A study of quantum groups by using the theory of unbounded operator algebras

利用无界算子代数理论研究量子群

基本信息

批准号：
10640222
负责人：
KUROSE Hideki
金额：
$ 1.79万
依托单位：
Fukuoka Universityq
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640222/
关键词：
quantum group Hopf algebra Hopf *-algebra locally compact quantum group quantum double double group construction Woronowicz unbound operator algebra unbound operator algebra ホップ*-代数 Doubule Group Construction 局所インパクト量子群

项目摘要

For Hopf (*-) algebras A and B, using an exchanging map S : B 【cross product】 A → A 【cross product】 B we defined a twisted tensor product of those. We showed that twisted tensor product is not but the bicrossed product of a matched pair of the Hopf (*-) algebras A and B and that the quantum double is a particular example of twisted tensor products. For any coquasitriangular Hopf (*-) algebra A, which is not necessarily finite dimensional, we defined a (*-) algebra homomorphism from quantum double of A and itself into the tensor product of A and its quantum enveloping algebra and showed that it is an isomorphism if A is factorizable. This is a generalization of a result by S. Majid for finite dimensional Hopf algebras. Passing through the above homomorphism, for any coquasitriangular Hopf (*-) algebra A, we prove that the quantum double of A and itself is paired with the quantum enveloping algebra of A and the opposite Hopf (*-) algebra of A and that the pairing nondegenerate iff A is factorizable. The result implies that the quantum enveloping algebra of the complex quantum group of a compact quantum Lie group A corresponds to the quantum double of the quantum enveloping algebra of A and the opposite Hopf algebra of A. The quantum Lorentz algebra is an example. We discussed about the modular theory of compact quantum groups at the level of Hopf *- algebra and defined the notion of algebraic Woronowicz algebras. Taking the dual we could think of the modular theory for discrete quantum groups as multiplier Hopf algebras. The modular structure is further defined for the quantum groups which are described as the quantum double or the double group of compact or discrete quantum groups. In the regular bounded representations of algebraic Woronowicz algebras, we got the corresponding Woronowicz algebras as those weak closures.

对于 Hopf (*-) 代数 A 和 B，使用交换映射 S : B 【叉积】 A → A 【叉积】 B 我们定义了它们的扭曲张量积我们证明扭曲张量积不是双叉的。一对匹配的 Hopf (*-) 代数 A 和 B 的乘积，并且量子双精度数是扭曲张量积的一个特定示例对于任何共拟三角 Hopf (*-) 代数 A，其中不一定是有限维的，我们将 A 及其自身的量子双精度定义为 A 及其量子包络代数的张量积的 (*-) 代数同态，并证明如果 A 可因式分解，则它是同构。这是的推广。 S. Majid 对有限维 Hopf 代数的结果通过上述同态，对于任何共拟三角 Hopf (*-) 代数 A，我们证明 A 及其自身的量子双精度数与 A 的量子包络代数和 A 的相反 Hopf (*-) 代数配对，并且配对非简并当且仅当 A 可因式分解。结果意味着复数的量子包络代数。紧量子李群 A 的量子群对应于 A 的量子包络代数和 A 的相反 Hopf 代数的量子对偶。量子洛伦兹代数就是一个例子。我们在Hopf*-代数的层面上讨论了紧量子群的模理论，并定义了代数Woronowicz代数的概念，我们可以将离散量子群的模理论视为乘子Hopf代数。模结构被进一步定义为量子群，在代数的正则有界表示中被描述为量子双群或紧致或离散量子群的双群。 Woronowicz 代数，我们得到了相应的 Woronowicz 代数作为那些弱闭包。