Computing with nilpotent orbits in simple Lie algebras of exceptional type

Fixed point properties of nilpotent and solvable Lie group actions on Hadamard manifolds

We will prove the following theorems. The first theorem posits the existence of a fixed point for the actions of nilpotent Lie groups on nonpositively curved compact manifolds. The second theorem states that actions of solvable Lie groups on nonpositively curved compact manifolds have either a fixed point or a 2-periodic point.

Computing Nilpotent Quotients in Finitely Presented Lie Rings

A nilpotent quotient algorithm for finitely presented Lie rings over Z (LieNQ) is described. The paper studies graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. The nilpotent presentation consists of generators for the abelian group and the products---expressed as linear combinations---for pairs formed by generators. Using that presentation the word problem is decidable in $L$. Provided that the Lie ring $L$ is graded, it is possible to determine the canonical presentation for a lower central factor…

类 $2$ 幂零群上的乘法 Lie 代数结构

Multiplicative Lie algebra structure on nilpotent group of class $2$

本文探讨了类 $2$ 幂零群上乘法 Lie 代数结构的性质，并提出了一种确定作为 Lie 环扩张的群（如亚循环群和类 $2$ 幂零群）上乘法 Lie 代数结构的方法。

Nilpotent Lie groups and hyperbolic automorphisms

A connected Lie group admitting an expansive automorphism is known to be nilpotent, but all nilpotent Lie groups do not admit expansive automorphism. In this article, we find sufficient conditions for a class of nilpotent Lie groups to admit expansive automorphism.

Proof of a Conjecture of Wiegold for nilpotent Lie algebras

In this short note we confirm an analog of a conjecture of James Wiegold for finite dimensional nilpotent Lie algebras.

On classification of (n+5)-dimensional nilpotent n-Lie algebra of class two

In this paper, we classify (n+5)-dimensional nilpotent n-Lie algebras of class two over the arbitrary field, when $n\ge 3$.

Lie algebras with nilpotent length greater than that of each of their subalgebras

The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$, and of nilpotent length $\leq k$, and {\em extreme} Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-${\mathcal N}$ Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index $\leq 3$.

On quasi-isometric nilpotent Lie groups

In this article we provide evidence for a well-known conjecture which states that quasi-isometric simply-connected nilpotent Lie groups are isomorphic. We do so by constructing new examples which are rigid in the sense that whenever they are quasi-isometric to any other simply-connected nilpotent Lie group, the groups are actually isomorphic.

C*-algebras with norm controlled dual limits and nilpotent Lie groups

Motivated by the description of the C*-algebras of 5 dimensional nilpotent Lie groups as algebras of operator fields defined over their spectra, we introduce the family of C* -algebras with norm controlled dual limits and we show that the C* -algebras of the 5 dimensional nilpotents Lie groups belong to this class.

Some examples of quasiisometries of nilpotent Lie groups

We construct quasiisometries of nilpotent Lie groups. In particular, for any simply connected nilpotent Lie group N, we construct quasiisometries from N to itself that is not at finite distance from any map that is a composition of left translations and automorphisms. We also construct biLipschitz maps of the Heisenberg groups that send vertical lines to non-vertical curves.

Casimir operators of Lie algebras with a nilpotent radical

We show that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. We give a different proof of this fact in the special and well-known case where the radical is abelian.

Affine actions on Nilpotent Lie groups

To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G)=dim(N) less than 6. Finally, we specialize to the case of abelian simply transitive affi…

The groups and nilpotent Lie rings of order $p^8$ with maximal class

We classify the nilpotent Lie rings of order $p^8$ with maximal class for $p \ge 5$. This also provides a classification of the groups of order $p^8$ with maximal class for $p \ge 11$ via the Lazard correspondence.

Discrete Cocompact Subgroups of the Five-Dimensional Connected and Simply Connected Nilpotent Lie Groups

The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if $G=N\times A$ is a connected, simply connected, nilpotent Lie group with an Abelian factor $A$, then every uniform subgroup of $G$ is the direct product of a uniform subgroup of $N$ and ${\mathbb Z}^r$ where $r=\dim A$.

Lie groups as permutation groups: Ulam's problem in the nilpotent case

Ulam asked whether every connected Lie group can be represented on a countable structure. This is known in the linear case. We establish it for the first family of non-linear groups, namely in the nilpotent case. Further context is discussed to illustrate the relevance of nilpotent groups for Ulam's problem.

nilpotent Lie algebras的gradings与smooth complex algebraic varieties的nilpotent fundamental groups相关

Gradings on nilpotent Lie algebras associated with the nilpotent fundamental groups of smooth complex algebraic varieties

本文研究了nilpotent Lie代数的gradings与smooth complex algebraic varieties的nilpotent fundamental groups之间的关系，并给出了晶格同构于基本群的必要条件。

Generic properties of 2-step nilpotent Lie algebras and torsion-free groups

To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained in a proper Zariski-closed subset. We compute the maximal dimension of an abelian subalgebra of a generic Lie algebra and give a sufficient condition for a generic Lie algebra to admit no surjective homomorphism onto a non-abelian Lie algebra of a given dimension. Also we consider analogous questions for finitely generated torsion free nilpotent groups of cl…

八维实非退化两步幂零李代数

Real non-degenerate two-step nilpotent Lie algebras of dimension eight

本文利用复数域结果，对实数域上八维非退化两步幂零李代数进行分类，并给出显式结构常数。