \hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+1}\right]={\alpha }_i{e}_{n+6},\hfill & 1\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+2}\right]={\beta }_i{e}_{n+6},\hfill & 1\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+3}\right]={\gamma }_i{e}_{n+6},\hfill & 1\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+1},e_{n+2}\right]={\chi }_{ij}{e}_{n+6},\hfill & 1\le i<j\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+1},e_{n+3}\right]={\delta }_{ij}{e}_{n+6},\hfill & 1\le i<j\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+2},e_{n+3}\right]={\lambda }_{ij}{e}_{n+6},\hfill & 1\le i<j\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,{\hat{e}}_k,\dots ,e_n,e_{n+1},e_{n+2},e_{n+3}\right]={\phi }_{ijk}{e}_{n+6},\hfill & 1\le i<j<k\le n,\hfill \\ \text{mtd}\hfill \end{array}$$

Regarding a suitable change of basis, one can assume that $\alpha =\beta =0$.

Since $\dim {\left(A/〈e_{n+4},e_{n+5}〉\right)}^2=1$, we have $A/〈e_{n+4},e_{n+5}〉\cong H\left(n,1\right)\oplus F(3)$. According to the structure of n-Lie algebra, we conclude that one of the coefficients

$$\begin{array}{l}{\alpha }_i,{\beta }_i,{\gamma }_i(1\le i\le n),{\chi }_{ij},{\delta }_{ij},{\lambda }_{ij}(1\le i<j\le n),\\ {\phi }_{ijk}(1\le i<j<k\le n,(i,j,k)\ne (\mathrm{1,2,3}))\end{array}$$

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

$$\begin{array}{l}\left[e_1,\dots ,e_n\right]=e_{n+5},\left[e_4,\dots ,e_{n+3}\right]=e_{n+4},\left[e_2,e_3,\dots ,e_{n+1}\right]=e_{n+6}\text{,}\\ \left[e_1,\dots ,e_n\right]=e_{n+5},\left[e_4,\dots ,e_{n+3}\right]=e_{n+4},\left[e_1,e_2,e_3,e_5,\dots ,e_{n+1}\right]=e_{n+6}\text{,}\\ \left[e_1,\dots ,e_n\right]=e_{n+5},\left[e_4,\dots ,e_{n+3}\right]=e_{n+4},\left[e_2,e_3,e_5,\dots ,e_{n+2}\right]=e_{n+6}\text{.}\end{array}$$

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

Theorem 3.7.

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

$$A_{n,n+\mathrm{6,7}},A_{n,n+\mathrm{6,8}},A_{n,n+\mathrm{6,9}},A_{n,n+\mathrm{6,10}},A_{n,n+\mathrm{6,11}},A_{n,n+\mathrm{6,12}}\mathit{and}{A}_{n,n+\mathrm{6,13}}\text{.}$$

Proof. Suppose that A is an $\left(n+6\right)$-dimensional generalized Heisenberg n-Lie algebra of rank 4 with basis $\left\{e_1,\dots ,e_{n+6}\right\}$, which $n\ge 4$. Also, suppose that $A^2=〈e_{n+3},e_{n+4},e_{n+5},e_{n+6}〉$. In this case, $A/〈e_{n+6}〉$ is an $\left(n+5\right)$-dimensional nilpotent n-Lie algebra of class 2 with derived algebra of dimension 3. By Theorem LABEL:?, we have three possibilities for $A/〈e_{n+6}〉$:

Case 4.

Let $A/〈e_{n+6}〉\cong A_{n,n+\mathrm{5,4}}$. Then the brackets in A can be written as

$$\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}+\alpha e_{n+6},\hfill & \hfill \\ \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}+\beta e_{n+6},\hfill & \hfill \\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5}+\gamma e_{n+6},\hfill & \hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+1}\right]={\alpha }_i{e}_{n+6},\hfill & 3\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+2}\right]={\beta }_i{e}_{n+6},\hfill & 1\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+1},e_{n+2}\right]={\chi }_{ij}{e}_{n+6},\hfill & 1\le i<j\le n.\hfill \end{array}$$

Regarding a suitable change of basis, one can assume that $\alpha =\beta =\gamma =0$.

