一类S-mKdV方程族及其扩展可积模型

由loop代数的一个子代数出发，构造了一个线性等谱问题，再利用屠格式计算出了一类Liouvelle意义下的可积系统及其双Hamilton结构，作为该可积系统的约化，得到了著名的Schrdinger方程和mKdV方程，因此称该系统为S-mKdV方程族.根据已构造的的子代数，又构造了维数为5的loop代数的一个新的子代数，由此出发设计了一个线性等谱形式，再利用屠格式求得了S-mKdV方程族的一类扩展可积模型.利用这种方法还可以求BPT方程族、TB方程族等谱系的扩展可积模型.因此本方法具有普遍应用价值.最后作为特例，求得了著名的Schrdinger方程和mKdV方程的可积耦合系统.     

耦合的AKNS-Kaup-Newell孤子方程族的一类可积扩展模型

基于Lie圈代数?1的子代数建立了一个等谱问题，导出了耦合的AKNS- Kaup- Newell孤子方程族。另外，利用扩展相应Lie代数于?2的方法，建立了该方程族的可积扩展模型，即可积耦合。     

一类NLS-mKdV方程族的扩展可积系统

利用代数变换，构造了与文献［５］中的loop代数₂的子代数等价的loop代数1的一个子 代数₁.再将₁扩展为一个高维的loop代数.利用设计了一个等谱问题 ，结合子代数间的直 和运算和同构关系，得到了NLS_mKdV方程族的一类扩展可积系统.作为约化情形，求得了著 名的Schrdinger方程与mKdV方程的可积耦合系统. 关键词： loop代数 可积系统 等谱问题     

AKNS方程族的一类扩展可积模型

首先构造了loop代数A～2的一个新的子代数,设计了一个等谱问题.应用屠格式求出了著名的AKNS方程族的一类扩展可积模型,即可积耦合.然后将这种求可积耦合的方法一般化,可用于一大类方程族,如KN族、GJ族、WKI族等谱系的可积耦合.提出的方法具有普遍应用价值.最后作为AKNS方程族的特例,求得了KdV方程和MKdV方程的可积耦合 关键词： 可积耦合 loop代数 AKNS方程族     

核类似铜酸盐超导模型的su_s(1,1)su_d(1,1)代数结构

由Iachello提出的核类似铜酸盐超导模型具有su(3)代数结构,其平均场近似下的Hamilton量可以写成su(3)生成元的线性组合.通过代数生成元的实现,该模型约化的Hamilton量具有Su_s(1,1)SU_d(1,1)代数结构,利用相干算子U(θ,φ)的幺正变换,可得到系统约化Hamilton量的能隙方程及本征值,发现应用不同代数结构求解会得到不同侧重点的分析结果.     

离散差分序列变质量Hamilton系统的Lie对称性与Noether守恒量

本文研究离散差分序列变质量Hamilton系统的Lie对称性与Noether守恒量. 构建了离散差分序列变质量Hamilton系统的差分动力学方程, 给出了离散差分序列变质量Hamilton系统差分动力学方程在无限小变 换群下的Lie对称性的确定方程和定义, 得到了离散力学系统Lie对称性导致Noether守恒量的条件及形式, 举例说明结果的应用. 关键词： 离散力学 Hamilton系统 Lie对称性 Noether守恒量     

离散差分变分Hamilton系统的Lie对称性与Noether守恒量

研究离散差分Hamilton系统的Lie对称性与Noether守恒量. 根据扩展的时间离散力学变分原理构建Hamilton系统的差分动力学方程.定义离散系统运动差分方程在无限小变换群下的不变性为Lie对称性, 导出由Lie对称性得到系统离散Noether守恒量的判据. 举例说明结果的应用. 关键词： 离散力学 差分Hamilton系统 Lie对称性 Noether守恒量     

Hamilton系统的Mei对称性、Noether对称性和Lie对称性

研究Hamilton系统的形式不变性即Mei对称性，给出其定义和确定方程.研究Hamilton系统的Mei对称性与Noether对称性、Lie对称性之间的关系，寻求系统的守恒量.给出一个例子说明本文结果的应用. 关键词： Hamilton系统 Mei对称性 Noether对称性 Lie对称性 守恒量     

机电动力系统的动量依赖对称性和非Noether守恒量

研究了Lagrange-Maxwell机电动力系统的Hamilton正则方程及动量依赖对称性的定义、判据、结构方程和守恒量的形式.研究表明，结构方程中的函数ψ只需是对称群的不变量.得到求解机电动力系统守恒量的新方法，并给出了应用实例. 关键词： 机电动力系统 Lie群分析 对称性 守恒量     

奇异系统Hamilton正则方程的Mei对称性、Noether对称性和Lie对称性

研究奇异系统Hamilton正则方程的形式不变性即Mei对称性，给出其定义、确定方程、限制方程和附加限制方程.研究奇异系统Hamilton正则方程的Mei对称性与Noether对称性、Lie对称性之间的关系，寻求系统的守恒量.给出一个例子说明结果的应用. 关键词： 奇异系统 Hamilton正则方程 约束 对称性 守恒量     

On the Structure of the Observable Algebra for QED on the Lattice

We prove that the matter field subalgebra of the observable algebra for QED on a finite lattice is isomorphic to the enveloping algebra of the Lie algebra sl(2N, C), factorized by a certain ideal. Using this result, we give a new proof of the decomposition of the physical Hilbert space into charge superselection sectors.     

Lie algebra structures for four-component Hamiltonian hydrodynamic type systems

Four-component Hamiltonian systems of hydrodynamic type induce, through the Haantjes tensor, a Lie algebra structure on tangent planes in the space of dependent variables. We show that this Lie algebra is either reductive or solvable with a nilpotent three-dimensional subalgebra. We demonstrate how the precise Lie algebraic structure is determined by the Hamiltonian structure of the system. An application to perturbations of the Benney system is presented.     

Strings fromN=2 gauged Wess-Zumino-Witten models

We present an algebraic approach to string theory. An embedding ofsl(2|1) in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of theN=2 superconformal algebra. The extension is completely determined by thesl(2|1) embedding. The realization of the superconformal algebra is determined by the grading. For a particular choice of grading, one obtains in this way, after twisting, the BRST structure of a string theory. We classify all embeddings ofsl(2|1) into Lie super algebras and give a detailed account of the branching of the adjoint representation. This provides an exhaustive classification and characterization of both all extendedN=2 superconformal algebras and all string theories which can be obtained in this way.     

The generalized nonlinear Schrödinger hierarchy, its integrable coupling system and their bi-Hamiltonian structure

Based on a subalgebra G of Lie algebra A₂, a new Lie algebra G^∗ is constructed. By making use of the Tu scheme, the generalized nonlinear Schrödinger hierarchy and its integrable coupling are both obtained with the help of their corresponding special loop algebras. At last, by means of the quadratic-form identity, their bi-Hamiltonian structures of the generalized nonlinear Schrödinger hierarchy and its integrable coupling system are worked out respectively. The approach presented in this Letter can be used in other integrable hierarchies.     

Lie Algebras and Integrable Systems

A 3? 3 matrix Lie algebra is first introduced, its subalgebras and the generated Lie algebras are obtained, respectively. Applications of a few Lie subalgebras give rise to two integrable nonlinear hierarchies of evolution equations from their reductions we obtain the nonlinear Schrödinger equations, the mKdV equations, the Broer-Kaup (BK) equation and its generalized equation, etc. The linear and nonlinear integrable couplings of one integrable hierarchy presented in the paper are worked out by casting a 3? 3 Lie subalgebra into a 2? 2 matrix Lie algebra. Finally, we discuss the elliptic variable solutions of a generalized BK equation.     

