Quantum Automorphism Groups of Connected Locally Finite Graphs and Quantizations of Discrete Groups

Abstract

Abstract We construct for every connected locally finite graph $\Pi $ the quantum automorphism group $\operatorname{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 $\operatorname{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 $, $\Pi ^{\prime}$ and prove that this implies monoidal equivalence of $\operatorname{QAut} \Pi $ and $\operatorname{QAut} \Pi ^{\prime}$.