ИСТИНА

It turns out that for many applications (say, if we study H-sub(co)modules, H-invariant ideals, polynomial H-identities...) it is not really important, which par- ticular Hopf algebra is (co)acting on a given H-(co)module algebra. Here we come naturally to the notion of (support) equivalence of Hopf (co)module structures on al- gebras which is the direct generalization of the notion of (weak) equivalence of group gradings. In addition, among all Hopf algebras that realize a given Hopf (co)module structure there are distinguished ones which we call universal Hopf algebras. One can develop a unified theory of so called V -universal bi/Hopf algebras that embraces the universal bi/Hopf algebras of given (co)module structures and the universal bi/Hopf algebras introduced by Sweedler, Manin and Tambara. On one hand, this enables to refine the existence conditions for the Manin–Tambara universal Hopf algebras. On the other hand, this approach makes it possible to establish a certain duality between the V -universal acting and coacting bi/Hopf algebras. In the talk we will discuss this theory as well as its possible applications and generalizations to arbitrary braided monoidal categories. Joint project with Ana Agore (Vrije Universiteit Brussel, Belgium) and Joost Vercruysse (Universite Libre de Bruxelles, Belgium).