Quantum graphs, subfactors and tensor categories I

TL;DR

This paper develops an equivariant framework for quantum graphs using tensor categories, modeling quantum sets via C*-algebra inclusions, and explores their substructures and symmetries with applications to quantum groupoids.

Contribution

It introduces a unifying tensor category approach to quantum graphs, linking subfactors, quantum symmetries, and graph theory in a novel way.

Findings

Finite-index subfactors can be viewed as quantum graphs.

All adjacency operators can be derived via quantum Fourier transform.

A quantum version of Frucht's Theorem is established.

Abstract

We develop an equivariant theory of graphs with respect to quantum symmetries and present a detailed exposition of various examples. We portray unitary tensor categories as a unifying framework encompassing all finite classical simple graphs, (quantum) Cayley graphs of finite (quantum) groupoids, and all finite-dimensional quantum graphs. We model a quantum set by a finite-index inclusion of C*-algebras and use the quantum Fourier transform to obtain all possible adjacency operators. In particular, we show every finite-index subfactor can be regarded as a complete quantum graph and describe how to find all its subgraphs. As applications, we prove a version of Frucht's Theorem for finite quantum groupoids, and introduce a version of path spaces for quantum graphs.