Anyonic quantum symmetries of finite spaces

TL;DR

This paper introduces a braided quantum permutation group acting on finite spaces, explores its properties, and demonstrates its application to symmetries of circulant graphs, extending previous results in quantum symmetry theory.

Contribution

It constructs a new braided quantum permutation group and applies it to finite spaces and circulant graphs, generalizing existing quantum symmetry results.

Findings

Existence of braided quantum symmetries for finite simple circulant graphs

Explicit computation of these symmetries for specific examples

Generalization of Banica's results on quantum symmetries

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.