Since $\dim {\left(A/〈e_{n+3},e_{n+4},e_{n+5}〉\right)}^2=1$, we have $A/〈e_{n+3},e_{n+4},e_{n+5}〉\cong H\left(n,1\right)\oplus F(2)$. According to the structure of n-Lie algebras, we conclude that one of the coefficients

$${\alpha }_i\left(3\le i\le n\right),{\beta }_i\left(1\le i\le n\right),{\chi }_{ij}\left(1\le i<j\le n\right)$$

is equal to one, and the others are zero. According to $Z(A)=\{e_{n+3},e_{n+4},e_{n+5},e_{n+6}\}$, the coefficients ${\alpha }_i\left(3\le i\le n\right)$ cannot be equal to one. We have three possibilities:

1. ${\beta }_1=1,{\beta }_i=0\left(2\le i\le n\right),{\alpha }_i=0\left(3\le i\le n\right),{\chi }_{ij}=0\left(1\le i<j\le n\right)$.

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

$$\begin{array}{l}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5},\left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+6}\text{,}\end{array}$$

which we denote it by $A_{n,n+\mathrm{6,7}}$.

1. ${\beta }_3=1,{\beta }_i=0\left(1\le i\le n,n\ne 3\right),{\alpha }_i=0\left(3\le i\le n\right),{\chi }_{ij}=0\left(1\le i<j\le n\right)$.

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

$$\begin{array}{l}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]e_{n+4}\text{,}\\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5},\left[e_1,e_2,e_4,\dots ,e_n,e_{n+2}\right]=e_{n+6}\text{,}\end{array}$$

which we denote it by $A_{n,n+\mathrm{6,8}}$.

1. Only one of ${\chi }_{ij}s\left(1\le i<j\le n\right)$ is equal to one and the others are zero. Up to isomorphism, we have the following algebras:

$$\begin{array}{l}\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5}\text{,}& \left[e_3,\dots ,e_{n+2}\right]=e_{n+6}\text{,}\end{array}\\ \{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5}\text{,}& \left[e_2,e_4,\dots ,e_{n+2}\right]=e_{n+6}\text{,}\end{array}\\ \{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5}\text{,}& \left[e_1,e_2,e_5,\dots ,e_{n+2}\right]=e_{n+6}\text{.}\end{array}\end{array}$$

One can easily see that the first and second algebras are isomorphic to $A_{n,n+\mathrm{6,7}}$ and $A_{n,n+\mathrm{6,8}}$, respectively. The third algebras is denoted by $A_{n,n+\mathrm{6,9}}$.

Case 5.

Let $A/〈e_{n+6}〉\cong A_{n,n+\mathrm{5,5}}$. Then the brackets in A can be written as

$$\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}+\alpha e_{n+6},\hfill & \hfill \\ \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}+\beta e_{n+6},\hfill & \hfill \\ \left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+5}+\gamma e_{n+6},\hfill & \hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+1}\right]={\alpha }_i{e}_{n+6},\hfill & 2\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+2}\right]={\beta }_i{e}_{n+6},\hfill & 2\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+1},e_{n+2}\right]={\chi }_{ij}{e}_{n+6},\hfill & 1\le i<j\le n.\hfill \end{array}$$

Regarding a suitable change of basis, one can assume that $\alpha =\beta =\gamma =0$.

Since $\dim {\left(A/〈e_{n+3},e_{n+4},e_{n+5}〉\right)}^2=1$, we have $A/〈e_{n+3},e_{n+4},e_{n+5}〉\cong H\left(n,1\right)\oplus F(2)$. According to the structure of n-Lie algebras, we conclude that one of the coefficients

$${\alpha }_i\left(3\le i\le n\right),{\beta }_i\left(1\le i\le n\right),{\chi }_{ij}\left(1\le i<j\le n\right)$$

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

1. Only one of ${\alpha }_i\left(3\le i\le n\right)$ and ${\beta }_i\left(1\le i\le n\right)$ is equal to one and the others are zero. Without loss of generality, we assume ${\alpha }_2=1$. Thus, the brackets in A can be written as

$$\begin{array}{l}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4},\\ \left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+5},\left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+6}.\end{array}$$

One can easily see that this algebra is isomorphic to $A_{n,n+\mathrm{6,7}}$.

1. Only one of ${\chi }_{ij}s\left(1\le i<j\le n\right)$ is equal to one and the others are zero. Up to isomorphism, we have the following algebras:

$$\begin{array}{l}\{\begin{array}{ll}[e_1,\dots ,e_n]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ [e_2,\dots ,e_n,e_{n+2}]=e_{n+5}\text{,}& \left[e_3,\dots ,e_{n+2}\right]=e_{n+6}\text{,}\end{array}\\ \{\begin{array}{ll}[e_1,\dots ,e_n]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ [e_2,\dots ,e_n,e_{n+2}]=e_{n+5}\text{,}& [e_1,e_4,\dots ,e_{n+2}]=e_{n+6}\text{.}\end{array}\end{array}$$

One can easily see that the first algebra is isomorphic to $A_{n,n+\mathrm{6,7}}$. The second algebra is denoted by $A_{n,n+\mathrm{6,10}}$.

