Metric n-Lie Algebras

An n-Lie algebra is said to be metric if it is endowed with an invariant, non-degenerate, symmetric bilinear form. We prove that any simple n-Lie algebra over an algebraically closed field of characteristic zero admits a unique metric structure and vice versa. Further, we present two metric n-Lie algebras, which are indecomposable but admit many more metric structures.     

Nilpotent n-Lie Algebras

We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I ² nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent.     

Some Properties On Isoclinism in n-Lie Algebras

The concept of n-Lie algebra were introduced by Filippov in 1987. One notes that not all properties of Lie algebras can be carried over to n-Lie algebras, as Williams showed in 2009. In the present article, among other results it is shown that the notions of isoclinism and isomorphism for two finite dimensional n-Lie algebras of the same dimension are equivalent. This was already done for ordinary Lie algebras by Moneyhun in 1994.     

The Socle of a Metric n-Lie Algebra

We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case.     

T *-extension of n-Lie algebras

We introduce a new technique, called the T?*-extension, of constructing metric n-Lie algebras from arbitrary ones and obtain some important properties on this extension. Finally, we investigate the metric n-Lie algebras that are isometric to certain T?*-extensions.     

A class of non-nilpotent 2-solvable n-Lie algebras

This article concerns a class of finite-dimensional minimal non-nilpotent 2-solvable n-Lie algebras. It is shown that if L is a finite-dimensional minimal non-nilpotent 2-solvable n-Lie algebra, then L can be decomposed into a semi-direct of an ideal A and an (n ? 1)-dimensional subalgebra H ₀ of L. Furthermore, H ₀ acts irreducibly on A/A ¹, and H ₀ + A ¹ is a self-normalizing maximal subalgebra of L with the core A ¹, the derived algebra of A.     

On Schur’s theorem and its converses for n-Lie algebras

An n-Lie algebra analogue of Schur’s theorem and its converse as well as a Lie algebra analogue of Baer’s theorem and its converse are presented. Also, it is shown that, an n-Lie algebra with finite dimensional derived subalgebra and finitely generated central factor is isoclinic to some finite dimensional n-Lie algebra.     

The Realization of 4-dimensional 3-Lie Algebras

In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.     

4维3-Lie代数的实现

In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.     

低维3-李代数的分类

该文研究了m≤5时特征为2的完备域上m-维3-李代数的分类,并给出了各不同构类的具体乘法表.     

Universal Associative Envelopes of (n+ 1)-Dimensional n-Lie Algebras

For n even, we prove Pozhidaev's conjecture on the existence of associative enveloping algebras for simple n-Lie (Filippov) algebras. More generally, for n even and any (n + 1)-dimensional n-Lie algebra L, we construct a universal associative enveloping algebra U(L) and show that the natural map L → U(L) is injective. We use noncommutative Gröbner bases to present U(L) as a quotient of the free associative algebra on a basis of L and to obtain a monomial basis of U(L). In the last section, we provide computational evidence that the construction of U(L) is much more difficult for n odd.     

n-Lie 代数的扩张

文章主要研究n-Lie 代数的扩张问题. 首先利用n-Lie 代数的模作n-Lie 代数的T_θ- 扩张与T_θ^*-扩张. 再利用模度量3-Lie 代数,做3-Lie 代数的双扩张. 文章最后利用4- 指标阵构造了m 维3-Lie代数的双扩张.     

n-Ary Mal'tsev Algebras

By analogy with n-Lie algebras, which are a natural generalization of Lie algebras to the case of n-ary multiplication, we define the concept of an n-ary Mal'tsev algerba. It is shown that exceptional algebras of a vector cross product are ternary central simple Mal'tsev algebras, which are not 3-Lie algebras if the characteristic of a ground field is distinct from 2 and 3. The basic result is that every n-ary algebra of the vector cross product is an n-ary central simple Mal'tsev algebra.     

Cartan Subalgebras of Leibniz n-Algebras

The present article is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz n-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz n-algebra and Cartan subalgebras and regular elements of the corresponding factor n-Lie algebra is established.     

On classification of n-Lie algebras

In this paper, we prove the isomorphic criterion theorem for (n+2)-dimensional n-Lie algebras, and give a complete classification of (n + 2)-dimensional n-Lie algebras over an algebraically closed field of characteristic zero.     

<Emphasis Type="Italic">n</Emphasis>-Lie Structures That Are Generated by Wronskians

We study the (k + 1)-Lie structures, k-left commutative and homotopy (k + 1)-Lie structures with multiplication generated by Wronskians and prove that the nontrivial structures of n-Lie algebras appear only in the case of small characteristic.Original Russian Text Copyright © 2005 Dzhumadil’daev A. S.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 759–773, July–August, 2005.     

Graded Multiple Analogs of Lie Algebras

Graded analogs of (n,k,r)-Lie algebras (in particular, of Nambu–Lie algebras), introduced by the authors, are defined and their general property are studied.     

3-李代数的辛结构

研究了3-李代数和度量3-李代数的辛结构.对任意3-李代数L,构造了无限多个度量辛3-李代数.证明了度量3-李代数(A,B)是度量辛3-李代数的充要条件,即存在可逆导子D,使得D∈Der_B(A).同时证明了每一个度量辛3-李代数(A,B,ω)是度量辛3-李代数(A,B,ω)的T_θ~*-扩张.最后,利用度量辛3-李代数经过特殊导子的双扩张得到了新的度量辛3-李代数.