Planar Algebras and the Structure of Subfactors

平面代数和子因子的结构

• 批准号: 9970511
• 负责人: Vaughan Jones
• 金额: $ 44.76万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970511&HistoricalAwards=false
• 关键词: Planar; Algebras; Structure; Subfactors

AbstractJones The main aim of this project is to explore the rich structure of planar algebras and their relations with von Neumann algebras. Planar algebras are best defined as algebras over the planar operad-the operad consisting of isotopy classes of planar tangles which are collections of disjoint discs inside a large disc connected up by smooth curves called strings. It is a remarkable theorem of Popa and myself that, together with suitable positivity conditions, a planar algebra is equivalent to a subfactor of finite index of a Murray-von Neumann type II_1 factor on a separable Hilbert space. The proof has analytic, algebraic and topological aspects and needs to be better understood but it is the wealth of examples and potential new examples that is the current reason for excitement. The quanum invariants of links and three manifolds admit a unified treatment as does the duality which allows one to locate critical points in various statistical mechanical models starting with the Ising model. The van Kampen diagrams of the theory of finitely generated come naturally from planar algebra considerations and Connes' cyclic category can be defined very simply as certain annular tangles. Bisch and myself discovered an entirely new kind of planar algebra called the Fuss-Catalan algebra by exploiting the connection with subfactors. This algebra has been used to construct a "new" solvable model in statistical mechanics. New algebras using variations of this method are just around the corner. The aim of this research project is to develop and exploit connections between various branches of science related to objects called "planar algebras". A planar algebra is a mathematical structure based on diagrams reminiscent of electronic circuits. There are inputs and outputs all connected with a system of wires that are not allowed to cross and so could be engraved on a surface. Models of statistical mechanics in two dimensions give examples of planar algebras and vice versa. Knots in three dimensional space can be projected onto the plane and the resulting picture consists of crossings - the inputs-tied together by non crossing strings. One obtains as output the theory of polynomial invariants of knots which have been used by molecular biologists in the study of DNA molecules. Indeed the planar diagrams resemble some of the models used to study DNA recombination. It is quite extraordinary that these planar algebras are, under the assumption of what physicists call "reflection positivity", equivalent to a theory of certain von Neumann algebras which themselves are collections of operators in Hilbert space devised by von Neumann to be used in quantum mechanics.

AbstractJones该项目的主要目的是探索平面代数的丰富结构及其与von Neumann代数的关系。平面代数最好定义为平面作战的代数 - 由平面缠结的同位素类别组成，它们是通过称为字符串的平滑曲线连接的大型圆盘内部的脱节盘的集合。 Popa和我本人的一个了不起的定理，与合适的阳性条件一起，平面代数等于在可分开的Hilbert空间上的Murray-Von Neumann II_1因子的有限指数的亚比例。该证明具有分析性，代数和拓扑方面，需要更好地理解，但是当前兴奋的原因是大量的例子和潜在的新例子。链接和三个流形的Quanum不变性允许统一处理，二元性也使人们可以在以ISING模型开头的各种统计机械模型中定位关键点。有限生成的理论的Van Kampen图自然来自平面代数考虑因素，Connes的环状类别可以非常简单地定义为某些环形缠结。 Bisch和我本人通过利用与子因子的联系来发现一种全新的平面代数，称为“大惊小怪”代数。该代数已用于构建统计力学中的“新”可解决的模型。使用此方法的变体的新代数临近。 该研究项目的目的是开发和利用与称为“平面代数”对象的各个科学分支之间的联系。平面代数是基于让电子电路的图表的数学结构。有一些输入和输出都与不允许交叉的电线系统连接，因此可以刻在表面上。 二维中的统计力学模型给出了平面代数的示例，反之亦然。在三维空间中的结中可以投射到平面上，由此产生的图片由交叉组成 - 通过非交叉字符串绑在一起的输入。一个人以输出结的多项式不变式理论，这些理论是由分子生物学家在DNA分子研究中使用的。实际上，平面图类似于用于研究DNA重组的一些模型。在物理学家所说的“反射阳性”的假设下，这些平面代数是非常非凡的，相当于某些von Neumann代数的理论，这些理论本身就是冯·诺伊曼（Von Neumann）设计的希尔伯特（Hilbert）空间中运营商的集合，用于量子力学。