李超代数B(0,1)的非齐次微分实现,Boson-Fermion实现及其表示

本文研究了李超代数B(0,1)在非齐次多项式空间上的微分实现(称为非齐次微分实现),和对应的非齐次Boson-Fermion实现.利用非齐次Boson-Fermion实现,在 Heisenberg-Weyl超代数的通用包络代数,其子空间和商空间上研究了B(0,1)的一类新的不可分解表示和不可约表示.B(0,1)的所有有限维不可约表示则作为特殊情况全部给出.     

关于李代数及其表示的玻色子实现和非齐性微分实现

本文分析和推广了作者以前建议的构造李代数不可分解表示的玻色子实现方法.不仅证明该方法构造的主表示可以在子空间上导出Gruber的主表示,而且讨论了该表示的商空间表示与相干态的联系.本文还建议了构造李代数非齐性玻色子实现的一般方法,由此得到了量子力学准精确可解问题中有用的李代数非齐性微分实现.     

量子(超)代数在qp=1时的循环表示

利用商模方法和q-boson实现方法研究了量子(超)代数在q^p=1时的循环表示.对于与任意有限维单李代数相关的量子代数给出了两种方法的一般理论,而且推广到了量子超代数U_qosp(1.2).通过构造q-Heisenberg-Wey1超代数的循环表示,q-boson实现方法推广到构造某些高秩量子超代数的循环表示.     

XXZ模型及其对称代数的结构

本文以具量子代数SL_q(2)对称性的XXZ Heisenberg自旋链为例,利用Temperley-Lieb代数结构,深入分析q^p=±1时SL_q(2)量子代数的有限维表示,讨论了可约而不可分解表示(Ⅰ型表示)出现的条件及其性质,并采用取极限办法得到新的状态,从而得到对应确定能量状态的完全集合.     

黑克、布劳、及伯曼-温采尔代数的诱导及分导系数和量子经典李代数的耦合与重新耦合系数

本文总结了计算黑克、布劳、及伯曼 温采尔代数在各种工数链下诱导及分导系数的线性方程方法(LEM)。特别强调了关于A,B,C,D型李代数及其量子情形与其中心代数之间的舒尔 魏尔 布劳双关性关系。这一关系使我们能够利用相应中心代数的诱导及分导系数计算出经典李代数及其量子情形的耦合与重新耦合系数。讨论了从该方法得到B,C,D型李代数不可约表示克罗内克积分解的应用。基于LEM还得到了处理对应于置换群CG系列问题的黑克代数张量积的方法。     

核磁共振波谱的李代数表示与计算方法

本文讨论NMR波谱与李代数表示论之间的联系及NMR波谱解析的李代数方法中的基本问题.若NMR中交换超算符方法中的算符集合生成的李代数为g,本文将讨论g的自然表示和伴随表示,g的典型性质,g的根系结构.提出了解析NMR波谱的三种李代数方法,列举了实例,显示了计算的具体步骤.     

黑克、布劳、及伯曼温采尔代数的诱导及分导系数和量子经典李代数的耦合与重新耦合系数(英文)

本文总结了计算黑克、布劳、及伯曼 温采尔代数在各种工数链下诱导及分导系数的线性方程方法(LEM)。特别强调了关于A,B,C,D型李代数及其量子情形与其中心代数之间的舒尔 魏尔 布劳双关性关系。这一关系使我们能够利用相应中心代数的诱导及分导系数计算出经典李代数及其量子情形的耦合与重新耦合系数。讨论了从该方法得到B,C,D型李代数不可约表示克罗内克积分解的应用。基于LEM还得到了处理对应于置换群CG系列问题的黑克代数张量积的方法。     

李超代数B(0,1)的非齐次微分实现,Boson-Fermion实现及其表示

本文研究了李超代数B(0,1)在非齐次多项式空间上的微分实现(称为非齐次微分实现),和对应的非齐次Boson—Fermion实现.利用非齐次Boson-Fermion实现,在Heisenberg-Weyl超代数的通用包络代数,其子空间和商空间上研究了B(0,1)的一类新的不可分解表示和不可约表示.B(0,1)的所有有限维不可约表示则作为特殊情况全部给出.     

李超代数B(0,1)的超相干态表示

本文反相干态的概念推广到李超代数的情形.我们具体地构造了李超代数B(0,1)的超相干态,计算了B(0,1)生成元在超相干态表示中的矩阵元,获得了B(0,1)代数的一种非齐次微分实现.这种非齐次微分实现对研究量子力学中的准精确可解问题有用.     

基于几何代数的彩色光流场计算

提出了一种新的基于几何代数的彩色光流场计算方法.从新的角度出发讨论了运用几何代数的概念来解决彩色图像序列的可行性和简便性.在几何代数域内对彩色图像序列用多重矢量表示,证明了彩色光流场分析时进行约束条件时的测量问题,通过多重矢量而非多通道的图像处理方法来表示图像连续特性,从而扩大了测量范围;然后依据物体色彩在运动过程中保持不变的原理,在几何代数域内推导了彩色图像序列的光流约束方程及其解法.结果表明,该算法能够显著提高各种情况下的彩色光流场计算能力.     

EPR Entangled States for Bipartite Kinematics and New Bosonic Representation of SU(2) Algebra

We find that the Einstein-Podolsky-Rosen (EPR) entangled state representation describing bipartite kinematics is closely related to a new Bose operator realization of SU(2) Lie algebra. By virtue of the new realization some Hamiltonian eigenfunction equation can be directly converted to the generalized confluent equation in the EPR entangled state representation and its solution is obtainable. This thus provides a new approach for studying dynamics of angular momentum systems.     

