Operator Algebras and Quantum Symmetry

自世纪前引入以来，SU（2）群和单李群的子群几乎分别演化。在算子代数中，子因子的几何结构以前没有被发现，而提出者发现子因子、量子SU（2）的子群与经典李群和量子李群之间的自然联系，表明构造简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个新的晶体学性质，其中例如六项精确序列对应于正六边形。 这些方法产生一个自然的基本建设的规范基础的量子包络代数的半单李群。 量子子群和根格之间的联系超出了SU（2）和经典李群。提出者在与一般量子子群相关联的权重格中发现了新的幺模根系统，这些量子子群与任何已知结构都没有联系。提出者认为发展与新的根系相对应的简单李群的高级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新数学对象，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造、分类和研究。当量子化参数是单位根时，半单李群的量子变形具有与经典李群的有限子群类似的子群。该结构的提出者多年来一直在引入这种结构，从算子代数的非交换伽罗瓦理论中的小指标子因子的代数结构的分类开始。这些结构的其他表现形式出现在拓扑量子场论中，它们提供了三维流形、共形场论和模不变量的数值不变量的边界扩展。SU（2）、SU（3）和SU（4）的量子子群现在被提出者分类，表明量子世界非常不同，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简单。该项目引入了与量子子群相关的几何结构，SU（2）的量子子群产生简单李群的根权和正则基，而其他量子子群在权格中产生了本质上新的广义根系统。 希望该项目产生的新结构可以在构建物理数量维度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.