Mathematical Sciences: Analytical and Combinatorial Aspects of Subfactors

数学科学：子因子的分析和组合方面

• 批准号: 9531566
• 负责人: Dietmar Bisch
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531566&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Analytical; Combinatorial; Aspects

9531566 Bisch This is a project for mathematical research in the theory of operator algebras that deals, in particular, with analytical and combinatorial aspects of the Jones theory of subfactors. New hierarchies of finite-dimensional algebras, generalizing the Temperley-Lieb algebras in a natural way, will be used to study the structure of subfactors. These algebras capture, for instance, the symmetry coming from an intermediate subfactor and can be viewed as a certain minimal product of the standard invariant associated to two subfactors. A variety of new and interesting combinatorial structures appears naturally in this process. The technique of studying a subfactor through its intermediate subfactors is very natural, since it is on the classical level (i.e., the group level) just the analysis of a given group through its subgroups. It is planned to extend and refine the available intermediate subfactors techniques by studying the finite-dimensional algebras canonically associated to these subfactors. Popa systems, a certain natural system of inclusions of finite-dimensional algebras of symmetries, will play a key role in this analysis. Computational aspects of these symmetries will be examined, and it is planned to construct and analyze explicit examples of subfactors and commuting squares. Possible new interrelations of the theory of subfactors with mathematical physics and low-dimensional topology will be investigated. John von Neumann introduced his algebras of operators on a Hilbert space to study a variety of mathematical structures, some of which arise naturally in theoretical physics. In the early 80's, Vaughan Jones initiated a Galois theory for inclusions of certain von Neumann algebras and discovered that these inclusions are extremely rigid. They turn out to have a surprising braid group symmetry, which led Jones to the discovery of his famous knot invariant, the Jones polynomial. Besides these profound connections of the theory of subfactors to knot theory and low-dimensional topology, there are also deep relations to statistical mechanics, conformal field theory, and algebraic quantum field theory that have been beneficial to the understanding of naturally occurring structures in these areas. ***

这是一个关于算子代数理论的数学研究项目，主要研究琼斯子因子理论的分析和组合方面。有限维代数的新层次，以一种自然的方式推广了Temperley-Lieb代数，将用于研究子因子的结构。例如，这些代数捕获了来自中间子因子的对称性，并且可以被视为与两个子因子相关的标准不变量的某个最小积。在这个过程中，各种新颖有趣的组合结构自然出现。通过中间子因子研究子因子的技术是非常自然的，因为它是在经典层面（即群层面）上，只是通过给定的群的子群来分析给定的群。计划通过研究通常与这些子因子相关的有限维代数来扩展和完善现有的中间子因子技术。Popa系统是有限维对称代数的包含体的一种自然系统，它将在这一分析中发挥关键作用。这些对称性的计算方面将被检查，并计划构造和分析子因子和交换平方的显式例子。子因子理论与数学物理和低维拓扑之间可能的新相互关系将被研究。约翰·冯·诺伊曼在希尔伯特空间中引入了他的算子代数来研究各种数学结构，其中一些是在理论物理中自然产生的。在80年代早期，Vaughan Jones为某些von Neumann代数的包涵体提出了伽罗瓦理论，并发现这些包涵体是非常刚性的。它们具有令人惊讶的辫群对称性，这使得琼斯发现了他著名的结不变量——琼斯多项式。除了这些子因子理论与结理论和低维拓扑的深刻联系之外，与统计力学、共形场论和代数量子场论也有深刻的关系，这些关系有助于理解这些领域中自然发生的结构。* * *