EPR Entangled States for Bipartite Kinematics and New Bosonic Representation of SU（2） Algebra

We find that the Einstein-Podolsky-Rosen (EPR) entangled state representation descr/bing bipartite kinematics is closely related to a new Bose operator realization of SU(2) Lie algebra. By virtue of the new realization some ttamiltonian eigenfunction equation can be directly converted to the generalized confluent equation in the EPR entangled state representation and its solution is obtainable. This thus provides a new approach for studying dynamics of angular momentum systems.     

Indecomposable Representations of the Square—Root Lie Algebras of Vector Type and Their Boson Realizations

The explicit expressions for indecomposable representations of nine square-root Lie algebras of vector type, R_νλ (ν, λ=0, ±1), are obtained on the space of universal enveloping algebra of two-state Heisenberg-Weyl algebra, the invariant subspaces and the quotient spaces. From Fock representations corresponding to these indecomposable representations, the inhomogeneous boson realizations of R_νλ are given. The expectation values of R_νλ in the angular momentum coherent states are calculated as well as the corresponding classical limits.     

Local Currents for a Deformed Heisenberg–Poincaré Lie Algebra of Quantum Mechanics,and Anyon Statistics

We set out to construct a Lie algebra of local currents whose space integrals, or “charges”, form a subalgebra of the deformed Heisenberg–Poincaré algebra of quantum mechanics discussed by Vilela Mendes, parameterized by a fundamental length scale ℓ. One possible technique is to localize with respect to an abstract single-particle configuration space having one dimension more than the original physical space. Then in the limit ℓ→0, the extra dimension becomes an unobservable, internal degree of freedom. The deformed (1+1)-dimensional theory entails self-adjoint representations of an infinite-dimensional Lie algebra of nonrelativistic, local currents modeled on (2+1)-dimensional space-time. This suggests a new possible interpretation of such representations of the local current algebra, not as describing conventional particles satisfying bosonic, fermionic, or anyonic statistics in two-space, but as describing systems obeying these statistics in a deformed one-dimensional quantum mechanics. In this context, we have an interesting comparison with earlier results of Hansson, Leinaas, and Myrheim on the dimensional reduction of anyon systems. Thus motivated, we introduce irreducible, anyonic representations of the deformed global symmetry algebra. We also compare with the technique of localizing currents with respect to the discrete positional spectrum.     

Applications of Polynomial Algebras to 2-Dimensional Deformed Oscillators

The polynomial algebra is a deformed su(2) algebra. Here, we use polynomial algebra as a method to solve a series of deformed oscillators. Thus, we find a series of physics systems corresponding with polynomial algebra with different highest orders.     

为什么轨道角动量不能是?/2?

讨论了一般角动量与轨道角动量的联系和区别，指出它们的联系在于SU（2）群和SO（3）群的局部性质即李代数相同（同构），它们的差别在于两群的整体性质不同．一般角动量的最小单位?/2由群的局部性质决定，亦即李代数的对易关系（相当于角动量的对易关系）决定，而轨道角动量的最小单位?由SO（3）群的整体性质决定 关键词：     

交换超算符方法的李代数研究

本文讨论了交换超算符方法的理论基础,结果表明由交换超算符所定义的算符集合g是一个李代数,交换超算符的定义就是李代数中内导子的定义,由此得出一些交换超算符间的代数关系。证明了g中所有算符诱导的超算符集合也是一个李代数,指出了与g对应的是由复盖群派生的,有内积定义的李群,而角动量超算符是由矢量场的内禀角动量和单位算符的直积所生成。结论是交换超算符方法的理论基础是李代数。     

Polynomial associative algebras of quantum superintegrable systems

The integrals of motion of classical two-dimensional superintegrable systems, with polynomial integrals of motion, close in a restrained polynomial Poisson algebra; the general form of the quadratic case is investigated. The polynomial Poisson algebra of the classical system is deformed into a quantum associative algebra of the corresponding quantum system, and the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. The finite-dimensional representations of the algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the roots of algebraic equations in the quadratic case.     

ON THE IRREDUCIBLE REPRESENTATIONS OF THE SIMPLE LIE GROUPS I——THE TENSOR BASIS FOR THE INFINITESIMAL GENERATORS OF THE CLASSICAL SIMPLE LIE GROUPS

In this paper, we analyse the commutation relations of the infinitesimal generatorsof all simple classical Lie groups and establish a new basis for these generators, calledthe tensor basis. In tensor basis, the infinitesimal, generators can be written as somescalar operators, some sets of angular momentum operators and some sets of irreducibletensor operators. The commutation relations, of these operators are very simple andhave many regularities. By means of the method that has been used in the earlier papers, "On the irre-ducible representations of the compact simple Lie groups of rank 2, I,II,III" and thetensor basis, all the irreducible representations of the classical simple Lie groups canbe calculated systematically.     

A representation of angular momentum (SU(2)) algebra

This paper seeks to construct a representation of the algebra of angular momentum (SU(2) algebra) in terms of the operator relations corresponding to Gentile statistics in which one quantum state can be occupied by n particles. First, we present an operator realization of Gentile statistics. Then, we propose a representation of angular momenta. The result shows that there exist certain underlying connections between the operator realization of the Gentile statistics and the angular momentum (SU(2)) algebra.