Easy quantum groups and quantum subgroups of a semi-direct product quantum group

Abstract

We consider homogeneous compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial symmetries with central support.We show that such quantum groups have a simple presentation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups with the above property by certain reflection groups.We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy homogeneous hyperoctahedral quantum groups. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for those easy quantum groups in the scope of this article.