Subfactors and Quantum Double

子因子和量子双精度

• 批准号: 0202613
• 负责人: Marta Asaeda
• 金额: $ 7.5万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-15 至 2004-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202613&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.

Orbifold子因子首先由Y. Kawahigashi作为副群理论的应用。 已知D_4n型和D_4n+2型的子因子可产生性质不同的双模融合代数，但这种差异的起因尚不清楚。 Asaeda一直在研究这个问题，并获得了阐明轨道子因子融合结构的线索。在这一研究中，经典李群表示理论与量子群表示理论（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，以及澄清WZW模型的轨道折叠，经典李群表示的结构和WZW模型产生的子因子的性质之间的关系。另一方面，她的研究延伸到子因子的研究，这些子因子不是从量子群或经典群中产生的。这些子因子称为外来子因子。她的重点是从奇异的子因子构建的量子双结构。算子代数理论是冯·诺依曼作为量子力学的数学背景而创立的。近年来，人们从与冯·诺依曼完全不同的角度来探讨算符代数、量子物理和低维拓扑之间的关系。对这一问题及其与其他学科关系的研究，将有助于数学和数学物理等多门学科的发展。