Case 6.

Let $A/〈e_{n+6}〉\cong A_{n,n+\mathrm{5,6}}$. Then the brackets in A can be written as

$$\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}+\alpha e_{n+6},\hfill & \hfill \\ \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}+\beta e_{n+6},\hfill & \hfill \\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5}+\gamma e_{n+6},\hfill & \hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+1}\right]={\alpha }_i{e}_{n+6},\hfill & 2\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+2}\right]={\beta }_i{e}_{n+6},\hfill & 1\le i\le n,\hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,{\hat{e}}_j,\dots ,e_n,e_{n+1},e_{n+2}\right]={\chi }_{ij}{e}_{n+6},\hfill & 1\le i<j\le n,\left(i,j\right)\ne \left(\mathrm{1,2}\right).\hfill \end{array}$$

Regarding a suitable change of basis, one can assume that $\alpha =\beta =\gamma =0$.

Since $\dim {\left(A/〈e_{n+3},e_{n+4},e_{n+5}〉\right)}^2=1$, we have $A/〈e_{n+3},e_{n+4},e_{n+5}〉\cong H\left(n,1\right)\oplus F(2)$. According to the structure of n-Lie algebras, we conclude that one of the coefficients

$${\alpha }_i\left(2\le i\le n\right),{\beta }_i\left(1\le i\le n\right),{\chi }_{ij}(1\le i<j\le n,\left(i,j\right)\ne \left(1,2\right))$$

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

$$\begin{array}{l}\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\text{,}& \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+6}\text{,}\end{array}\\ \{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\text{,}& \left[e_1,e_2,e_4,\dots ,e_{n+1}\right]=e_{n+6}\text{,}\end{array}\\ \{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\text{,}& \left[e_1,e_3,\dots ,e_n,e_{n+2}\right]=e_{n+6}\text{,}\end{array}\\ \{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\text{,}& \left[e_1,e_2,e_4,\dots ,e_n,e_{n+2}\right]=e_{n+6}\text{,}\end{array}\\ \{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+3}\text{,}& \left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\text{,}& \left[e_1,e_2,e_5,\dots ,e_{n+2}\right]=e_{n+6}\text{.}\end{array}\end{array}$$

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

Theorem 3.8.

The only $\left(n+6\right)$-dimensional nilpotent n-Lie algebras of class 2 where $n\ge 4$ are

$$\{\begin{array}{l}H\left(n,1\right)\oplus F(5)\left(n\ge 4\right),H\left(4,2\right)\oplus F(1),H\left(5,2\right)\text{,}\hfill \\ A_{n,n+\mathrm{5,1}}\oplus F(1),A_{n,n+\mathrm{4,1}}\oplus F(2),A_{n,n+\mathrm{3,1}}\oplus F(3),A_{n,n+\mathrm{5,4}}\oplus F(1)\hfill \\ A_{n,n+\mathrm{5,5}}\oplus F(1),A_{n,n+\mathrm{4,3}}\oplus F(2),A_{n,n+\mathrm{5,6}}\oplus F(1),and{A}_{n,n+6,i}\left(1\le i\le 14\right)\text{.}\hfill \end{array}$$

Proof. Assume that A is an $\left(n+6\right)$-dimensional nilpotent n-Lie algebra of class 2, where $n\ge 4$ and $A=〈e_1,\dots ,e_{n+6}〉$. If $\text{dim}{A}^2=1$, then by Theorem 2.2, A is isomorphic to one of the following algebras:

$$H\left(n,1\right)\oplus F(5)\left(n\ge 4\right),H\left(4,2\right)\oplus F(1),H\left(5,2\right)\text{.}$$

Now, assume that $\text{dim}{A}^2\ge 2$ and that $〈e_{n+5},e_{n+6}〉\subset A^2$. Therefore, $A/〈e_{n+6}〉$ is an $\left(n+5\right)$-dimensional nilpotent n-Lie algebra of class 2. It follows from Theorem 2.5 that $A/〈e_{n+6}〉$ is one of the following forms:

$$H\left(n,1\right)\oplus F(4),H\left(4,2\right),A_{n,n+5,i}\left(1\le i\le 7\right)\text{.}$$

