The quest of a finite purely quantum group

An important open problem is whether there exists a finite quantum group which cannot be cooked up from (classical) finite groups. A finite purely quantum group would be a finite dimensional Hopf C*-algebra (Kac algebra) K such that the unitary integral fusion category Rep(K) is not weakly group-theoretical, and admits no such fusion subcategory other than the trivial one. This talk will expose the first results in the quest of such an object, through joint works with Zhengwei Liu, Yunxiang Ren and Jinsong Wu, involving subfactor planar algebras, quantum Fourier analysis and fusion categories.