Quantum automorphism groups of connected locally finite graphs and quantizations of discrete 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 $\text{QAut}\ \Pi'$.