A quantum Frucht's theorem and quantum automorphisms of quantum Cayley graphs
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866911215608922112 |
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| author | Brannan, Michael Gromada, Daniel Matsuda, Junichiro Skalski, Adam Wasilewski, Mateusz |
| author_facet | Brannan, Michael Gromada, Daniel Matsuda, Junichiro Skalski, Adam Wasilewski, Mateusz |
| contents | We establish a quantum version of Frucht's Theorem, proving that every finite quantum group is the quantum automorphism group of an undirected finite quantum graph. The construction is based on first considering several quantum Cayley graphs of the quantum group in question, and then providing a method to systematically combine them into a single quantum graph with the right symmetry properties. We also show that the dual $\widehat Γ$ of any non-abelian finite group $Γ$ is ``quantum rigid''. That is, $\widehat Γ$ always admits a quantum Cayley graph whose quantum automorphism group is exactly $\widehat Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11149 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quantum Frucht's theorem and quantum automorphisms of quantum Cayley graphs Brannan, Michael Gromada, Daniel Matsuda, Junichiro Skalski, Adam Wasilewski, Mateusz Operator Algebras Quantum Algebra Primary 46L89, Secondary 05C25, 20G42 We establish a quantum version of Frucht's Theorem, proving that every finite quantum group is the quantum automorphism group of an undirected finite quantum graph. The construction is based on first considering several quantum Cayley graphs of the quantum group in question, and then providing a method to systematically combine them into a single quantum graph with the right symmetry properties. We also show that the dual $\widehat Γ$ of any non-abelian finite group $Γ$ is ``quantum rigid''. That is, $\widehat Γ$ always admits a quantum Cayley graph whose quantum automorphism group is exactly $\widehat Γ$. |
| title | A quantum Frucht's theorem and quantum automorphisms of quantum Cayley graphs |
| topic | Operator Algebras Quantum Algebra Primary 46L89, Secondary 05C25, 20G42 |
| url | https://arxiv.org/abs/2503.11149 |