Abstract

I will report on an on-going project with Michael Brannan, Daniel Gromada, Junichiro Matsuda, and Adam Skalski.   A classical result of Frucht says that every finite group can be realized as an automorphism group of a finite graph. Due to Banica and McCarthy, the following analogue does not hold: not every finite quantum group is a quantum automorphism group of a finite graph, e.g. the dual of the permutation group on three generators. Nevertheless we obtained a version of Frucht’s theorem utilizing quantum graphs: every finite quantum group is a quantum automorphism group of a finite quantum graph. Moreover, the argument is more efficient than the original one in the case of classical groups. For a given finite quantum group we also tackled the following question: when can we find a quantum Cayley graph, whose quantum automorphism group is the original finite quantum group. I will offer some answers, mostly for duals of classical groups.