If $A/〈e_{n+6}〉$ is isomorphic to $H\left(n,1\right)\oplus F(4)$ or $H\left(4,2\right)$, then $\text{dim}{A}^2=2$. According to Lemma 2.7, we can write $A=H\oplus F$, 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 $H_0,H_1\oplus F(1),H_2\oplus F(2),H_3\oplus F(3)$, where $H_0,H_1,H_2,H_3$ 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:

$$A_{n,n+\mathrm{6,1}},A_{n,n+\mathrm{5,1}}\oplus F(1),A_{n,n+\mathrm{4,1}}\oplus F(2),A_{n,n+\mathrm{3,1}}\oplus F(3)\text{.}$$

If $A/〈e_{n+6}〉$ is isomorphic to $A_{n,n+\mathrm{5,1}},A_{n,n+\mathrm{5,2}}$ or $A_{n,n+\mathrm{5,3}}$, then $\text{dim}{A}^2=3$. According to Lemma 2.7, we can write $A=H\oplus F$, where H is a generalized Heisenberg n-Lie algebra of rank 3 and F is abelian. According to ?, these algebras read as follows:

$$\begin{array}{l}{A}_{n,n+\mathrm{6,2}},A_{n,n+\mathrm{6,3}},A_{n,n+\mathrm{6,4}},A_{n,n+\mathrm{6,5}},A_{n,n+\mathrm{6,6}}\text{,}\\ A_{n,n+\mathrm{5,4}}\oplus F(1),A_{n,n+\mathrm{5,5}}\oplus F(1),A_{n,n+\mathrm{4,3}}\oplus F(2)\text{.}\end{array}$$

Also, If $A/〈e_{n+6}〉$ is isomorphic to $A_{n,n+\mathrm{5,4}},A_{n,n+\mathrm{5,5}}$ or $A_{n,n+\mathrm{5,6}}$, then $\text{dim}{A}^2=4$. According to Lemma 2.7, we can write $A=H\oplus F$, where H is a generalized Heisenberg n-Lie algebra of rank 4 and F is abelian. According to ?, these algebras read as follows:

$$\begin{array}{l}{A}_{n,n+\mathrm{6,7}},A_{n,n+\mathrm{6,8}},A_{n,n+\mathrm{6,9}},A_{n,n+\mathrm{6,10}}\text{,}\\ A_{n,n+\mathrm{6,11}},A_{n,n+\mathrm{6,12}},A_{n,n+\mathrm{6,13}},A_{n,n+\mathrm{5,6}}\oplus F(1)\text{.}\end{array}$$

Finally, If $A/〈e_{n+6}〉\cong A_{n,n+\mathrm{5,7}}$, then $A^2=Z(A)=〈e_{n+1},e_{n+3},e_{n+4},e_{n+5},e_{n+6}〉$. The brackets in A can be written as

$$\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+1}+\alpha e_{n+6}\text{,}\hfill & \hfill \\ \left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+3}+\beta e_{n+6}\text{,}\hfill & \hfill \\ \left[e_1,e_3,\dots ,e_n,e_{n+2}\right]=e_{n+4}+\gamma e_{n+6}\text{,}\hfill & \hfill \\ \left[e_1,e_2,e_4,\dots ,e_n,e_{n+2}\right]=e_{n+5}+\phi e_{n+6}\text{,}\hfill & \hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+2}\right]={\beta }_i{e}_{n+6}\text{,}\hfill & 4\le i\le n\text{.}\hfill \end{array}$$

With a suitable change of basis, one can assume that $\alpha =\beta =\gamma =\phi =0$. Thus, the brackets in A are

$$\{\begin{array}{ll}\left[e_1,\dots ,e_n\right]=e_{n+1}\text{,}\hfill & \hfill \\ \left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+3}\text{,}\hfill & \hfill \\ \left[e_1,e_3,\dots ,e_n,e_{n+2}\right]=e_{n+4}\text{,}\hfill & \hfill \\ \left[e_1,e_2,e_4,\dots ,e_n,e_{n+2}\right]=e_{n+5}\text{,}\hfill & \hfill \\ \left[e_1,\dots ,{\hat{e}}_i,\dots ,e_n,e_{n+2}\right]={\beta }_i{e}_{n+6}\text{,}\hfill & 4\le i\le n\text{.}\hfill \end{array}$$

