Subfactors and Symmetry

这个项目的主要目标是更深入地了解子因子及其标准不变量的结构。新的平面代数技术将被用来调查和构造子因子，通过一个发电机和关系的方法相关联的平面代数。这将建立在以前的工作Bisch和琼斯，其中一个完整的分类单生成平面代数受到一定的自然维数条件。例如，Bisch和Jones的Fuss-Catalan代数，其表示已被用于在统计力学中构建新的可积格模型，似乎是这个分类程序的重要组成部分。 环形Fuss-Catalan代数将被研究，最终目标是用它们来回答与中间子因子相关的问题。 分析了子因子构成的障碍和子因子的刚性特性。子因子的思想和技术在统计力学和量子信息理论中的潜在应用将被研究。来自算子代数和非交换几何的思想在共形场论、弦理论和量子信息理论中越来越重要。很明显，非对易空间和量子物理系统的对称性不再能够仅仅通过经典数学对象（例如群）来理解。 子因子可以被看作是一个数学对象，它捕捉了这些新的对称性，算子代数技术可以用来分析和理解它们。在这种背景下自然出现的一个特别有趣的结构是平面代数，这种结构允许人们通过简单地操纵平面图拓扑来计算非常抽象的数学对象。这样的形式主义是量身定制的统计力学和新的可解模型的基础上，这些想法已经构建的应用。算子代数理论的创始人之一是约翰·冯·诺依曼，他在20世纪30年代发现希尔伯特空间上的某些算子代数是理解量子物理系统对称性的自然框架。他的思想在量子力学和自然的基本定律中发挥了重要作用，例如海森堡不确定性原理，这是冯·诺依曼的抽象理论的自然结果。理论的子因子，这是根据冯诺依曼的概念，产生了重要的影响，在几个领域的数学和物理的深刻应用结理论，低维拓扑，统计力学和共形场理论仅举几例。

Abstract BischThe main goal of this project is to gain a deeper understanding of the structure of subfactors and their standard invariants. The novel planar algebra techniques will be used to investigate and construct subfactors through a generators and relations approach to the associated planar algebras. This will build on prior work of Bisch and Jones in which a complete classification of singly generated planar algebras subject to a certain natural dimension condition was given. For instance, the Fuss-Catalan algebras of Bisch and Jones, whose representations have been used to construct new integrable lattice models in statistical mechanics, appear as an important part of this classification program. Annular Fuss-Catalan algebras will be investigated with the ultimate goal of using them to answer questions related to intermediate subfactors. Obstructions for compositions of subfactors and rigidity properties of subfactors will be analyzed. Potential applications of subfactor ideas and techniques to statistical mechanics and quantum information theory will be investigated.Ideas from operator algebras and non-commutative geometry have gained increasing importance in conformal field theory, string theory and quantum information theory. It has become clear that the symmetries of non-commutative spaces and quantum physical systems can no longer be understood through classical mathematical objects alone (such as groups). A subfactor can be viewed as a mathematical object that captures these new symmetries and operator algebra techniques can be used to analyze and understand them. A particular intriguing structure that appears naturally in this context are the planar algebras, a structure that allows one to compute with very abstract mathematical objects by simply manipulating planar diagrams topologically. Such a formalism is tailor-made for applications in statistical mechanics and new solvable models based on these ideas have already been constructed. One of the founders of the theory of operator algebras was John von Neumann who in the 1930's discovered that certain algebras of operators on a Hilbert space are the natural framework for understanding symmetries of quantum physical systems. His ideas play an important role in quantum mechanics and fundamental laws of nature such as the Heisenberg uncertainty principle appear as a natural consequence of von Neumann's abstract theory. The theory of subfactors, which is based on von Neumann's concepts, had an important impact across several fields in mathematics and physics with profound applications to knot theory, low dimensional topology, statistical mechanics and conformal field theory to mention just a few.