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

Abstract

We construct for every connected locally finite graph $Π$ the quantum automorphism group $\text{QAut}\ Π$ as a locally compact quantum group. When $Π$ is vertex transitive, we associate to $Π$ a new unitary tensor category $\mathcal{C}(Π)$ and this is our main tool to construct the Haar functionals on $\text{QAut}\ Π$. When $Π$ 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 $Π$, $Π'$ and prove that this implies monoidal equivalence of $\text{QAut}\ Π$ and $\text{QAut}\ Π'$.

Figures (12)

• Figure 1: The bi-labeled graph $\mathcal{K} \in \mathcal{G}(1,1)$ with $T^{\mathcal{K}}_{ii} = \deg i$ for all $i \in I$
• Figure 2: The bi-labeled graph $\mathcal{K} \in \mathcal{G}(1,1)$ such that $T^{\mathcal{K}}$ is the adjacency matrix of $\Pi$
• Figure 3: The bi-labeled graph $\mathcal{R} \in \mathcal{P}(2n,1)$ for $n=3$
• Figure 4: The bi-labeled graph $\mathcal{R}_n \in \mathcal{P}(2n,0)$ for $n=3$
• Figure 5: The bi-labeled graph $\mathcal{S}_n \in \mathcal{P}(2n+1,1)$ for $n=3$

Theorems & Definitions (107)

• Theorem 1
• Definition 2
• Theorem 3
• Definition 4
• Theorem 5: See Theorem \ref{['thm.quantum-iso-criteria']}
• Definition 2.1
• Lemma 2.2
• Definition 2.3
• proof : Proof of Lemma \ref{['lem.crucial-technical-Mor-lemma']}
• Definition 2.4