Mathematical Sciences: Analytical and Combinatorial Aspects of Subfactors

小行星9531566 这是一个数学研究项目， 算子代数，特别是与分析和组合方面的琼斯理论的子因子。 新的有限维代数的层次，推广的Temperley-Lieb代数在一个自然的方式，将被用来研究子因子的结构。 这些代数捕捉，例如，来自中间子因子的对称性，可以被视为与两个子因子相关的标准不变量的某个最小乘积。 在这一过程中，各种新颖有趣的组合结构自然出现。 通过其中间子因子来研究子因子的技术是非常自然的，因为它是在经典水平上的（即，组级）只是通过其子组对给定组进行分析。 计划通过研究与这些子因子相关的有限维代数，扩展和改进现有的中间子因子技术。 Popa系统，有限维对称代数的包含体的某种自然系统，将在这一分析中发挥关键作用。 这些对称性的计算方面将进行检查，并计划构建和分析明确的例子，子因子和通勤广场。 可能的新的相互关系的子因子理论与数学物理和低维拓扑将进行调查。 约翰·冯·诺依曼在希尔伯特空间上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，沃恩·琼斯提出了一个关于某些冯·诺依曼代数的包含体的伽罗瓦理论，并发现这些包含体是非常刚性的。 他们原来有一个令人惊讶的辫子群对称性，这导致琼斯发现他著名的结不变量，琼斯多项式。 除了子因子理论与纽结理论和低维拓扑学的这些深刻联系之外，子因子理论与统计力学、共形场论和代数量子场论也有着深刻的关系，这些关系有助于理解这些领域中自然发生的结构。 ***

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. ***