A quantum Frucht's theorem and quantum automorphisms of quantum Cayley graphs

Abstract

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 $\hat{\Gamma }$ of any non-abelian finite group $\Gamma$ is ``quantum rigid''. That is, $\hat{\Gamma }$ always admits a quantum Cayley graph whose quantum automorphism group is exactly $\hat{\Gamma }$.

33 pages, v2 modifies a few minor points and adds more details in Section 6. The final version of the paper will appear in Proceedings of the London Mathematical Society