Quantum groups acting on 4 points

Abstract

We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group ${\mathcal{Q}}_4$. Our main tool is a new presentation for the algebra $\mathrm{C}({\mathcal{Q}}_4)$, corresponding to an isomorphism of type ${\mathcal{Q}}_4\simeq S{O}_{-1}(3)$. The quantum subgroups of ${\mathcal{Q}}_4$ are subject to a McKay type correspondence, that we describe at the level of algebraic invariants.

34 pages. Comments added in last section