Subfactors and Symmetry

摘要：该项目的主要目标是对子因子及其标准不变量的结构有更深入的了解。新的平面代数技术将通过相关平面代数的生成和关系方法来研究和构造子因子。这将建立在Bisch和Jones先前工作的基础上，其中给出了受一定自然维数条件约束的单生成平面代数的完整分类。例如，Bisch和Jones的fuss - catalalan代数，其表示已用于构建统计力学中的新的可积晶格模型，是该分类程序的重要组成部分。环状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。分析了子因子组成的障碍和子因子的刚度特性。子因子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。算子代数和非交换几何的思想在共形场论、弦论和量子信息理论中越来越重要。很明显，非交换空间和量子物理系统的对称性不能再仅仅通过经典数学对象（如群）来理解。子因子可以被视为捕获这些新对称性的数学对象，并且可以使用算子代数技术来分析和理解它们。在这种情况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从拓扑上操纵平面图来计算非常抽象的数学对象。这种形式是为统计力学的应用而量身定做的，基于这些思想的新的可解模型已经被构建出来了。算符代数理论的创始人之一是约翰·冯·诺伊曼他在20世纪30年代发现希尔伯特空间上的某些算符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基本定律中发挥了重要作用，如海森堡的不确定性原理，是冯·诺伊曼抽象理论的自然结果。基于冯·诺伊曼概念的子因子理论在数学和物理的多个领域产生了重要影响，在结理论、低维拓扑、统计力学和共形场论等领域都有深刻的应用。

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.