Operator Algebras and Symmetry

摘要自一个世纪前提出以来，SU(2)的子群和单李群几乎是分开演化的。在算子代数中，以前没有发现子因子的几何结构。支持者发现了子因子、量子SU(2)的子群与经典李群和量子李群之间的自然联系，表明构造单李群的信息自然来自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同调理论的一种新的结晶学性质，例如，六项精确序列对应于正六边形。这些方法给出了半单李群的量子包络代数的标准基的自然初等构造。量子子群和根格之间的联系超出了SU(2)和经典李群。支持者在与一般量子子群相关的权格中发现了新的么模根系统，而一般量子子群并不与任何已知结构相连。支持者认为，发展与新的根系相对应的单李群的更高类似物是一个优先事项。这些很可能是具有自然多对一定律的本质上的新的数学对象，它们在物理(3或4)维的构造性QFT中具有潜在的应用，而通常的二元律产生自然的二维场理论。该项目围绕李群的量子子群的性质和表现的构造、分类和研究。当量子化参数为单位根时，半单李群的量子形变具有类似于经典李群的有限子群的子群。从算子代数的非对易伽罗瓦理论中小指标子因子的代数结构的分类开始，提出者多年来已经引入了这种结构。这些结构的其他表现形式出现在拓扑量子场理论中，其中它们提供了三维流形、保形场论和模不变量的边界扩展。SU(2)、SU(3)和SU(4)的量子子群现在被支持者分类，表明量子世界是非常不同的，显然与经典世界几乎无关，例外量子子群的情况明显比相应的经典子群的情况简单。这个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标准基，而其他量子子群则给出了本质上新的加权格上的广义根系统。人们希望，该项目产生的新结构可以在构建多个物理维度的量子场论模型方面发挥作用。

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.