论文标题
来自离散子因子的紧凑型超组
Compact Hypergroups from Discrete Subfactors
论文作者
论文摘要
冯·诺伊曼（Von Neumann）代数环境中描述了手性结合田间理论的共形夹杂物，或更常见的量子场理论夹杂物，由亚因子的网络纳入von Neumann代数设置，如果考虑到非理性理论，则可能使用无限琼斯指数。考虑到这种情况，我们在纯粹的亚比例理论上下文中研究某些类别的编织离散子因子具有附加的通勤性约束，我们称之为局部性，并且对应于量子场理论中类似空间距离的现场距离的换向操作员之间的换向关系。这种类型的子因子的示例来自对一个因素的最小作用，并考虑了固定点的亚词法。我们向每个不可理用的本地离散子因子$ \ Mathcal {n} \ subset \ subset \ Mathcal \ Mathcal {m} $ of Type $ {i \！i \！i \！i} $都有相关的规范紧凑型超组（用于子因子的不变性），该组对$ \ Mathcal cop and thit and Intial（由MAPS MAPS MAPS）MAPS MAPS MAPS {m} MAPS MAPS {m} MAPS {m} maaps {m} maaps {m} maaps {m} $ \ MATHCAL {N} $作为固定点。为了证明这一点，我们在$ \ Mathcal {M} $上的所有$ \ Mathcal {n} $的集合之间建立了二元性配对，并在$ \ Mathcal {M} $上与某些交换性Unital $ C^*$ - 代数 - 我们与紧凑型超级分类相关的频谱。如果子因子具有深度2，则紧凑型高群是一个紧凑的组。这排除了紧凑型\ emph {量子}组的发生，在局部保形场理论中充当全局仪表对称性。
Conformal inclusions of chiral conformal field theories, or more generally inclusions of quantum field theories, are described in the von Neumann algebraic setting by nets of subfactors, possibly with infinite Jones index if one takes non-rational theories into account. With this situation in mind, we study in a purely subfactor theoretical context a certain class of braided discrete subfactors with an additional commutativity constraint, that we call locality, and which corresponds to the commutation relations between field operators at space-like distance in quantum field theory. Examples of subfactors of this type come from taking a minimal action of a compact group on a factor and considering the fixed point subalgebra. We show that to every irreducible local discrete subfactor $\mathcal{N}\subset\mathcal{M}$ of type ${I\!I\!I}$ there is an associated canonical compact hypergroup (an invariant for the subfactor) which acts on $\mathcal{M}$ by unital completely positive (ucp) maps and which gives $\mathcal{N}$ as fixed points. To show this, we establish a duality pairing between the set of all $\mathcal{N}$-bimodular ucp maps on $\mathcal{M}$ and a certain commutative unital $C^*$-algebra, whose spectrum we identify with the compact hypergroup. If the subfactor has depth 2, the compact hypergroup turns out to be a compact group. This rules out the occurrence of compact \emph{quantum} groups acting as global gauge symmetries in local conformal field theory.
