Extension of quantum theory of angular momentum and Yang-Baxter relations

我们广泛地推广了角动量理论，即旋转群的表示理论，称为Wigner-Racah代数。数学工具主要致力于量子（q-）群代数。我们所关心的物理空间是所谓的q-空间，一个非交换空间，它是普遍的交换空间的一个扩展。我们已经成功地在q-协变/逆变形式的框架内规定了所有的量及其关系。它沿着我们的假设，即任何可观察量（转移概率）都不应该依赖于q空间的某种线性坐标变换（q线性变换）。根据q-协方差，所有的量都被适当地分类为q-张量（q-标量，q-向量等）。我们有了著名的Wigner和Eckart中心定理的推广--q-Wigner-Eckart定理和非常自然的q-单位张量的概念，研究了各种基函数，如q-旋转函数，q-Clebsh-Gordan系数，q-Racah系数（或一般的q-n-j符号），并建立了它们之间的各种关系。着重指出这些q-函数构成了构成面模型和顶点模型的Yang-Baxter关系的一类基函数，进而找到了一种系统的方法来确定q-玻色子和q-费米子的广义创生-湮灭算符之间的q-对易关系。在q-协变理论中，有几种方法可以指定q-产生和湮灭算子来形成旋量。例如，我们描述了一个q-旋量的一对创造和湮灭运营商获得q-模拟BCS Bogoliubov形式主义。利用这个结果，我们成功地推广了所谓的辛代数，并将上述考虑的一部分推广到U_q（n）的情形。我们还发现了非常新的q-扩展Young图（Schur函数）。

We have widely extended the theory of angular momentum i.e.representation theory of rotation group, called Wigner-Racah algebras. Mathematical devices are devoted mainly to quantum (q-) group algebras. Physical space of concern is so-called q-space, a non-commutative one, which is an extension of the prevalent commutative space.We have succeeded to specify all the quantities and their relations in the framework of q-covariant/contravariant forms. It is along the line of our postulate that any observable quantity (transition probability) should not rely on a certain linear coordinate transformation (q-linear transformation) of the q-space. According to the q-covariance, all the quantities are properly classified as q-tensors (q-scalar, q-vector, etc.). We have q-Wigner-Eckart theorem, which is an extension of well-known central theorem by Wigner and Eckart, and notions of q-unit tensors in a very natural way.We have investigated kinds of basis functions, such as q-rotation functions, q-Clebsh-Gordan coefficients, q-Racah coefficients (or, generally q-n-j symbols), and have established various relationship among them. Emphasis has been put on the point that these q-functions form a class of basis functions constituting Yang-Baxter relations of face models and of vertex models.Further, we have found a systematic way to specify q-commutation relations among extended creation-annihilation operators of q-bosons and q-fermions. There are several ways to assign the q-creation and annihilation operators to form spinors in the q-covariant theory. For example, we describe a q-spinor in terms of a pair of creation and annihilation operators to obtain q-analog BCS Bogoliubov formalism. Using this result, we have succeeded to extend so-called symplecton algebras.We have generalized a part of the above consideration to the case of U_q(n). We have found also very new kinds of q-extended Young diagrams (Schur functions).