Subfactors and Quantum Double

子因子和量子双精度

• 批准号: 0411628
• 负责人: Marta Asaeda
• 金额: $ 3.34万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-11-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0411628&HistoricalAwards=false
• 关键词: Subfactors; Quantum; Double

Orbifold subfactors have been first described by Y. Kawahigashi as an application of paragroup theory. It is known that subfactors of type D_4n and D_4n+2 give rise to fusion algebras of bimodules with different natures, but the origin of the difference is unknown. Asaeda has been working on this problem, and obtained the clue to clarify the fusion structure of orbifold subfactors. On this research, the relation between the representation theory of classical Lie group and that of quantum group (WZW model) plays a crucial role. Her research includes further research on this matter, and clarification of the relation between orbifold of WZW model, structure of the representation of the classical Lie groups, and the nature of subfactors arising from WZW model. On the other hand, her research extends to the study of subfactors which are not arising from quantum groups or classical groups. These subfactors are called exotic subfactors. Her focus is on the structure of quantum double constructed from exotic subfactors. The theory of operator algebras was founded by von Neumann as a mathematical background of quantum mechanics. Lately the relation among operator algebras, quantum physics, and low dimensional topology has been sought from the aspect that is totally different from that of von Neumann. The research on this subject and it's relation with other subjects will contribute the development of many subjects in mathematics and mathematical physics.

作为共群理论的一种应用，Y.Kawahigashi首先描述了Orbiold子因子。众所周知，D_4n和D_4n+2型的子因子导致性质不同的双模的融合代数，但这种差异的起源尚不清楚。早稻田一直致力于这一问题，并获得了阐明奥比福德亚因子融合结构的线索。在这项研究中，经典李群的表示理论和量子群的表示理论(WZW模型)之间的关系起着至关重要的作用。她的研究包括对这一问题的进一步研究，并澄清了WZW模型的奥比福尔德、经典李群的表示的结构和WZW模型产生的子因子的性质之间的关系。另一方面，她的研究扩展到不是来自量子群或经典群的子因子的研究。这些次因素被称为异域次因素。她的重点是由奇异的子因子构成的量子双重结构。作为量子力学的数学背景，冯·诺伊曼创立了算符代数理论。最近，人们从与von Neumann完全不同的角度来寻找算子代数、量子物理和低维拓扑之间的关系。对该学科及其与其他学科关系的研究，将有助于数学和数学物理多学科的发展。