Mathematical Sciences: Analytical and Combinatorial Aspects of Subfactors

9531566比希这是一个算符代数理论的数学研究项目，特别涉及琼斯次因子论的分析和组合方面。新的有限维代数族是Temperley-Lieb代数的自然推广，将被用来研究子因子的结构。例如，这些代数捕捉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种极小乘积。在这个过程中自然而然地出现了各种新的有趣的组合结构。通过中间子因子研究子因子的技术是非常自然的，因为它是在经典水平上(即，群水平)，只是通过它的子群对给定的群进行分析。它计划通过研究与这些子因子相关的有限维代数来扩展和提炼可用的中间子因子技术。POPA系统是有限维对称代数的自然包含系统，它将在这一分析中起到关键作用。我们将研究这些对称性的计算方面，并计划构建和分析子因子和通勤平方的显式例子。我们将研究亚因子论与数学物理和低维拓扑学之间可能存在的新的相互关系。约翰·冯·诺伊曼引入了他在希尔伯特空间上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。80年代初，S提出了某些von Neumann代数的包含的伽罗瓦理论，并发现这些包含是非常严格的。它们具有令人惊讶的辫子群对称性，这导致琼斯发现了他著名的纽结不变量--琼斯多项式。除了亚因子论与纽结理论和低维拓扑学的这些深刻联系外，还与统计力学、保形场论和代数量子场论有着深刻的联系，这些联系有助于理解这些领域中自然产生的结构。***

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. ***