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Generalized boson algebras : their properties and applications to mathematics and physics

广义玻色子代数：它们的性质及其在数学和物理中的应用

基本信息

批准号：
15540132
负责人：
AIZAWA Naruhiko
金额：
$ 2.05万
依托单位：
Osaka Prefecture University (2005)Osaka Women's University (2003-2004)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

The purposes of this research project are three fold. Firstly, seeking possibilities of generalizing the boson algebra in the framework of Hopf algebra. Secondly, investigating properties of the generalized boson algebras. Then, applying the generalized boson algebras to some mathematical or physical problems. One of the generalized boson algebras, already known, is regarded as bosonization of the quantum algebra U_q[osp(1/2)]. We, thus, started our investigation with the algebra (denoted by U) and U_q[osp(1/2)]. The main results of the present research are summarized as follows.1.The algebra dual to U was obtained. The duality is expressed in the form of universal T-matrix.2.The single-particle and bipartite coherent states for U were constructed. It was shown that the bipartite coherent state had entanglement which disappears in the classical limit. An analytic proof of orthogonality and completeness for the single-particle coherent state was given.3.Irreducible representations of th … More e algebra dual to U were obtained explicitly and it was shown that the representation matrices are related to little q-Jacobi polynomials.4.Tensor product of the representations of U is decomposed into irreducible ones. It was shown that the decomposition was carried out with q-Hahn polynomials. The result No.3 and 4 imply that the algebra U gives a new algebraic background for basic hypergeometri functions.5.A general method for constructing noncommutative space with supersymmetric nature, which means that the space is covariant under the action of quantum group OSp_q(1/2), was introduced. By the method, 3-dimensional noncommutative flat superspace and 5-dimensional noncommutative supersphere were constructed and their properties, as well as relations to boson algebras, were studied.6.Differential geometry on noncommutative spaces has been developed by Dubois-Viollete et al. In this research, the geometry was extended to noncommutative superspaces. As an example, covariant derivative, curvature etc on 3-dimensional noncommutative superspace, which is a covariant algebra of the Jordanian quantum group OSp_h(1/2), were computed. The computation shows that the superspace is not physical, since the covariant derivative is not compatible with the metric. Less

这个研究项目的目的有三个方面。第一，在Hopf代数的框架下寻求推广玻色子代数的可能性。其次，研究了广义玻色子代数的性质。然后，将广义玻色子代数应用于一些数学或物理问题。已知的广义玻色子代数之一被认为是量子代数U_q[osp（1/2）]的玻色子化。因此，我们从代数（记为U）和U_q[osp（1/2）]开始研究。本文的主要研究成果如下：1.得到了U的对偶代数。构造了U的单粒子相干态和二体相干态。结果表明，两体相干态存在纠缠，但在经典极限下纠缠消失。给出了单粒子相干态的正交性和完备性的解析证明 ...更多信息 e代数的对偶，并证明其表示矩阵与小q-Jacobi多项式有关。4.将U的表示的张量积分解为不可约的张量积。结果表明，分解是用q-Hahn多项式进行的。结果3和4表明，代数U为基本超几何函数提供了新的代数背景。5.给出了构造具有超对称性质的非对易空间的一般方法，即在量子群OSp_q（1/2）作用下空间是协变的。利用这种方法，构造了三维非对易平坦超空间和5维非对易超球，并研究了它们的性质以及与玻色子代数的关系。6.非对易空间上的微分几何是由Dubois-Viollete等人发展起来的，本文将微分几何推广到非对易超空间上。作为例子，计算了Jordan量子群OSph（1/2）的协变代数--三维非对易超空间上的协变导数、曲率等.计算表明，超空间不是物理的，因为协变导数与度量不相容。少