Some Evolution Hierarchies Derived from Self-dual Yang-Mills Equations

We develop in this paper a new method to construct two explicit Lie algebras E and F. By using a loop algebra \bar{E} of the Lie algebra E and the reduced self-dual Yang-Mills equations, we obtain an expanding integrable model of the Giachetti-Johnson (GJ) hierarchy whose Hamiltonian structure can
also be derived by using the trace identity. This provides a much simplier construction method in comparing with the tedious variational identity approach. Furthermore, the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra g_N. As an application, we apply the loop algebra \tilde{E} of the Lie algebra E to obtain a kind of expanding integrable model of the Kaup-Newell (KN) hierarchy which, consisting of two arbitrary parametersα andβ, can be reduced to two nonlinear evolution equations. In addition, we use a loop algebra \tilde{F} of the Lie algebra F to obtain an
expanding integrable model of the BT hierarchy whose Hamiltonian structure is the same as using the trace identity. Finally, we deduce five integrable systems in R³ based on the self-dual Yang-Mills equations, which include Poisson structures, irregular lines, and the reduced equations.     

sl(N) Onsager's algebra and integrability

We define ansl(N) analog of Onsager's algebra through a finite set of relations that generalize the Dolan-Grady defining relations for the original Onsager's algebra. This infinite-dimensional Lie algebra is shown to be isomorphic to a fixed-point subalgebra ofsl(N) loop algebra with respect to a certain involution. As the consequence of the generalized Dolan-Grady relations a Hamiltonian linear in the generators ofsl(N) Onsager's algebra is shown to posses an infinite number of mutually commuting integrals of motion.     

A Lie algebraic condition for exponential stability of discrete hybrid systems and application to hybrid synchronization

A Lie algebraic condition for global exponential stability of linear discrete switched impulsive systems is presented in this paper. By considering a Lie algebra generated by all subsystem matrices and impulsive matrices, when not all of these matrices are Schur stable, we derive new criteria for global exponential stability of linear discrete switched impulsive systems. Moreover, simple sufficient conditions in terms of Lie algebra are established for the synchronization of nonlinear discrete systems using a hybrid switching and impulsive control. As an application, discrete chaotic system's synchronization is investigated by the proposed method.     

-Realization of Lie Algebras: Application to so(4,2) and Algebras

The property of some finite -algebras to appear as the commutant of a particular subalgebra in a simple Lie algebra is exploited for the obtention of new -realizations from a “canonical” differential one. The method is applied to the conformal algebra so(4,2) and therefore yields also results for its Poincar\'e subalgebra. Unitary irreducible representations of these algebras are recognized in this approach, which is naturally compared -or associated- to the induced representation technique. Received: 10 June 1996 / Accepted: 8 October 1996     

Upon Generating (2+1)-dimensional Dynamical Systems

Under the framework of the Adler-Gel’fand-Dikii(AGD) scheme, we first propose two Hamiltonian operator pairs over a noncommutative ring so that we construct a new dynamical system in 2+1 dimensions, then we get a generalized special Novikov-Veselov (NV) equation via the Manakov triple. Then with the aid of a special symmetric Lie algebra of a reductive homogeneous group G, we adopt the Tu-Andrushkiw-Huang (TAH) scheme to generate a new integrable (2+1)-dimensional dynamical system and its Hamiltonian structure, which can reduce to the well-known (2+1)-dimensional Davey-Stewartson (DS) hierarchy. Finally, we extend the binormial residue representation (briefly BRR) scheme to the super higher dimensional integrable hierarchies with the help of a super subalgebra of the super Lie algebra sl(2/1), which is also a kind of symmetric Lie algebra of the reductive homogeneous group G. As applications, we obtain a super 2+1 dimensional MKdV hierarchy which can be reduced to a super 2+1 dimensional generalized AKNS equation. Finally, we compare the advantages and the shortcomings for the three schemes to generate integrable dynamical systems.