Research Article| December 11 2020

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

Mehdi Jamshidi;

Farshid Saeedi;

Hamid Darabi

Author & Article Information

Hamid Darabi can be contacted at: darabi@esfarayen.ac.ir

Publisher: Emerald Publishing on behalf of King Saud University

Received: September 24 2020

Revision Received: September 26 2020

Accepted: September 26 2020

Online ISSN: 2588-9214

Print ISSN: 1319-5166

2020

Mehdi Jamshidi, Farshid Saeedi and Hamid Darabi

Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode

Arab Journal of Mathematical Sciences (2021) 27 (2): 139–150.

https://doi.org/10.1108/AJMS-09-2020-0075

Purpose

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

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 $n \geq 4$ ⁠, 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.

1. Introduction

In 1985, Filippov [1] introduced the concept of n-Lie (Filippov) algebras, as an n-ary multilinear and skew-symmetric operation $[x_{1}, \dots, x_{n}]$ ⁠, which satisfies the following generalized Jacobi identity

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

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

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

\frac{Z_{i} (A)}{Z_{i - 1} (A)} = Z (\frac{A}{Z_{i - 1} (A)})

for all $i \geq 1$ ⁠. 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 $(n + 3)$ -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 $n \geq 4$ ⁠. 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 $dim A^{2} = k$ ⁠. 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 $m n + 1$ for some natural number m, and it is isomorphic to

H (n, m) = 〈 x, x_{1}, \dots, x_{n m} : [x_{n (i - 1) + 1}, x_{n (i - 1) + 2}, \dots, x_{n i}] = x, i = 1, \dots, m 〉 .

Theorem 2.2.

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

A ≅ H (n, m) \oplus F (d - m n - 1) .

Theorem 2.3.

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

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

Theorem 2.5.

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

Theorem 2.6.

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

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 $A^{2} \subseteq Z (A)$ ⁠. 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 $2 n + 3$ ⁠.

Theorem 3.1.

[18] Let A be a nilpotent n-Lie algebra of dimension $d = n + k$ for $3 \leq k \leq n + 1$ such that $A^{2} = Z (A)$ and $\dim A^{2} = 2$ ⁠. Then

A ≅ 〈 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} 〉 .

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

A ≅ 〈 e_{1}, e_{2}, e_{3}, e_{4}, e_{5} : [e_{1}, e_{2}] = e_{4}, [e_{2}, e_{3}] = e_{5} 〉 .

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 $2 n + 2$ ⁠. Then

A ≅ A_{n, 2 n + 2,1} = 〈 e_{1}, \dots, e_{2 n + 2} : [e_{1}, \dots, e_{n}] = e_{2 n + 1}, [e_{n + 1}, \dots, e_{2 n}] = e_{2 n + 2} 〉 .

Theorem 3.3.

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

A_{n, 2 n + 3,1} = 〈 e_{1}, \dots, e_{2 n + 3} : [e_{1}, \dots, e_{n}] = e_{2 n + 3}, [e_{2}, \dots, e_{n + 1}] = [e_{n + 2}, \dots, e_{2 n + 1}] = e_{2 n + 2} 〉 .

A_{n, 2 n + 3,2} = 〈 e_{1}, \dots, e_{2 n + 3} : [e_{1}, \dots, e_{n}] = [e_{n + 1}, \dots, e_{2 n}] = e_{2 n + 3}, [e_{2}, \dots, e_{n + 1}] = [e_{n + 2}, \dots, e_{2 n + 1}] = e_{2 n + 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 $A = H \oplus F$ ⁠, 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.

The only $(n + 4)$ -dimensional generalized Heisenberg n-Lie algebra of rank 3 is $A_{n, n + 4,3}$ ⁠.
The only $(n + 5)$ -dimensional generalized Heisenberg n-Lie algebras of rank 3 are $A_{n, n + 5,4}$ and $A_{n, n + 5,5}$ ⁠.
The only $(n + 5)$ -dimensional generalized Heisenberg n-Lie algebra of rank 4 is $A_{n, n + 5,6}$ ⁠.

