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.

Generalized Kinematics Analysis of Hybrid Mechanisms Based on Screw Theory and Lie Groups Lie Algebras

Advanced mathematical tools are used to conduct research on the kinematics analysis of hybrid mechanisms,and the generalized analysis method and concise kinematics transfer matrix are obtained.In this study,first,according to the kinematics analysis of serial mechanisms,the basic principles of Lie groups and Lie algebras are briefly explained in dealing with the spatial switching and differential operations of screw vectors.Then,based on the standard ideas of Lie operations,the method for kinematics analysis of parallel mechanisms is derived,and Jacobian matrix and Hessian matrix are formulated recursively and in a closed form.Then,according to the mapping relationship between the parallel joints and corresponding equivalent series joints,a forward kinematics analysis method and two inverse kinematics analysis methods of hybrid mechanisms are examined.A case study is performed to verify the calculated matrices wherein a humanoid hybrid robotic arm with a parallel-series-parallel configuration is considered as an example.The results of a simulation experiment indicate that the obtained formulas are exact and the proposed method for kinematics analysis of hybrid mechanisms is practically feasible.

Lie point symmetry algebras and finite transformation groups of the general Broer-Kaup system

Using a new symmetry group theory, the transformation groups and symmetries of the general Broer-Kaup system are obtained. The results are much simpler than those obtained via the standard approaches.

Contractions of Certain Lie Algebras in the Context of the DLF-Theory

Contractions of the Lie algebras d = u(2), f = u(1 ,1) to the oscillator Lie algebra l are realized via the adjoint action of SU(2,2) when d, l, f are viewed as subalgebras of su(2,2). Here D, L, F are the corresponding (four-dimensional) real Lie groups endowed with bi-invariant metrics of Lorentzian signature. Similar contractions of (seven-dimensional) isometry Lie algebras iso(D), iso(F) to iso(L) are determined. The group SU(2,2) acts on each of the D, L, F by conformal transformation which is a core feature of the DLF-theory. Also, d and f are contracted to T, S-abelian subalgebras, generating parallel translations, T, and proper conformal transformations, S (from the decomposition of su(2,2) as a graded algebra T + Ω + S, where Ω is the extended Lorentz Lie algebra of dimension 7).

Entanglement of E8E8 Exceptional Lie Symmetry Group Dark Energy, Einstein’s Maximal Total Energy and the Hartle-Hawking No Boundary Proposal as the Explanation for Dark Energy

The present note is concerned with two connected and highly important fundamental questions of physics and cosmology, namely if E8E8 Lie symmetry group describes the universe and where cosmic dark energy comes from. Furthermore, we reason following Wheeler, Hartle and Hawking that since the boundary of a boundary is an empty set which models the quantum wave of the cosmos, then it follows that dark energy is a fundamental physical phenomenon associated with the boundary of the holographic boundary. This leads directly to a clopen universe which is its own Penrose tiling-like multiverse with energy density in full agreement with COBE, WMAP and Type 1a supernova cosmic measurements.

TAYLOR POLYNOMIAL STEPWISE REFINEMENTALGORITHM FOR LIE AND HIGH SYMMETRIES OF PARTIAL DIFFERENTIAL EQUATIONS

In this article, based on the Taylor expansions of generating functions and stepwise refinement procedure, authors suggest a algorithm for finding the Lie and high (generalized) symmetries of partial differential equations (PDEs). This algorithm transforms the problem having to solve over-determining PDEs commonly encountered and difficulty part in standard methods into one solving to algebraic equations to which one easy obtain solution. so, it reduces significantly the difficulties of the problem and raise computing efficiency. The whole procedure of the algorithm is carried out automatically by using any computer algebra system. In general, this algorithm can yields many more important symmetries for PDEs.

Introduction to Lie Algebras and Their Representations

This paper is made up of a desire for me to contribute to this beautiful field of mathematics that I have encountered in recent years. In addition, I would like to mention that I am not aware that there are papers in our Balkans on Lie algebra, although this is only an introductory part for which in the near future in collaboration with several professors from abroad I will do a book in our mother tongue on Lie groups and algebras. The main content of this paper is similar to the books that have been published regarding Lie algebras, from basic definition and example, structure, killing form, classification to root system. In my opinion, this paper is important in relation to Lie algebras, because it will be helpful to all those who write papers on algebra, as well as the fact that the paper will be written in Montenegrin, which is understood by almost more than 70 percent of the population. For me, this work has the significance of being useful to all who need it.

Representations of Lie Groups

In this paper, the most important liner groups are classified. Those that we often have the opportunity to meet when studying linear groups as well as their application in left groups. In addition to the introductory part, we have general linear groups, special linear groups, octagonal groups, symplicit groups, cyclic groups, dihedral groups: generators and relations. The paper is summarized with brief deficits, examples and evidence as well as several problems. When you ask why this paper, I will just say that it is one of the ways I contribute to the community and try to be a part of this little world of science.

Using Tangent Boost along a Worldline and Its Associated Matrix in the Lie Algebra of the Lorentz Group

In order to generalize the relativistic notion of boost to the case of non inertial particles and to general relativity, we look closer into the definition of the Lie group of Lorentz matrices and its Lie algebra and we study how this group acts on the Minskowski space. We thus define the notion of tangent boost along a worldline. This very general notion gives a useful tool both in special relativity (for non inertial particles or/and for non rectilinear coordinates) and in general relativity. We also introduce a matrix of the Lie algebra which, together with the tangent boost, gives the whole dynamical description of the considered system (acceleration and Thomas rotation). After studying the properties of Lie algebra matrices and their reduced forms, we show that the Lie group of special Lorentz matrices has four one-parameter subgroups. These tools lead us to introduce the Thomas rotation in a quite general way. At the end of the paper, we present some examples using these tools and we consider the case of an electron rotating on a circular orbit around an atom nucleus. We then discuss the twin paradox and we show that when the one who made a journey into space in a high-speed rocket returns home, he is not only younger than the twin who stayed on Earth but he is also disorientated because his gyroscope has turned with respect to earth referential frame.

