Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups

TL;DR

This paper constructs quantum automorphism groups for connected locally finite graphs, introduces a tensor category framework, and explores quantum isomorphisms and their implications for group quantizations.

Contribution

It develops a method to construct quantum automorphism groups for graphs and links these to quantum group representations and group quantizations.

Findings

Quantum automorphism groups are constructed for all connected locally finite graphs.

A new tensor category is associated with vertex transitive graphs to study Haar functionals.

Quantum isomorphism implies monoidal equivalence of quantum automorphism groups.

Abstract

We construct for every connected locally finite graph $\Pi$ the quantum automorphism group $\text{QAut}\ \Pi$ as a locally compact quantum group. When $\Pi$ is vertex transitive, we associate to $\Pi$ a new unitary tensor category $\mathcal{C}(\Pi)$ and this is our main tool to construct the Haar functionals on $\text{QAut}\ \Pi$. When $\Pi$ is the Cayley graph of a finitely generated group, this unitary tensor category is the representation category of a compact quantum group whose discrete dual can be viewed as a canonical quantization of the underlying discrete group. We introduce several equivalent definitions of quantum isomorphism of connected locally finite graphs $\Pi$, $\Pi'$ and prove that this implies monoidal equivalence of $\text{QAut}\ \Pi$ and $\text{QAut}\ \Pi'$.