Partial actions of a Hopf algebra on its base field and the corresponding partial smash product algebra

Abstract

In this paper, we introduce the concept of a [Formula: see text]-Hopf algebra as a Hopf algebra obtained as the partial smash product algebra of a Hopf algebra and its base field, and show that every Hopf algebra is a [Formula: see text]-Hopf algebra. Moreover, a method to compute partial actions of a given Hopf algebra on its base field is developed and, as an application, we exhibit all partial actions of such type for some families of Hopf algebras.