Operator Algebras and Symmetry

摘要：SU(2)群和单李群的子群自一个世纪前被提出以来，几乎是各自独立的演化。在算子代数中，以前没有发现子因子的几何结构。作者发现了子因子、量子SU(2)的子群与经典李群和量子李群之间的自然联系，表明构建简单李群的信息自然来源于SU(2)的量子子群表示上的融合结构。这两个研究领域之间的桥梁是一种新的晶体学性质的同调理论，其中，例如六项精确序列对应于正六边形。这些方法产生了半单李群的量子包络代数的正则基的自然初等构造。量子子群和根格之间的联系超越了SU(2)和经典李群。倡议者在与一般量子子群相关的权重格中发现了新的单模根系，这些根不与任何已知结构相连。支持者认为发展与新根系相对应的简单李群的高级类似物是优先事项。这些可能本质上是具有自然多对一定律的新数学对象，它们在物理（3或4）维数的构造性QFT中有潜在的应用，而通常的二进制定律产生自然的二维场论。该项目围绕李群的量子子群的性质和表现的构建、分类和研究展开。当量子化参数为单位根时，半单李群的量子变形具有类似于经典李群的有限子群的子群。从算子代数非交换伽罗瓦理论中小指标子因子的代数结构分类开始，倡议者多年来一直在介绍这种结构。这些结构的其他表现形式出现在拓扑量子场论中，它们为3流形、共形场论和模不变量提供了数值不变量的边界扩展。SU(2)， SU(3)和SU(4)的量子子群现在由支持者分类，并表明量子世界非常不同，显然几乎与经典世界无关，异常量子子群的情况明显比相应的经典子群简单。该项目引入了与量子子群相关的几何结构，其中SU(2)的量子子群产生了单李群的根权和正则基，而其他量子子群则在权格中提出了本质上新的广义根系统。希望该项目产生的新结构能够在构建物理维度上的量子场论模型中发挥作用。

AbstractOcneanuSince their introduction a century ago, subgroups of SU(2) and simple Lie groups have evolved almost separately. In operator algebras no geometrical structure of subfactors has been previously found. The proponent has found the natural link between subfactors, the subgroups of quantum SU(2) and the classical and quantum Lie groups, showing that the information for building a simple Lie group comes naturally from the fusion structure on representations of a quantum subgroup of SU(2). The bridge between these two areas of research is a new crystallographic property of homology theory, wherein for instance six term exact sequences correspond to regular hexagons. These methods yield a natural elementary construction of a canonical basis of the quantum enveloping algebra of the semisimple Lie groups. The link found between quantum subgroups and root lattices extends beyond SU(2) and the classical Lie groups. The proponent found new unimodular root systems in weight lattices associated to general quantum subgroups, which are not connected to any known structures. The proponent considers the development of the higher analogs of simple Lie groups corresponding to the new root systems a priority. These are likely to be essentially new mathematical objects with natural many-to-one laws, which have potential applications in constructive QFT in a physical (3 or 4) number of dimensions, while the usual binary laws produce naturally 2-dimensional field theories.The project is centered around the construction, classification and study of the properties and manifestations of the quantum subgroups of Lie groups. The quantum deformations of the semisimple Lie groups have, when the quantization parameter is a root of unity, subgroups which are the analogs of the finite subgroups of the classical Lie groups. The proponent has introduced this structure over the years, starting with the classification of the algebraic structure of small index subfactors in the noncommutative Galois theory for operator algebras. Other manifestations of these structures appear in topological quantum field theory, where they provide boundary extensions of numerical invariants for 3-manifolds , conformal field theory, and modular invariants. The quantum subgroups of SU(2), SU(3) and SU(4) are now classified by the proponent, and show that the quantum world is very different and apparently nearly unrelated to the classical world, with a markedly simpler situation for the exceptional quantum subgroups than for the corresponding classical subgroups. The project introduces geometrical structures associated to quantum subgroups, with the quantum subgroups of SU(2) producing the roots weights and canonical bases for the simple Lie groups, while the other quantum subgroups give raise to essentially new generalized root systems in weight lattices. It is hoped that the new structures produced by the project could play a role in constructing models of quantum field theory in a physical number of dimensions.