Quantum graphs, subfactors and tensor categories I

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.

40 pages including references, many figures, comments welcomed!

References in corpus (12)

• A compositional approach to quantum functions
• Orientation of quantum Cayley trees and applications
• Bigalois extensions and the graph isomorphism game
• The Morita theory of quantum graph isomorphisms
• Q-system completion for C* 2-categories
• Quantum Fourier Analysis
• The quantum-to-classical graph homomorphism game
• Minimal index and dimension for 2-$C^{\ast }$-categories with finite-dimensional centers
• Some examples of quantum graphs
• Classification of Quantum Graphs on $M_2$ and their Quantum Automorphism Groups
• K-theoretic classification of inductive limit actions of fusion categories on AF-algebras
• Covariant quantum combinatorics with applications to zero-error communication