Galois Reconstruction of Finite Quantum Groups

Let be a (small) category and let F: → algf be a functor, where algf 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 : → coalgf(Aaut(F)), where coalgf(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 coalgf(A) and the forgetful functor ω: coalgf(A) → algf: 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.