Subfactors, bimodules and planar algebras

子因子、双模和平面代数

• 批准号: 1001560
• 负责人: Dietmar Bisch
• 金额: $ 17.6万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-10-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001560&HistoricalAwards=false
• 关键词: Subfactors; bimodules; planar; algebras

This project on subfactors and planar algebras aims at developing a wide range of techniques to better understand the structure of subfactors and discover new examples. One of the main goals is to understand the technique of composing subfactors in terms of their associated planar algebras. This will lead to new methods of constructing subfactors and planar algebras, in particular new examples with infinite principal graphs which are essential for progress in the structure theory. Moreover, every such subfactor will provide a potentially very interesting fusion category of bimodules, which can be computed explicitly. As a first step, those compositions will be studied that arise as planar subalgebras of the tensor product of two Temperley-Lieb planar algebras. For more general compositions, a cohomology theory for subfactors and planar algebras has to be developed which will replace group cohomology. A very rich theory is likely to emerge from these investigations. In particular, it is expected that new classification results for subfactors with Jones index two times the golden ratio squared will be obtained.A subfactor can be viewed as a mathematical object which captures quantum symmetries of a mathematical or quantum physical system. These symmetries play a key role in understanding the behavior of these complex systems, and the theory of subfactors provides effective tools to manipulate and study them. This theory has had many profound and surprising applications to numerous areas of mathematics and physics, such as conformal field theory, statistical mechanics, low dimensional topology and combinatorics. Subfactors have contributed in an important way to the understanding of naturally occurring structures in these a priori quite distinct areas of mathematics and physics. It is expected that the project will make important contributions to some of these areas of basic science. Exciting applications of planar algebras and their associated fusion categories to solid state physics and to topological quantum computing are possible. Furthermore, the project will involve graduate students and postdoctoral researchers and contribute to their training as researchers in mathematics.

这个关于子因子和平面代数的项目旨在开发广泛的技术，以更好地理解子因子的结构并发现新的例子。其中一个主要目标是了解的技术组成的子因子在其相关的平面代数。这将导致新的方法来构建子因子和平面代数，特别是新的例子与无限的主图是必不可少的进展，在结构理论。此外，每个这样的子因子将提供一个潜在的非常有趣的双模融合范畴，它可以显式计算。作为第一步，这些成分将被研究所产生的张量积的两个Temperley-Lieb平面代数的平面子代数。对于更一般的组成，一个上同调理论的子因子和平面代数已制定这将取代群上同调。从这些研究中可能会产生一个非常丰富的理论。特别是，对于琼斯指数为黄金分割率平方的2倍的子因子，可以得到新的分类结果。子因子可以被看作是一个数学对象，它捕获了数学或量子物理系统的量子对称性。这些对称性在理解这些复杂系统的行为中起着关键作用，子因子理论为操纵和研究它们提供了有效的工具。这个理论在数学和物理的许多领域都有着深刻而令人惊讶的应用，如共形场论、统计力学、低维拓扑学和组合学。子因子以一种重要的方式对理解数学和物理学中这些先验的、相当不同的领域中自然发生的结构做出了贡献。预计该项目将对其中一些基础科学领域作出重要贡献。平面代数及其相关的融合范畴在固态物理和拓扑量子计算中有着令人兴奋的应用。此外，该项目将涉及研究生和博士后研究人员，并有助于他们作为数学研究人员的培训。