Lie Symmetries,One-Dimensional Optimal System and Optimal Reduction of(2+1)-Coupled nonlinear Schrodinger Equations

For a class of (1 + 2)-dimensional nonlinear Schrodinger equations, the infinite dimensional Lie algebra of the classical symmetry group is found and the one-dimensional optimal system of an 8-dimensional subalgebra of the infinite Lie algebra is constructed. The reduced equations of the equations with respect to the optimal system are derived. Furthermore, the one-dimensional optimal systems of the Lie algebra admitted by the reduced equations are also constructed. Consequently, the classification of the twice optimal symmetry reductions of the equations with respect to the optimal systems is presented. The reductions show that the (1 + 2)-dimensional nonlinear Schrodinger equations can be reduced to a group of ordinary differential equations which is useful for solving the related problems of the equations.

On the Construction and Classification of the Common Invariant Solutions for Some P(1,4) -Invariant Partial Differential Equations

We consider the following (1 + 3)-dimensional P(1,4)-invariant partial differential equations (PDEs): the Eikonal equation, the Euler-Lagrange-Born-Infeld equation, the homogeneous Monge-Ampère equation, the inhomogeneous Monge-Ampère equation. The purpose of this paper is to construct and classify the common invariant solutions for those equations. For this aim, we have used the results concerning construction and classification of invariant solutions for the (1 + 3)-dimensional P(1,4)-invariant Eikonal equation, since this equation is the simplest among the equations under investigation. The direct checked allowed us to conclude that the majority of invariant solutions of the (1 + 3)-dimensional Eikonal equation, obtained on the base of low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1,4), satisfy all the equations under investigation. In this paper, we present obtained common invariant solutions of the equations under study as well as the classification of those invariant solutions.

On the A-extended Lie Rinehart Algebras

The purpose of this paper is to give a brief introduction to the category of Lie Rinehart algebras and introduces the concept of smooth manifolds associated with a unitary, commutative, associative algebra A. It especially shows that the A-extended algebra as well as the action algebra can be realized as the space of A-left invariant vector fields on a Lie group, analogous to the well known relationship of Lie algebras and Lie groups.

Geometry Algorisms of Dynkin Diagrams in Lie Group Machine Learning

This paper uses the geometric method to describe Lie group machine learning(LML)based on the theoretical framework of LML,which gives the geometric algorithms of Dynkin diagrams in LML.It includes the basic conceptions of Dynkin diagrams in LML,the classification theorems of Dynkin diagrams in LML,the classification algorithm of Dynkin diagrams in LML and the verification of the classification algorithm with experimental results.

Solutions of the D-dimensional Schrdinger equation with Killingbeck potential:Lie algebraic approach

Algebraic solutions of the D-dimensional Schrodinger equation with Killingbeck potential are investigated using the Lie algebraic approach within the framework of quasi-exact solvability. The spectrum and wavefunctions of the system are reported and the allowed values of the potential parameters are obtained through the sl（2） algebraization.

Lie Algebras Associated with Group U（n）

Starting from the subgroups of the group U（n）, the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.

QED-Lie Algebra and Their £-Modules in Superconductivity

It’s created a canonical Lie algebra in electrodynamics with all the “nice” algebraic and geometrical properties of an universal enveloping algebra with the goal of can to obtain generalizations in quantum electrodynamics theory of the TQFT, and the Universe based in lines and twistor bundles to the obtaining of irreducible unitary representations of the Lie groups SO(4) ?andO(3,1) , based in admissible representations of U(1) , and SU(n)? . The obtained object haves the advantages to be an algebraic or geometrical space at the same time. This same space of ￡-modules can explain and model different electromagnetic phenomena in superconductor and quantum processes where is necessary an organized transformation of the electromagnetic nature of the space- time and obtain nanotechnologies of the space-time and their elements.

Operator Methods and SU(1,1) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials

Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.

基于Lie群的机器学习理论框架

借用具有良好数学结构的Lie群来研究机器学习,提出了基于Lie群的机器学习(ML)基本概念、对偶空间学习概念等,形成了基于Lie群的学习理论框架.该理论框架可以用代数和几何的方法来描述机器学习系统,弥补了原有机器学习理论的不足.

基于Lie群的刚体动力学建模及数值计算方法研究

基于Lie群和Lie代数之间的指数映射等价关系,推导了基于Lie群的自由刚体连续动力学方程.结合离散变分原理,推导了其Lie群离散变分积分子.通过证明可知连续和离散动力学系统都具有动量守恒性.对连续动力学方程进行同维化处理,使其变为常规非线性方程组的形式,利用Runge-Kutta法进行求解;基于Runge-Kutta基本理论,推导了直接用于Lie群的Runge-Kutta法,从而使Runge-Kutta法可用于求解变维非线性方程组;通过Lie代数变换,利用Kelly变换和Newton迭代对Lie群离散变分积分子进行求解.仿真对比结果表明,3种算法下的计算结果高度吻合,且能高精度地保持系统的结构守恒和动量守恒性.