Galois reconstruction of finite quantum groups

Let C be a (small) category and let F : C −→ Malgf be a functor, where Malgf is the category of finite-dimensional measured algebras over a field k (or Frobenius algebras). We construct a universal Hopf algebra Aaut(F ) such that F factorizes through a functor F : C −→ Mcoalgf(Aaut(F )), where Mcoalgf(Aaut(F )) is the category of finite-dimensional measured Aaut(F )-comodule algebras. This general reconstruction result allows us to recapture a finite-dimensional Hopf algebra A from the category Mcoalgf(A) and the forgetful functor ω : Mcoalgf(A) −→ Malgf : we have A ∼= Aaut(ω). Our universal construction is also done in a C ∗-algebra framework, and we get compact quantum groups in the sense of Woronowicz.