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

DOI10.1093/IMRN/RNAD099arXiv2209.03770MaRDI QIDQ6508015

Lukas Rollier, Stefaan Vaes

Abstract: We construct for every connected locally finite graph $Pi$ the quantum automorphism group $extQAutPi$ as a locally compact quantum group. When $Pi$ is vertex transitive, we associate to $Pi$ a new unitary tensor category $mathcalC(Pi)$ and this is our main tool to construct the Haar functionals on $extQAutPi$. 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 $Pi$, $P{i}^{\prime }$ and prove that this implies monoidal equivalence of $extQAutPi$ and $extQAutP{i}^{\prime }$.

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