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

这个研究项目的目的有三个方面。首先，在Hopf代数的框架中寻找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。已知的一种广义玻色子代数被认为是量子代数U_q[osp(1/2)]的玻色子化。因此，我们从代数（用U表示）和U_q[osp(1/2)]开始我们的研究。本文的主要研究成果总结如下：得到了U的对偶代数。对偶用泛t矩阵的形式表示。构建了U的单粒子相干态和二部相干态。证明了二部相干态存在纠缠态，而纠缠态在经典极限下消失。给出了单粒子相干态正交性和完备性的解析证明。明确地得到了对U对偶的更多的e代数的不可约表示，并证明了这些表示矩阵与小q-Jacobi多项式有关。将U表示的张量积分解为不可约的张量积。结果表明，用q-Hahn多项式进行分解。结果3和4暗示了代数U为基本超几何函数提供了一个新的代数背景。给出了构造具有超对称性质的非交换空间的一般方法，即在量子群OSp_q（1/2）作用下空间是协变的。利用该方法构造了三维非交换平面超空间和五维非交换超球，研究了它们的性质及其与玻色子代数的关系。非交换空间上的微分几何是由Dubois-Viollete等人提出的。在本研究中，将几何扩展到非交换超空间。以jordan量子群OSp_h（1/2）的协变代数为例，计算了三维非对易超空间上的协变导数、曲率等。计算表明，由于协变导数与度规不相容，超空间不是物理的。少

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