Ações diagonais de categorias de Hopf

Abstract

The study of actions and coactions of Hopf algebras on algebras has been one of the central goals of the theory of Hopf algebras since the 1970s. More recently, in 2016, Batista, Caenepeel and Vercruysse introduced a generalization of Hopf algebras, called Hopf categories. Thus, it is natural to ask whether the definitions and results already known for actions and coactions of Hopf algebras, can also be generalized to Hopf categories. In 2018, Caenepeel and Fieremans provide some answers in this regard by developing a Galois theory for Hopf categories, however several questions remain open. In this work, using the adjoint action of a Hopf algebra as inspiration, we were able to obtain a definition of action for Hopf categories. As a consequence, we managed to obtain an adjoint action for Hopf categories, a smash product which we showed to be a Galois extension according to the theory of Caenepeel and Fieremans, and a connection with the classical theory involving actions by groupoids. Finally, a duality theorem has been constructed for Hopf categories, unifying the theories developed so far.