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。     

多项式角动量代数的代数表示及实现

本文利用三参数李群求代数表示的方法求出多项式角动量代数的代数表示及其酉表示，找到一个能同时承载李代数及相对应的多项式角动量代数的基底，并在该基底下求出两种代数之间的联系，利用该联系则也可求出多项式角动量代数的代数表示.最后求出多项式角动量代数的单玻色实现及其在有限维多项式函数空间的微分实现. 关键词： 多项式角动量代数 Higgs代数 su(2)代数     

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.     

推广的一类Lie代数及其相关的一族可积系统

对已知的Lie代数An-1作直接推广得到一类新的Lie代数gl(n,C).为应用方便，本文只考虑Lie代数gl(3,C)情形.构造了gl(3,C)的一个子代数，通过对阶数的规定，得到了一类新的loop代数.作为其应用，设计了一个新的等谱问题，得到了一个新的Lax对.利用屠格式获得了一族新的可积系统，具有双Hamilton结构,且是Liouville可积系.作为该方程族的约化情形，得到了新的耦合广义Schrdinger方程. 关键词： Lie代数 可积系 Hamilton结构     

SU(3)线性非自治量子系统的代数动力学求解

利用代数动力学方法，得到了量子物理中十分重要的SU（3）线性非自治量子系统的严格解及其不变Cartan算子，并建立起量子解与经典解之间的对应关系．同时计算了周期系统的非绝热Berry相因子 关键词：     

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.     

变耦合系数非线性三波导定向耦合器特性研究

报道了对高斯型及指数型变耦合系数三波导耦合器的一些重要的全光开关特性进行的研究。利用四阶龙格一库塔方法对指数型和高斯型两类变耦合系数三波导耦合器进行了数值计算。数值计算结果表明：对于变耦合系数三波导耦合器而言,功率可在波导1与波导3之间100％转换,而波导2则不可能达到100％的功率输出。与双波导变耦合系数耦合器相比,在相同的最大耦合系数情况下三波导变耦合系数耦合器开关曲线要更陡一些．即具有更好的开关特性。与平行三波导耦合器相比,变耦合系数三波导耦合器作为光开关的最大优点在于开关曲线中不存在振荡。     

外腔半导体激光器中反馈耦合系数及介持吸收系数的测量

在室温下研究了外腔反馈对ＧａＡｌＡｓ量子阱半导体激光器阈值的影响，提出了测量实际反馈量及半导体激光器增益介质吸收系数的方法。利用所测反馈系数汲阈值得到了反馈耦合因子ｋ，驱动电流和增益之间的系数ζ以及实验所用半导体激光器增益介质的吸收系数。     

C6H6可见光与红外跃迁强度的代数方法计算

在Iachello Oss的代数模型中 ,采用对称化基 ,利用在能谱计算中得到的分子的波函数的基础上 ,计算了C6H6和C6D6分子拉伸振动模式的可见光与红外跃迁强度 ,给出了拉曼跃迁的计算式 .跃迁算符的形式由分子对称性决定 .其中对称化基的构造 ,采用了对称化玻色表象方法 ,大大简化了计算 .计算结果与实验观测符合的相当好 .这表明其它振动模式与拉伸振动模式之间的耦合 ,或者很弱 ,或者可通过等效参数考虑进来 .研究表明 ,代数模型和对称化玻色表象方法的结合是解决分子振动问题的强有力的工具     

Graded quasi-Lie algebras

This paper is concerned with a new class of graded algebras naturally uniting within a single framework various deformations of the Witt, Virasoro and other Lie algebras based on twisted and deformed derivations, as well as color Lie algebras and Lie superalgebras. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. Supported by the Liegrits network Supported by the Crafoord foundation     

On representations of Hecke algebras

In this report we review some facts about representation theory of Hecke algebras. For Hecke algebras we adapt the approach of A. Okounkov and A. Vershik [Selecta Math., New Ser., 2 (1996) 581], which was developed for the representation theory of symmetric groups. We justify explicit construction of idempotents for Hecke algebras in terms of Jucys-Murphy elements. Ocneanu's traces for these idempotents (which can be interpreted as q-dimensions of corresponding irreducible representations of quantum linear groups) are presented. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. This work was supported in part by the grants INTAS 03-51-3350 and RFBR 05-01-01086-a.     

Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras . On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra an abstract quantum Lie algebra independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras are isomorphic to an abstract quantum Lie algebra . In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same are isomorphic, 2) the quantum Lie product of any is q-antisymmetric. We also describe a construction of which establishes their existence. Received: 23 May 1996 / Accepted: 17 October 1996     

Fine Gradings on Non-simple Lie Algebras: Example of o(4,C)

On any Lie algebra L, it is of significant convenience to have at one's disposal all the possible fine gradings of L, since they reflect the basic structural properties of the Lie algebra. They also provide useful bases of the representations of the algebra -- namely such bases that are preserved by the commutator.We list all the six fine gradings on the non-simple Lie algebra o(4,C) and we explain their relation to the fine gradings of the Lie algebra sl(2,C) where relevant. The existence of such relation is not surprising, since o(4,C) is in fact a product of two specimen of sl(2,C). The example of o(4,C) is especially important due to the fact that one of its fine gradings is not generated by any MAD-group. This proves that, unlike in the case of classical simple Lie algebras over C, on the non-simple classical Lie algebras over C there can exist a fine grading that is not generated by any MAD-group on the Lie algebra.     

Dynamical algebras for Pöschl-Teller Hamiltonian hierarchies

The dynamical algebras of the trigonometric and hyperbolic symmetric Pöschl-Teller Hamiltonian hierarchies are obtained. A kind of discrete-differential realizations of these algebras are found which are isomorphic to so(3, 2) Lie algebras. In order to get them, first the relation between ladder and factor operators is investigated. In particular, the action of the ladder operators on normalized eigenfunctions is found explicitly. Then, the whole dynamical algebras are generated in a straightforward way.     

Invariant functions and contractions of certain types of Lie algebras of lower dimensions

In this paper, we deal with contractions of Lie algebras. We use two invariant functions of Lie algebras as a tool, named ψ and ? function, respectively, which have a great application in Physics due to their remarkable properties. We focus the study of these functions in the frame of the filiform Lie algebras, trying to extend to these algebras some of the properties of such functions over semi-simple Lie algebras.     

Infinitesimal deformations of filiform Lie algebras of order 3

The Lie algebras of order$F$ have important applications for the fractional supersymmetry, and on the other hand the filiform Lie (super)algebras have very important properties into the Lie Theory. Thus, the aim of this work is to study filiform Lie algebras of order$F$ which were introduced in Navarro (2014). In this work we obtain new families of filiform Lie algebras of order 3, in which the complexity of the problem rises considerably respecting to the cases considered in Navarro (2014).     

Quantum counterparts of three-dimensional real Lie algebras over harmonic oscillator

Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of three-dimensional (3D) real Lie algebras in Bianchi classification. The Jacobi operators of the quantum algebras are found.     

Quantizer–dequantizer operators as a tool for formulating the quantization procedure

We consider the quantization procedure and investigate the application of the quantizer–dequantizer method and star-product technique to construct associative products and the associative algebras formed by the quantizer–dequantizer operators and their symbols. The corresponding Lie algebras are also constructed. We study the case where the quantizer–dequantizer operators form a self-dual system and show that the structure constants of the Lie algebras satisfy some identity, in addition to the Jacobi identity. Using tomographic quantizer–dequantizer operators and their symbols, we construct the continuous associative algebra and the corresponding Lie algebra.