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 ≅ A_{n, n + 6,1} = 〈 e_{1}, \dots, e_{n + 6} : [e_{1}, \dots, e_{n}] = e_{n + 5}, [e_{5}, \dots, e_{n + 4}] = e_{n + 6} 〉 .

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

Theorem 3.6.

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

A_{n, n + 6,2}, A_{n, n + 6,3}, A_{n, n + 6,4}, A_{n, n + 6,5}, a n d A_{n, n + 6,6} (ε) .

Proof. Suppose that A is an $(n + 6)$ -dimensional generalized Heisenberg n-Lie algebra of rank 3 with basis ${e_{1}, \dots, e_{n + 6}}$ ⁠, which $n \geq 4$ ⁠. Also, suppose that $A^{2} = 〈 e_{n + 4}, e_{n + 5}, e_{n + 6} 〉$ ⁠. In this case, $A / 〈 e_{n + 6} 〉$ 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 $A / 〈 e_{n + 6} 〉$ ⁠: Case 1: Let $A / 〈 e_{n + 6} 〉 ≅ A_{n, n + 5,1}$ ⁠. Then the brackets in A can be written as

{\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 4} + α e_{n + 6}, \\ [e_{2}, \dots, e_{n + 1}] = e_{n + 5} + β e_{n + 6}, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 1}] = α_{i} e_{n + 6}, & 2 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 2}] = β_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 3}] = γ_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 1}, e_{n + 2}] = χ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 1}, e_{n + 3}] = δ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 2}, e_{n + 3}] = λ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, \\ [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}] = φ_{i j k} e_{n + 6}, & 1 \leq i < j < k \leq n . \end{array}

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

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

λ_{i j} (1 \leq i < j \leq n), φ_{i j k} (1 \leq i < j < k \leq n)

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

$λ_{12} = 1, λ_{i j} = 0 (1 \leq i < j \leq n, (i, j) \neq (1, 2)), and φ_{i j k} = 0 (1 \leq i < j < k \leq n)$ ⁠. In this case, the brackets in A can be written as

[e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{3}, \dots, e_{n}, e_{n + 2}, e_{n + 3}] = e_{n + 6},

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

$λ_{23} = 1, λ_{i j} = 0 (1 \leq i < j \leq n, (i, j) \neq (2, 3)), and φ_{i j k} = 0 (1 \leq i < j < k \leq n)$ ⁠. In this case, the brackets in A can be written as

[e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{1}, e_{4}, \dots, e_{n}, e_{n + 2}, e_{n + 3}] = e_{n + 6},

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

$λ_{i j} = 0 (1 \leq i < j \leq n), φ_{123} = 1, and φ_{i j k} = 0 (1 \leq i < j < k \leq n, (i, j, k) \neq (1, 2, 3))$ ⁠. In this case, the brackets in A can be written as

[e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{4}, \dots, e_{n + 3}] = e_{n + 6} .

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

$λ_{i j} = 0 (1 \leq i < j \leq n), φ_{234} = 1, φ_{i j k} = 0 (1 \leq i < j < k \leq n), (i, j, k) \neq (2,3,4))$ ⁠. In this case, the brackets in A can be written as

[e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{1}, e_{5}, \dots, e_{n + 3}] = e_{n + 6},

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

Case 2.

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

{\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 4} + α e_{n + 6}, \\ [e_{2}, \dots, e_{n + 1}] = e_{n + 5} + β e_{n + 6}, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 1}] = α_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 2}] = β_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 3}] = γ_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 1}, e_{n + 2}] = χ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, \\ (i, j) \neq (1,2), \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 1}, e_{n + 3}] = δ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 2}, e_{n + 3}] = λ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, \\ [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}] = φ_{i j k} e_{n + 6}, & 1 \leq i < j < k \leq n . \end{array}

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

Since $\dim {(A / 〈 e_{n + 4}, e_{n + 5} 〉)}^{2} = 1$ ⁠, we have $A / 〈 e_{n + 4}, e_{n + 5} 〉 ≅ H (n, 1) \oplus F (3)$ ⁠. 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

γ_{i} (1 \leq i \leq n), δ_{i j} (1 \leq i < j \leq n),

λ_{i j} (1 \leq i < j \leq n), φ_{i j k} (1 \leq i < j < k \leq n)

