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

TL;DR

This paper introduces the concept of λ-Hopf algebras as partial smash product algebras of Hopf algebras with their base fields, providing a method to compute and classify such partial actions.

Contribution

It defines λ-Hopf algebras, shows all Hopf algebras are λ-Hopf algebras, and develops a method to compute their partial actions on the base field.

Findings

Every Hopf algebra is a λ-Hopf algebra.

A method to compute partial actions on the base field.

Classification of partial actions for specific Hopf algebra families.

Abstract

We introduce the concept of a $\lambda$-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 $\lambda$-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.