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VOL. 80 | 2019 The hypergroupoid of boundary conditions for local quantum observables
Marcel Bischoff, Karl-Henning Rehren

Editor(s) Masaki Izumi, Yasuyuki Kawahigashi, Motoko Kotani, Hiroki Matui, Narutaka Ozawa

## Abstract

We review the definition of hypergroups by Sunder, and we associate a hypergroup to a type III subfactor $N \subset M$ of finite index, whose canonical endomorphism $\gamma \in \mathrm{End}(M)$ is multiplicity-free. It is realized by positive maps of $M$ that have $N$ as fixed points. If the depth is $\gt 2$, this hypergroup is different from the hypergroup associated with the fusion algebra of $M$-$M$ bimodules that was Sunder's original motivation to introduce hypergroups.

We explain how the present hypergroup, associated with a suitable subfactor, controls the composition of transparent boundary conditions between two isomorphic quantum field theories, and that this generalizes to a hypergroupoid of boundary conditions between different quantum field theories sharing a common subtheory.

## Information

Published: 1 January 2019
First available in Project Euclid: 21 August 2019

zbMATH: 07116420
MathSciNet: MR3966581

Digital Object Identifier: 10.2969/aspm/08010023