Partial Hopf actions, partial invariants and a Morita context

Abstract

Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to assure the existence of enveloping actions [1]. This allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we explore some properties of the fixed point subalgebra with relation to a partial action of a Hopf algebra. We also construct, for partial actions of finite dimensional Hopf algebras a Morita context relating the fixed point subalgebra and the partial smash product. This is a generalization of a well known result in the theory of Hopf algebras [9]  for the case of partial actions. Finally, we study Hopf-Galois extensions and reobtain some classical results in the partial case.