Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913228174393344 |
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| author | Rollier, Lukas Vaes, Stefaan |
| author_facet | Rollier, Lukas Vaes, Stefaan |
| contents | 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 $\text{QAut}\ Π'$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_03770 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups Rollier, Lukas Vaes, Stefaan Quantum Algebra Operator Algebras 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 $\text{QAut}\ Π'$. |
| title | Quantum automorphism groups of connected locally finite graphs and quantizations of discrete groups |
| topic | Quantum Algebra Operator Algebras |
| url | https://arxiv.org/abs/2209.03770 |