Admissibility conjecture for quantum group representations

A long-standing open problem in the representation theory of quantum groups is to decide whether the following statement is true or false: Every finite dimensional, unitary representation of (locally) compact quantum group ''factors'' in a suitable sense, through a representation of some matrix quantum group. This conjecture came up first in a work of P. Sołtan, and later on it kept featuring in many subsequent works on representation theory of (locally) compact quantum groups. We will give a partial solution to this conjecture, through proving that for a large class of (locally) compact quantum groups, this is true. Based on a joint work with P. Salmi and M. Daws.