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- Title
Quantum Automorphism Groups of Connected Locally Finite Graphs and Quantizations of Discrete Groups.
- Authors
Rollier, Lukas; Vaes, Stefaan
- 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}$|.
- Subjects
AUTOMORPHISM groups; DISCRETE groups; QUANTUM groups; COMPACT groups; ISOMORPHISM (Mathematics); CAYLEY graphs
- Publication
IMRN: International Mathematics Research Notices, 2024, Vol 2024, Issue 3, p2219
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnad099