Planar Algebras and the Structure of Subfactors

摘要：本课题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操作符上的代数——操作符由平面缠结的同位素类组成，这些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。在适当的正性条件下，平面代数等价于可分离Hilbert空间上Murray-von Neumann型II_1因子有限指数的子因子，这是Popa和我的一个重要定理。证明有解析的、代数的和拓扑的方面，需要更好地理解，但它的丰富的例子和潜在的新例子是目前令人兴奋的原因。连杆和三流形的量子不变量可以统一处理，对偶性也可以统一处理，对偶性允许人们在从Ising模型开始的各种统计力学模型中定位临界点。有限生成理论的van Kampen图自然来源于平面代数的考虑，Connes的循环范畴可以很简单地定义为某些环形缠结。Bisch和我发现了一种全新的平面代数叫做Fuss-Catalan代数通过利用子因子的联系。该代数已被用于构建统计力学中的“新”可解模型。使用这种方法的变体的新代数即将出现。这个研究项目的目的是开发和利用与“平面代数”相关的各种科学分支之间的联系。平面代数是一种基于图形的数学结构，使人联想到电子电路。所有的输入和输出都与不允许交叉的电线系统相连，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。三维空间中的结点可以投射到平面上，最终的图像由交叉组成——输入被非交叉的弦绑在一起。得到了分子生物学家在研究DNA分子时所使用的结点的多项式不变量理论。事实上，平面图类似于一些用于研究DNA重组的模型。在物理学家所谓的“反射正性”的假设下，这些平面代数与某些冯·诺伊曼代数的理论相当，这些代数本身就是冯·诺伊曼设计的希尔伯特空间中的算子集合，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.