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

\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{2}, \dots, e_{n}, e_{n + 3}] = e_{n + 6}, \\ [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{1}, e_{2}, e_{4}, \dots, e_{n}, e_{n + 3}] = e_{n + 6}, \\ [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{2}, e_{4}, \dots, e_{n + 1}, e_{n + 3}] = e_{n + 6}, \\ [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{1}, e_{2}, e_{5}, \dots, e_{n + 1}, e_{n + 3}] = e_{n + 6} . \end{array}

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

\begin{array}{l} A_{n, n + 6,5} = 〈 e_{1}, \dots, e_{n + 6} : [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{2}, e_{4}, \dots, e_{n + 1}, e_{n + 3}] = e_{n + 6} 〉, \\ A_{n, n + 6,6} = 〈 e_{1}, \dots, e_{n + 6} : [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, [e_{1}, e_{2}, e_{5}, \dots, e_{n + 1}, e_{n + 3}] = e_{n + 6} 〉, \end{array}

Case 3.

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

{\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 4} + α e_{n + 6}, \\ [e_{2}, \dots, e_{n + 1}] = e_{n + 5} + β e_{n + 6}, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 1}] = α_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 2}] = β_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 3}] = γ_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 1}, e_{n + 2}] = χ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 1}, e_{n + 3}] = δ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 2}, e_{n + 3}] = λ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, \\ [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}] = φ_{i j k} e_{n + 6}, & 1 \leq i < j < k \leq n, \\ (i, j, k) \neq (1,2,3) . \end{array}

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

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

\begin{array}{l} α_{i}, β_{i}, γ_{i} (1 \leq i \leq n), χ_{i j}, δ_{i j}, λ_{i j} (1 \leq i < j \leq n), \\ φ_{i j k} (1 \leq i < j < k \leq n, (i, j, k) \neq (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} [e_{1}, \dots, e_{n}] = e_{n + 5}, [e_{4}, \dots, e_{n + 3}] = e_{n + 4}, [e_{2}, e_{3}, \dots, e_{n + 1}] = e_{n + 6}, \\ [e_{1}, \dots, e_{n}] = e_{n + 5}, [e_{4}, \dots, e_{n + 3}] = e_{n + 4}, [e_{1}, e_{2}, e_{3}, e_{5}, \dots, e_{n + 1}] = e_{n + 6}, \\ [e_{1}, \dots, e_{n}] = e_{n + 5}, [e_{4}, \dots, e_{n + 3}] = e_{n + 4}, [e_{2}, e_{3}, e_{5}, \dots, e_{n + 2}] = e_{n + 6} . \end{array}

One can easily see that these algebras are isomorphic to $A_{n, n + 6,3}, A_{n, n + 6,4}$ and $A_{n, n + 6,6}$ ⁠, 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

A_{n, n + 6,7}, A_{n, n + 6,8}, A_{n, n + 6,9}, A_{n, n + 6,10}, A_{n, n + 6,11}, A_{n, n + 6,12} and A_{n, n + 6,13} .

Proof. Suppose that A is an $(n + 6)$ -dimensional generalized Heisenberg n-Lie algebra of rank 4 with basis ${e_{1}, \dots, e_{n + 6}}$ ⁠, which $n \geq 4$ ⁠. 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 $(n + 5)$ -dimensional nilpotent n-Lie algebra of class 2 with derived algebra of dimension 3. By Theorem LABEL:?, we have three possibilities for $A / 〈 e_{n + 6} 〉$ ⁠:

Case 4.

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

{\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3} + α e_{n + 6}, \\ [e_{2}, \dots, e_{n + 1}] = e_{n + 4} + β e_{n + 6}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5} + γ e_{n + 6}, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 1}] = α_{i} e_{n + 6}, & 3 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 2}] = β_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 1}, e_{n + 2}] = χ_{i j} e_{n + 6}, & 1 \leq i < j \leq n . \end{array}

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

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

α_{i} (3 \leq i \leq n), β_{i} (1 \leq i \leq n), χ_{i j} (1 \leq i < j \leq n)

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 $α_{i} (3 \leq i \leq n)$ cannot be equal to one. We have three possibilities:

$β_{1} = 1, β_{i} = 0 (2 \leq i \leq n), α_{i} = 0 (3 \leq i \leq n), χ_{i j} = 0 (1 \leq i < j \leq n)$ ⁠.

