Noncommutative Geometry for Quantum Groups

量子群的非交换几何

基本信息

批准号：
09640210
负责人：
NAKAGAMI Yoshiomi
金额：
$ 1.92万
依托单位：
Yokohama City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640210/
关键词：
quantum group von Neumann algebra Kac algebra Woronowicz algebra Hopf algebra Lorentz group C^* -algebra Lorents群 von Neumarsu環 Hopf環 Lorentz群

项目摘要

In 1997 : Let's consider the quantum Lorentz group as an example of non-compact quantum groups. Since the Lie algebra sl(2, C) is identified with the complexification of the Lie algebra su(2), the quatum Lorentz group SL_q(2, C) is considered to be the quantum double of the quantum group SL_q(2). We apply this idea to obtain a non-compact Woronowicz algebras corresponding to the quantum Lorentz group. Using the analysis on Woronowicz algebra, we obtain fundamental relations which hold for generators of the quantum enveloping algebra U_q(sl(2, C)). We also find the natural pairing between the quantum group and the quantum enveloping algebra separating to each other. The definition of the quantum enveloping algebra is found to be isomorphic to the definition obtained from the regular functions in the dual space of the coordinate ring for the quantum Lorentz group.In 1998 : Since the Woronowicz algebra is described in the framework of von Neumann algebras, the action in general is not continuous but measurable, we encounter some troubles in some applications such as non-commutative geometry and so forth. For such reasons we are obliged to define it in the framework of C^*-algebras. While we have known the general operator a1gebraic structure for the quantum groups through the study of Woronowicz algebras, we must solve several non trivial technical difficulties when we go into the argument using the C^*-algebras. More than 5 years has passed since we began to consider such problems together with Woronowicz and Masuda. Now, we can succeed to solve all the mathematical difficulties. In the last few years we spend times to simplify or to revise the manuscript repeatedly from several respects.

1997 年：让我们考虑量子洛伦兹群作为非紧量子群的一个例子。由于李代数 sl(2, C) 等同于李代数 su(2) 的复化，因此量子洛伦兹群 SL_q(2, C) 被认为是量子群 SL_q(2) 的量子双倍。我们应用这个想法来获得对应于量子洛伦兹群的非紧沃罗诺维奇代数。通过对 Woronowicz 代数的分析，我们得到了量子包络代数 U_q(sl(2, C)) 的生成元成立的基本关系。我们还发现量子群和量子包络代数之间的自然配对彼此分离。量子包络代数的定义被发现与从量子洛伦兹群坐标环对偶空间中的正则函数获得的定义同构。 1998年：自从沃罗诺维奇代数在冯·诺依曼代数的框架中描述以来，一般来说，作用不是连续的，而是可测量的，我们在某些应用中遇到一些麻烦，例如非交换几何等。由于这些原因，我们有义务在 C^* 代数的框架中定义它。虽然我们通过对 Woronowicz 代数的研究已经知道了量子群的一般算子代数结构，但当我们使用 C^* 代数进行论证时，我们必须解决几个重要的技术难题。自从我们开始与沃罗诺维奇和增田一起考虑这些问题以来，已经过去了五年多了。现在，我们可以成功解决所有数学难题了。几年来，我们花时间从几个方面反复简化或修改稿件。