Planar Algebras and the Structure of Subfactors

这个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。平面代数最好被定义为平面算子上的代数，平面算子由平面缠结的保序类组成，平面缠结是由称为弦的光滑曲线连接的大圆盘中不相交圆盘的集合。Popa和我的一个值得注意的定理是，在适当的正性条件下，平面代数等价于可分Hilbert空间上的Murray-von Neumann型II_1因子的有限指数的子因子。证明包括分析、代数和拓扑学方面，需要更好地理解，但目前令人兴奋的原因是大量的例子和潜在的新例子。链节和三个流形的量子不变量允许统一处理，对偶性也是如此，它允许人们从伊辛模型开始在各种统计力学模型中定位临界点。有限生成理论的范坎本图自然源于平面代数的考虑，而Connes的循环范畴可以非常简单地定义为某些环形纠缠。比希和我利用与子因子的联系，发现了一种全新的平面代数，称为Fuss-Catalan代数。这个代数被用来构造统计力学中的一个“新的”可解模型。使用这种方法变体的新代数即将问世。这项研究项目的目的是发展和利用与被称为“平面代数”的对象相关的不同科学分支之间的联系。平面代数是一种基于使人联想到电子电路的图表的数学结构。输入和输出都与一个不允许交叉的导线系统相连，因此可以雕刻在表面上。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，结果图像由不交叉的线连接在一起的交叉点-输入-组成。作为输出，我们得到了分子生物学家在DNA分子研究中所使用的纽结多项式不变量理论。事实上，这些平面图类似于一些用于研究DNA重组的模型。非常不寻常的是，这些平面代数在物理学家所称的“反射正性”的假设下，等价于某些冯·诺伊曼代数的理论，这些代数本身就是冯·诺伊曼在希尔伯特空间中设计的用于量子力学的算符的集合。

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.