Discussion on the Complex Structure of Nilpotent Lie Groups Gk

Consider the real, simply-connected, connected, s-step nilpotent Lie group G endowed with a left-invariant, integrable almost complex structure J, which is nilpotent. Consider the simply-connected, connected nilpotent Lie group Gk, defined by the nilpotent Lie algebra g/ak, where g is the Lie algebra of G, and ak is an ideal of g. Then, J gives rise to an almost complex structure Jk on Gk. The main conclusion obtained is as follows: if the almost complex structure J of a nilpotent Lie group G is nilpotent, then J can give rise to a left-invariant integrable almost complex structure Jk on the nilpotent Lie group Gk, and Jk is also nilpotent.

Lie Groups Actions on Non Orientable <i>n</i>-Dimensional Complex Manifolds

Analytic atlases on can be easily defined making it an n-dimensional complex manifold. Then with the help of bi-Möbius transformations in complex coordinates Abelian groups are constructed making this manifold a Lie group. Actions of Lie groups on differentiable manifolds are well known and serve different purposes. We have introduced in previous works actions of Lie groups on non orientable Klein surfaces. The purpose of this work is to extend those studies to non orientable n-dimensional complex manifolds. Such manifolds are obtained by factorizing with the two elements group of a fixed point free antianalytic involution of . Involutions h(z) of this kind are obtained linearly by composing special Möbius transformations of the planes with the mapping . A convenient partition of is performed which helps defining an internal operation on and finally actions of the previously defined Lie groups on the non orientable manifold are displayed.

紧致齐性空间G/T上复结构的若干性质

设G为连通单连通s阶实幂零李群,且其上有左不变可积近复结构J,T为李群G的最大秩的格;考虑紧致齐性空间G/T上是否有幂零复结构;运用归纳法证明了复结构J可诱导出紧致齐性空间G/T的可积复结构J且是幂零的.

On the PoincaréAlgebra in a Complex Space-Time Manifold

We extend the Poincaré group to the complex Minkowski space-time. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the generalizations of the two Casimir operators.

复Poisson李群

文章将复李群上的典型复结构扩展到任意的k重向量场上,又结合复李群和Po isson李群的特点,定义了复Pos-sion李群的概念,得到了一些有趣的结论。

复域上几个过极限环积分流形的几何结构

研究了几个多项式自治系统在复域上过其极限环积分流形的复杂的几何结构,得到了在积分流形碰到无穷远奇点后黎曼曲面的4种变化趋向,并且从李群角度上证明了这些系统具有不可积性.

基于李群李代数的复杂机械系统的实时动力学

针对日益复杂的机械系统,提出用李群李代数符号描述正向和反向递推动力学,研究其控制和仿真问题。运用李群李代数符号表示了伴随变换和伴随算子,基于树形拓扑结构系统,推导得到其递推动力学方法,讨论了其闭环系统和柔性系统的高效动力学算法。实例验证表明:李群李代数描述的动力学算法计算效率高,形式简洁,易于编程,对于实时动力学的研究具有重要意义。

A Note on <i>m</i>-Möbius Transformations

Lie groups of bi-Möbius transformations are known and their actions on non orientable n-dimensional complex manifolds have been studied. In this paper, m-Möbius transformations are introduced and similar problems as those related to bi-Möbius transformations are tackled. In particular, it is shown that the subgroup generated by every m-Möbius transformation is a discrete group. Cyclic subgroups are exhibited. Vector valued m-Möbius transformations are also studied.

Weyl群的长度函数的几个性质

讨论了复半单李代数的Ｗｅｙｌ群的长度函数的性质。Ｗｅｙｌ群的长度函数与根系的关系，以及有某一长度的Ｗｅｙｌ群元素的个数等。这些性质在讨论李代数的不可约表示中是很有用的。

On Ellipsoids Attached to Root Systems

For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.