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 $\Pi$, ${\Pi }^{\prime }$ and prove that this implies monoidal equivalence of $\text{QAut}\Pi$ and $\text{QAut}{\Pi }^{\prime }$.

arXiv commentv3: final version to appear in International Mathematics Research Notices. In this final version v3, there are several small changes and also the new proposition 3.4 providing a connection to the quantum isometry groups of arXiv:1002.2551

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