# The hypergroupoid of boundary conditions for local quantum observables

@article{Bischoff2016TheHO, title={The hypergroupoid of boundary conditions for local quantum observables}, author={M. Bischoff and K. Rehren}, journal={arXiv: Mathematical Physics}, year={2016} }

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 $>2$, 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.
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