Planar Algebras and the Structure of Subfactors

本项目的主要目的是探索平面代数的丰富结构及其与冯诺依曼代数的关系。平面代数最好定义为平面运算上的代数，平面运算由平面缠结的合痕类组成，平面缠结是由称为弦的光滑曲线连接起来的大圆盘内的不相交圆盘的集合。Popa和我的一个著名定理是：在适当的正性条件下，平面代数等价于可分Hilbert空间上Murray-von Neumann型II_1因子的有限指标子因子.证明有分析，代数和拓扑方面，需要更好地理解，但它是丰富的例子和潜在的新例子，这是目前令人兴奋的原因。量子不变量的链接和三个流形承认一个统一的治疗一样的对偶性，它允许一个定位临界点在各种统计力学模型开始的伊辛模型。生成的理论的货车坎彭图自然来自平面代数的考虑和Connes的循环范畴可以被定义为非常简单的某些环形缠结。比施和我发现了一种全新的平面代数称为Fuss-Catalan代数利用与子因子的联系。这个代数已经被用来构造统计力学中的一个“新”可解模型。使用这种方法的变体的新代数即将出现。 该研究项目的目的是开发和利用与称为“平面代数”的对象相关的各种科学分支之间的联系。平面代数是一种基于图的数学结构，让人联想到电子电路。有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。 二维统计力学模型给出了平面代数的例子，反之亦然。三维空间中的节点可以投影到平面上，结果图像由交叉点组成-输入由非交叉线连接在一起。人们得到的输出理论的多项式不变量的结已被使用的分子生物学家在研究DNA分子。事实上，平面图类似于用于研究DNA重组的一些模型。这是相当不寻常的是，这些平面代数，假设下什么物理学家所谓的“反射积极性”，相当于理论的某些冯诺依曼代数本身是集合的运营商在希尔伯特空间设计的冯诺依曼被用于量子力学。

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.