Given a finite dimensional Hopf algebra [math]A[/math], prove that its dual [math]A^*[/math] is a Hopf algebra too, with structural maps as follows:

Also, check that [math]A[/math] is commutative if and only if [math]A^*[/math] is cocommutative, and also discuss what happens in the cases [math]A=C(G)[/math] and [math]A=C^*(H)[/math], with [math]G,H[/math] being finite groups.

Prove that the compact quantum groups [math]G[/math] which are finite, in the sense that [math]\dim C(G) \lt \infty[/math], coincide with the discrete quantum groups [math]\Gamma[/math] which are finite, in the sense that [math]\dim C^*(\Gamma) \lt \infty[/math], and coincide as well with the finite quantum groups.

Clarify the discrete quantum group formulation of the various compact quantum group product operations, namely taking subgroups, quotients, dual free products, free complexifications and projective versions.

Prove that the free complexification embedding

is an isomorphism at the level of the associated diagonal tori.

Find the complexification operation [math]O_N\to U_N[/math], by using linear algebra, or Lie algebras, or whatever other means.