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We introduce the concept of a $\lambda$-Hopf algebra as a Hopf algebraobtained as the partial smash product algebra of a Hopf algebra and its basefield, and show that every Hopf algebra is a $\lambda$-Hopf algebra.  Moreover, a method to compute partial actions of a given Hopf algebra on itsbase field is developed and, as an application, we exhibit all partial actionsof such type for some families of Hopf algebras.

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