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

\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5}, [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 6}, \end{array}

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

$β_{3} = 1, β_{i} = 0 (1 \leq i \leq n, n \neq 3), α_{i} = 0 (3 \leq i \leq n), χ_{i j} = 0 (1 \leq i < j \leq n)$ ⁠.

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

\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] e_{n + 4}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5}, [e_{1}, e_{2}, e_{4}, \dots, e_{n}, e_{n + 2}] = e_{n + 6}, \end{array}

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

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

\begin{array}{l} {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5}, & [e_{3}, \dots, e_{n + 2}] = e_{n + 6}, \end{array} \\ {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5}, & [e_{2}, e_{4}, \dots, e_{n + 2}] = e_{n + 6}, \end{array} \\ {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5}, & [e_{1}, e_{2}, e_{5}, \dots, e_{n + 2}] = e_{n + 6} . \end{array} \end{array}

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

Case 5.

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

{\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3} + α e_{n + 6}, \\ [e_{2}, \dots, e_{n + 1}] = e_{n + 4} + β e_{n + 6}, \\ [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 5} + γ e_{n + 6}, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 1}] = α_{i} e_{n + 6}, & 2 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 2}] = β_{i} e_{n + 6}, & 2 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 1}, e_{n + 2}] = χ_{i j} e_{n + 6}, & 1 \leq i < j \leq n . \end{array}

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

α_{i} (3 \leq i \leq n), β_{i} (1 \leq i \leq n), χ_{i j} (1 \leq i < j \leq n)

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

Only one of $α_{i} (3 \leq i \leq n)$ and $β_{i} (1 \leq i \leq n)$ is equal to one and the others are zero. Without loss of generality, we assume $α_{2} = 1$ ⁠. Thus, the brackets in A can be written as

\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 5}, [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 6} . \end{array}

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

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

\begin{array}{l} {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 5}, & [e_{3}, \dots, e_{n + 2}] = e_{n + 6}, \end{array} \\ {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 5}, & [e_{1}, e_{4}, \dots, e_{n + 2}] = e_{n + 6} . \end{array} \end{array}

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

Case 6.

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

{\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3} + α e_{n + 6}, \\ [e_{2}, \dots, e_{n + 1}] = e_{n + 4} + β e_{n + 6}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5} + γ e_{n + 6}, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 1}] = α_{i} e_{n + 6}, & 2 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 2}] = β_{i} e_{n + 6}, & 1 \leq i \leq n, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, {\hat{e}}_{j}, \dots, e_{n}, e_{n + 1}, e_{n + 2}] = χ_{i j} e_{n + 6}, & 1 \leq i < j \leq n, (i, j) \neq (1,2) . \end{array}

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

α_{i} (2 \leq i \leq n), β_{i} (1 \leq i \leq n), χ_{i j} (1 \leq i < j \leq n, (i, j) \neq (1, 2))

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

\begin{array}{l} {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, & [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 6}, \end{array} \\ {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, & [e_{1}, e_{2}, e_{4}, \dots, e_{n + 1}] = e_{n + 6}, \end{array} \\ {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, & [e_{1}, e_{3}, \dots, e_{n}, e_{n + 2}] = e_{n + 6}, \end{array} \\ {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, & [e_{1}, e_{2}, e_{4}, \dots, e_{n}, e_{n + 2}] = e_{n + 6}, \end{array} \\ {\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 3}, & [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, & [e_{1}, e_{2}, e_{5}, \dots, e_{n + 2}] = e_{n + 6} . \end{array} \end{array}

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

Theorem 3.8.

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

{\begin{array}{l} H (n, 1) \oplus F (5) (n \geq 4), H (4, 2) \oplus F (1), H (5, 2), \\ A_{n, n + 5,1} \oplus F (1), A_{n, n + 4,1} \oplus F (2), A_{n, n + 3,1} \oplus F (3), A_{n, n + 5,4} \oplus F (1) \\ A_{n, n + 5,5} \oplus F (1), A_{n, n + 4,3} \oplus F (2), A_{n, n + 5,6} \oplus F (1), a n d A_{n, n + 6, i} (1 \leq i \leq 14) . \end{array}

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

H (n, 1) \oplus F (5) (n \geq 4), H (4, 2) \oplus F (1), H (5, 2) .

