Anyonic quantum symmetries of finite spaces

Abstract

We construct a braided analogue of the quantum permutation group and show that it is the universal braided compact quantum group acting on a finite space in the category of $\mathbb{Z}/N\mathbb{Z}$-$\textrm{C}^*$-algebras with a twisted monoidal structure. As an application, we prove the existence of braided quantum symmetries of finite, simple, undirected, circulant graphs, explicitly compute it for several examples, and obtain a generalization of a result of Banica in this direction. Finally, in an appendix, we briefly describe the irreducible representations of this braided analogue of the quantum permutation group and their fusion rules.