Operator Algebraic Approach to Quantum Groups

量子群的算子代数方法

基本信息

批准号：
12640216
负责人：
NAKAGAMI Yoshiomi
金额：
$ 2.3万
依托单位：
JAPAN WOMEN'S UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640216/
关键词：
operator algebra von Neumann algebra C^*-algebra quantum group locally compact quantum group duality multiplicative unitary Tomita-Takesaki theory Womonowicz環荷重付きHopf C^*環従順性二重群構成法 Quantum Group Woronowicz algebra C^*-algebr

项目摘要

More than 6 years has passed since we began to attack the problem of quantization of locally compact groups within the framework of C^*-algebras. While we could penetrate the global framework when we started our project and announced a part of our results under the name of "weighted Hopf C^*-algebra", we devoted enough time to solve technical problems occurring from the detail arguments and to pursuit the substantiality of the contents. During the time an analogous object as ours was announced by Kustermann and Vaes under the name of "locally compact quantum group" and we were obliged to play second fiddle to it However the originality of our arguments and the difference of outword looks of defining axioms made us convince that our results would be worth to be published and had to be continued. The most of our results are the C^*-versions of our previous results formulated in the framework of von Neumann algebras, under the name of "Woronowicz algebras", almost 10 years ago. These theory have a common serious problem that we must assume the existence of the Haar weight contrary to the classical case. Therefore our next main problem is to remove the existence of the Haar weights from our axioms.On the way to the completion for the fundamental theory of the weighted Hopf C^*-algebras, the analysis of the dual object of the quantum groups were left open as one of the problems to be solved. While no deep results in this area have not been obtained even in the classical case, this area seems to be a core part of a noncommutative theory. To clarify the part corresponding to the part known at least in the classical case, we consider first the amenability for the weighted Hopf C^*-algebra. As a results we find that there exist two candidate for the definition of amenability contrary to the classical case and mat the relationship between discrete quantum groups and nuclear C^*-algebras are different from the case of Kac algebras.

自从我们开始在 C^* 代数框架内解决局部紧群的量化问题以来，已经过去了 6 年多了。虽然当我们开始我们的项目并以“加权Hopf C^*-代数”的名义公布我们的部分成果时，我们可以渗透到全球框架中，但我们投入了足够的时间来解决细节论证中出现的技术问题并追求内容的实质性。在库斯特曼和维斯以“局部紧量子群”的名义宣布了与我们类似的物体期间，我们不得不对它进行次要的处理，但是我们论证的独创性和定义公理的外在外观的差异使得我们相信我们的结果值得发表并且必须继续下去。我们的大部分结果都是我们之前结果的 C^* 版本，是在大约 10 年前在冯诺依曼代数框架下以“Woronowicz 代数”的名义制定的。这些理论有一个共同的严重问题，即我们必须假设哈尔权重的存在，这与经典情况相反。因此我们的下一个主要问题是从我们的公理中去除Haar权的存在。在完成带权Hopf C^*-代数的基本理论的过程中，留下了量子群对偶对象的分析开放作为需要解决的问题之一。虽然即使在经典情况下也没有获得该领域的深入结果，但该领域似乎是非交换理论的核心部分。为了澄清与至少在经典情况下已知的部分相对应的部分，我们首先考虑加权 Hopf C^* 代数的适用性。结果我们发现存在两个与经典情况相反的顺从性定义的候选者，并且离散量子群和核C^*-代数之间的关系与Kac代数的情况不同。