Now, assume that $dim A^{2} \geq 2$ and that $〈 e_{n + 5}, e_{n + 6} 〉 \subset A^{2}$ ⁠. Therefore, $A / 〈 e_{n + 6} 〉$ is an $(n + 5)$ -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 (n, 1) \oplus F (4), H (4, 2), A_{n, n + 5, i} (1 \leq i \leq 7) .

If $A / 〈 e_{n + 6} 〉$ is isomorphic to $H (n, 1) \oplus F (4)$ or $H (4, 2)$ ⁠, then $dim A^{2} = 2$ ⁠. 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 + 6,1}, A_{n, n + 5,1} \oplus F (1), A_{n, n + 4,1} \oplus F (2), A_{n, n + 3,1} \oplus F (3) .

If $A / 〈 e_{n + 6} 〉$ is isomorphic to $A_{n, n + 5,1}, A_{n, n + 5,2}$ or $A_{n, n + 5,3}$ ⁠, then $dim A^{2} = 3$ ⁠. 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 + 6,2}, A_{n, n + 6,3}, A_{n, n + 6,4}, A_{n, n + 6,5}, A_{n, n + 6,6}, \\ A_{n, n + 5,4} \oplus F (1), A_{n, n + 5,5} \oplus F (1), A_{n, n + 4,3} \oplus F (2) . \end{array}

Also, If $A / 〈 e_{n + 6} 〉$ is isomorphic to $A_{n, n + 5,4}, A_{n, n + 5,5}$ or $A_{n, n + 5,6}$ ⁠, then $dim A^{2} = 4$ ⁠. 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 + 6,7}, A_{n, n + 6,8}, A_{n, n + 6,9}, A_{n, n + 6,10}, \\ A_{n, n + 6,11}, A_{n, n + 6,12}, A_{n, n + 6,13}, A_{n, n + 5,6} \oplus F (1) . \end{array}

Finally, If $A / 〈 e_{n + 6} 〉 ≅ A_{n, n + 5,7}$ ⁠, 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}{l} [e_{1}, \dots, e_{n}] = e_{n + 1} + α e_{n + 6}, \\ [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 3} + β e_{n + 6}, \\ [e_{1}, e_{3}, \dots, e_{n}, e_{n + 2}] = e_{n + 4} + γ e_{n + 6}, \\ [e_{1}, e_{2}, e_{4}, \dots, e_{n}, e_{n + 2}] = e_{n + 5} + φ e_{n + 6}, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 2}] = β_{i} e_{n + 6}, & 4 \leq i \leq n . \end{array}

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

{\begin{array}{l} [e_{1}, \dots, e_{n}] = e_{n + 1}, \\ [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 3}, \\ [e_{1}, e_{3}, \dots, e_{n}, e_{n + 2}] = e_{n + 4}, \\ [e_{1}, e_{2}, e_{4}, \dots, e_{n}, e_{n + 2}] = e_{n + 5}, \\ [e_{1}, \dots, {\hat{e}}_{i}, \dots, e_{n}, e_{n + 2}] = β_{i} e_{n + 6}, & 4 \leq i \leq n . \end{array}

By $dim Z (A)$ ⁠, we must have $β_{i} \neq 0$ for some $4 \leq i \leq n$ ⁠. Without loss of generality, assume that $β_{4} \neq 0$ ⁠. By applying the transformations

e_{4}^{'} = e_{4} + \sum_{j = 5}^{n} {(- 1)}^{j} \frac{β_{i}}{β_{4}} e_{j}, e_{i}^{'} = e_{i} (1 \leq i \leq n + 5, i \neq 4), e_{n + 6}^{'} = β_{4} e_{n + 6},

we conclude that

A = 〈 e_{1}, \dots, e_{n + 6} : [e_{1}, \dots, e_{n}] = e_{n + 1}, [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 3}, = e_{n + 4}, [e_{1}, e_{2}, e_{4}, \dots, e_{n}, e_{n + 2}] = e_{n + 5}, = e_{n + 6} 〉,

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

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.