By $\text{dim}Z(A)$, we must have ${\beta }_i\ne 0$ for some $4\le i\le n$. Without loss of generality, assume that ${\beta }_4\ne 0$. By applying the transformations

$${{\displaystyle e}}_4^{\prime }=e_4+{\displaystyle \sum _{j=5}^n{{\displaystyle \left(-1\right)}}^j}\frac{{\beta }_i}{{\beta }_4}{e}_j,{{\displaystyle e}}_i^{\prime }=e_i\left(1\le i\le n+5,i\ne 4\right),{{\displaystyle e}}_{n+6}^{\prime }={\beta }_4{e}_{n+6}\text{,}$$

we conclude that

$$A=〈e_1,\dots ,e_{n+6}:\left[e_1,\dots ,e_n\right]=e_{n+1},\left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+3},=e_{n+4},\left[e_1,e_2,e_4,\dots ,e_n,e_{n+2}\right]=e_{n+5},=e_{n+6}〉\text{,}$$

which we denote it by $A_{n,n+\mathrm{6,14}}$. ∎

In Table 1, we show all $\left(n+4\right)$-dimensional and $\left(n+5\right)$-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
$A_{n,n+\mathrm{4,1}}$	$\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}$
$A_{n,n+\mathrm{4,2}}$	$\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_3,\dots ,e_{n+2}\right]=e_{n+4}\left(n\ge 3\right)$
$A_{n,n+\mathrm{4,3}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+1},\left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+3}\text{,}\\ \left[e_1,e_3,\dots ,e_n,e_{n+2}\right]=e_{n+4}\end{array}$
$A_{n,n+\mathrm{5,1}}$	$\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5}$
$A_{n,n+\mathrm{5,2}}$	$\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\left(n\ge 3\right)$
$A_{n,n+\mathrm{5,3}}$	$\left[e_1,\dots ,e_n\right]=e_{n+5},\left[e_4,\dots ,e_{n+3}\right]=e_{n+4}\left(n\ge 3\right)$
$A_{n,n+\mathrm{5,4}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5}\end{array}$
$A_{n,n+\mathrm{5,5}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+5}\end{array}$
$A_{n,n+\mathrm{5,6}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\end{array}$
$A_{n,n+\mathrm{5,7}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+1},\left[e_1,e_2,e_4,\dots ,e_n,e_{n+2}\right]=e_{n+5}\text{,}\\ \left[e_1,e_3,\dots ,e_n,e_{n+2}\right]=e_{n+4},\left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+3}\end{array}$

Table 2

Nilpotent n-Lie algebras of class 2	Nonzero multiplications
$A_{n,n+\mathrm{6,1}}$	$\left[e_1,\dots ,e_n\right]=e_{n+5},\left[e_5,\dots ,e_{n+4}\right]=e_{n+6}$
$A_{n,n+\mathrm{6,2}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5}\text{,}\\ \left[e_3,\dots ,e_n,e_{n+2},e_{n+3}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,3}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5}\text{,}\\ \left[e_1,e_4,\dots ,e_n,e_{n+2},e_{n+3}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,4}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_2,\dots ,e_{n+1}\right]=e_{n+5}\text{,}\\ \left[e_1,e_5,\dots ,e_{n+3}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,5}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\text{,}\\ \left[e_2,e_4,\dots ,e_{n+1},e_{n+3}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,6}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+4},\left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\text{,}\\ \left[e_1,e_2,e_5,\dots ,e_{n+1},e_{n+3}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,7}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5},\left[e_3,\dots ,e_{n+2}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,8}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5},\left[e_2,e_4,\dots ,e_{n+2}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,9}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_1,e_3,\dots ,e_{n+1}\right]=e_{n+5},\left[e_1,e_2,e_5,\dots ,e_{n+2}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,10}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+3},[e_2,\dots ,e_{n+1}]=e_{n+4}\text{,}\\ \left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+5},\left[e_1,e_4,\dots ,e_{n+2}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,11}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5},\left[e_1,e_3,\dots ,e_n,e_{n+2}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,12}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+3},\left[e_2,\dots ,e_{n+1}\right]=e_{n+4}\text{,}\\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5},\left[e_1,e_2,e_4,\dots ,e_n,e_{n+2}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,13}}$	$\begin{array}{c}[e_1,\dots ,e_n]=e_{n+3},[e_2,\dots ,e_{n+1}]=e_{n+4},\\ \left[e_3,\dots ,e_{n+2}\right]=e_{n+5},\left[e_1,e_2,e_5,\dots ,e_{n+2}\right]=e_{n+6}\end{array}$
$A_{n,n+\mathrm{6,14}}$	$\begin{array}{c}\left[e_1,\dots ,e_n\right]=e_{n+1},\left[e_2,\dots ,e_n,e_{n+2}\right]=e_{n+3}\text{,}\\ \left[e_1,e_3,\dots ,e_n,e_{n+2}\right]=e_{n+4},\left[e_3,\dots ,e_{n+2}\right]=e_{n+5}\text{,}\\ \left[e_1,e_3,\dots ,e_n,e_{n+2}\right]=e_{n+6}\end{array}$