Deformation of finite dimensional C*-quantum groupoids

Abstract

In this work we prove, in full details, that any finite dimensional $C^{\ast }$-quantum groupoid can be deformed in order that the square of the antipode is the identity on the base. We also prove that for any $C^{\ast }$-quantum groupoid with non abelian base, there is uncountably many $C^{\ast }$-quantum groupoids with the same underlying algebra structure but which are not isomorphic to it. In fact, the $C^{\ast }$-quantum groupoids are closed in an analog of the procedure presented by D.Nikshych ([N] 3.7) in a more general situation.