References

Filippov

n-Lie algebras

Sib. Math. J.

1985

;

879

Nilpotent n-Lie algebras of class 2	Nonzero multiplications
$A_{n, n + 4,1}$	$[e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}$
$A_{n, n + 4,2}$	$[e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{3}, \dots, e_{n + 2}] = e_{n + 4} (n \geq 3)$
$A_{n, n + 4,3}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 1}, [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 3}, \\ [e_{1}, e_{3}, \dots, e_{n}, e_{n + 2}] = e_{n + 4} \end{matrix}$
$A_{n, n + 5,1}$	$[e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}$
$A_{n, n + 5,2}$	$[e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{3}, \dots, e_{n + 2}] = e_{n + 5} (n \geq 3)$
$A_{n, n + 5,3}$	$[e_{1}, \dots, e_{n}] = e_{n + 5}, [e_{4}, \dots, e_{n + 3}] = e_{n + 4} (n \geq 3)$
$A_{n, n + 5,4}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5} \end{matrix}$
$A_{n, n + 5,5}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 5} \end{matrix}$
$A_{n, n + 5,6}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5} \end{matrix}$
$A_{n, n + 5,7}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 1}, [e_{1}, e_{2}, e_{4}, \dots, e_{n}, e_{n + 2}] = e_{n + 5}, \\ [e_{1}, e_{3}, \dots, e_{n}, e_{n + 2}] = e_{n + 4}, [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 3} \end{matrix}$

Nilpotent n-Lie algebras of class 2	Nonzero multiplications
$A_{n, n + 6,1}$	$[e_{1}, \dots, e_{n}] = e_{n + 5}, [e_{5}, \dots, e_{n + 4}] = e_{n + 6}$
$A_{n, n + 6,2}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, \\ [e_{3}, \dots, e_{n}, e_{n + 2}, e_{n + 3}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,3}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, \\ [e_{1}, e_{4}, \dots, e_{n}, e_{n + 2}, e_{n + 3}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,4}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{2}, \dots, e_{n + 1}] = e_{n + 5}, \\ [e_{1}, e_{5}, \dots, e_{n + 3}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,5}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, \\ [e_{2}, e_{4}, \dots, e_{n + 1}, e_{n + 3}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,6}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 4}, [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, \\ [e_{1}, e_{2}, e_{5}, \dots, e_{n + 1}, e_{n + 3}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,7}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5}, [e_{3}, \dots, e_{n + 2}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,8}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5}, [e_{2}, e_{4}, \dots, e_{n + 2}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,9}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{1}, e_{3}, \dots, e_{n + 1}] = e_{n + 5}, [e_{1}, e_{2}, e_{5}, \dots, e_{n + 2}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,10}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 5}, [e_{1}, e_{4}, \dots, e_{n + 2}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,11}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, [e_{1}, e_{3}, \dots, e_{n}, e_{n + 2}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,12}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, [e_{1}, e_{2}, e_{4}, \dots, e_{n}, e_{n + 2}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,13}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 3}, [e_{2}, \dots, e_{n + 1}] = e_{n + 4}, \\ [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, [e_{1}, e_{2}, e_{5}, \dots, e_{n + 2}] = e_{n + 6} \end{matrix}$
$A_{n, n + 6,14}$	$\begin{matrix} [e_{1}, \dots, e_{n}] = e_{n + 1}, [e_{2}, \dots, e_{n}, e_{n + 2}] = e_{n + 3}, \\ [e_{1}, e_{3}, \dots, e_{n}, e_{n + 2}] = e_{n + 4}, [e_{3}, \dots, e_{n + 2}] = e_{n + 5}, \\ [e_{1}, e_{3}, \dots, e_{n}, e_{n + 2}] = e_{n + 6} \end{matrix}$

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

1. Introduction

2. Preliminaries

3. Main results

References

Data & Figures

Contents

Supplements

References

Related

New and popular articles

Email Alerts

Suggested Reading

Related Chapters

Recommended for you

Cited By

Languages

Sharing Unavailable