Solvable Indecomposable Extensions of Two Nilpotent Lie Algebras

A pair of sequences of nilpotent Lie algebras denoted by N_{n, 7} and N_{n, 16} are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that N_{n, 7} and N_{n, 16} serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

中心为4的8维幂零李代数的分类

In this paper, we give a complete classification of eight dimensional nilpotent Lie algebras with four-dimensional center by using the method of Skjelbred and Sund.     

Degenerations of 8-Dimensional 2-step Nilpotent Lie Algebras

Algebras and Representation Theory - In this work, we consider degenerations between 8-dimensional 2-step nilpotent Lie algebras over ? and obtain the geometric classification of the variety...     

5维幂零李代数的导子代数的结构

对维数为5的所有幂零李代数的导子代数进行了研究,按照其分类分别给出了9种互不同构的导子代数的结构。     

Classification of 2-Step Nilpotent Lie Algebras of Dimension 8 with 2-Dimensional Center

In this article, by choosing appropriate basis, we give a complete classification of 2-step nilpotent Lie algebras of dimension 8 with 2-dimensional center.     

A Characterisation of Nilpotent Lie Algebras by Invertible Leibniz-Derivations

Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques. We only consider finite-dimensional Lie algebras over a fields of characteristic zero.     

Automorphisms of Relatively Free Nilpotent Lie Algebras

Let L be a relatively free nilpotent Lie algebra over ? of rank n and class c, with n ≥ 2; freely generated by a set 𝒵. Give L the structure of a group, denoted by R, by means of the Baker–Campbell–Hausdorff formula. Let G be the subgroup of R generated by the set 𝒵 and N _Aut(L)(G) the normalizer in Aut(L) of the set G. We prove that the automorphism group of L is generated by GL _n (?) and N _Aut(L)(G). Let H be a subgroup of finite index in Aut(G) generated by the tame automorphisms and a finite subset X of IA-automorphisms with cardinal s. We construct a set Y consisting of s + 1 IA-automorphisms of L such that Aut(L) is generated by GL _n (?) and Y. We apply this particular method to construct generating sets for the automorphism groups of certain relatively free nilpotent Lie algebras.     

Codimension Growth of Strong Lie Nilpotent Associative Algebras

We study Lie nilpotent varieties of associative algebras. We explicitly compute the codimension growth for the variety of strong Lie nilpotent associative algebras. The codimension growth is polynomial and found in terms of Stirling numbers of the first kind. To achieve the result we take the free Lie algebra of countable rank L(X), consider its filtration by the lower central series and shift it. Next we apply generating functions of special type to the induced filtration of the universal enveloping algebra U(L(X)) = A(X).     

Nilpotent Lie Algebras with a Small Second Derived Quotient 1

Here we find the structure of nilpotent Lie algebras L with dim(L′/L″) = 3 and L″ ≠ 0. Following the pattern of results of Csaba Schneider in p-groups, we show that L is the central direct sum of ideals H and U, where U is the direct sum of a generalized Heisenberg Lie algebra and an abelian Lie algebra. We then find over the complex numbers that H falls into one of fourteen isomorphism classes.     

The Moduli Space of Six-Dimensional Two-Step Nilpotent Lie Algebras

We determine the moduli space of metric two-step nilpotent Lie algebras of dimension up to 6. This space is homeomorphic to a cone over a four-dimensional contractible simplicial complex. Moreover, we exhibit standard metric representatives of the seven isomorphism types of six-dimensional two-step nilpotent Lie algebras within our picture. Mathematics Subject Classifications (2000): Primary 22E25, 53C30, 22E60     

On Orbits of Action of 5-Dimensional Non-Solvable Lie Algebras in Three-Dimensional Complex Space

Doklady Mathematics - In 1932, E. Cartan described holomorphically homogeneous real hypersurfaces of two-dimensional complex spaces, but a similar study in the three-dimensional case remains...     

低维幂零李代数的Yang-Baxter算子

在特征零的代数闭域F上,利用3维和4维幂零李代数的分类,通过计算刻画了3维和4维幂零李代数的Yang-Baxter算子.     

Maximal Abelian Dimensions in Some Families of Nilpotent Lie Algebras

This paper deals with the maximal abelian dimension of a Lie algebra, that is, the maximal value for the dimensions of its abelian Lie subalgebras. Indeed, we compute the maximal abelian dimension for every nilpotent Lie algebra of dimension less than 7 and for the Heisenberg algebra $\mathfrak{H}_k$ , with $k\in\mathbb{N}$ . In this way, an algorithmic procedure is introduced and applied to compute the maximal abelian dimension for any arbitrary nilpotent Lie algebra with an arbitrary dimension. The maximal abelian dimension is also given for some general families of nilpotent Lie algebras.     

Strengthened Wiegold Conjecture in the Theory of Nilpotent Lie Algebras

Mathematical Notes - In the present paper, we strengthen the assertion of the Wiegold conjecture for nilpotent Lie algebras over an infinite field by proving that if there exists a subset of a...     

复数域C上的一类非退化幂零李代数

本文讨论了复数域Ｃ上的有限维非退化李代数．并将它们与正交代数Ｂ_ｌ和 Ｄ_ｌ联系起来,从而得出它们的若干性质,同时也讨论了一类非退化幂零李代数并彻底 将它们刻划清楚．