Mathematical Sciences: Quantum Groups and C*-Algebras

9623008 PI：堪萨斯州阿尔伯特·休大学将继续他对C*-代数与几何之间相互作用的研究，重点是量子群的群态C*-代数方法和形变量子化。所研究的课题包括量子群作为群类C*-代数的实现，具体Poisson李群的保叶和保群形变量化，全纯分层和形变量子化，以及群C*-代数的抵消问题。在这项研究中使用的主要工具将来自不同的数学分支，如C*-代数、李群、微分几何和伪微分算子。这项研究的动机是最近全世界对几何学的非对易方面的兴趣，这一方面源于量子化的概念，量子化将数学和量子物理的几个分支联系在一起。在物理科学中，研究量在各种对称群作用下的不变性是很重要的。量子群理论推广了经典的对称群理论，为量子物理的研究提供了新的、有前途的方向。通过以一种新的方式研究量子群的结构并探索相关问题，提出者希望有助于理解这一相对较新和令人兴奋的数学领域。

DMS-9623008 PI: Albert Sheu University of Kansas The proposer will continue his research on the interaction between C*-algebras and geometry, with emphasis on the groupoid C*-algebra approach to quantum groups and deformation quantization. Topics to be studied include the realization of quantum groups as groupoid C*-algebras, leaf-preserving and group-preserving deformation quantizations of concrete Poisson Lie groups, holomorphic foliation and deformation quantization, and the cancellation problem for group C*-algebras. The main tools to be used in this research will be from various mathematical branches like C*-algebras, Lie groups, differential geometry, and pseudodifferential operators. This research is motivated by the recent worldwide interest in the noncommutative aspect of geometry originated from the concept of quantization which links together several branches of mathematics and quantum physics. In the physical sciences, it is important to study the invariance of quantities under various symmetry group actions. The theory of quantum groups generalizes the classical theory of symmetry groups and provides new and promising directions in the study of quantum physics. By studying the structure of quantum groups in a new way and by exploring related questions, the proposer hopes to contribute to the understanding of this relatively new and exciting area of mathematics.